text stringlengths 4 2.78M |
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abstract: 'The volume of news content has increased significantly in recent years and systems to process and deliver this information in an automated fashion at scale are becoming increasingly prevalent. One critical component that is required in such systems is a method to automatically determine how notable a certain news story is, in order to prioritize these stories during delivery. One way to do so is to compare each story in a stream of news stories to a notable event. In other words, the problem of detecting notable news can be defined as a ranking task; given a trusted source of notable events and a stream of candidate news stories, we aim to answer the question: “Which of the candidate news stories is most similar to the notable one?”. We employ different combinations of features and learning to rank (LTR) models and gather relevance labels using crowdsourcing. In our approach, we use structured representations of candidate news stories (triples) and we link them to corresponding entities. Our evaluation shows that the features in our proposed method outperform standard ranking methods, and that the trained model generalizes well to unseen news stories.'
author:
- 'Antonia Saravanou[^1]'
- Giorgio Stefanoni
- Edgar Meij
bibliography:
- 'refs.bib'
title: Identifying Notable News Stories
---
Introduction {#sec:intro}
============
Problem Statement {#sec:problem_statement}
=================
Method {#sec:method}
======
Experimental Setup {#sec:setup}
==================
Results and Discussion {#sec:discussion}
======================
Conclusion and Future Work
==========================
[^1]: This work was done whilst interning at Bloomberg.
|
---
abstract: 'In [@Rains:Transformations], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric ($q$-hypergeometric) integral identities as limits from the elliptic level.'
author:
- 'Eric M. Rains'
date: 'September 5, 2007'
title: Limits of elliptic hypergeometric integrals
---
Introduction
============
In [@Rains:Transformations], the author proved and extended a pair of multivariate elliptic hypergeometric integrals conjectured by van Diejen and Spiridonov [@vanDiejenJF/SpiridonovVP:2001], elliptic analogues of integrals due to Gustafson [@GustafsonRA:1992]. In a follow-up paper [@vanDiejenJF/SpiridonovVP:2005], van Diejen and Spiridonov proved “hyperbolic” degenerations of these integrals (after the hyperbolic gamma function of Ruijsensaars [@RuijsenaarsSNM:1999], and results of Stokman [@StokmanJV:2005] at the univariate level). Unfortunately, the asymptotic estimates they had available were insufficient to derive these hyperbolic integrals as limits of the elliptic integrals; instead, they were forced to degenerate the elliptic proofs. The objective of the present paper is to put these (and other) degenerations on a sounder footing by showing that they are indeed limiting cases of the elliptic integral. It is to be hoped that a better understanding of the relation between the elliptic and other integrals will lead to new results at various levels, both by clarifying how arguments at low levels extend to higher levels and by providing new results at low levels as limits of the elliptic identities. The latter hope has already been fulfilled to some extent; see, for instance, Corollary \[cor:triglim\_IC12\] below.
A full study of degenerations of the elliptic hypergeometric integrals is beyond the scope of the present paper. Here, we focus only on the top level of each of the five types of limit (hyperbolic, trigonometric, elliptic, rational, and classical); degenerations within each type will be deferred to future work with F. van de Bult. We also do not consider in any detail the degenerations between types; rather, we obtain each case directly as a limit from the elliptic level. (Note, however, that in several cases our estimates are sufficiently uniform that one could view certain between-level limits as special cases of the limits from the elliptic level.) Similarly, we do not consider the discrete degenerations, a.k.a. hypergeometric sums; aside from finite sums (which have already been considered), the only interesting discrete degenerations appear to arise via lower-level (e.g., hyperbolic or trigonometric) integrals. Finally, we comment that at the univariate level, there are important relations between the trigonometric and hyperbolic integrals ([@StokmanJV:2005], also [@vdBultF/RainsEM/StokmanJV:2007]); presumably, these have analogues at the multivariate level, but the existing techniques do not appear to apply. (However, see [@StokmanJV:2007] for recent progress on a slightly degenerate, but multivariate case.)
The plan of the paper is as follows. In Section 2, we define the relevant gamma functions (elliptic, hyperbolic, trigonometric, and rational), and prove a number of asymptotic results relating them. In order to make the derivation of limit integrals as simple as possible, we have tried to make these estimates as uniform as possible, and the relevant error estimates similarly strong; since our techniques also apply equally well to higher-order multiple gamma functions, we prove our estimates in that level of generality. Even for the ordinary (i.e., $r=2$) elliptic and hyperbolic gamma functions, our results are new, as the results in the literature were nonuniform (compare, for instance, Theorem \[thm:hyper\_to\_rat\] to its special cases Corollaries \[cor:hyper\_to\_rat\] and \[cor:hyper\_to\_rat2\], which were proved (for $r=2$, and not uniform) in [@RuijsenaarsSNM:1999]). Corollary \[cor:trig\_to\_rat\], a uniform version of the $q\to 1$ asymptotics of $q$-symbols, may also be of independent interest.
Section 3 considers the inequalities required to pinpoint where the integrands of interest are maximized. It turns out that these all follow from a single master inequality (Lemma \[lem:gen\_tri\_ell\]), which in turn follows from an asymptotic analysis of an elliptic analogue of the Cauchy determinant. The result is an inequality stating that certain combinations of elliptic gamma functions are exponentially small unless their arguments alternate around the unit circle. It may be worth investigating other elliptic determinant and pfaffian identities to see whether they give rise to interesting inequalities.
Section 4 begins the study of limit integrals with the hyperbolic case. This case turns out to be relatively straightforward, given the estimates and inequalities already established; in each case, a standard tail-exchange argument gives the desired limit. In fact, not only do we obtain the hyperbolic integrals as limits of the corresponding elliptic integrals, but moreover obtain exponentially small error estimates. We also discuss the extra arguments required to include the case in which the integrands are multiplied by appropriate abelian functions (the interpolation and biorthogonal functions of [@Rains:Transformations]).
Section 5 considers the trigonometric (i.e., basic hypergeometric) case. It turns out that in addition to the integrals of Gustafson that originally motivated van Diejen and Spiridonov, there are additional limiting cases. That these should exist is suggested by the fact that in the transformations of [@Rains:Transformations], only one side typically has a straightforward limit as $p\to 0$. It turns out, however, that if one breaks the symmetry of the integrand in a suitable way, one can arrange for both sides of the transformations to have reasonable limits. As a special case (violating our rule of considering only the top level of each case), we find that not only are the ordinary Macdonald polynomials limits of the biorthogonal abelian functions, but in such a way that their orthogonality follows as well; in particular, the Macdonald “conjectures” (proved in [@MacdonaldIG:1995]) for ordinary Macdonald polynomials are limiting cases of the corresponding identities for biorthogonal abelian functions.
The remaining two cases correspond to the case $q\to 1$ with $p$ fixed. In the rational case, considered in Section 6, the parameters all tend to 1, and one obtains a hybrid of the hyperbolic and trigonometric cases. In particular, to obtain the full spectrum of possibilities, one must first break the symmetry of the integrand, and then do a tail-exchange asymptotic argument. The arguments are still fairly straightforward, although the resulting estimates are much weaker.
Finally, in the classical case, considered in Section 7, the parameters behave in such a way that the integrand is exponentially small unless certain inequalities are satisfied. This gives rise to multivariate analogues of the ordinary beta integral; notably those of Dixon [@DixonAL:1905] and Selberg [@SelbergA:1944]. Curiously, the most natural forms of these limits remain elliptic in nature, with the integrand involving powers of theta functions, although these can be removed with a suitable change of variables.
The author would like to thank P. Forrester, J. Stokman and F. van de Bult for motivating conversations regarding the trigonometric and hyperbolic cases, and R. Askey for suggesting the use of the modular transformation to derive classical limits (as in [@NasrallahB/RahmanM:1985]), which led the author to consider the paper [@NarukawaA:2004]; the author would also like to thank an anonymous referee for pointing out that the original version of Theorem \[thm:biorth\_lim\] was badly stated. The author was supported in part by NSF Grant No. DMS-0401387.
### Conventions {#conventions .unnumbered}
We use standard conventions for big $O$ notation in uniform estimates; that is, we say that $f(v,z)=O(g(v,z))$ as $v\to 0^+$ uniformly over the region $z\in D(v)$ (where $D(v)$ is a set depending on $v$) if there exist constants $\delta>0$ and $C>0$ such that $$|f(v,z)|<C |g(v,z)|$$ whenever $0<v<\delta$, $z\in D(v)$; and similarly for limits $v\to
\pm\infty$ (which can be viewed as limits $\pm 1/v\to 0^+$).
All logarithms are taken on the principal branch, with branch cut along the negative real axis. All powers are determined correspondingly; in particular, square roots are chosen with positive real part. (The obvious exception is $\sqrt{-1}$, which is always taken to have positive imaginary part; we refrain from denoting it by $i$ so as to retain $i$ as an indexing variable.)
Finally, for a real number $x$, $\{x\}$ denotes the fractional part of $x$; i.e., the unique representative in $[0,1)$ of $x+{\mathbb Z}$.
Asymptotics of multiple gamma functions
=======================================
For integers $r\ge 0$, let $\phi^{(r)}(w;x;\omega_1,\dots,\omega_r)$ denote the function $$\phi^{(r)}(w;x;\omega_1,\dots,\omega_r)
:=
\frac{e(xw)}{\prod_{1\le i\le r} (e(\omega_i w)-1)},$$ where $e(x):=\exp(2\pi\sqrt{-1}x)$, and define a family of polynomials $P^{(r)}_n(x;\omega_1,\dots,\omega_r)$ in $x$ by the Laurent series expansion $$\phi^{(r)}(w;x;\omega_1,\dots,\omega_r)
=
\sum_{n\ge -r} w^n P^{(r)}_n(x;\omega_1,\dots,\omega_r)$$ valid in a punctured neighborhood of $w=0$; we also set $$Q^{(r)}_n(x;\omega_1,\dots,\omega_r)
:=
P^{(r+1)}_n(x;\omega_1,\dots,\omega_r,-1).$$ By convention, if $n$ is omitted, then $n=0$. Note that these are related to the multiple Bernoulli polynomials of [@NarukawaA:2004] by $$P^{(r)}_n(x;\omega_1,\dots,\omega_r)
=
\frac{(2\pi\sqrt{-1})^n}{(n+r)!} B_{r,n+r}(x|\omega_1,\dots,\omega_r).$$ Note the special cases $$\begin{aligned}
P^{(0)}(x;) &= 1\notag\\
P^{(1)}(x;\omega) &= \frac{x-\omega/2}{\omega}\notag\\
P^{(2)}(x;\omega_1,\omega_2)
&=
\frac{12(x-\omega_1/2-\omega_2/2)^2-\omega_1^2-\omega_2^2}{24\omega_1\omega_2}
\notag\end{aligned}$$ and $$\begin{aligned}
Q^{(0)}(x;) &= -(x+1/2)\notag\\
Q^{(1)}(x;\omega)
&=
\frac{-12(x-\omega/2+1/2)^2+\omega^2+1}{24\omega}
\notag\\
Q^{(2)}(x;\omega_1,\omega_2)
&=
\frac{
(x-\omega_1/2-\omega_2/2+1/2)
(-4(x-\omega_1/2-\omega_2/2+1/2)^2+\omega_1^2+\omega_2^2+1)
}{24\omega_1\omega_2}.\notag\end{aligned}$$ We will also need a third polynomial $$\begin{aligned}
R^{(r)}(x;\omega_1,\dots,\omega_r)
&=
Q^{(r)}(x;\omega_1,\dots,\omega_r)
+\frac{1}{2}
P^{(r)}(x;\omega_1,\dots,\omega_r)\notag\\
&=
\frac{
Q^{(r)}(x;\omega_1,\dots,\omega_r)
+
Q^{(r)}(x-1;\omega_1,\dots,\omega_r)
}{2}\notag\end{aligned}$$ and note $$\begin{aligned}
R^{(0)}(x;) &= -x\notag\\
R^{(1)}(x;\omega) &= \frac{-12(x-\omega/2)^2+\omega^2-2}{24\omega}\notag\\
R^{(2)}(x;\omega_1,\omega_2) &=
\frac{
(x-\omega_1/2-\omega_2/2)
(-4(x-\omega_1/2-\omega_2/2)^2+\omega_1^2+\omega_2^2-2)
}{24\omega_1\omega_2}.
\notag\end{aligned}$$
We can then define the [*hyperbolic gamma function*]{} ${\Gamma_{\!h}}^{(r)}$ as follows. For all integers $r\ge 1$, and $0<\Im(x)<\sum_{1\le i\le
r}\Im(\omega_i)$, $${\Gamma_{\!h}}^{(r)}(x;\omega_1,\omega_2,\dots,\omega_r)
:=
\exp(
{\text{\rm PV}\!\!\int}_{\mathbb R}\phi^{(r)}(w;x;\omega_1,\dots,\omega_r) \frac{dw}{w}
),$$ where by the principal value integral ${\text{\rm PV}\!\!\int}_{\mathbb R}$ we mean the average of the integral over two contours agreeing with ${\mathbb R}$ away from 0; one contour passes to the left of $0$, and the other passes to the right of $0$. This differs slightly from the corresponding definitions of hyperbolic gamma functions in the literature; in particular, we have $${\Gamma_{\!h}}^{(r)}(x;\omega_1,\dots,\omega_r)
=
S_r(x|\omega_1,\dots,\omega_r)^{(-1)^{r-1}},$$ where $S_r$ is the multiple sine function (see, for instance, [@NarukawaA:2004]), and $${\Gamma_{\!h}}^{(2)}(x;\omega_1,\omega_2)
=
G(-\omega_1\sqrt{-1},-\omega_2\sqrt{-1};x-\omega_1/2-\omega_2/2)$$ in terms of Ruijsenaars’ hyperbolic gamma function [@RuijsenaarsSNM:1999]. Since $${\text{\rm PV}\!\!\int}_{\mathbb R}\phi^{(r)}(w;x;\omega_1,\dots,\omega_r)
\frac{dw}{w}
=
\frac{1}{2}
{\text{\rm PV}\!\!\int}_{\mathbb R}\left(\phi^{(r)}(w;x;\omega_1,\dots,\omega_r)
-
\phi^{(r)}(-w;x;\omega_1,\dots,\omega_r)\right)
\frac{dw}{w}$$ and $${\text{\rm PV}\!\!\int}_{\mathbb R}w^k dw = 0$$ for integers $k<-1$, we can express this as the ordinary integral over ${\mathbb R}$ of $$\frac{e(xw)+(-1)^{r-1} e((\sum_{1\le i\le r}\omega_i-x)w)}
{2w\prod_{1\le i\le r} (e(\omega_iw)-1)}
-
\sum_{1\le k\le (r+1)/2} P^{(r)}_{1-2k}(x;\omega_1,\dots,\omega_r) w^{-2k},$$ or by symmetry as twice the integral over $[0,\infty)$, thus recovering the definitions of [@RuijsenaarsSNM:1999] and [@NarukawaA:2004].
When $r=1$, we have $${\text{\rm PV}\!\!\int}_{\mathbb R}\frac{e(wx)}{(e(w\omega)-1)} \frac{dw}{w}
=
\log(2\sin(\pi x/\omega))$$ for $0<\Im(x)<\Im(\omega)$ (the branch with value $\log(2)$ at $x=\omega/2$), which then gives an analytic continuation of ${\Gamma_{\!h}}^{(1)}$ to all $x$, $\omega$ such that $\omega\ne 0$: $${\Gamma_{\!h}}^{(1)}(x;\omega) = 2\sin(\pi x/\omega).$$ (This is a fairly standard definite integral; it can be shown, for instance , by moving the contour infinitely far to the left (for $\Im(-x/\omega)>0$; if $\Im(-x/\omega)<0$, the contour should be moved to the right, and the case $-x/\omega$ real follows by analytic continuation), and observing that the resulting sum of residues (compare Theorem \[thm:residue\_sum\] below) is $$\pi\sqrt{-1}(x/\omega)
-\pi\sqrt{-1}/2
-
\sum_{k\ge 1} e(-kx/\omega)/k
=
\log(2\sin(\pi x/\omega)),$$ as required.) In general, we have $${\Gamma_{\!h}}^{(r)}(x+\omega_r;\omega_1,\dots,\omega_r)
=
{\Gamma_{\!h}}^{(r)}(x;\omega_1,\dots,\omega_r)
{\Gamma_{\!h}}^{(r-1)}(x;\omega_1,\dots,\omega_{r-1})$$ in the domain of definition, and thus by induction have a meromorphic continuation to all $x$. Since ${\Gamma_{\!h}}^{(r)}$ was defined via its logarithm, we in fact have a nearly canonical choice of branch for $\log{\Gamma_{\!h}}^{(r)}$; more precisely, for each $\omega_0$ with $\Im(\omega_0)>0$, there is a unique analytic continuation of $$\log{\Gamma_{\!h}}^{(r)}(x;\omega_1,\dots,\omega_r)
:=
{\text{\rm PV}\!\!\int}_{\mathbb R}\phi^{(r)}(w;x;\omega_1,\dots,\omega_r) \frac{dw}{w}$$ to the set ${\cal C}^{(r)}(\omega_0;\omega_1,\dots,\omega_r)$ obtained from ${\mathbb C}$ by removing the countable unions of rays $$\omega_0{\mathbb R}_{\ge 0}+\sum_{1\le i\le r} \omega_i{\mathbb Z}_{\ge 1}
\quad\text{and}\quad
\omega_0{\mathbb R}_{\le 0}+\sum_{1\le i\le r} \omega_i{\mathbb Z}_{\le 0}.$$ That is, for each zero and pole of ${\Gamma_{\!h}}^{(r)}$, we cut along a ray in the direction $\pm\omega_0$, as appropriate. Also of importance is the analogous analytic continuation of $$\log((x/\omega_0)^{(-1)^r}{\Gamma_{\!h}}^{(r)}(x;\omega_1,\dots,\omega_r))$$ to the domain ${\cal C}^{\prime(r)}(\omega_0;\omega_1,\cdots,\omega_r)$ which differs from ${\cal C}^{(r)}(\omega_0;\omega_1,\dots,\omega_r)$ only in that the ray $\omega_0{\mathbb R}_{\le 0}$ has not been cut; this continuation exists since the zero/pole at 0 has been cancelled. We also by convention take ${\Gamma_{\!h}}^{(0)}(x;)=-1$, to make the functional equation valid for $r=1$ as well. For the functional equation to hold for the logarithm, we must take $$\log{\Gamma_{\!h}}^{(0)}(x;) := -\pi\sqrt{-1}\operatorname{sgn}(\Im(x/\omega_0)),$$ defined on the set ${\cal C}^{(0)}(\omega_0;)={\mathbb C}\setminus \omega_0{\mathbb R}$.
Using the fact that $${\Gamma_{\!h}}^{(r)}(cx;c\omega_1,\dots,c\omega_r)
=
{\Gamma_{\!h}}^{(r)}(x;\omega_1,\dots,\omega_r)$$ we can further extend ${\Gamma_{\!h}}^{(r)}$ to arbitrary $\omega_i$, so long as there exists a constant $c$ with $\Im(c\omega_i)>0$ for all $i$. We will not be using this extension in the sequel, although the ability to rescale within the upper half-plane will be quite useful in the proofs.
In [@NarukawaA:2004 Prop. 5], Narukawa derived a product expansion for ${\Gamma_{\!h}}^{(r)}$, based on the observation that the integral over a suitably chosen sequence of contours $\Im(w)=a$ can be made to tend to 0 as $a\to\infty$, and thus the integral expands as a sum of residues. We will need a more precise form of the bound on the integral.
\[lem:hyperb\_asympt\] Fix $\epsilon>0$, and let $a$, $\Im(x)$, $\omega_i$ range over the domain $0<\Im(x)<\sum_i\Im(\omega_i)$, and for $1\le i\le r$, $\Im(\omega_i)>0$ and $|a-\Im(-n/\omega_i)|>\epsilon$, all integers $n\ge 0$. Then $$\left|\int_{\Im(w)=a} \phi^{(r)}(w;x;\omega_1,\dots,\omega_r) \frac{dw}{w}\right|
\le
a^{-1}
\left[
\prod_i C(\epsilon|\omega_i|)^{-1}
\right]
\left[
\frac{\exp(-2\pi a\Re(x))}
{2\pi\Im(x)}
+
\frac{\exp(-2\pi a\Re(x-\sum_i\omega_i))}
{2\pi\Im(\sum_i\omega_i-x)}
\right],$$ where $C(x) = \min_{d(y,{\mathbb Z})\ge x} |e(y)-1|.$ In particular, the integral is uniformly $O(\exp(-2\pi a\Re(x)))$ over any compact subset of the domain.
We have the estimates $$\begin{aligned}
|e(wx)| &= \exp(-2\pi \Re(w)\Im(x))\exp(-2\pi a\Re(x))\notag\\
|w|^{-1} &= 1/\sqrt{\Re(w)^2+a^2}\le 1/a\notag\\
|e(w\omega_i)-1|^{-1} &\le C(\epsilon|\omega_i|)^{-1}\notag\\
|e(w\omega_i)-1|^{-1} &\le C(\epsilon|\omega_i|)^{-1}
\exp(2\pi\Re(w)\Im(\omega_i))\exp(2\pi a \Re(\omega_i))\notag\end{aligned}$$ where we note that $|w\omega_i-n|>\epsilon|\omega_i|$. If we use the third estimate for $\Re(w)>0$ and the fourth estimate for $\Re(w)<0$, we obtain the stated bound.
Narukawa then uses the fact that for $\Re(x)>\max(0,\Re(\sum_i\omega_i))$, the above bound tends to 0 as $a\to \infty$; for our purposes, it is more convenient to fix $a$ and obtain an asymptotic series.
\[thm:residue\_sum\] Let $a$, $\arg(x)$, $\omega_i$ range over the domain $a>0$, $\Im(e(-\arg(x)/2\pi)\omega_i)>0$ for $1\le i\le r$. Then as $x\to\infty$, $$\begin{aligned}
-\pi\sqrt{-1}P^{(r)}(x;\omega_1,\dots,\omega_r)
-
\sum_{0<\Im(e(\arg(x)/2\pi)y)\le a}
\operatorname{Res}_{w=y}(\phi^{(r)}(w;x;\omega_1,\dots,\omega_r) \frac{dw}{w})
+\log&{\Gamma_{\!h}}^{(r)}(x;\omega_1,\dots,\omega_r)
\notag\\
&
{}=O(\exp(-2\pi a |x|)),\notag\end{aligned}$$ uniformly over any compact subset of the domain.
Clearly replacing $\arg(x)$ by $\arg(x-b\sum_i\omega_i)$ for $0<b<1$ will have no effect on the validity of the bound. We may thus take $$x=e(\tau)|y|+b\sum_i\omega_i.$$ Since $$\log{\Gamma_{\!h}}^{(r)}(e(\phi)|y| + b\sum_i\omega_i;\omega_1,\dots,\omega_r)
=
\log{\Gamma_{\!h}}^{(r)}(|y| + b\sum_i
e(-\phi)\omega_i;e(-\phi)\omega_1,\dots,e(-\phi)\omega_r),$$ we find that uniformity in $a,\tau,\omega_i$ will follow from uniformity in $a,\omega_i$ with $\tau=0$.
Thus assume $\Im(x)=b\sum_i\Im(\omega_i)$ for $0<b<1$. For every point in the domain, there exists $a'\ge a$ and $\epsilon>0$ such that the previous lemma applies, and such that there are no poles with imaginary part in $(a,a']$; by compactness, we can cover the domain by a finite number of such choices. Since $O(\exp(-2\pi a'\Re(x))) = O(\exp(-2\pi a\Re(x)))$, it suffices to consider the case $a'=a$. The result follows by residue calculus.
Note that the cut lines for $\log{\Gamma_{\!h}}^{(r)}$ can be taken along any direction $\omega_0$ in the convex cone generated by $\omega_1$,…,$\omega_r$, and the argument is valid for $x$ in the complement of the cones $$\omega_1+\cdots+\omega_r + {\mathbb R}_{\le 0}\langle \omega_1,\dots,\omega_r\rangle
\quad\text{and}\quad
-{\mathbb R}_{\le 0}\langle \omega_1,\dots,\omega_r\rangle,$$ and not approaching $\infty$ parallel to the boundary of the cone.
In particular, we obtain the following estimate.
\[cor:hyperb\_asympt\] Let $a$, $\arg(x)$, $\omega_i$ range over the domain $0<a<\min_i\Im(-e(\arg(x)/2\pi)/\omega_i)$. Then as $x\to\infty$, we have the estimates $$\begin{aligned}
-\pi\sqrt{-1}P^{(r)}(x;\omega_1,\dots,\omega_r)
+\log{\Gamma_{\!h}}^{(r)}(x;\omega_1,\dots,\omega_r)
&=
O(\exp(-2\pi a |x|)),\notag\\
\pi\sqrt{-1}P^{(r)}(-x;\omega_1,\dots,\omega_r)
+\log{\Gamma_{\!h}}^{(r)}(-x;\omega_1,\dots,\omega_r)
&=
O(\exp(-2\pi a |x|)),\notag\end{aligned}$$ uniformly on compact subsets.
The first estimate follows immediately from the theorem. The second estimate then follows using the facts $$P^{(r)}(x;\omega_1,\dots,\omega_r)
=
(-1)^r P^{(r)}(\sum_i\omega_i-x;\omega_1,\dots,\omega_r)$$ and $$\log{\Gamma_{\!h}}^{(r)}(x;\omega_1,\dots,\omega_r)
=
(-1)^{r-1}\log{\Gamma_{\!h}}^{(r)}(\sum_i\omega_i-x;\omega_1,\dots,\omega_r)$$
For $r=2$, this result is essentially due to Ruijsenaars [@RuijsenaarsSNM:1999 App. 2]. In general, on the domain $\Im(-e(\arg(x)/2\pi)/\omega_i)>0$, it should be possible to improve the error term to $$O(|x|^{r-1}\exp(-2\pi |x|\min_i\Im(-e(\arg(x)/2\pi)/\omega_i)),$$ by taking $a=\min_i \Im(-e(\arg(x)/2\pi)/\omega_i)$ and bounding the leading residues. In particular, if the poles with minimum imaginary part are all simple in the given compact subset of parameter space, then one easily has the bound $$O(\exp(-2\pi |x|\min_i\Im(-e(\arg(x)/2\pi)/\omega_i)))$$ on their residues, and thus on the error term.
We will also need an extension of this to the case that one of the moduli tends to 0. For convenience, write $$\gamma^{(r)}_h(x;\omega_1,\dots,\omega_r)
=
-\pi\sqrt{-1}P^{(r)}(x;\omega_1,\dots,\omega_r)
+\log{\Gamma_{\!h}}^{(r)}(x;\omega_1,\dots,\omega_r).$$
\[thm:gasympt:xbig\_vsmall\] Let $a$, $\arg(x)$, $\omega_i$,$\alpha$,$\beta$ range over the domain $$0<a<\min_{1\le i\le r-1}\Im(-e(\arg(x)/2\pi)/\omega_i),\quad
\Im(e(-\arg(x)/2\pi)/\omega_r)>0.$$ Then as $x\to\infty$, $v\to 0^+$, $$\begin{aligned}
\gamma_h^{(r)}(x+v\alpha\omega_r;\omega_1,\dots,\omega_{r-1},v\omega_r)
-\gamma_h^{(r)}(x+v\beta\omega_r;\omega_1,\dots,\omega_{r-1},v\omega_r)
-(\alpha-\beta)\gamma^{(r-1)}_h&(x;\omega_1,\dots,\omega_{r-1})\notag\\
&{}=
O(v\exp(-2\pi a|x|)),\notag\end{aligned}$$ uniformly over compact subsets of the domain.
For $r=1$, this is immediate, so take $r\ge 2$. As before, we may restrict our attention to the case $0<\Im(x)<\Im(\sum_{1\le i\le r-1}\omega_i)$. Then the left-hand side can be expressed as $$\int_{\Re(w)=a}
\left(
\frac{e(v\alpha\omega_r w)-e(v\beta\omega_r w)}{e(v\omega_r w)-1}
-
(\alpha-\beta)
\right)
\phi^{(r-1)}(w;x;\omega_1,\dots,\omega_{r-1})
\frac{dw}{w}$$ The desired estimate then follows as in Lemma \[lem:hyperb\_asympt\], using the fact that as $v\to 0^+$, $$\frac{1}{v}
\left(
\frac{e(v\alpha\omega_r w)-e(v\beta\omega_r w)}{e(v\omega_r w)-1}
-
(\alpha-\beta)
\right)$$ is $O(\exp(\epsilon |\Re(w)|))$ for any $\epsilon>0$, uniformly in $w$ and on compact subsets of parameter space.
\[cor:hyper\_to\_rat\] Let $x$, $\omega_1$,…,$\omega_r$, $\alpha$, $\beta$ range over the domain $\Im(\omega_i)>0$, $x\in {\cal
C}^{(r-1)}(\omega_r;\omega_1,\dots,\omega_{r-1})$. Then as $v\to 0^+$, we have the estimate $$\begin{aligned}
\log
\frac{{\Gamma_{\!h}}^{(r)}(x+v\alpha\omega_r;\omega_1,\dots,\omega_{r-1},v\omega_r)}
{{\Gamma_{\!h}}^{(r)}(x+v\beta\omega_r;\omega_1,\dots,\omega_{r-1},v\omega_r)}
-(\alpha-\beta)\log{\Gamma_{\!h}}^{(r-1)}(x;\omega_1,\dots,\omega_{r-1})
=
O(v),\notag\end{aligned}$$ uniformly over compact subsets of the domain.
Since $${\cal
C}^{(r-1)}(\omega_r;\omega_1,\dots,\omega_{r-1})
=
{\cal C}^{(r)}(\omega_r;\omega_1,\dots,\omega_r),$$ there is a canonical choice of branch for the left-hand side, and the proof of Theorem \[thm:gasympt:xbig\_vsmall\] gives the desired estimate on compact subsets of the complement of the cones $$\omega_1+\cdots+\omega_r + {\mathbb R}_{\le 0}\langle \omega_1,\dots,\omega_r\rangle
\quad\text{and}\quad
-{\mathbb R}_{\le 0}\langle \omega_1,\dots,\omega_r\rangle.$$ Since the estimate is consistent with the functional equation for ${\Gamma_{\!h}}^{(r)}$, the result follows.
In both cases, this can be extended to a uniform asymptotic series $$\log
\frac{{\Gamma_{\!h}}^{(r)}(x+v\alpha\omega_r;\omega_1,\dots,\omega_{r-1},v\omega_r)}
{{\Gamma_{\!h}}^{(r)}(x+v\beta\omega_r;\omega_1,\dots,\omega_{r-1},v\omega_r)}
=
\sum_{1\le k\le n}
\frac{B_k(\alpha)-B_k(\beta)}{k!}
\left(\omega_r v\frac{d}{dx}\right)^{k-1}
\log{\Gamma_{\!h}}^{(r-1)}(x;\omega_1,\dots,\omega_{r-1})
+
O(v^n),$$ with error $O(v^n \exp(-2\pi a|x|))$ if also $x\to\infty$; here $B_k(x)$ is the $k$th (ordinary) Bernoulli polynomial.
In fact, with care, we can give an estimate valid on the larger domain ${\cal C}^{\prime(r-1)}(\omega_r;\omega_1,\dots,\omega_{r-1})$, and thus in particular in a neighborhood of 0. The point is that we can identify the zeros and poles of ${\Gamma_{\!h}}^{(r)}$ and ${\Gamma_{\!h}}^{(r-1)}$ that give rise to the cut line $\omega_r{\mathbb R}_{\le 0}$ above, and using the ordinary gamma function, cancel them out. The resulting asymptotics can then still be computed via Stirling’s formula. It will be convenient to use a slightly renormalized form of the ordinary gamma function; we define $${\Gamma_{\!r}}(x;\omega) := \frac{\Gamma(x/\omega)}{\sqrt{2\pi}},$$ with the convention that $\omega=1$ if omitted, and the usual convention on multiple arguments. As a justification for this convention, note that the reflection identity for the ordinary gamma function becomes $${\Gamma_{\!r}}(x,\omega-x;\omega) = {\Gamma_{\!h}}^{(1)}(x;\omega)^{-1}.$$ We also let $\log{\Gamma_{\!r}}(x;\omega)$ denote the standard branch on ${\mathbb C}\setminus \omega{\mathbb R}_{\le 0}$, and note the following version of Stirling’s formula.
Let $x\notin {\mathbb R}_{\le 0}$. Then for all $m\ge 1$, $$\log{\Gamma_{\!r}}(x)
=
(x\log(x)-x) - \frac{1}{2}\log(x)
+
\sum_{1\le i<m} \frac{B_{2i} x^{1-2i}}{2i(2i-1)}
+
O(d(x,{\mathbb R}_{\le 0})^{1-2m}),$$ uniformly in $x$ as $d(x,{\mathbb R}_{\le 0})\to\infty$. More generally, for $\alpha$ ranging over any compact subset of ${\mathbb C}$, $$\log{\Gamma_{\!r}}(x+\alpha)
=
(x\log(x)-x) +B_1(\alpha)\log(x)
+
\sum_{1\le i<m} \frac{(-1)^{i+1} B_{i+1}(\alpha)}{i(i+1) x^i}
+
O(d(x,{\mathbb R}_{\le 0})^{-m}),$$ uniformly as $d(x,{\mathbb R}_{\le 0})\to\infty$.
The claim for general $\alpha$ follows from the claim for $\alpha=0$ by straightforward algebraic manipulation. For $\alpha=0$, we observe that by [@OlverFWJ:1974 §8.4] or [@AndrewsGE/AskeyR/RoyR:1999 Thm. 1.4.2], the error term is $$\frac{B_{2m} x^{1-2m}}{2m(2m-1)}
-
\frac{1}{2m}
\int_0^\infty
\frac{B_{2m}(\{t\})}{(x+t)^{2m}}
dt.$$ The first term certainly has the correct asymptotics; for the second term, we have $$\frac{1}{2m}
\int_0^\infty
\frac{B_{2m}(\{t\})}{(x+t)^{2m}}
dt
=
O(\int_0^\infty |x+t|^{-2m} dt).$$ For $\Re(x)\ge 0$, $|x+t|^{-2m}\le (|x|^2+t^2)^{-m}$, and thus the integral has order $|x|^{1-2m}$. For $\Re(x)\le 0$, the integral is still bounded above by $$\int_{-\infty}^{\infty} (\Im(x)^2+t^2)^{-m} dx = O(|\Im(x)|^{1-2m}).$$
More generally, [@OlverFWJ:1974 Ex. 8.4.4] gives the error term $$\frac{(-1)^{m+1} B_{m+1}(\alpha) x^{-m}}{2m(2m+1)}
-
\frac{1}{m+1}
\int_0^\infty
\frac{B_{m+1}(\{t-\alpha\})}{(x+t)^{m+1}}
dt$$ for $0\le \alpha\le 1$.
Let $x,\omega$,$\alpha$,$\beta$ range over the domain $\omega\ne 0$, $x\in
{\mathbb C}\setminus \omega{\mathbb R}_{\le 0}$. Then as $v\to 0^+$, $$\begin{aligned}
\log
\frac{{\Gamma_{\!r}}(x+v\alpha\omega;v\omega)}
{{\Gamma_{\!r}}(x+v\beta\omega;v\omega)}
={}&
(B_1(\alpha)-B_1(\beta))\log(x/v\omega)
+
\sum_{2\le i<m} \frac{(-1)^i (B_i(\alpha)-B_i(\beta)) (x/v\omega)^{1-i}}{i(i-1)}\notag\\
&+
O(d(x,\omega{\mathbb R}_{\le 0})^{-m}v^m),\notag\end{aligned}$$ uniformly in $x$ and over compacta in $\omega$, $\alpha$, $\beta$.
\[thm:hyper\_to\_rat\] Let $x$, $\omega_1$,…,$\omega_r$, $\alpha$,$\beta$ range over the domain $\Im(\omega_i)>0$, $x\in {\cal
C}^{\prime (r-1)}(\omega_r;\omega_1,\dots,\omega_{r-1})$. Then as $v\to 0^+$, we have the estimate $$\begin{aligned}
\log
\frac{{\Gamma_{\!h}}^{(r)}(x+v\alpha\omega_r;\omega_1,\dots,\omega_{r-1},v\omega_r)}
{{\Gamma_{\!h}}^{(r)}(x+v\beta\omega_r;\omega_1,\dots,\omega_{r-1},v\omega_r)}
&{}-
(-1)^r
\log\frac{{\Gamma_{\!r}}(x+v\alpha\omega_r;v\omega_r)}
{{\Gamma_{\!r}}(x+v\beta\omega_r;v\omega_r)}\notag\\
&{}-(\alpha-\beta)
\left(
\log{\Gamma_{\!h}}^{(r-1)}(x;\omega_1,\dots,\omega_{r-1})
-(-1)^r\log(x/v\omega_r)
\right)
=
O(v),\notag\end{aligned}$$ uniformly over compact subsets of the domain.
We first observe that $$\log{\Gamma_{\!h}}^{(r)}(x+v\alpha\omega_r;\omega_1,\dots,\omega_{r-1})
-(-1)^r\log{\Gamma_{\!r}}(x+v\alpha\omega_r;v\omega_r)$$ and $$\log{\Gamma_{\!h}}^{(r-1)}(x;\omega_1,\dots,\omega_{r-1})
-(-1)^r\log(x/v\omega_r)$$ are analytic on the given domain, and thus the overall left-hand side is analytic. Moreover, the stated estimate holds on the smaller domain ${\cal
C}^{(r-1)}(\omega_r;\omega_1,\dots,\omega_{r-1})$. Using the functional equation, we may immediately extend this to the full domain ${\cal
C}^{\prime(r-1)}(\omega_r;\omega_1,\dots,\omega_{r-1})$, except when $r=2$, where the point $x=0$ must still be excluded (since the only points related to $x=0$ via the functional equation are also in the cut set). But in that case, we may simply use Cauchy’s theorem to deduce a uniform estimate on a neighborhood of $0$ from a uniform estimate on the boundary of the neighborhood.
More generally one has an asymptotic series in which the coefficient of the $k$th term depends on the $(k-1)$-st derivative of $$\log{\Gamma_{\!h}}^{(r-1)}(x;\omega_1,\dots,\omega_{r-1})
-(-1)^r\log(x/v\omega_r).$$
If $r=2$, we have $$\lim_{x\to 0} \log{\Gamma_{\!h}}^{(2)}(x;\omega_1,\omega_2)-\log{\Gamma_{\!r}}(x;\omega_2)
=
\frac{\log(\omega_1)-\log(\omega_2)-\log(2\pi)}{2},$$ which gives rise to a nice corollary by taking $x=\beta=0$ above.
[@RuijsenaarsSNM:1999]\[cor:hyper\_to\_rat2\] As $v\to 0^+$, $$\log{\Gamma_{\!h}}^{(2)}(v\alpha\omega_2;\omega_1,v\omega_2)
-\log{\Gamma_{\!r}}(\alpha)
-(\alpha-1/2)\log(2\pi v\omega_2/\omega_1)
=
O(v)$$ uniformly over compact subsets of the region $\Im(\omega_1),\Im(\omega_2)>0$, $\alpha\in {\mathbb C}$.
Now, consider the elliptic gamma function, defined as $${\Gamma_{\!e}}^{(r)}(z;p_1,p_2,\dots,p_r)
=
\prod_{0\le k_1,k_2,\dots,k_r}
(1-p_1^{k_1+1}p_2^{k_2+1}\cdots p_r^{k_r+1}/z)
(1-p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r} z)^{(-1)^{r-1}}.$$ For $|p_1p_2\cdots p_r|<|z|<1$, we have $$\log{\Gamma_{\!e}}^{(r)}(z;p_1,p_2,\dots,p_r)
=
\sum_{1\le k}
\frac{(-1)^r z^k-(p_1p_2\cdots p_r/z)^k}
{k\prod_{1\le i\le r} (1-p_i^k)};$$ this, then, for any $\omega_0$ with $\Im(\omega_0)>0$, defines a branch of $\log{\Gamma_{\!e}}^{(r)}$ on the region ${\cal
C}^{(r)}_e(\omega_0;p_1,\dots,p_r)$ obtained from ${\mathbb C}^*={\mathbb C}\setminus
\{0\}$ by removing the countable union of logarithmic spirals $$p_1^{{\mathbb Z}_{\ge 1}}p_2^{{\mathbb Z}_{\ge 1}}\cdots p_r^{{\mathbb Z}_{\ge 1}}e(\omega_0{\mathbb R}_{\ge
0})
\quad\text{and}\quad
p_1^{{\mathbb Z}_{\le 0}}p_2^{{\mathbb Z}_{\le 0}}\cdots p_r^{{\mathbb Z}_{\le 0}}e(\omega_0{\mathbb R}_{\le
0}).$$ (We define a region ${\cal C}^{\prime(r)}_e$ analogously.) And, of course, we have the functional equation $$\log{\Gamma_{\!e}}^{(r)}(p_rz;p_1,p_2,\dots,p_r)
-
\log{\Gamma_{\!e}}^{(r)}(z;p_1,p_2,\dots,p_r)
=
\log{\Gamma_{\!e}}^{(r-1)}(z;p_1,p_2,\dots,p_{r-1}),$$ with $$\log{\Gamma_{\!e}}^{(0)}(z;)=\log(-1/z),\quad \log{\Gamma_{\!e}}^{(0)}(-1;)=0.$$
Narukawa [@NarukawaA:2004 Theorem 14] gives the following “product” expansion of the elliptic gamma function in terms of the hyperbolic gamma function: $$\begin{aligned}
-2\pi\sqrt{-1}Q^{(r)}(x;\omega_1,\dots,\omega_r)
+
{}&\log{\Gamma_{\!e}}^{(r)}(e(x);e(\omega_1),\dots,e(\omega_r))\notag\\
&{}=
\sum_{0\le k}
-\pi\sqrt{-1}P^{(r)}(x+k+1;\omega_1,\dots,\omega_r)
+\log{\Gamma_{\!h}}^{(r)}(x+k+1;\omega_1,\dots,\omega_r)\notag\\
&+\sum_{0\le k}
\pi\sqrt{-1}P^{(r)}(x-k;\omega_1,\dots,\omega_r)
+\log{\Gamma_{\!h}}^{(r)}(x-k;\omega_1,\dots,\omega_r)\notag\end{aligned}$$ Note that each term in the infinite sums converges uniformly exponentially to 0 as $k\to\infty$, so the sums converge uniformly and absolutely.
Using this expansion and the asymptotics of the hyperbolic gamma function, we obtain the following estimates for the elliptic gamma function. First, the hyperbolic limit $p_1,\dots,p_r\to 1$.
\[prop:ell\_to\_hyper2\] Let $A$, $\omega_1,\dots,\omega_r$, $x$ range over the domain $$0<A<\min_{1\le i\le r} \Im(-1/\omega_i)-|\Im(x/\omega_i)|.$$ Then as $v\to 0^+$, we have the estimate $$-2\pi\sqrt{-1}R^{(r)}(x;v\omega_1,\dots,v\omega_r)
+\log{\Gamma_{\!e}}^{(r)}(e(x);e(v\omega_1),\dots,e(v\omega_r))
-\log{\Gamma_{\!h}}^{(r)}(x;v\omega_1,\dots,v\omega_r)
=
O(\exp(-2\pi A/v)),$$ uniform over compact subsets of the domain.
If we do not care about the choice of $A$, the constraint on the domain is simply $$|\Im(x/\omega_i)|<\Im(-1/\omega_i), 1\le i\le r;$$ this is a parallelogram, two of the vertices of which are $\pm 1$.
\[prop:ell\_to\_hyper1\] Let $A$, $\omega_1,\dots,\omega_r$, $x$ range over the domain $$0<A<\min_{1\le i\le r} \min(\Im(-x/\omega_i),\Im((x-1)/\omega_i)).$$ Then as $v\to 0^+$, we have the estimate $$-2\pi\sqrt{-1}Q^{(r)}(x;v\omega_1,\dots,v\omega_r)
+\log{\Gamma_{\!e}}^{(r)}(e(x);e(v\omega_1),\dots,e(v\omega_r))
=
O(\exp(-2\pi A/v)),$$ uniform over compact subsets of the domain.
Using the “product” expansion and the identity $$\log{\Gamma_{\!h}}^{(r)}(z;v\omega_1,\dots,v\omega_r)
=
\log{\Gamma_{\!h}}^{(r)}(z/v;\omega_1,\dots,\omega_r),$$ we can express each left-hand side as a sum over functions to which Corollary \[cor:hyperb\_asympt\] applies, giving the desired uniform asymptotics.
Using this limit, we can obtain the following bound.
\[prop:ell\_bound\_hyper\] Fix a compact subset $S$ of the set of $\omega_i$ such that $0<\Im(\omega_i)$, and constants $0<\epsilon,C_1,C_2$. Then as $v\to 0^+$, we have the following estimate, uniform over the product of $S$ with the region $-1-vC_1\le \Re(x)\le vC_1$, $|\Im(x)|\le v C_2$, apart from a hole of radius $\epsilon v$ around every pole of the left-hand side: $${\Gamma_{\!e}}^{(r)}(e(x);e(v\omega_1),\dots,e(v\omega_r))^{\pm 1}
=
O(e(\pm Q^{(r)}(x;v\omega_1,\dots,v\omega_r))).$$
We consider the $+$ case; the $-$ case is completely analogous. Choose $1/2<D<1$ and $0<a<\min_i\Im(-1/\omega_i)$. Proposition \[prop:ell\_to\_hyper2\] gives $$e(-Q^{(r)}(x;v\omega_1,\dots,v\omega_r))
{\Gamma_{\!e}}^{(r)}(e(x);e(v\omega_1),\dots,e(v\omega_r))
=
O(
e(\frac{1}{2}P^{(r)}(x;\omega_1,\dots,\omega_r))
{\Gamma_{\!h}}^{(r)}(x;v\omega_1,\dots,v\omega_r)
)$$ away from the poles, in the subregion $-D\le \Re(x)\le vC_1$. Since $$e(\frac{1}{2}P^{(r)}(x;v\omega_1,\dots,v\omega_r))
{\Gamma_{\!h}}^{(r)}(x;v\omega_1,\dots,v\omega_r)$$ is uniformly bounded in that region (we have excluded neighborhoods of the poles, and it converges uniformly to 1 for $-\Re(x)/v$ large), we have the uniform estimate $$e(-Q^{(r)}(x;v\omega_1,\dots,v\omega_r))
{\Gamma_{\!e}}^{(r)}(e(x);e(v\omega_1),\dots,e(v\omega_r))
=
O(1)$$ in this region. A similar application of Proposition \[prop:ell\_to\_hyper2\] gives $$\begin{aligned}
e(-Q^{(r)}(x+1;v\omega_1,\dots,v\omega_r))
&{\Gamma_{\!e}}^{(r)}(e(x);e(v\omega_1),\dots,e(v\omega_r))
\notag\\
&=
O(
e(\frac{1}{2}P^{(r)}(x+1;v\omega_1,\dots,v\omega_r))
{\Gamma_{\!h}}^{(r)}(x+1;v\omega_1,\dots,v\omega_r)
)
\notag\end{aligned}$$ uniformly over the subset $-vC_1\le \Re(x)+1\le D$. Since $$Q^{(r)}(x+1;v\omega_1,\dots,v\omega_r)
=
Q^{(r)}(x;v\omega_1,\dots,v\omega_r)
-
P^{(r)}(x+1;v\omega_1,\dots,v\omega_r),$$ we find $$\begin{aligned}
e(-Q^{(r)}(x;v\omega_1,\dots,v\omega_r))
&{\Gamma_{\!e}}^{(r)}(e(x);e(v\omega_1),\dots,e(v\omega_r))
\notag\\
&=
O(
e(-\frac{1}{2}P^{(r)}(x+1;v\omega_1,\dots,v\omega_r))
{\Gamma_{\!h}}^{(r)}(x+1;v\omega_1,\dots,v\omega_r)
)
=
O(1)
\notag\end{aligned}$$ on this region as well.
Similarly, we can obtain asymptotics of ${\Gamma_{\!e}}^{(r)}$ in the “rational” limit $p_r\to 1$, $p_1$,…$p_{r-1}$ fixed.
\[thm:ell\_to\_rat1\] Let $x$, $p_1$,…,$p_{r-1}$, $\omega_r$, $\alpha$, $\beta$ range over the domain $0<|p_1|,\dots,|p_{r-1}|<1$, $\Im(\omega_r)>0$, and $$x\in e^{-1}({\cal C}^{(r-1)}_e(\omega_r;p_1,\dots,p_{r-1}))
\cup
(\omega_r{\mathbb R}_{\ge 0}\cap e^{-1}({\cal C}^{\prime
(r-1)}_e(\omega_r;p_1,\dots,p_{r-1})).$$ Then as $v\to 0^+$, we have the estimate $$\begin{aligned}
\log
\frac{{\Gamma_{\!e}}^{(r)}(e(x+v\alpha\omega_r);p_1,\dots,p_{r-1},e(v\omega_r))}
{{\Gamma_{\!e}}^{(r)}(e(x+v\beta\omega_r);p_1,\dots,p_{r-1},e(v\omega_r))}
&{}-
(-1)^r
\log\frac{{\Gamma_{\!r}}(x+v\alpha\omega_r;v\omega_r)}
{{\Gamma_{\!r}}(x+v\beta\omega_r;v\omega_r)}\notag\\
{}-(\alpha-\beta)&
\left(
\log{\Gamma_{\!e}}^{(r-1)}(e(x);p_1,\dots,p_{r-1})
-(-1)^r\log(x/v\omega_r)
\right)
=
O(v),\notag\end{aligned}$$ uniformly over compact subsets of the domain.
If we expand the elliptic gamma functions via the product representation and group corresponding terms, we find by Corollary \[cor:hyper\_to\_rat\] that all but one term of the result is uniformly $O(v)$. Moreover, it follows from Theorem \[thm:gasympt:xbig\_vsmall\] that the coefficient of $v$ in the estimates is exponentially small as $k\to \infty$, and thus the error terms are summable. The only surviving term can be estimated using Theorem \[thm:hyper\_to\_rat\], giving the desired result.
\[cor:ell\_to\_rat2\] As $v\to 0^+$, $$\log
\frac{
{\Gamma_{\!e}}^{(2)}(e(\alpha v\omega);p,e(v\omega))
}{
{\Gamma_{\!r}}(\alpha)
}
=
\frac{\pi\sqrt{-1}}{12v\omega}
+
(\alpha-1/2)
\log(2\pi v\omega (p;p)^2/\sqrt{-1})
+
O(v),$$ uniformly over compact subsets of the region $\Im(\omega)>0$, $0<|p|<1$, $\alpha\in {\mathbb C}$.
Similarly, one has the following.
\[cor:ell\_to\_rat3\] As $v\to 0^+$, $$\frac{
{\Gamma_{\!e}}^{(r)}(e(\alpha v\omega) z;p_1,\dots,p_{r-1},e(v\omega))
}{
{\Gamma_{\!e}}^{(r)}(e(\beta v\omega) z;p_1,\dots,p_{r-1},e(v\omega))
}
=
{\Gamma_{\!e}}^{(r-1)}(z;p_1,\dots,p_{r-1})^{\alpha-\beta}
(1+O(v))$$ uniformly over compact subsets of the domain $0<|p_1|,\dots,|p_{r-1}|<1$, $\Im(\omega)>0$, $|p_1\cdots p_{r-1}|<z\le \min_{1\le i\le r-1}
|p_i|^{-1}$, $z\ne 1$, $\alpha,\beta\in {\mathbb C}$.
\[cor:ell\_bound\_rat\] As $v\to 0^+$, the function $$\frac{
{\Gamma_{\!e}}^{(r)}(e(v\alpha\omega) z;p_1,\dots,p_r,e(v\omega))
}{
{\Gamma_{\!e}}^{(r)}(e(v\beta\omega) z;p_1,\dots,p_r,e(v\omega))
}$$ is uniformly bounded over compact subsets of the domain $0<|p_1|,\dots,|p_{r-1}|<1$, $\Im(\omega)>0$, $|p_1\cdots p_{r-1}|<z<1$, $\alpha,\beta\in{\mathbb C}$. On compact subsets of the domain $0<|p_1|,\dots,|p_{r-1}|<1$, $\Im(\omega)>0$, $|p|<z\le 1$, $\alpha,\beta\in {\mathbb C}$, it is uniformly $$O(v^{\min(\Re(\alpha-\beta),0)}).$$
There are corresponding estimates for the trigonometric gamma function $${\Gamma_{\!t}}(z;q) := \prod_{0\le k} (1-q^k z)^{-1} = \frac{1}{(z;q)_\infty},$$ with the corresponding analytic continuation of its logarithm (the branch with $\log{\Gamma_{\!t}}(0;q)=0$).
[@KoornwinderTH:1990]\[lem:trig\_to\_rat0\] Let $z$, $\omega$, $\alpha$, $\beta$ range over the domain $\Im(\omega)>0$, $0\le |z|<1$. Then as $v\to 0$, we have the estimate $$\log
\frac{{\Gamma_{\!t}}(e(v\alpha\omega)z;e(v\omega))}
{{\Gamma_{\!t}}(e(v\beta\omega)z;e(v\omega))}
-
(\alpha-\beta)\log(1-z)
=
O(vz),$$ uniformly over compact subsets of the domain.
Indeed, for $|z|<1$, the left-hand side is given by the sum $$\sum_{k\ge 1}
\frac{z^k}{k}
\left(
\frac{e(kv\alpha\omega)-e(kv\beta\omega)}
{1-e(kv\omega)}
+
\alpha-\beta
\right)$$ and the quantity in parentheses is uniformly $O(v\exp(\epsilon k))$ for all $\epsilon>0$.
This gives rise to a trigonometric analogue of Theorem \[thm:ell\_to\_rat1\].
For $r>1$, let $z$, $p_1$,…,$p_{r-1}$, $\omega$, $\alpha$, $\beta$ range over the domain $0\le |p_1|,\dots,|p_{r-1}|<1$, $\Im(\omega)>0$, and $z\in {\cal C}^{\prime(r-1)}_e(\omega;p_1,\dots,p_{r-1})$. Then as $v\to 0^+$, we have the estimate $$\begin{aligned}
\log
\frac{{\Gamma_{\!e}}^{(r)}(e(v\alpha\omega)z;p_1,\dots,p_{r-1},e(v\omega))}
{{\Gamma_{\!e}}^{(r)}(e(v\beta\omega)z;p_1,\dots,p_{r-1},e(v\omega))}
&{}-
(-1)^r
\log\frac{{\Gamma_{\!t}}(e(v\alpha\omega)z;e(v\omega))}
{{\Gamma_{\!t}}(e(v\beta\omega)z;e(v\omega))}\notag\\
{}-(\alpha-\beta)&
\left(
\log{\Gamma_{\!e}}^{(r-1)}(z;p_1,\dots,p_{r-1})
-(-1)^r\log(1-z)
\right)
=
O(v),\notag\end{aligned}$$ uniformly over compact subsets of the domain.
For $|p_1p_2\cdots p_{r-1}|<z<\min_i |p_i|^{-1}$, this follows immediately from the expansion $$\log{\Gamma_{\!e}}^{(r)}(z;p_1,\dots,p_{r-1},q)
=
-
\sum_{0\le k_1,k_2,\dots,k_{r-1}}
{\Gamma_{\!t}}(p_1^{k_1+1}\cdots p_{r-1}^{k_{r-1}+1}/z;q)
+(-1)^{r-1}{\Gamma_{\!t}}(p_1^{k_1}\cdots p_{r-1}^{k_{r-1}}z;q)$$ and the asymptotics of ${\Gamma_{\!t}}$.
The general case follows from the functional equation.
Comparing this to Theorem \[thm:ell\_to\_rat1\] gives the following result, a uniform version of the results of the appendices of [@KoornwinderTH:1990].
\[cor:trig\_to\_rat\] Let $x$, $\omega$, $\alpha$, $\beta$ range over the domain $\Im(\omega)>0$, $$x\in \bigl({\mathbb C}\setminus ({\mathbb Z}+\omega{\mathbb R}_{\ge 0})\bigr)\cup \omega {\mathbb R}_{\ge 0}.$$ Then as $v\to 0^+$, we have the estimate $$\log
\frac{{\Gamma_{\!t}}(e(x+v\alpha\omega);e(v\omega))}
{{\Gamma_{\!t}}(e(x+v\beta\omega);e(v\omega))}
-
\log
\frac{{\Gamma_{\!r}}(x+v\alpha\omega;v\omega)}
{{\Gamma_{\!r}}(x+v\beta\omega;v\omega)}
-
(\alpha-\beta)
\bigl(\log(1-e(x))-\log(x/v\omega)\bigr)
=
O(v),$$ uniformly over compact subsets of the domain.
Note that the validity of the theorem for $r=2$, $|p|<z<|p|^{-1}$ is enough to give the corollary in general, and in turn give the theorem in general, without using the functional equation. This also implies that Theorem \[thm:ell\_to\_rat1\] and its corollaries continue to hold even without the constraint $0<|p_i|$ on the domain, and further implies that Lemma \[lem:trig\_to\_rat0\] holds on the domain $z\notin e(\omega{\mathbb R}_{\ge 0})$.
[@KoornwinderTH:1990] Let $\omega$, $\alpha$ range over the domain $\Im(\omega)>0$, $\alpha\in {\mathbb C}$. Then as $v\to 0^+$, we have the estimate $$\log
\frac{{\Gamma_{\!t}}(e(v\alpha\omega);e(v\omega))}
{{\Gamma_{\!r}}(\alpha)}
=
\frac{\pi\sqrt{-1}}{12 v\omega}
+
(\alpha-1/2)\log(2\pi v\omega/\sqrt{-1})
+
O(v),$$ uniformly over compact subsets of the domain.
Generalized triangle inequalities
=================================
When taking limits of elliptic hypergeometric integrals, the first step is naturally to determine which part of the contour makes the most significant contribution to the integral. We first note the following consequence of Proposition \[prop:ell\_bound\_hyper\]. Since in the sequel, we will only be using gamma functions for $r\le 3$, we will write $\theta$ for $\Gamma^{(1)}_e$, ${\Gamma_{\!e}}$ for $\Gamma^{(2)}_e$, and ${\Gamma_{\!e}}^+$ for $\Gamma^{(3)}_e$, and similarly (with a subscript $h$) for the hyperbolic versions; we will also omit the superscript $(r)$ on $P$, $Q$, and $R$.
\[cor:ell\_bound\_hyper\] For any parameters $\mu$, $\nu$, and any real number $x$, we have, as $v\to
0^+$, the estimates (uniform over compact subsets avoiding the poles) $$\begin{aligned}
\frac{{\Gamma_{\!e}}(e(v\mu+x),e(v\nu-x);e(v\omega_1),e(v\omega_2))}
{e(R(v\mu;v\omega_1,v\omega_2)+R(v\nu;v\omega_1,v\omega_2))}
&=
O(e(\frac{\mu+\nu-\omega_1-\omega_2}{2v\omega_1\omega_2}\vartheta(x))),\notag\\
\frac{e(R(v\nu;v\omega_1,v\omega_2))
{\Gamma_{\!e}}(e(v\mu+x);e(v\omega_1),e(v\omega_2))}
{e(R(v\mu;v\omega_1,v\omega_2))
{\Gamma_{\!e}}(e(v\nu+x);e(v\omega_1),e(v\omega_2))}
&=
O(e(\frac{\mu-\nu}{2v\omega_1\omega_2}\vartheta(x))),\notag\\
\frac{e(R(v\mu;v\omega_1,v\omega_2)+R(v\nu;v\omega_1,v\omega_2))}
{{\Gamma_{\!e}}(e(v\mu+x),e(v\nu-x);e(v\omega_1),e(v\omega_2))}
&=
O(e(\frac{\omega_1+\omega_2-\mu-\nu}{2v\omega_1\omega_2}\vartheta(x))),\notag\end{aligned}$$ where $\vartheta(x)$ is the continuous, even, periodic function defined by $$\vartheta(x)=\{x\}(1-\{x\})=\{x\}\{-x\}.$$ Similarly, $$e(-R(v\mu;v\omega))
\theta(e(v\mu+x);e(v\omega))
=
O(e(\vartheta(x)/2v\omega)),$$ and likewise for the reciprocal (avoiding the poles).
Note that in the above bounds, we can ignore $O(1)$ terms in $R$, and may thus replace $$\begin{aligned}
R(v\mu;v\omega) &\mapsto -1/12v\omega\notag\\
R(v\mu;v\omega_1,v\omega_2) &\mapsto
(\omega_1+\omega_2-2\mu)/24v\omega_1\omega_2.\notag\end{aligned}$$ Also, although we assume $x$ real, the above estimates are clearly still valid if we make a $O(v)$ perturbation to $x$ on the left-hand sides.
We will thus require some inequalities involving this quantity $\vartheta$.
\[lem:gen\_tri\_ell\] For any sequences $c_1$,…,$c_n$, $d_1$,…,$d_n$ of real numbers, we have the inequality $$\sum_{1\le i,j\le n} \vartheta(c_i-d_j)
-\sum_{1\le i<j\le n} \vartheta(c_i-c_j)
-\sum_{1\le i<j\le n} \vartheta(d_i-d_j)
\ge
\vartheta(\sum_{1\le i\le n} c_i-d_i),$$ with equality if and only if the sequences interlace in ${\mathbb R}/{\mathbb Z}$; that is, iff they can be permuted so that either $$\{c_1\}\le \{d_1\}\le \{c_2\}\le\cdots\le\{d_{n-1}\}\le \{c_n\}\le \{d_n\}$$ or $$\{d_1\}\le \{c_1\}\le \{d_2\}\le\cdots\le\{c_{n-1}\}\le \{d_n\}\le \{c_n\}.$$
First, observe that if two elements $c_i$, $d_j$ agree modulo ${\mathbb Z}$, then their contributions to the inequality cancel, and the result thus follows by induction. We may therefore assume that $c_i\ne d_j$ for all $1\le
i,j\le n$.
Now, consider the asymptotics as $v\to 0^+$ of the case $\tau=\omega/2$ of the determinant identity [@FrobeniusG:1882]: $$\begin{aligned}
&\det_{1\le i,j\le n}(
\frac{e(-1/12v\omega)\theta(e(v\tau+c'_i-d'_j);e(v\omega))}
{\theta(e(v\tau),e(c'_i-d'_j);e(v\omega))}
)\notag\\
&\qquad=
(-1)^{n(n-1)/2}
\frac{\theta(e(v\tau+\sum_i c'_i-d'_i);e(v\omega))}
{e(n/12v\omega)\theta(e(v\tau);e(v\omega))}
\frac{\prod_{1\le i<j\le n} e(c'_j-d'_i)\theta(e(c'_i-c'_j),e(d'_i-d'_j);e(v\omega))}
{\prod_{1\le i,j\le n} \theta(e(c'_i-d'_j);e(v\omega))},
\notag\end{aligned}$$ where real constants $(c'_i-c_i)/v$, $(d'_i-d_i)/v$ are chosen so that the $2n$ quantities $c'_i$, $d'_i$ are all distinct for sufficiently small $v$. Now, since $c_i\ne d_j$, we have $$\lim_{v\to 0^+}
\frac{e(-1/12v\omega)\theta(e(v\omega/2+c'_i-d'_j);e(v\omega))}
{\theta(e(v\omega/2),e(c'_i-d'_j);e(v\omega))}
=
\frac{1}{2}\operatorname{sgn}(\{c_i\}-\{d_j\})e(-(\{c_i\}-\{d_j\})/2+1/4).$$ In particular, the determinant converges to a well-defined limit. Moreover, this limit is nonzero iff the sequences interlace, as follows by considering the rescaled determinant $\det_{1\le i,j\le
n}(\operatorname{sgn}(\{c_i\}-\{d_j\}))$. (Indeed, if the sequences fail to interlace, two rows or columns must agree; otherwise, the $n$ distinct rows are easily verified to be linearly independent.)
On the other hand, we have the estimates $$\begin{aligned}
\frac{\theta(e(v\omega/2+x);e(v\omega))}
{\theta(e(v\omega/2);e(v\omega))}
&=
\Theta(e(\vartheta(x)/2v\omega))\notag\\
e(1/12v\omega)\theta(e(x);e(v\omega))
&=\Theta(e(\vartheta(x)/2v\omega))
\qquad \text{assuming}\ |\{x\}|,|\{1-x\}|=\Omega(v),
\notag\end{aligned}$$ giving the estimate $$\Theta(e((\vartheta(\sum_i c_i-d_i)+\sum_{1\le i<j\le n}
(\vartheta(c_i-c_j)+\vartheta(d_i-d_j))-\sum_{1\le i,j\le n}\vartheta(c_i-d_j))/2v\omega))$$ for the right-hand side. Since $\Im(-1/\omega)>0$, this is bounded as $v\to 0^+$ iff $$\vartheta(\sum_i c_i-d_i)
+\sum_{1\le i<j\le n}(\vartheta(c_i-c_j)+\vartheta(d_i-d_j))
-\sum_{1\le i,j\le n}\vartheta(c_i-d_j)
\le 0,$$ and is bounded away from 0 iff equality holds. The result follows.
More precise asymptotic calculations give the following well-known (and easy) identity (valid for $n\ge 1$) as a limit of the elliptic Cauchy determinant: $$\det_{1\le i,j\le n}(\operatorname{sgn}(x_i-y_j))
=
2^{n-1}
(-1)^{n(n-1)}
\prod_{1\le i,j\le n} \operatorname{sgn}(x_i-y_j)
\prod_{1\le i<j\le n} \operatorname{sgn}(x_i-x_j)\operatorname{sgn}(y_i-y_j)$$ for interlacing sequences with distinct elements, and 0 otherwise. Similarly, for more general values of $\tau$, we obtain the identity $$\det_{1\le i,j\le n}(x^{\operatorname{sgn}(c_i-d_j)})
=
(-1)^{n(n-1)/2} (x-1/x)^{n-1}
\operatorname{sgn}(\sum_i c_i-d_i) x^{\operatorname{sgn}(\sum_i c_i-d_i)}
\frac{\prod_{1\le i<j\le n} \operatorname{sgn}(c_i-c_j)\operatorname{sgn}(d_i-d_j)}
{\prod_{1\le i,j\le n} \operatorname{sgn}(c_i-d_j)}.$$
We can also obtain a version with hyperoctahedral symmetry.
\[lem:gen\_tri\_ellC\] For any sequences $c_0$,…,$c_n$, $d_1$,…,$d_n$ of real numbers, we have the inequality $$\sum_{0\le i\le n,1\le j\le n} \vartheta( c_i\pm d_j)
-\sum_{0\le i<j\le n} \vartheta(c_i\pm c_j)
-\sum_{1\le i<j\le n} \vartheta(d_i\pm d_j)
-\sum_{1\le i\le n} \vartheta(2d_i)
\ge
0,$$ with equality iff the sequences can be permuted so that $$\min(\{\pm c_0\})
\le
\min(\{\pm d_1\})
\le
\min(\{\pm c_1\})
\le\cdots\le
\min(\{\pm c_{n-1}\})
\le
\min(\{\pm d_n\})
\le
\min(\{\pm c_n\}).$$ Here $\vartheta(x\pm y):=\vartheta(x+y)+\vartheta(x-y)$ and $\min(\{\pm
x\}):=\min(\{x\},\{-x\})$.
Apply the preceding lemma to the sequences $\pm c_i$ and $0,\pm d_i,1/2$, and use the identity $$\vartheta(2x)=2(\vartheta(x)+\vartheta(x+1/2)-\vartheta(1/2)).$$ The given equality condition simply restates the condition that $\pm c_i$ and $0,\pm d_i,1/2$ interlace.
\[cor:gen\_tri\_ellC\] For any integer $n\ge 1$ and any sequence $e_1,\cdots,e_n$ of real numbers, we have the inequality $$2\sum_{1\le i<j\le n} \vartheta(e_i\pm e_j)
-(n-1)\sum_{1\le i\le n} \vartheta(2e_i)
\ge
0$$ with equality iff the sequence $\min(\{\pm e_i\})$ is constant.
The case $c_0=0$, $n=1$ of Lemma \[lem:gen\_tri\_ellC\] implies that $$\vartheta(x\pm y) + 2(\vartheta(x)-\vartheta(y))-\vartheta(2x)\ge 0$$ with equality iff $\min(\{\pm x\})\le \min(\{\pm y\})$. Adding all specializations of the form $(x,y)\mapsto (e_i,e_j)$ with $i\ne j$ gives the desired result.
If we rescale $c_i,d_i\mapsto v c_i,vd_i$ and take $v\to 0^+$, the fact that $\vartheta(vx)=v|x|-v^2 x^2$ for sufficiently small $v$ gives us the following limit.
\[lem:gen\_tri2\] For any sequences $c_1,\dots,c_n$, $d_1,\dots,d_n$, of real numbers, we have the following inequality: $$\sum_{\substack{1\le i\le n\\1\le j\le n}} |c_i-d_j|
-\sum_{1\le i<j\le n} |c_i-c_j|
-\sum_{1\le i<j\le n} |d_i-d_j|
\ge
|\sum_{1\le i\le n} c_i-\sum_{1\le i\le n} d_i|,$$ with equality iff the sequences can be permuted so that $$c_1\le d_1\le\cdots\le c_n\le d_n$$ or $$d_1\le c_1\le\cdots\le d_n\le c_n.$$
In particular, we have the following fact.
\[lem:gen\_tri1\] For any sequences $c_0,\dots,c_n$, $d_1,\dots,d_n$, of real numbers, we have the following inequality: $$\sum_{\substack{0\le i\le n\\1\le j\le n}} |c_i-d_j|
-\sum_{0\le i<j\le n} |c_i-c_j|
-\sum_{1\le i<j\le n} |d_i-d_j|
\ge 0,$$ with equality iff the sequences can be permuted so that $$c_0\le d_1\le c_1\le\cdots\le c_{n-1}\le d_n\le c_n.$$
Choose a number $d_0$ such that $d_0<\min(c_0,\dots,c_n,d_1,\dots,d_n,\sum_i
c_i-\sum_{i>0} d_i)$, and apply Lemma \[lem:gen\_tri2\].
The case $n=1$ is of course just the usual triangle inequality in ${\mathbb R}$, thus justifying the title of this section.
Hyperbolic limits
=================
Using the above asymptotic estimates for the hyperbolic and elliptic gamma functions, we can obtain corresponding estimates for the various elliptic hypergeometric integrals of [@Rains:Transformations] in the hyperbolic limit. In particular, in each case, it will turn out that up to an explicit exponential factor, the elliptic integral converges exponentially quickly to the hyperbolic integral.
Let us first consider the case of the Type I (perhaps better named “elliptic Dixon”, see Corollary \[cor:Dixon\] below and [@DixonAL:1905]) integral with $BC_n$ symmetry, defined for all nonnegative integers $m$, $n$, and parameters $p$, $q$, $u_0$…$u_{2m+2n+3}$ satisfying $$0<|p|,|q|<1,\quad \prod_{0\le r\le 2m+2n+3} u_r = (pq)^{m+1}$$ by the integral $$I^{(m)}_{BC_n}(u_0,u_1,\dots ;p,q)
:=
\frac{(p;p)^n(q;q)^n}{2^n n!}
\int_{C^n}
\frac{\prod_{1\le i\le n}\prod_{0\le r\le 2m+2n+3} {\Gamma_{\!e}}(u_r z_i^{\pm 1};p,q)}
{\prod_{1\le i<j\le n} {\Gamma_{\!e}}(z_i^{\pm 1}z_j^{\pm 1};p,q)
\prod_{1\le i\le n} {\Gamma_{\!e}}(z_i^{\pm 2};p,q)}
\prod_{1\le i\le n} \frac{dz_i}{2\pi\sqrt{-1}z_i},$$ where the contour is chosen to contain all points of the form $p^i q^j
u_r$, $0\le i,j$, and exclude their reciprocals.
In the hyperbolic limit $p,q,u_r\to 1$, this gives rise to the following limit.
Let $\mu_0$, $\mu_1$,…, $\mu_{2m+2n+3}$, $\omega_1$, $\omega_2$ be parameters such that $$\Im(\omega_1),\Im(\omega_2)>0,\quad \sum_r \mu_r = (m+1)(\omega_1+\omega_2).$$ Then as $v\to 0^+$, $$e(-2n\sum_r R(v\mu_r;v\omega_1,v\omega_2)+(2 n^2+n) R(0;v\omega_1,v\omega_2))
I^{(m)}_{BC_n}(e(v\mu_0),e(v\mu_1),\dots;e(v\omega_1),e(v\omega_2))$$ converges uniformly exponentially (over compact subsets) to $$\frac{1}{(\sqrt{-\omega_1\omega_2})^n 2^n n!}
\int_{C^n}
\frac{\prod_{1\le i\le n}\prod_{0\le r\le 2m+2n+3} {\Gamma_{\!h}}(\mu_r\pm x_i;\omega_1,\omega_2)}
{\prod_{1\le i<j\le n} {\Gamma_{\!h}}(\pm x_i\pm x_j;\omega_1,\omega_2)
\prod_{1\le i\le n} {\Gamma_{\!h}}(\pm 2x_i;\omega_1,\omega_2)}
\prod_{1\le i\le n} dx_i,$$ where the contour agrees with ${\mathbb R}$ outside a compact set, and is chosen to contain all points of the form $\mu_r+i\omega_1+j\omega_2$, $i,j\ge 0$ and exclude their negatives.
We first observe that $$\begin{aligned}
e(R(0;v\omega_1,v\omega_2))
(e(v\omega_1);e(v\omega_1))
(e(v\omega_2);e(v\omega_2))
&=
\lim_{z\to 0}
\frac{e(R(vz;v\omega_1,v\omega_2))}
{(1-e(vz)){\Gamma_{\!e}}(e(vz);e(v\omega_1),e(v\omega_2))}\notag\\
&\sim
\lim_{z\to 0}
\frac{1}{(1-e(vz)){\Gamma_{\!h}}(z;\omega_1,\omega_2)}\notag\\
&=
\frac{1}{v\sqrt{-\omega_1\omega_2}},\notag\end{aligned}$$ with uniform exponentially small relative error as $v\to 0^+$.
For the remaining factors, we first assume that $\Im(\mu_r)>0$ for all $r$, and thus the elliptic contour may be taken to be the unit circle. Now, in the elliptic integral, introduce the change of variables $z_i=e(x_i)$, and thus $dz_i/2\pi\sqrt{-1}z_i = dx_i$; this replaces the unit circle by the cube $[-1/2,1/2]^n$. We next claim that if we restrict to the smaller cube $[-1/4,1/4]^n$, the resulting error is uniformly exponentially small. Indeed, we can use Corollary \[cor:ell\_bound\_hyper\] to bound the integrand on the full cube. The $\mu$ factors satisfy $$e(-2R(v\mu_r;v\omega_1,v\omega_2))
{\Gamma_{\!e}}(e(v\mu_r\pm x_i);e(v\omega_1),e(v\omega_2))
=
O(
e(\frac{-\omega_1-\omega_2+2\mu_r}{2v\omega_1\omega_2} \vartheta(x_i))
)$$ and thus, using the balancing condition, $$\prod_{0\le r\le 2m+2n+3}
e(-2R(v\mu_r;v\omega_1,v\omega_2))
{\Gamma_{\!e}}(e(v\mu_r\pm x_i);e(v\omega_1),e(v\omega_2))
=
O(
e(
\frac{-\omega_1-\omega_2}{2v\omega_1\omega_2} 2(n+1)\vartheta(x_i)
)
).$$ Similarly, the remaining univariate factors satisfy $$e(2R(0;v\omega_1,v\omega_2))
{\Gamma_{\!e}}(e(\pm 2x_i);e(v\omega_1),e(v\omega_2))^{-1}
=
O(
e(\frac{-\omega_1-\omega_2}{2v\omega_1\omega_2} (-\vartheta(2x_i)))
)$$ and, for $i<j$, the cross factors satisfy $$e(4R(0;v\omega_1,v\omega_2))
{\Gamma_{\!e}}(e(\pm x_i\pm x_j);e(v\omega_1),e(v\omega_2))^{-1}
=
O(
e(\frac{-\omega_1-\omega_2}{2v\omega_1\omega_2} (-\vartheta(x_i\pm x_j)))
)$$ Combining these bounds, we find that the integrand is uniformly $$O(
e(
\frac{-\omega_1-\omega_2}{2v\omega_1\omega_2}
\bigl(
\sum_{1\le i\le n} (2n+2)\vartheta(x_i)
-\sum_{1\le i\le n} \vartheta(2x_i)-\sum_{1\le i<j\le n} \vartheta(x_i\pm
x_j)
\bigr)
))$$ Since $$\Im(\frac{-\omega_1-\omega_2}{\omega_1\omega_2})=\Im(-1/\omega_1)+\Im(-1/\omega_2)>0,$$ the bound is maximized when $$\sum_{1\le i\le n} (2n+2)\vartheta(x_i)
-\sum_{1\le i\le n} \vartheta(2x_i)-\sum_{1\le i<j\le n} (\vartheta(x_i+
x_j)+\vartheta(x_i-x_j))$$ is minimized, which in turn happens when $x_1=x_2=\cdots=x_n=0$, by Lemma \[lem:gen\_tri\_ellC\] applied to the case $c_i\equiv 0$, $d_i=x_i$. In particular, the integrand is exponentially small everywhere else, and thus restricting to $|x_i|\le 1/4$ introduces an exponentially small error.
At this point, using Proposition \[prop:ell\_to\_hyper2\] allows us to replace the gamma functions in the integrand with hyperbolic gamma functions (times an exponential factor that turns out to be trivial). The factor $v^{-n}$ from $((p;p)(q;q))^n$ can be absorbed in rescaling the variables of integration; we thus obtain the restriction of the desired integral to the cube $[-1/4v,1/4v]$. But again we can bound the integrand, this time using Corollary \[cor:hyperb\_asympt\], and find the uniform bound $$O(
e(
\frac{-\omega_1-\omega_2}{2\omega_1\omega_2}
\left(
\sum_{1\le i\le n} (2n+2)|x_i|
-\sum_{1\le i\le n} |2x_i|-\sum_{1\le i<j\le n} (|x_i+x_j|+|x_i-x_j|)
\right)
)
),$$ so the omitted tail is again uniformly exponentially small.
For the general case, we note that if $C$ is a valid choice of contour for the hyperbolic integral, then for sufficiently small $v$, the image of the subcontour $[-1/2v,1/2v]$ under $x\mapsto e(vx)$ is a valid choice of contour for the elliptic integral. The result agrees with the unit circle outside a neighborhood of size $O(v)$ of 1; as a result, the difference from the unit circle has no effect on the asymptotics.
If we denote the above integral by $I^{(m)}_{BC_n;h}$, we have the following corollary, obtained as the limit of the corresponding identity for the elliptic case; note that we do not need to compare the exponential factors on both sides, since both sides must agree throughout and have generically nonzero limits.
Let $\mu_0$, $\mu_1$,…, $\mu_{2m+2n+3}$, $\omega_1$, $\omega_2$ be parameters such that $$\Im(\omega_1),\Im(\omega_2)>0,\quad \sum_r \mu_r = (m+1)(\omega_1+\omega_2).$$ Then $$I^{(m)}_{BC_n;h}(\dots,\mu_r,\dots;\omega_1,\omega_2)
=
\prod_{0\le r<s\le 2m+2n+3} {\Gamma_{\!h}}(\mu_r+\mu_s;\omega_1,\omega_2)
I^{(n)}_{BC_m;h}(\dots,\frac{\omega_1+\omega_2}{2}-\mu_r,\dots;\omega_1,\omega_2),$$ and in particular $$I^{(0)}_{BC_n;h}(\dots,\mu_r,\dots;\omega_1,\omega_2)
=
\prod_{0\le r<s\le 2n+3} {\Gamma_{\!h}}(\mu_r+\mu_s;\omega_1,\omega_2).$$
As van Diejen and Spiridonov [@vanDiejenJF/SpiridonovVP:2005] observed for the Type II evaluation, one can also prove hyperbolic results by simply replacing the arguments of [@Rains:Transformations] by appropriate limits, rather than taking limits directly. Those arguments depend strongly on the fact that the set $p^{\mathbb Z}q^{\mathbb Z}$ generically has finite limit points (in fact, is generically dense), which makes analytic continuation trivial. In the hyperbolic setting, the corresponding set ${\mathbb Z}\omega_1+{\mathbb Z}\omega_2$ is never dense, and only has a limit point when $\omega_1/\omega_2$ is real irrational, so an additional analytic continuation argument is needed to extend to generic moduli.
A similar argument will work in the other cases; some technical issues do arise, however, so it is worth discussing those cases as well.
For the Type II (again, the name “elliptic Selberg” might be better) integral, the main complication is that without an additional condition on the parameters, the integrand is not maximized near $z_i\equiv 1$. We have the following result. Define a family of integrals $$\begin{aligned}
{\mathord{I\!I}}^{(m)}_{BC_n}&(u_0,u_1,\dots,u_{2m+5};t;p,q)\notag\\
&:=
\frac{(p;p)^n(q;q)^n{\Gamma_{\!e}}(t;p,q)^n}{2^n n!}
\int_{C^n}
\prod_{1\le i<j\le n} \frac{{\Gamma_{\!e}}(t z_i^{\pm 1} z_j^{\pm
1};p,q)}{{\Gamma_{\!e}}(z_i^{\pm 1} z_j^{\pm 1};p,q)}
\prod_{1\le i\le n}
\frac{\prod_{0\le r\le 2m+5} {\Gamma_{\!e}}(u_r z_i^{\pm 1};p,q)}{{\Gamma_{\!e}}(z_i^{\pm
2};p,q)}
\frac{dz_i}{2\pi\sqrt{-1}z_i},
\notag\end{aligned}$$ on the domain $t^{2n-2}\prod_r u_r = (pq)^{m+1}$, $0<|p|,|q|,|t|<1$, where the contour $C$ satisfies $C=C^{-1}$, and for all $i,j\ge 0$, contains the points $p^i q^j u_r$ as well as the contour $p^i q^j t C$.
Let $\mu_0$, $\mu_1$,…, $\mu_{2m+5}$, $\tau$, $\omega_1$, $\omega_2$ be parameters such that $$\Im(\tau),\Im(\omega_1),\Im(\omega_2)>0,\quad (2n-2)\tau+\sum_r \mu_r = (m+1)(\omega_1+\omega_2),$$ and satisfying the convergence condition $$\Im(\frac{-(n-1)\tau-\omega_1-\omega_2}{\omega_1\omega_2})>0.$$ Then as $v\to 0^+$, $$\begin{aligned}
e(-2n\sum_r R(v\mu_r;v\omega_1,v\omega_2)+2n^2R(0;v\omega_1,v\omega_2)
-2n(n-1)R(v\tau;v\omega_1,v\omega_2))\qquad&\notag\\
\cdot{\mathord{I\!I}}^{(m)}_{BC_n}(e(v\mu_0),e(v\mu_1),\dots;e(v\tau);e(v\omega_1),e(v\omega_2))&
\notag\end{aligned}$$ converges uniformly exponentially (over compact subsets) to $$\frac{{\Gamma_{\!h}}(\tau;\omega_1,\omega_2)^n}{(\sqrt{-\omega_1\omega_2})^n 2^n n!}
\int_{C^n}
\prod_{1\le i<j\le n}
\frac{{\Gamma_{\!h}}(\tau\pm x_i\pm x_j;\omega_1,\omega_2)}
{{\Gamma_{\!h}}(\pm x_i\pm x_j;\omega_1,\omega_2)}
\prod_{1\le i\le n}
\frac{\prod_{0\le r\le 2m+5} {\Gamma_{\!h}}(\mu_r\pm x_i;\omega_1,\omega_2)}
{{\Gamma_{\!h}}(\pm 2x_i;\omega_1,\omega_2)}
dx_i,$$ where the contour $C=-C$ agrees with ${\mathbb R}$ outside a compact set and for all $i,j\ge 0$ contains the points $i\omega_1+j\omega_2+\mu_r$ as well as the contour $i\omega_1+j\omega_2+\tau+C$.
Again we change variables to $z_i=e(x_i)$ and integrate over the cube $[-1/2,1/2]^n$; we may also freely assume $n>1$, as the case $n=1$ has already been dealt with. In this case, we find that the integrand is uniformly bounded by $$\begin{aligned}
O(e(&
\frac{-\omega_1-\omega_2}{2(n-1)v\omega_1\omega_2}
\Bigl(
2\sum_{1\le i<j\le n} \vartheta(x_i\pm x_j)
-(n-1)\sum_{1\le i\le n}\vartheta(2x_i)
\Bigr)\notag\\
{}+{}&
\frac{-(n-1)\tau-\omega_1-\omega_2}{(n-1)v\omega_1\omega_2}
\Bigl(
2(n-1)\sum_{1\le i\le n}\vartheta(x_i)
-
\sum_{1\le i<j\le n} \vartheta(x_i\pm x_j)
\Bigr)
)).
\notag\end{aligned}$$ By Corollary \[cor:gen\_tri\_ellC\], the first $\vartheta$ sum is $\ge 0$, with equality iff the sequence $|x_i|$ is constant; by the case $d_i\equiv
0$ of Lemma \[lem:gen\_tri\_ellC\], the second $\vartheta$ sum is $\ge 0$, with equality iff at most one of the $x_i$ is nonzero. It follows that the integrand is exponentially small unless both conditions are satisfied; i.e., unless $x_i\equiv 0$. The remainder of the proof is as above.
Note that it also follows from the above proof that the convergence condition is necessary for the integrand to be localized. One can readily arrange for equality to hold in the first sum, but not the second, at which point if $\Im(-((n-1)\tau+\omega_1+\omega_2)/\omega_1\omega)<0$, the integrand is exponentially larger than its value near $x_i\equiv 0$.
Let ${\mathord{I\!I}}^{(m)}_{BC_n;h}(\mu_0,\dots;\tau;\omega_1,\omega_2)$ denote the above hyperbolic integral, as a meromorphic function on the domain $$\Im(\tau),\Im(\omega_1),\Im(\omega_2),\Im(\frac{-(n-1)\tau-\omega_1-\omega_2}{\omega_1\omega_2})>0,\qquad
(2n-2)\tau+\sum_r \mu_r = (m+1)(\omega_1+\omega_2).$$
Let $\mu_0$, $\mu_1$,…, $\mu_5$, $\tau$, $\omega_1$, $\omega_2$ be parameters such that $$\Im(\tau),\Im(\omega_1),\Im(\omega_2)>0,\quad
(2n-2)\tau+\sum_r \mu_r = \omega_1+\omega_2,$$ and satisfying the convergence condition $$\Im(\frac{-(n-1)\tau-\omega_1-\omega_2}{\omega_1\omega_2})>0.$$ Then $${\mathord{I\!I}}^{(0)}_{BC_n;h}(\mu_0,\dots,\mu_5;\tau;\omega_1,\omega_2)
=
\prod_{0\le i<n} {\Gamma_{\!h}}((i+1)\tau;\omega_1,\omega_2)
\prod_{0\le r<s\le 5} {\Gamma_{\!h}}(i\tau+\mu_r+\mu_s;\omega_1,\omega_2).$$
For parameters $\mu_0,\dots,\mu_7$, $\tau$, $\omega_1$, $\omega_2$ such that there exists an integer $n$ (necessarily unique) with $$(2n+2)\tau+\sum_r \mu_r = 2(\omega_1+\omega_2)$$ and $$\Im(\tau),\Im(\omega_1),\Im(\omega_2),\Im(\frac{-(n-1)\tau-\omega_1-\omega_2}{\omega_1\omega_2})>0,$$ define $$\tilde{{\mathord{I\!I}}}_h(\mu_0,\dots,\mu_7;\tau;\omega_1,\omega_2)
:=
\bigl(\prod_{0\le r<s\le 7}
{\Gamma_{\!h}}^+(\tau+\mu_r+\mu_s;\tau,\omega_1,\omega_2)
\bigr)
{\mathord{I\!I}}^{(1)}_h(\tau/2+\mu_0,\dots,\tau/2+\mu_7;\tau;\omega_1,\omega_2).$$ Then $\tilde{{\mathord{I\!I}}}_h$ is invariant under the natural action of the Weyl group $E_7$; in other words, it satisfies the identities $$\tilde{{\mathord{I\!I}}}_h(\mu_0,\dots,\mu_7;\tau;\omega_1,\omega_2)
=
\tilde{{\mathord{I\!I}}}_h(\mu_0+\nu,\dots,\mu_3+\nu,\mu_4-\nu,\dots,\mu_7-\nu;\tau;\omega_1,\omega_2)$$ where $\nu=(\mu_4+\mu_5+\mu_6+\mu_7-\mu_0-\mu_1-\mu_2-\mu_3)/4$; $$\tilde{{\mathord{I\!I}}}_h(\mu_0,\dots,\mu_7;\tau;\omega_1,\omega_2)
=
\tilde{{\mathord{I\!I}}}_h(\nu-\mu_0,\dots,\nu-\mu_3,\nu'-\mu_4,\dots,\nu'-\mu_7;\tau;\omega_1,\omega_2),$$ where $\nu=(\mu_0+\mu_1+\mu_2+\mu_3)/2$, $\nu'=(\mu_4+\mu_5+\mu_6+\mu_7)/2$; and $$\tilde{{\mathord{I\!I}}}_h(\mu_0,\dots,\mu_7;\tau;\omega_1,\omega_2)
=
\tilde{{\mathord{I\!I}}}_h(\nu-\mu_0,\dots,\nu-\mu_7;\tau;\omega_1,\omega_2),$$ where $\nu=(\mu_0+\mu_1+\dots+\mu_7)/2$; as well as invariance under permutations of $\mu_0$ through $\mu_7$.
Similarly, the other double coset of $E_7$ in $E_8$ that gives rise to (dimension-altering) transformations of the elliptic integral also gives rise to transformations of the hyperbolic integral; we omit the obvious details. The key observation is that the overall exponential factor that arises when taking the limit is, once one solves for $n$, a function of $\sum_i \mu_i^2$, and is thus $E_8$-invariant. The work of [@Rains:Recurrences] on recurrences also descends to the hyperbolic case; in particular, for $\tau=\omega_2$, one obtains a tau-function for a hyperbolic analogue of the elliptic Painlevé equation.
For the $A_n$ integral, the difficulty is that the elliptic integral has a condition $\prod_i z_i = 1$, which in $x_i$ coordinates, becomes $\sum_i
x_i\in {\mathbb Z}$; this introduces extra complications when maximizing the integrand. Recall that the $A_n$ integral is defined by $$\begin{aligned}
I^{(m)}_{A_n}(u_0,\dots u_{m+n+1};&v_0,\dots v_{m+n+1};p,q)\notag\\
&:=
\frac{(p;p)^n(q;q)^n}{(n+1)!}
\int_{\prod_{0\le i\le n} z_i=1}
\frac{
\prod_{0\le i\le n}
\prod_{0\le r<m+n+2} {\Gamma_{\!e}}(u_r z_i,v_r/z_i;p,q)
}{
\prod_{0\le i<j\le n} {\Gamma_{\!e}}(z_i/z_j,z_j/z_i;p,q)
}
\prod_{1\le i\le n} \frac{dz_i}{2\pi\sqrt{-1}z_i},\notag\end{aligned}$$ for $0<|p|,|q|<1$, $0<|u_0|,\dots,|u_{m+n+1}|,|v_0|,\dots,|v_{m+n+1}|<1$, $\prod_i u_iv_i=(pq)^{m+1}$. (It follows from general principles that this can be extended to a meromorphic function on the domain $0<|p|,|q|<1$, $\prod_i u_iv_i=(pq)^{m+1}$, but the condition on the contour is complicated.)
Let $\mu_0$, $\mu_1$,…, $\mu_{m+n+1}$, $\nu_0$, $\nu_1$,…, $\nu_{m+n+1}$, $\omega_1$, $\omega_2$ be parameters in the upper half-plane such that $$\sum_r \mu_r+\nu_r = (m+1)(\omega_1+\omega_2)$$ Then as $v\to 0^+$, $$\begin{aligned}
e(-(n+1)\sum_r (R(v\mu_r;v\omega_1,v\omega_2)+R(v\nu_r;v\omega_1,v\omega_2))
+(n^2+2n)R(0;v\omega_1,v\omega_2))\qquad&\notag\\
\cdot I^{(m)}_{A_n}(e(v\mu_0),e(v\mu_1),\dots;e(v\nu_0),e(v\nu_1),\dots;e(v\omega_1),e(v\omega_2))&
\notag\end{aligned}$$ converges uniformly exponentially (over compact subsets) to $$\frac{1}{(\sqrt{-\omega_1\omega_2})^n(n+1)!}
\int_{\sum_{0\le i\le n} x_i = 0}
\frac{
\prod_{0\le i\le n}
\prod_{0\le r<m+n+2} {\Gamma_{\!h}}(\mu_r+x_i,\nu_r-x_i;\omega_1,\omega_2)
}{
\prod_{0\le i<j\le n} {\Gamma_{\!h}}(x_i-x_j,x_j-x_i;\omega_1,\omega_2)
}
\prod_{1\le i\le n} dx_i.$$
If we perform the change of variables $z_i=e(x_i)$ in the elliptic integral, the result is an integral over the domain $$-1/2\le x_0,x_1,x_2,\dots,x_n\le 1/2;\quad \sum_{0\le i\le n} x_i\in Z,$$ a disjoint union of polytopes. Over the entire cube, we find that the integrand is uniformly $$O(e(
\frac{-\omega_1-\omega_2}{2v\omega_1\omega_2}
\bigl(
\sum_{0\le i\le n} (n+1)\vartheta(x_i)
-
\sum_{0\le i<j\le n} \vartheta(x_i-x_j)
\bigr)
)).$$ Now, we find from the case $d_i\equiv 0$ of Lemma \[lem:gen\_tri\_ell\] that $$\sum_{0\le i\le n} (n+1)\vartheta(x_i)
-
\sum_{0\le i<j\le n} \vartheta(x_i-x_j)
\ge
\vartheta(\sum_i x_i)
\ge
0$$ with equality iff $x_0,\dots,x_n$ interlaces with $0,\dots,0$ and $\sum_i
x_i\in {\mathbb Z}$; i.e., iff $x_i\equiv 0$. We thus conclude that the integral over the polytope $$-1/4\le x_0,x_1,\dots,x_n\le 1/4;\quad \sum_i x_i = 0$$ is uniformly exponentially close to the original integral. Thus, as above, the theorem reduces to showing that the hyperbolic integral decays exponentially. This in turn reduces to the identity $$(n+1)\sum_{0\le i\le n} |x_i| - \sum_{0\le i<j\le n} |x_i-x_j|
\ge 0$$ with equality only when $x_1=x_2=\cdots x_n=0$.
The remaining issue in degenerating [@Rains:Transformations] to the hyperbolic level is the degeneration of the biorthogonal functions constructed there. The primary difficulty is that the construction of those functions in [@Rains:Transformations] does not give rise to good uniform asymptotics. However, we can still establish the following.
\[thm:biorth\_lim\] Let the parameters $\tau_0$, $\tau_1$, $\tau_2$, $\tau_3$, $\mu_0$, $\mu_1$, $\tau$, $\omega_1$, $\omega_2$ be parameters with $\tau$, $\omega_1$, $\omega_2$ in the upper half-plane such that $$(2n-2)\tau+\tau_0+\tau_1+\tau_2+\tau_3+\mu_0+\mu_1 = \omega_1+\omega_2.$$ Then for any partition pair ${{\boldsymbol\lambda}}$, and for generic values of the parameters, the biorthogonal function $$\tilde{\cal
R}^{(n)}_{{{\boldsymbol\lambda}}}(\dots,e(x_i),\dots;e(v\tau_0){:}e(v\tau_1),e(v\tau_2),e(v\tau_3);e(v\mu_0),e(v\mu_1);e(v\tau);e(v\omega_1),e(v\omega_2))$$ is uniformly bounded for $(x_1,\dots,x_n)\in D(v)^n$, where $D(v)$ is a region of the form $-1-vC_1\le \Re(x)\le vC_1$, $|\Im(x)|\le vC_2$, and excluding a hole of radius $\epsilon v$ around every pole of the biorthogonal function. Moreover, there exists a function $$\tilde{\cal
R}^{(n)}_{{{\boldsymbol\lambda}};h}(\dots,x_i,\dots;\tau_0{:}\tau_1,\tau_2,\tau_3;\mu_0,\mu_1;\tau;\omega_1,\omega_2)$$ such that as $v\to 0^+$, $$\begin{aligned}
&\tilde{\cal
R}^{(n)}_{{{\boldsymbol\lambda}}}(\dots,e(x_i),\dots;e(v\tau_0){:}e(v\tau_1),e(v\tau_2),e(v\tau_3);e(v\mu_0),e(v\mu_1);e(v\tau);e(v\omega_1),e(v\omega_2))\notag\\
{}-{}&
\tilde{\cal
R}^{(n)}_{{{\boldsymbol\lambda}};h}(\dots,x_i/v,\dots;\tau_0{:}\tau_1,\tau_2,\tau_3;\mu_0,\mu_1;\tau;\omega_1,\omega_2)
\notag\end{aligned}$$ converges exponentially to 0, uniformly for $x$ in a compact subset of the domain $-1<\Re(x)<1$.
We first observe that the claims of the theorem are certainly true if we replace $\tilde{\cal R}^{(n)}_{{{\boldsymbol\lambda}}}$ by a product of functions of the form $$e(2n(R(v\beta;v\omega_1)-R(v\alpha;v\omega_1)))\prod_{1\le i\le n}
\frac{\theta(e(v\alpha) z_i^{\pm 1};e(v\omega_1))}{\theta(e(v\beta) z_i^{\pm 1};e(v\omega_1))},$$ or similarly for $\omega_2$. In particular, it was established in [@Rains:Transformations] that there exist functions $F^{(n)}_{{{\boldsymbol\lambda}}}(\mu_0{:};\tau;\omega_1,\omega_2)$ of the above form such that there exists an expansion $$\tilde{\cal R}^{(n)}_{{{\boldsymbol\lambda}}} = \sum_{{{\boldsymbol\mu}}\subset{{\boldsymbol\lambda}}}
C_{{{\boldsymbol\lambda}}{{\boldsymbol\mu}}} F_{{{\boldsymbol\lambda}}}$$ for some coefficients $C_{{{\boldsymbol\lambda}}{{\boldsymbol\mu}}}$ independent of $z_i$. It thus remains only to show that for generic parameters, these coefficients $C_{{{\boldsymbol\lambda}}{{\boldsymbol\mu}}}$ converge exponentially. Moreover, the action of the integral operators of [@Rains:Transformations] can be computed explicitly in the $F_{{{\boldsymbol\lambda}}}$ basis, and the coefficients of the corresponding matrices converge exponentially (to a triangular matrix with generically nonzero diagonal). Thus the generalized eigenvalue equations satisfied by $\tilde{\cal R}^{(n)}_{{{\boldsymbol\lambda}}}$ set up linear equations in the $C_{{{\boldsymbol\lambda}}{{\boldsymbol\mu}}}$ with exponentially converging coefficients. Since the limits of the generalized eigenvalues are generically distinct, the limiting linear equations are generically nonsingular, and the result follows.
The unviariate hyperbolic biorthogonal function $$\tilde{\cal R}^{(1)}_{{{\boldsymbol\lambda}};h}(x;\tau_0{:}\tau_1,\tau_2,\tau_3;\mu_0,\mu_1;\tau;\omega_1,\omega_2)$$ was discussed in [@SpiridonovVP:2005 §8.3].
Note in particular that if we multiply the integrand of either $BC_n$ integral by a function satisfying such convergence properties, the resulting integral will also converge exponentially (assuming the unadorned integral so converges). Also, a similar argument works for the interpolation functions (which as special cases of the biorthogonal functions do not quite fall under the above generic result, but again satisfy suitable integral equations). As a result, every identity of [@Rains:Transformations] involving such functions converges exponentially (possibly with an explicit factor of the form $\exp(a+bv)$) to a corresponding hyperbolic limit. One should note (as observed in [@SpiridonovVP:2005 §8.3]) that further degeneration of the parameters can lead to convergence issues, as without the moderating effect of the poles, the biorthogonal functions grow exponentially as $|\Re(x)|\to\infty$.
Trigonometric limits
====================
The main difficulty with the trigonometric limit $p\to 0$ is that the general case of the transformations involves parameters tending to infinity, making the contour ill-behaved in the limit. This can be fixed at the expense of breaking the symmetry of the integrand.
Recall that for the type $I$ $BC_n$ integral, the parameters are constrained to satisfy the balancing condition $$\prod_{0\le r\le 2n+2m+3} u_r = (pq)^{m+1}.$$ The natural way to satisfy this in the $p\to 0$ limit is for $2n+m+3$ of the parameters to be $\Theta(1)$, while the remaining $m+1$ parameters are $\Theta(p)$. This then makes the $p\to 0$ limit of the integral trivial to compute.
\[thm:triglim\_IC1\] For any parameters $u_0,\dots,u_{2n+m+2}$, $v_0,\dots,v_m$, $q$ satisfying $$|q|<1,\quad \prod_{0\le r\le 2n+m+2} u_r = \prod_{0\le r\le m} v_r,$$ we have the limit $$\begin{aligned}
\lim_{p\to 0}
I^{(m)}_{BC_n}&(u_0,\dots,u_{2n+m+2},pq/v_0,\dots,pq/v_m;p,q)\notag\\
&=
\frac{(q;q)^n}{2^n n!}
\int_{C^n}
\prod_{1\le i<j\le n} {\Gamma_{\!t}}(z_i^{\pm 1}z_j^{\pm 1};q)^{-1}
\prod_{1\le i\le n}
\frac{ \prod_{0\le r\le 2n+m+2} {\Gamma_{\!t}}(u_r z_i^{\pm 1};q)}
{{\Gamma_{\!t}}(z_i^{\pm 2};q) \prod_{0\le r\le m} {\Gamma_{\!t}}(v_r z_i^{\pm 1};q)}
\frac{dz_i}{2\pi\sqrt{-1}z_i},\notag\end{aligned}$$ where the contour contains all points of the form $p^iq^ju_r$, $i,j\ge 0$, and excludes their reciprocals.
This follows immediately from the facts that as $p\to 0$, $$\begin{aligned}
{\Gamma_{\!e}}(x;p,q)^{\pm 1} &= {\Gamma_{\!t}}(x;q)^{\pm 1}(1+O(p))\notag\\
{\Gamma_{\!e}}(pq/x;p,q)^{\pm 1} &= {\Gamma_{\!t}}(x;q)^{\mp 1}(1+O(p)),\notag\end{aligned}$$ uniformly in $x$ away from the poles.
Unfortunately, the right-hand side of the type I transformation involves parameters $$(pq)^{1/2}/u_0,\dots,(pq)^{1/2}/u_{2n+m+2},(pq)^{-1/2}v_0,\dots,(pq)^{-1/2}v_m,$$ which as mentioned above gives an apparently ill-behaved limit. The primary difficulty is that the divergent parameters not only deform the contour, but in fact pinch the contour in the limit, making it approach both 0 and infinity. It turns out, however, that there is a way to break the symmetry in such a way as to eliminate half of the offending poles, thus allowing the contour to be renormalized, giving a well-behaved limit.
The key fact is the following identity of $q$-elliptic functions. Here $R(z_i)$ denotes the operator such that $R(z_i)f(z_i)=f(1/z_i)$.
\[lem:diffop\_ICn\] For any parameters $u_0,\dots,u_{n+1}$, $q$ we have the identity $$\prod_{1\le i\le n} (1+R(z_i))
\frac{
\theta(\prod_{0\le r\le n+1} u_r/\prod_{1\le i\le n} z_i;q)
\prod_{1\le i\le n} \prod_{0\le r\le n+1} \theta(u_r z_i;q)
}
{\prod_{1\le i\le j\le n} \theta(z_iz_j;q)}
=
\prod_{0\le r<s\le n+1} \theta(u_ru_s;q).$$
The left-hand side can be expressed as a sum of $2^n$ terms, all of which are elliptic functions in $z$ with respect to multiplication by $q$, and thus the sum is also an elliptic function. Moreover, since the original function is invariant under permutations, the sum is invariant under the action of $BC_n$. In particular, the order of the sum along each reflection hyperplane must be even; since the summands have at most simple poles there, it follows that the sum is constant. The constant can be recovered by taking $z_i = u_i$, making all but one summand vanish.
\[lem:nonsym\_ICn\] For any nonzero parameters $t_0,\dots,t_n$, $u_0,\dots,u_{n+m+1}$, $v_0,\dots,v_m$, $p$, $q$ with $$0<|p|,|q|<1,\quad
\prod_{0\le r\le n} t_r = \prod_{0\le r\le n+m+1}u_r\prod_{0\le r\le m}v_r$$ and any complex parameters $a,w\ne 0$, we have the identity $$\begin{aligned}
\prod_{0\le r<s\le n} \theta(t_rt_s/a;q)
I^{(m)}_{BC_n}&(t_0/a^{1/2},\dots,t_n/a^{1/2},
a^{1/2}/u_0,\dots,a^{1/2}/u_{m+n+1},
pq/a^{1/2}v_0,\dots,pq/a^{1/2}v_m
;p,q)\notag\\
=
\frac{(p;p)^n(q;q)^n}{n!}
\int_{C^n}&
\frac{\theta(\prod_{0\le r\le n} t_r/w\prod_{1\le i\le n} z_i;q)
\prod_{1\le i\le n} \theta(z_i/w;q)}
{\prod_{0\le r\le n} \theta(t_r/w;q)}
\frac{\prod_{1\le i\le j\le n} \theta(pz_iz_j/a;p)}
{\prod_{1\le i<j\le n} {\Gamma_{\!e}}((z_i/z_j)^{\pm 1};p,q)}\notag\\
&\prod_{1\le i\le n}
\prod_{0\le r\le n} {\Gamma_{\!e}}(p t_rz_i/a,t_r/z_i;p,q)
\prod_{0\le r\le m+n+1} \frac{{\Gamma_{\!e}}(z_i/u_r;p,q)}{{\Gamma_{\!e}}(pqz_iu_r/a;p,q)}
\notag\\
&\phantom{\prod_{1\le i\le n}}
\prod_{0\le r\le m} \frac{{\Gamma_{\!e}}(pq z_i/av_r;p,q)}{{\Gamma_{\!e}}(z_iv_r;p,q)}
\frac{dz_i}{2\pi\sqrt{-1}z_i}
\notag\end{aligned}$$ where the contour contains all points of the form $p^iq^jt_r$, $p^iq^ja/u_r$, $p^{i+1}q^{j+1}/v_r$, $i,j\ge 0$, and excludes all points of the form $p^{-1-i}q^{-j}a/t_r$, $p^{-i}q^{-j}u_r$, $p^{-i-1}q^{-j-1}av_r$, $i,j\ge 0$.
If we multiply the integrand on the left-hand side by the case $u_r = a^{-1/2} v_r$, $0\le r\le n$, $u_{n+1}=a^{1/2}/w$ of the lemma, the symmetry of the integrand implies that $\prod_{1\le i\le
n}(1+R(z_i))$ can be replaced by $2^n$. Shifting the variables of integration by $z_i\to a^{-1/2}z_i$ gives the right-hand side, up to a shift in contour with no effect on the integral.
Note that for specific choices of $w$, the contour condition may conceivably be weakened; the point is that the $w$-dependent factors can cancel out poles of the integrand, making the corresponding constraints on the contour superfluous. In particular, for certain specializations of the parameters, it can be the case that the contour conditions for generic $w$ are inconsistent, but a suitable choice of $w$ makes the integral well-defined.
If we multiply the integrand on the left by a symmetric function $f$ (adjusting the contour conditions accordingly), the effect is to multiply the nonsymmetric integrand by $f(\dots a^{-1/2}z_i\dots)$, with suitable contour conditions.
This makes the limit $p\to 0$ trivial again, as long as $|pq|\le |a|<1$. Taking $a=pq$ gives the following.
\[thm:triglim\_IC2\] For any nonzero parameters $u_0,\dots,u_{2n+m+2}$, $v_0,\dots,v_m$, $q$ with $$0<|q|<1,\quad \prod_{0\le r\le 2n+m+2}u_r = \prod_{0\le r\le m} v_r,$$ we have the limit $$\begin{aligned}
\lim_{p\to 0}
\prod_{0\le r<s\le m} \theta(v_rv_s/pq;q)
I^{(n)}_{BC_m}&((pq)^{1/2}/u_0,\dots,(pq)^{1/2}/u_{2n+m+2},(pq)^{-1/2}v_0,\dots,(pq)^{-1/2}v_m;p,q)\notag\\
{}=
\frac{(q;q)^m}{m!}
\int_{C^m}&
\frac{
\theta(\prod_{0\le r\le m} v_r/w\prod_{1\le i\le m} z_i;q)
\prod_{1\le i\le m} \theta(z_i/w;q)
}
{\prod_{0\le r\le m} \theta(v_r/w;q)}
\frac{\prod_{1\le i\le j\le m} (1-z_iz_j/q)}
{\prod_{1\le i<j\le m} {\Gamma_{\!t}}((z_i/z_j)^{\pm 1};q)}\notag\\
&\prod_{1\le i\le m}
\prod_{0\le r\le 2n+m+2} \frac{{\Gamma_{\!t}}(z_i/u_r;q)}{{\Gamma_{\!t}}(z_iu_r;q)}
\prod_{0\le r\le m} {\Gamma_{\!t}}(v_r z_i/q,v_r/z_i;q)
\frac{dz_i}{2\pi\sqrt{-1}z_i},
\notag\end{aligned}$$ where the contour contains all points of the form $q^j v_r$, $j\ge 0$, and excludes all points of the form $q^{-j}u_r$, $q^{1-j}/v_r$, $j\ge 0$.
\[cor:triglim\_IC12\] The trigonometric integral of Theorem \[thm:triglim\_IC1\] is equal to $$\prod_{0\le r<s\le 2n+m+2} {\Gamma_{\!t}}(u_ru_s;q)
\prod_{\substack{0\le r\le 2n+m+2\\0\le s\le m}} {\Gamma_{\!t}}(v_s/u_r;q)^{-1}
\prod_{0\le r<s\le m} {\Gamma_{\!t}}(v_rv_s/q;q)^{-1}$$ times the trigonometric integral of Theorem \[thm:triglim\_IC2\].
The univariate cases $n=1$, $m=0$ and $n=0$, $m=1$ are the Nasrallah-Rahman integral and an integral identity of Gasper (equations (6.4.1) and (4.11.4) of [@GasperG/RahmanM:2004]); the general $m=0$ case is due to Gustafson [@GustafsonRA:1992].
We also obtain a nontrivial transformation by taking $a\sim p^\alpha$ for $0<\alpha<1$, say $a=(pq)^{1/2}$ for symmetry.
\[thm:triglim\_IC3\] For any nonzero parameters $t_0,\dots,t_n$, $u_0,\dots,u_{n+m+1}$, $v_0,\dots,v_m$, $q$ with $$0<|q|<1,\quad
\prod_{0\le r\le n} t_r = \prod_{0\le r\le n+m+1}u_r\prod_{0\le r\le m}v_r$$ and any complex parameter $w$, we have the limit $$\begin{aligned}
\lim_{p\to 0}&
\prod_{0\le r<s\le n} \theta((pq)^{-1/2}t_rt_s;q)
I^{(m)}_{BC_n}(\frac{t_0}{(pq)^{1/4}},\dots,\frac{t_n}{(pq)^{1/4}},
\frac{(pq)^{1/4}}{u_0},\dots,\frac{(pq)^{1/4}}{u_{m+n+1}},
\frac{(pq)^{3/4}}{v_0},\dots,\frac{(pq)^{3/4}}{v_m}
;p,q)\notag\\
&{}=
\frac{(q;q)^n}{n!}
\int_{C^n}
\frac{\theta(\prod_{0\le r\le n} t_r/w\prod_{1\le i\le n} z_i;q)
\prod_{1\le i\le n} \theta(z_i/w;q)}
{\prod_{0\le r\le n} \theta(t_r/w;q)}
\prod_{1\le i<j\le n} {\Gamma_{\!t}}((z_i/z_j)^{\pm 1};q)^{-1}\notag\\
&\phantom{{}= \frac{(q;q)^n}{n!} \int_{C^n}}
\prod_{1\le i\le n}
\frac{\prod_{0\le r\le n} {\Gamma_{\!t}}(t_r/z_i;q)
\prod_{0\le r\le m+n+1} {\Gamma_{\!t}}(z_i/u_r;q)}
{\prod_{0\le r\le m} {\Gamma_{\!t}}(z_iv_r;q)}
\frac{dz_i}{2\pi\sqrt{-1}z_i}
\notag\end{aligned}$$ where the contour contains all points of the form $q^it_r$, $i\ge 0$, and excludes all points of the form $q^{-i}u_r$, $i\ge 0$.
The trigonometric integral of Theorem \[thm:triglim\_IC3\] is independent of $w$, and if multiplied by $$\prod_{0\le r\le n} \prod_{0\le s\le n+m+1} {\Gamma_{\!t}}(t_r/u_s;q)^{-1},$$ is invariant under the involution $$(m,n;\dots,t_r,\dots;\dots,u_r,\dots;\dots,v_r\dots)
\to
(n,m;\dots,v_r\dots;\dots,u_r^{-1},\dots;\dots,t_r,\dots).$$
This can also be obtained as a limit of Corollary \[cor:triglim\_IC12\] after first breaking the symmetry of the left-hand side as in Lemma \[lem:nonsym\_ICn\]. In particular, this should perhaps be thought of as a degeneration rather than a direct limit; we mention it to point out that that distinction is somewhat artificial (any degeneration should be obtainable as a limit directly from the elliptic level), but more importantly because the Type II analogue has important consequences.
For the Type II integral, we again have a trivial limit in one case.
\[thm:triglim\_IIC1\] For any parameters $u_0,\dots,u_{m+4}$, $v_0,\dots,v_m$, $q$ satisfying $$|q|<1,\quad t^{2n-2}\prod_{0\le r\le m+4} u_r = \prod_{0\le r\le m} v_r,$$ we have the limit $$\begin{aligned}
\lim_{p\to 0}
{\mathord{I\!I}}^{(m)}_{BC_n}&(u_0,\dots,u_{m+4},pq/v_0,\dots,pq/v_m;t;p,q)\notag\\
&=
\frac{(q;q)^n{\Gamma_{\!t}}(t;q)^n}{2^n n!}
\int_{C^n}
\prod_{1\le i<j\le n}
\frac{{\Gamma_{\!t}}(t z_i^{\pm 1} z_j^{\pm 1};q)}
{{\Gamma_{\!t}}( z_i^{\pm 1}z_j^{\pm 1};q)}
\prod_{1\le i\le n}
\frac{\prod_{0\le r\le m+4} {\Gamma_{\!t}}(u_r z_i^{\pm 1};q)}
{{\Gamma_{\!t}}(z_i^{\pm 2};q) \prod_{0\le r\le m} {\Gamma_{\!t}}(v_r z_i^{\pm 1};q)}
\frac{dz_i}{2\pi\sqrt{-1}z_i},\notag\end{aligned}$$ where the contour $C$ satisfies $C=C^{-1}$, and for all $i\ge 0$, contains the points $q^i u_r$ as well as the contour $q^i t C$.
We recall from [@Rains:Transformations] the following identity, which plays the role of Lemma \[lem:diffop\_ICn\] for the type II $BC_n$ integral. $$\prod_{1\le i\le n} (1+R(z_i))
\frac{\theta(u_0 z_i,u_1 z_i,u_2 z_i,t^{n-1}u_0u_1u_2/z_i;q)}
{\theta(z_i^2;q)}
\prod_{1\le i<j\le n} \frac{\theta(tz_iz_j;q)}
{\theta(z_iz_j;q)}
=
\prod_{0\le i<n} \theta(t^i u_0u_1,t^i u_0u_2,t^i u_1u_2;q).$$
\[lem:nonsym\_IICn\] For any nonzero parameters $t_0,t_1$, $u_0,\dots,u_{m+1}$, $v_0,\dots,v_m$, $p$, $q$ with $$0<|p|,|q|<1,\quad
t^{2n-2}t_0t_1 = \prod_{0\le r\le m+2}u_r\prod_{0\le r\le m}v_r$$ and any complex parameters $a,w\ne 0$, we have the identity $$\begin{aligned}
\prod_{0\le i<n} &\theta(t^i t_0t_1/a;q)
{\mathord{I\!I}}^{(m)}_{BC_n}(t_0/a^{1/2},t_1/a^{1/2},
a^{1/2}/u_0,\dots,a^{1/2}/u_{m+2},
pq/a^{1/2}v_0,\dots,pq/a^{1/2}v_m
;t;p,q)\notag\\
{}={}&
\frac{(p;p)^n(q;q)^n{\Gamma_{\!e}}(t;p,q)^n}{n!}
\int_{C^n}
\prod_{1\le i\le n} \frac{\theta(z_i/w,t^{n-1}t_0t_1/wz_i;q)}
{\theta(t^{i-1} t_0/w,t^{i-1} t_1/w;q)}
\prod_{1\le i<j\le n}
\frac{{\Gamma_{\!e}}(tp z_iz_j/a,pq z_iz_j/ a,t (z_i/z_j)^{\pm 1};p,q)}
{{\Gamma_{\!e}}( p z_iz_j/a,pq z_iz_j/ta, (z_i/z_j)^{\pm 1};p,q)}\notag\\
&
\phantom{\frac{(p;p)^n(q;q)^n{\Gamma_{\!e}}(t;p,q)^n}{n!}\int_{C^n}}
\prod_{1\le i\le n}
\theta(p z_i^2/a;p)
{\Gamma_{\!e}}(p t_0 z_i/a,p t_1 z_i/a,t_0/z_i,t_1/z_i;p,q)\notag\\
&\phantom{\frac{(p;p)^n(q;q)^n{\Gamma_{\!e}}(t;p,q)^n}{n!}\int_{C^n}\prod_{1\le
i\le n}}
\prod_{0\le r\le m+2} \frac{{\Gamma_{\!e}}(z_i/u_r;p,q)}
{{\Gamma_{\!e}}(pq z_iu_r/a;p,q)}
\prod_{0\le r\le m} \frac{{\Gamma_{\!e}}(pq z_i/a v_r;p,q)}
{{\Gamma_{\!e}}(z_iv_r;p,q)}
\frac{dz_i}{2\pi\sqrt{-1}z_i},
\notag\end{aligned}$$ where the contour $C$ is chosen so that for all $i,j\ge 0$, it contains the points and contours $$p^i q^j t_0,p^i q^j t_1,p^i q^j a/u_r,p^{i+1}q^{j+1}/v_r,
\quad p^i q^j t C,p^i q^j t a/C$$ and excludes the points and contours $$a/p^{i+1}q^jt_0,a/p^{i+1}q^jt_1,u_r/p^iq^j,av_r/p^{i+1}q^{j+1},
\quad
a/p^{i+1}q^jtC,C/p^iq^jt.$$
It is possible to choose a contour of the given form satisfying $C=aC^{-1}$, namely $a^{-1}C_0$ where $C_0$ is a suitable contour for the left-hand side.
\[thm:triglim\_IIC2\] For any nonzero parameters $u_0,\dots,u_{2m+3}$, $v_0$, $v_1$, $q$ with $$0<|q|<1,\quad
\prod_{0\le r\le 2m+3}u_r = t^{2n-2}v_0v_1$$ and any complex parameters $w\ne 0$, we have the identity $$\begin{aligned}
\lim_{p\to 0}
\prod_{0\le i<n} &\theta(t^i v_0v_1/pq;q)
{\mathord{I\!I}}^{(m)}_{BC_n}(v_0/(pq)^{1/2},v_1/(pq)^{1/2},
(pq)^{1/2}/u_0,\dots,(pq)^{1/2}/u_{2m+3}
;t;p,q)\notag\\
{}={}&
\frac{(q;q)^n{\Gamma_{\!t}}(t;q)^n}{n!}
\int_{C^n}
\prod_{1\le i\le n} \frac{\theta(z_i/w,t^{n-1}v_0v_1/wz_i;q)}
{\theta(t^{i-1} v_0/w,t^{i-1} v_1/w;q)}
\prod_{1\le i<j\le n}
\frac{{\Gamma_{\!t}}(tz_iz_j/q,z_iz_j ,t (z_i/z_j)^{\pm 1};q)}
{{\Gamma_{\!t}}( z_iz_j/q,z_iz_j/t, (z_i/z_j)^{\pm 1};q)}\notag\\
&
\phantom{\frac{(q;q)^n{\Gamma_{\!t}}(t;q)^n}{n!}\int_{C^n}}
\prod_{1\le i\le n}
(1-z_i^2/q)
{\Gamma_{\!t}}(v_0 z_i/q,v_1 z_i/q,v_0/z_i,v_1/z_i;q)
\prod_{0\le r\le 2m+3} \frac{{\Gamma_{\!t}}(z_i/u_r;q)}
{{\Gamma_{\!t}}(z_i u_r;q)}
\frac{dz_i}{2\pi\sqrt{-1}z_i}
\notag\end{aligned}$$ where the contour $C$ is chosen so that for all $i\ge 0$, it contains the points and contours $$q^i v_0,q^i v_1, \quad q^i t C,$$ and excludes the points and contours $$q^{1-i}/v_0,q^{1-i}/v_1,u_r/q^i, \quad q^{1-i}/tC,q^{-i}C/t,$$ assuming such a contour exists.
It is easy to verify that there exist choices of the parameters for which a circular contour of radius $q^{1/2}$ satisfies the given conditions, and thus the integral on the right has a well-defined meromorphic extension to general parameters (and the limit will continue to hold); the only question is whether this can be obtained from a domain of integration of the form $C^n$.
When $m=0$, the above trigonometric integral evaluates to $$\prod_{0\le i<n}
\frac{{\Gamma_{\!t}}(t^{i+1},t^i v_0 v_1/q;q)
\prod_{0\le r\le 3} {\Gamma_{\!t}}(t^i v_0/u_r,t^i v_1/u_r;q)}
{\prod_{0\le r<s\le 3} {\Gamma_{\!t}}(t^{-i} u_ru_s;q)}$$
We also obtain a transformation.
When $m=1$, the trigonometric integral of Theorem \[thm:triglim\_IIC1\] is equal to $$\prod_{0\le i<n}
\frac{
\prod_{0\le r<s\le 5} \Gamma(t^i u_ru_s;p,q)}
{\prod_{0\le r\le 5} \Gamma(v_0/t^i u_r,v_1/t^i u_r;p,q)
\Gamma(v_0v_1/t^i q;p,q)}$$ times the image of the trigonometric integral of Theorem \[thm:triglim\_IIC2\] under the specialization $u_i\mapsto t^{(n-1)/2}
u_i$, $v_i\to t^{-(n-1)/2}v_i$.
There are other transformations relating these integrals, but all can be obtained by applying the above transformation to one or both sides of a transformation of the integral of Theorem \[thm:triglim\_IIC1\] alone.
Similarly, taking $a=p^{1/2}$ above, we obtain the limit
\[thm:nonsym\_IICn\] For any nonzero parameters $t_0,t_1$, $u_0,\dots,u_{m+1}$, $v_0,\dots,v_m$, $p$, $q$ with $$0<|p|,|q|<1,\quad
t^{2n-2}t_0t_1 = \prod_{0\le r\le m+2}u_r\prod_{0\le r\le m}v_r$$ and any complex parameter $w\ne 0$, we have the identity $$\begin{aligned}
\lim_{p\to 0}
\prod_{0\le i<n}\theta(t^i t_0t_1/p^{1/2};q)
{\mathord{I\!I}}^{(m)}_{BC_n}&(t_0/p^{1/4},t_1/p^{1/4},
p^{1/4}/u_0,\dots,p^{1/4}/u_{m+2},
p^{3/4}q/v_0,\dots,p^{3/4}q/v_m
;t;p,q)\notag\\
{}={}
\frac{(q;q)^n{\Gamma_{\!t}}(t;q)^n}{n!}
\int_{C^n}&
\prod_{1\le i\le n} \frac{\theta(z_i/w,t^{n-1}t_0t_1/wz_i;q)}
{\theta(t^{i-1} t_0/w,t^{i-1} t_1/w;q)}
\prod_{1\le i<j\le n}
\frac{{\Gamma_{\!t}}(t (z_i/z_j)^{\pm 1};q)}
{{\Gamma_{\!t}}( (z_i/z_j)^{\pm 1};q)}\notag\\
&\prod_{1\le i\le n}
\frac{
{\Gamma_{\!t}}(t_0/z_i,t_1/z_i;q)
\prod_{0\le r\le m+2} {\Gamma_{\!t}}(z_i/u_r;p,q)}
{\prod_{0\le r\le m}{\Gamma_{\!t}}(z_iv_r;p,q)}
\frac{dz_i}{2\pi\sqrt{-1}z_i},
\notag\end{aligned}$$ where the contour $C$ is chosen so that for all $j\ge 0$, it contains the points and contours $$q^j t_0,q^j t_1,
\quad q^j t C$$ and excludes the points and contours $$u_r/q^j, \quad C/q^jt.$$
Of course, with the Type II integral, we are particularly interested in the effect of multiplying the integrand for $m=0$ by the biorthogonal functions. Note that since the integral is taken over a compact curve in each case, the limiting relation will continue to hold as long as the limit of biorthogonal functions exists, and (more difficult) the revised contour conditions are satisfiable in the limit. The primary constraint is that we may only consider $p$-abelian biorthogonal functions, since otherwise the contour must contain at least one point converging to $\infty$ as $p\to
0$. For the first two limits, there is no difficulty with convergence of the biorthogonal function. Indeed, the $p$-abelian biorthogonal functions satisfy the further identities $$\begin{aligned}
\tilde{\cal R}^{(n)}_{0\lambda}&(\dots
p^{\pm 1/2}z_i\dots;p^{1/2}t_0{:}p^{1/2}t_1,p^{-1/2}t_2,p^{-1/2}t_3;p^{1/2}u_0,p^{-1/2}u_1;t;p,q)\notag\\
&{}=
\tilde{\cal R}^{(n)}_{0\lambda}(\dots
p^{\pm 1/2}z_i\dots;p^{-1/2}t_0{:}p^{-1/2}t_1,p^{1/2}t_2,p^{1/2}t_3;p^{1/2}u_0,p^{-1/2}u_1;t;p,q)\notag\\
&{}=
\tilde{\cal R}^{(n)}_{0\lambda}(\dots
z_i\dots;t_0{:}t_1,t_2,t_3;u_0,u_1;t;p,q)
\notag\end{aligned}$$ and thus in each case the relevant limit of biorthogonal functions is the same.
For the $a=\sqrt{p}$ limit, the situation is more delicate, but we find that if $t^{2n-2}a_0a_1b_0b_1cd=q$, then we have a well-defined limit $$\begin{aligned}
R^{(n)}_{\lambda;AS\text{-}I}&(\dots,z_i,\dots;a_0{:}a_1,b_0,b_1;c,d;q,t;p)
\notag\\
&:=
\lim_{p\to 0}
\tilde{\cal R}^{(n)}_{0\lambda}(\dots p^{-1/4}z_i\dots;
p^{-1/4}a_0{:}p^{-1/4}a_1,p^{1/4}b_0,p^{1/4}b_1;
p^{1/4} c,p^{3/4} d;t;p,q)\notag\\
&=
\lim_{p\to 0}
\tilde{\cal R}^{(n)}_{0\lambda}(\dots p^{-1/4}/z_i\dots;
p^{1/4}a_0{:}p^{1/4}a_1,p^{-1/4}b_0,p^{-1/4}b_1;
p^{3/4} c,p^{1/4} d;t;p,q).
\notag\end{aligned}$$ This is a multivariate analogue of the biorthogonal rational functions of Al-Salam and Ismail [@AlSalamWA/IsmailMEH:1994]. More precisely, by specializing the $a=\sqrt{p}$ limit appropriately (in particular, $w=c^{-1}$), we find that the functions $$R^{(n)}_{\lambda;AS\text{-}I}(\dots,z_i,\dots;a_0{:}a_1,b_0,b_1;c,d;q,t;p)
\text{ and }
R^{(n)}_{\mu;AS\text{-}I}(\dots,1/z_i,\dots;b_0{:}b_1,a_0,a_1;d,c;q,t;p)$$ are biorthogonal with respect to the density $$\prod_{1\le i<j\le n}
\frac{{\Gamma_{\!t}}(t (z_i/z_j)^{\pm 1};q)}{{\Gamma_{\!t}}((z_i/z_j)^{\pm 1};q)}
\prod_{1\le i\le n}
\frac{
{\Gamma_{\!t}}(a_0/z_i,a_1/z_i,b_0 z_i,b_1 z_i;q)
}{
{\Gamma_{\!t}}(q/cz_i,qz_i/d,t^{n-1}a_0a_1c/z_i,t^{n-1}b_0b_1dz_i;q)
},$$ which becomes Al-Salam and Ismail’s density when $t^{n-1}a_0a_1c=q^{1/2}=t^{n-1}b_0b_1d$ and $n=1$. The constraints on the contour are independent of $\lambda$, $\mu$, $c$ and $d$, and are simply that $C$ must contain the points $q^i a_r$ and the contours $q^i t C$, and exclude the points $1/q^i b_r$ and the contours $C/q^i t$. If we then take $a_1,b_1\to
0$ and set $a_0=b_0=q^{1/2}$, we obtain polynomials biorthogonal with respect to the density $$\prod_{1\le i<j\le n}
\frac{{\Gamma_{\!t}}(t (z_i/z_j)^{\pm 1};q)}{{\Gamma_{\!t}}((z_i/z_j)^{\pm 1};q)};$$ these are, of course, simply the ordinary Macdonald polynomials [@MacdonaldIG:1995], up to a suitable normalization. That these arise as limits of the biorthogonal functions is not particularly new (since they are limits of Koornwinder polynomials); what [*is*]{} new is that a limit exists that respects the inner product.
It should be possible to obtain similar limits in the $A_n$ case; since the contour conditions are significantly more complicated in that case, however, we mention only the identity which presumably plays the role of Lemma \[lem:diffop\_ICn\] in this case: $$\begin{aligned}
\text{symm}_{S_{n+1}}
\frac{\prod_{0\le i\le n}
\theta(x z_i/\prod_{0\le j<i} v_j\prod_{i<j\le n}u_j;q)
\prod_{0\le j<i} \theta(v_j z_i;q)
\prod_{i<j\le n} \theta(u_j z_i;q)}
{\prod_{0\le i<j\le n} z_j\theta(z_i/z_j;q)u_j\theta(v_i/u_j;q)}
&\notag\\
{}=
\theta(x\prod_{0\le j\le n}z_j;q)
\prod_{1\le i\le n} &\theta(x/\prod_{0\le j<i} v_j\prod_{i\le j\le n}u_j;q),
\notag\end{aligned}$$ a special case of Theorem 4.4 of [@RosengrenH/SchlosserM:2005].
Rational limits
===============
The rational limit is most naturally viewed as a combination of the hyperbolic and trigonometric limits, and thus in particular requires both the asymptotic calculations from the hyperbolic case and the symmetry breaking from the trigonometric case. In addition, it can be reached by taking $\omega_2\to 0$ in the hyperbolic case, or $q\to 1$ in the trigonometric or elliptic cases. We consider the limit from the elliptic level, as the other levels introduce no further complications. In each case, the integrand factors as a product of $q$-theta functions and functions to which Corollary \[cor:ell\_to\_rat3\] applies; the exponential behaviour of the integrand comes only from the former.
\[thm:ratlim\_IC1\] For $\Im(\omega)>0$, $\sum_{0\le r\le 2n+m+2}\mu_r = \sum_{0\le r\le m}\nu_r$, $$\begin{aligned}
\lim_{v\to 0^+}&
e(-n(2n+3)/24v\omega)
\left((p;p)\sqrt{2\pi v\omega/\sqrt{-1}}\right)^{n(2n+3)}\notag\\
&I^{(m)}_{BC_n}(e(v\mu_0),\dots,e(v\mu_{2n+m+2}),
pe(v(\omega-\nu_0)),\dots,pe(v(\omega-\nu_m));p,e(v\omega))\notag\\
&=
\frac{(\sqrt{2\pi}\omega/\sqrt{-1})^{-n}}{2^n n!}
\int_{C^n}
\prod_{1\le i<j\le n} {\Gamma_{\!r}}(\pm x_i\pm x_j;\omega)^{-1}
\prod_{1\le i\le n}
\frac{\prod_{0\le r\le 2n+m+2} {\Gamma_{\!r}}(\mu_r\pm x_i;\omega)}
{{\Gamma_{\!r}}(\pm 2x_i;\omega)
\prod_{0\le r\le m} {\Gamma_{\!r}}(\nu_r\pm x_i;\omega)}
dx_i,
\notag\end{aligned}$$ where $C$ is a contour agreeing with ${\mathbb R}$ outside a compact set, and separating the points of the form $\mu_r+j\omega$, $j\ge 0$ from the points of the form $-\mu_r-j\omega$, $j\ge 0$.
We consider the case in which the original contour is the unit circle; deformed cases are analogous. As in the hyperbolic case, we make the change of variables $z_i=e(x_i)$ and integrate over $[-1/2,1/2]^n$. If we divide the integrand by $$\prod_{1\le i\le j\le n} \theta(e(x_i+x_j);e(v\omega))
\prod_{1\le i<j\le n} \theta(e(x_i-x_j);e(v\omega))
\prod_{1\le i\le n} \theta(e(v\omega/2+x_i);e(v\omega))^{-2n-2}$$ the remaining factors of the integrand are controlled by Corollary \[cor:ell\_to\_rat3\] to have at worst polynomial growth in $v$, and thus the exponential behavior of the theta functions dominates. In particular, up to polynomial factors, the integrand satisfies the bound $$O(e((-1/2v\omega)[
\sum_{1\le i\le n} 2(n+1)\vartheta(x_i)
-
\sum_{1\le i<j\le n} \vartheta(x_i\pm x_j)
-
\sum_{1\le i\le n} \vartheta(2x_i)
]))$$ which decays exponentially unless $x_1$,…,$x_n=o(1)$. We can thus restrict the integral to $[-1/4,1/4]^n$ and rescale the variables by $v$. The result then follows from Corollary \[cor:ell\_to\_rat2\].
The other case is more complicated, in that the exponential contribution to the asymptotics is not enough to properly localize the integral.
\[thm:ratlim\_IC2\] For $\Im(\omega)>0$, $\sum_{0\le r\le 2n+m+2}\mu_r = \sum_{0\le r\le m}\nu_r$, $$\begin{aligned}
&\lim_{v\to 0^+}
e(-(m^2+2m)/24v\omega)
\left((p;p)\sqrt{2\pi v\omega/\sqrt{-1}}\right)^{2m^2+3m}
\prod_{0\le r<s\le m} \theta(e(v(\nu_r+\nu_s-\omega))/p;e(v\omega))\notag\\
&\phantom{\lim_{v\to 0^+}}I^{(n)}_{BC_m}(\sqrt{p}e(v(\omega/2-\mu_0)),\dots,\sqrt{p}e(v(\omega/2-\mu_{2n+m+2})),
\frac{e(v(\nu_0-\omega/2))}{\sqrt{p}},\dots,\frac{e(v(\nu_m-\omega/2))}{\sqrt{p}};p,e(v\omega))\notag\\
&\qquad=
\frac{(\sqrt{2\pi}\omega/\sqrt{-1})^{-m}}{m!}
\int_{C^m}
\frac{\theta_h(\sum_{0\le r\le m} \nu_r-w-\sum_{1\le i\le m} x_i;\omega)}
{\prod_{1\le i\le m} \theta_h(x_i-w;\omega)^{-1}\prod_{0\le r\le m} \theta_h(\nu_r-w;\omega)}
\frac{\prod_{1\le i\le j\le m} ((x_i+x_j)/\omega-1)}
{\prod_{1\le i<j\le m} {\Gamma_{\!r}}(\pm (x_i-x_j);\omega)}\notag\\
&\phantom{\qquad=\frac{(\omega/\sqrt{-1})^{-m}}{m!}\int_{C^m}{}}
\prod_{1\le i\le m}
\prod_{0\le r\le 2n+m+2} \frac{{\Gamma_{\!r}}(x_i-\mu_r;\omega)}
{{\Gamma_{\!r}}(x_i+\mu_r;\omega)}
\prod_{0\le r\le m} {\Gamma_{\!r}}(\nu_r+x_i-\omega,\nu_r-x_i;\omega)
dx_i,
\notag\end{aligned}$$ where $C$ is a contour agreeing with ${\mathbb R}$ outside a compact set, and separating the points of the form $\nu_r+j\omega$, $j\ge 0$ from the points of the form $\mu_r-j\omega$, $-\nu_r-(j-1)\omega$, $j\ge 0$.
We begin with the integral of Lemma \[lem:nonsym\_ICn\], replacing the extra parameter by $e(vw)$. The exponential factor in the asymptotics of the resulting elliptic integrand is $$e((
\vartheta(\sum_{1\le i\le m} x_i)
+\sum_{1\le i<j\le m}\vartheta(x_i-x_j)
-m\sum_{1\le i\le m}\vartheta(x_i)
)/2v\omega),$$ and thus the integrand is exponentially small unless the sequence $x_1,\dots,x_m$ in ${\mathbb R}/{\mathbb Z}$ interlaces (or nearly interlaces) with the all-zero sequence. More precisely, if we split the integral into $2^m$ integrals based on the decomposition ${\mathbb R}/{\mathbb Z}=[-1/4,1/4]\cup[1/4,3/4]$, then any piece with more than one $[1/4,3/4]$ factor contributes an exponentially small amount. For the pieces with exactly one $[1/4,3/4]$ factor, we find that upon rescaling the $[-1/4,1/4]$ variables, the resulting integrand has order $O(v^2)$ and is integrable; thus those pieces again contribute a negligible amount to the limit. We may thus restrict our attention to $[-1/4,1/4]^n$, or equivalently (up to $O(v)$), the integral over $[-1/4v,1/4v]$ of the rational limit integrand. The omitted tails are again either exponentially small or have integral of order $O(v^2)$, so the result follows.
The rational integral of Theorem \[thm:ratlim\_IC1\] is equal to $$\prod_{0\le r<s\le 2n+m+2} {\Gamma_{\!r}}(\mu_r+\mu_s;\omega)
\prod_{\substack{0\le r\le 2n+m+2\\0\le s\le m}} {\Gamma_{\!r}}(\nu_s-\mu_r;\omega)^{-1}
\prod_{0\le r<s\le m} {\Gamma_{\!r}}(\nu_r+\nu_s-\omega;\omega)^{-1}$$ times the rational integral of Theorem \[thm:ratlim\_IC2\].
For the Type II integral, again the first case is straightforward.
For any parameters $\mu_0,\dots,\mu_{m+4}$, $\nu_0,\dots,\nu_m$, $\omega$, $\tau$ satisfying $$\Im(\omega),\Im(\tau)>0,
\quad
(2n-2)\tau+\prod_{0\le r\le m+4} \mu_r = \prod_{0\le r\le m} \nu_r,$$ we have the limit $$\begin{aligned}
\lim_{v\to 0^+}
&
e(-n/4v\omega)
\left((p;p)\sqrt{2\pi v\omega/\sqrt{-1}}\right)^{2n(2n-3)\tau/\omega+6n}
\notag\\
&
{\mathord{I\!I}}^{(m)}_{BC_n}(e(v\mu_0),\dots,e(v\mu_{m+4}),
pe(v(\omega-\nu_0)),\dots,pe(v(\omega-\nu_m));
e(v\tau);p,e(v\omega))\notag\\
&{}=
\frac{\Gamma_r(\tau;\omega)^n}{(\sqrt{2\pi}\omega/\sqrt{-1})^n 2^n n!}
\int_{C^n}
\prod_{1\le i<j\le n}
\frac{\Gamma_r(\tau\pm x_i\pm x_j;\omega)}{\Gamma_r(\pm x_i\pm x_j;\omega)}
\prod_{1\le i\le n}
\frac{\prod_{0\le r\le m+4} \Gamma_r(\mu_r\pm x_i;\omega)}
{\Gamma_r(\pm 2x_i;\omega)\prod_{0\le r\le m} \Gamma_r(\nu_r\pm
x_i;\omega)}
dx_i
\notag\end{aligned}$$ where the contour $C$ agrees with ${\mathbb R}$ outside a compact set, satisfies $C=-C$ and for all $i\ge 0$, contains the points $i\omega+\mu_r$ as well as the contour $i\omega+\tau+C$.
The exponential factor in the asymptotics of the integrand is $$O(e(\sum_i (\vartheta(2x_i)-4\vartheta(x_i))/2v\omega)),$$ which is exponentially small unless $x_i\equiv 0$. The limit follows as above.
The nonsymmetric Type II integral has even worse behavior than the nonsymmetric Type I case, however.
Let $\mu_0,\dots,\mu_{2m+3}$, $\nu_0$, $\nu_1$, $\omega$, $\tau$, $p$ be parameters such that $|p|<1$, $\Im(\omega),\Im(\tau)>0$, and $$(2n-2)\tau+\nu_0+\nu_1=\sum_r \mu_r,$$ as well as the convergence condition $\Re(\tau/\omega)>-1/n$. Then we have the limit $$\begin{aligned}
\lim_{v\to 0}&
e(-n/4v\omega)
\left((p;p)\sqrt{2\pi v\omega/\sqrt{-1}}\right)^{2n(2n-3)\tau/\omega+6n}
\prod_{0\le i<n} \theta(e(v(i\tau+\nu_0+\nu_1-\omega)/p);q)\notag\\
&{\mathord{I\!I}}^{(m)}_{BC_n}(
\frac{e(v(\nu_0-\omega/2)}{\sqrt{p}},\frac{e(v(\nu_1-\omega/2)}{\sqrt{p}},
\sqrt{p}e(v(\omega/2-\mu_0)),\dots,\sqrt{p}e(v(\omega/2-\mu_{2m+3}));
e(v\tau);p,e(v\omega))\notag\\
&{}=
\frac{\Gamma_r(\tau;\omega)^n}{(\sqrt{2\pi}\omega/\sqrt{-1})^n n!}
\int_{C^n}
\prod_{1\le i<j\le n}
\frac{\Gamma_r(\tau+x_i+x_j-\omega,x_i+x_j,\tau+x_i-x_j,\tau+x_j-x_i;\omega)}
{\Gamma_r(x_i+x_j-\omega,x_i+x_j-\tau,x_i-x_j,x_j-x_i;\omega)}\notag\\
&\phantom{
{}=
\frac{\Gamma_r(\tau;\omega)^n}{(\sqrt{2\pi}\omega/\sqrt{-1})^n n!}
\int_{C^n}}
\prod_{1\le i\le n}
(2x_i/\omega-1)
\Gamma_r(\nu_0+x_i-\omega,\nu_1+x_i-\omega,\nu_0-x_i,\nu_1-x_i;\omega)
\notag\\
&\phantom{
{}=
\frac{\Gamma_r(\tau;\omega)^n}{(\sqrt{2\pi}\omega/\sqrt{-1})^n n!}
\int_{C^n}
\prod_{1\le i\le n}}
\frac{\theta_h(x_i-w,(n-1)\tau+\nu_0+\nu_1-w-x_i;\omega)}
{\theta_h((i-1)\tau+\nu_0-w,(i-1)\tau+\nu_1-w;\omega)}
\prod_{0\le r\le 2m+3} \frac{{\Gamma_{\!r}}(x_i-\mu_r;\omega)}{{\Gamma_{\!r}}(x_i+\mu_r;\omega)}
dx_i,
\notag\end{aligned}$$ where the contour $C$ is chosen so that for all $i\ge 0$, it contains the points and contours $$i\omega+\nu_0,i\omega+\nu_1, \quad i\omega+\tau+C,$$ and excludes the points and contours $$(1-i)\omega-\nu_0,(1-i)\omega-\nu_1,\mu_r-i\omega, \quad
(1-i)\omega-\tau-C,C-i\omega-\tau,$$ assuming such a contour exists.
The exponential factors in the asymptotics of the elliptic integrand actually cancel completely, with the result that the integrand has polynomial asymptotics. The result will follow from dominated convergence if we can show that the rational integrand converges.
For the rational tails, we find that (assuming $C=\omega/2+{\mathbb R}$ for simplicity) the integrand converges iff the integral $$\int_{{\mathbb R}^n}
\prod_{1\le i<j\le n} |x_i^2-x_j^2|^{2\Re(\tau/\omega)}
\prod_{1\le i\le n} |x_i+\sqrt{-1}\epsilon|^{-4(n-1)\Re(\tau/\omega)-3} dx_i$$ converges for $\epsilon>0$; equivalently, via the change of variables $y_i
= x_i^2/\epsilon^2$, the theorem reduces to the convergence of $$\int_{[0,\infty)^n}
\prod_{1\le i<j\le n} |y_i-y_j|^{2\Re(\tau/\omega)}
\prod_{1\le i\le n} |1+y_i|^{-2(n-1)\Re(\tau/\omega)-2} dy_i.$$ Up to linear fractional transformation, this is an instance of the Selberg integral ([@SelbergA:1944], stated as Corollary \[cor:Selberg\] below), and thus converges as long as $\Re(\tau/\omega)>-1/n$, as required.
Classical limits
================
The final limit we consider is that corresponding to the usual beta integral. Although the beta integral itself is generally viewed as the bottom level, this is in fact a somewhat misleading view, as the integrals we obtain are in fact still elliptic (involving powers of theta functions). For the $BC_n$ cases, a suitable change of variables exists that essentially eliminates the dependence on $p$, but the corresponding change of variables for the $A_n$ integral is much less obvious (if it exists at all). Furthermore, even for the beta case, the classical transformation analogue can only easily be reached by degenerating either the hyperbolic or elliptic levels; the symmetry breaking of the trigonometric and rational cases introduces unnecessary complications.
Since the integrand involves powers of theta functions, there are in general some subtle issues involving choices of branch. It will thus be convenient to restrict to the case $p$ real, where the phases are easier to control. We have the following.
Choose $p$ and $z$ such that $-1<p<1$ and $|z|=1$. Then the standard branch of $\log\theta(z;p)$ satisfies $$\begin{aligned}
\log\theta(z;p) &= \log(-z)/2 + \log|\theta(z;p)|,\notag\\
\log\theta(p^{1/2}z;p) &= \log|\theta(p^{1/2}z;p)|.\notag\end{aligned}$$
We have $$\log\theta(z;p) = \log(1-z) + \sum_{1\le i} \log(1-p^i z)+\log(1-p^i/z),$$ taking the principal branch of the logarithm on the right-hand side. Now, $$\log(1-p^i z)+\log(1-p^i/z) = 2\log|1-p^i z|,$$ so it suffices to show that $$\log(1-z) = \log(-z)/2 + \log|1-z|,$$ which follows from the observation $(1-z)/\sqrt{-z}=|1-z|$. Similarly, $$\log\theta(p^{1/2}z;p) = \sum_{0\le i} 2\log|1-p^{i+1/2}z|.$$
We will thus assume $-1<p<1$ in the sequel; note, however, that the case of more general $p$ can be obtained by replacing $$\begin{aligned}
|\theta(z;p)|^{\kappa}&\mapsto (-z)^{\kappa/2}
\theta(z;p)^\kappa\notag\\
|\theta(p^{1/2}z;p)|^{\kappa}&\mapsto \theta(p^{1/2}z;p)^\kappa.\notag\end{aligned}$$ In any event, we need all parameters to have absolute value $1$, $|p|^{1/2}$, or $|p|$ within $1+o(1)$ for this to work.
For the Type I $BC_n$ integral, it is particularly natural to take $2m+2$ parameters to have norm $|p|^{1/2}$, at which point both sides of the transformation take the same form. More general cases could be considered, but appear to give rise to the same limiting identities, so we will restrict our attention to the simplest case.
Given points $x,y,z$ on the unit circle with $x,z$ distinct, $y\in [x,z]$ or equivalently $x\le y\le z$ indicates that $y$ is on the closed counterclockwise arc from $x$ to $z$, and similarly for open arcs.
Let $a_0,\dots,a_n$, $b_0,\dots,b_m$ be points on the unit circle with $$1\le a_0<a_1<\cdots<a_n\le -1,$$ let $\omega$ be a point in the upper half-plane, and let $\alpha^{\pm}_0,\dots,\alpha^{\pm}_n$, $\beta^{\pm}_0,\dots,\beta^{\pm}_m$ be parameters such that $\Re(\alpha^{\pm}_r)>0$ and $$\sum_{0\le r\le n} \alpha^+_r+\alpha^-_r=\sum_{0\le r\le m} \beta^+_r+\beta^-_r.$$ Then, writing $q=e(v\omega)$, $\alpha_r=\alpha^+_r+\alpha^-_r$, $\beta_r=\beta^+_r+\beta^-_r$, we have $$\begin{aligned}
\lim_{v\to 0^+}&
\frac{{\Gamma_{\!e}}(q^{\sum_r \alpha_r};p,q)}
{
\prod_{0\le r\le n} {\Gamma_{\!e}}(q^{\alpha_r};p,q)
\prod_{0\le r<s\le n} {\Gamma_{\!e}}(q^{\alpha^+_r+\alpha^+_s} a_r a_s,q^{\alpha^+_r+\alpha^-_s} a_r/a_s,q^{\alpha^-_r+\alpha^+_s} a_s/a_r,q^{\alpha^-_r+\alpha^-_s}/a_r/a_s;p,q)
}
\notag\\
&I^{(m)}_{BC_n}(\dots,q^{\alpha^+_r} a_r,q^{\alpha^-_r}/a_r,\dots,
\dots,p^{1/2} q^{1/2-\beta^+_r}/b_r,p^{1/2} q^{1/2-\beta^-_r} b_r,\dots;p,q)\notag\\
&=
\prod_{0\le r<s\le n} |\theta(a_r a_s^{\pm 1};p)|^{1-\alpha_r-\alpha_s}
\frac{\Gamma(\sum_{0\le r\le n} \alpha_r)}{\prod_{0\le r\le n} \Gamma(\alpha_r)}
(2\pi(p;p)^2)^n\notag\\
&\phantom{{}={}}
\int_{z_i\in [a_{i-1},a_i]}
\prod_{1\le i<j\le n} |\theta(z_i z_j^{\pm 1};p)|
\prod_{1\le i\le n}
\frac{\prod_{0\le r\le n} |\theta(a_r z_i^{\pm 1};p)|^{\alpha_r-1}}
{\prod_{0\le r\le m} |\theta(p^{1/2} b_r z_i^{\pm 1};p)|^{\beta_r}}
\frac{|\theta(z_i^2;p)|dz_i}{2\pi\sqrt{-1}z_i}.
\notag\end{aligned}$$
Using Lemma \[lem:gen\_tri\_ellC\], we find that the integral decays exponentially outside the stated product of arcs (or images under the hyperoctahedral group). The result then follows by dominated convergence.
If we define $\phi(z) = -\theta(z;p)^2/\theta(-z;p)^2$, then $$\begin{aligned}
\phi(z)-\phi(w) &= \frac{\theta(-1;p)^2 z\theta(wz^{\pm
1};p)}{\theta(-z;p)^2 \theta(-w;p)^2}\notag\\
\phi'(z) &= \frac{\theta(-1;p)^2 (p;p)^2 \theta(z^2;p)}{\theta(-z;p)^4},
\notag\end{aligned}$$ so for $z$ in the arc $[1,-1]$, $$d\phi(z) = 2\pi (p;p)^2 \frac{\theta(-1;p)^2}{|\theta(-z;p)|^4}
\frac{|\theta(z^2;p)|dz}{2\pi\sqrt{-1}z}$$ and $$\begin{aligned}
|\theta(a z^{\pm 1};p)|^\kappa
&=
|\phi(z)-\phi(a)|^\kappa
\frac{|\theta(-z;p)|^{2\kappa}|\theta(-a;p)|^{2\kappa}}{|\theta(-1;p)|^{2\kappa}}
\notag\\
|\theta(p^{1/2} b z^{\pm 1};p)|^\kappa
&=
|\phi(z)-\phi(p^{1/2}b)|^\kappa
\frac{|\theta(-z;p)|^{2\kappa} |\theta(-p^{1/2}b;p)|^{2\kappa}}
{|\theta(-1;p)|^{2\kappa} }
\notag\end{aligned}$$ Consequently, we can make a change of variables in the resulting transformation to obtain the following result of Dixon [@DixonAL:1905].
\[cor:Dixon\] For any parameters $a_0,\dots,a_n$, $b_0,\dots,b_m$, $\alpha_0,\dots,\alpha_n$, $\beta_0,\dots,\beta_m$ such that $$-b_m<-b_{m-1}<\dots<-b_0<a_0<a_1<\dots<a_n,\quad
0<\Re(\alpha_r),\Re(\beta_r),\quad
\sum_{0\le r\le n}\alpha_r = \sum_{0\le r\le m}\beta_r$$ we have the identity $$\begin{aligned}
\frac{\prod_{0\le i<j\le n} |a_i-a_j|^{1-\alpha_i-\alpha_j}}
{\prod_{0\le i\le n} \Gamma(\alpha_i)\prod_{0\le j\le m} |a_i+b_j|^{\alpha_i}}
\int_{x_i\in [a_{i-1},a_i]}&
\prod_{1\le i<j\le n} |x_i-x_j|
\prod_{1\le i\le n}
\frac{\prod_{0\le j\le n} |a_j-x_i|^{\alpha_j-1}}
{\prod_{0\le j\le m} |x_i+b_j|^{\beta_j}}
dx_i\notag\\
{}=
\frac{\prod_{0\le i<j\le m} |b_i-b_j|^{1-\beta_i-\beta_j}}
{\prod_{0\le i\le m} \Gamma(\beta_i)\prod_{0\le j\le n} |b_i+a_j|^{\beta_i}}
&
\int_{x_i\in [b_{i-1},b_i]}
\prod_{1\le i<j\le m} |x_i-x_j|
\prod_{1\le i\le m}
\frac{\prod_{0\le j\le m} |b_j-x_i|^{\beta_j-1}}
{\prod_{0\le j\le n} |x_i+a_j|^{\alpha_j}}
dx_i.
\notag\end{aligned}$$
A similar argument gives the type II analogue.
\[thm:ell\_selberg\_lim\] Let $a_0,a_1$, $b_0,\dots,b_m$ be points on the unit circle with $$1\le a_0<a_1\le -1,$$ let $\omega$ be a point in the upper half-plane, and let $\alpha^{\pm}_0,\alpha^{\pm}_1$, $\beta^{\pm}_0,\dots,\beta^{\pm}_m$, $\tau$ be parameters such that $$\Re(\alpha^{\pm}_0),\Re(\alpha^{\pm}_1),\Re(\tau)>0,\quad
2(n-1)\tau+\alpha_0+\alpha_1=\sum_{0\le r\le m} \beta_r,$$ where $\alpha_r=\alpha^+_r+\alpha^-_r$, $\beta_r=\beta^+_r+\beta^-_r$. Writing $q=e(v\omega)$, we have $$\begin{aligned}
\lim_{v\to 0^+}
&\prod_{0\le i<n}
\frac{{\Gamma_{\!e}}(q^{2(n-1)\tau + \alpha_0+\alpha_1-i\tau};p,q)}
{{\Gamma_{\!e}}(q^{(i+1)\tau},q^{i\tau+\alpha_0},q^{i\tau+\alpha_1},
q^{i\tau+\alpha^+_0+\alpha^+_1}a_0a_1,
q^{i\tau+\alpha^+_0+\alpha^-_1}a_0/a_1,
q^{i\tau+\alpha^-_0+\alpha^+_1}a_1/a_0,
q^{i\tau+\alpha^-_0+\alpha^-_1}/a_0a_1;p,q)}
\notag\\
&{\mathord{I\!I}}^{(m)}_{BC_n}(q^{\alpha^+_0} a_0,q^{\alpha^-_0}/a_0,q^{\alpha^+_1} a_1,q^{\alpha^-_1}/a_1,
\dots,p^{1/2} q^{1/2-\beta^+_r}/b_r,p^{1/2} q^{1/2-\beta^-_r}
b_r,\dots;q^\tau,p,q)
\notag\\
&{}=
|\theta(a_0 a_1^{\pm 1};p)|^{n-n(n-1)\tau-n\alpha_0-n\alpha_1}
\prod_{0\le i<n}
\frac{\Gamma(2(n-1)\tau + \alpha_0+\alpha_1-i\tau)\Gamma(\tau)}
{\Gamma((i+1)\tau)\Gamma(i\tau+\alpha_0)\Gamma(i\tau+\alpha_1)}
\notag\\
&\phantom{{}={}}\frac{(2\pi(p;p)^2)^n}{n!}
\int_{[a_0,a_1]^n}
\prod_{1\le i<j\le n} |\theta(z_i z_j^{\pm 1};p)|^{2\tau}
\prod_{1\le i\le n}
\frac{|\theta(a_0 z_i^{\pm 1};p)|^{\alpha_0-1}
|\theta(a_1 z_i^{\pm 1};p)|^{\alpha_1-1}}
{\prod_{0\le j\le m} |\theta(p^{1/2} b_j z_i^{\pm 1};p)|^{\beta_j}}
\frac{|\theta(z_i^2;p)|dz_i}{2\pi\sqrt{-1}z_i}
\notag\end{aligned}$$
\[cor:Selberg\] [@SelbergA:1944] For any real numbers $a_0$, $a_1$, $b$ with $-b<a_0<a_1$, and parameters $\alpha_0$, $\alpha_1$, $\tau$ with positive real part, $$\begin{aligned}
\frac{1}{n!}
\int_{[a_0,a_1]^n}&
\prod_{1\le i<j\le n} |x_i-x_j|^{2\tau}
\prod_{1\le i\le n}
\frac{|a_0-x_i|^{\alpha_0-1}
|a_1-x_i|^{\alpha_1-1}}
{|b+x_i|^{2(n-1)\tau+\alpha_0+\alpha_1}}
dx_i
\notag\\
&=
\frac{|a_0-a_1|^{n(n-1)\tau+n\alpha_0+n\alpha_1-n}}
{|a_0+b|^{n(n-1)\tau+n\alpha_1}|a_1+b|^{n(n-1)\tau+n\alpha_0}}
\prod_{0\le i<n}
\frac{\Gamma((i+1)\tau)\Gamma(i\tau+\alpha_0)\Gamma(i\tau+\alpha_1)}
{\Gamma(2(n-1)\tau + \alpha_0+\alpha_1-i\tau)\Gamma(\tau)}
\notag\end{aligned}$$
In fact (as observed in [@SelbergA:1944]), the constraint that $\Re(\tau)>0$ is too strict, as can be seen from the fact that the right-hand side remains finite and positive as long as $$\Re(\tau)>-1/n,-\Re(\alpha_0)/(n-1),-\Re(\alpha_1)/(n-1).$$ One can presumably weaken the conditions of Theorem \[thm:ell\_selberg\_lim\] correspondingly.
For any real numbers $a_0$, $a_1$, $b_0$, $b_1$ with $-b_1<-b_0<a_0<a_1$, and parameters $\alpha_0$, $\alpha_1$, $\beta_0$, $\beta_1$, $\tau$ with positive real part such that $\alpha_0+\alpha_1=\beta_0+\beta_1$, we have the transformation $$\begin{aligned}
\int_{[a_0,a_1]^n}
\prod_{1\le i<j\le n} |x_i-x_j|^{2\tau}
\prod_{1\le i\le n}&
\frac{|a_0-x_i|^{\alpha_0-1}|a_1-x_i|^{\alpha_1-1}}
{|b_0+x_i|^{(n-1)\tau+\beta_0}|b_1+x_i|^{(n-1)\tau+\beta_1}}
dx_i\notag\\
=
C&
\int_{[b_0,b_1]^n}
\prod_{1\le i<j\le n} |x_i-x_j|^{2\tau}
\prod_{1\le i\le n}
\frac{|b_0-x_i|^{\beta_0-1}|b_1-x_i|^{\beta_1-1}}
{|a_0+x_i|^{(n-1)\tau+\alpha_0}|a_1+x_i|^{(n-1)\tau+\alpha_1}}
dx_i
\notag\end{aligned}$$ where $$C
=
\prod_{0\le i<n}
\frac{
|a_0+b_0|^{\alpha_0-\beta_0}
|a_0+b_1|^{\alpha_0-\beta_1}
|a_1+b_0|^{\alpha_1-\beta_0}
|a_1+b_1|^{\alpha_1-\beta_1}
}
{
|a_0-a_1|^{1-(n-1)\tau-\alpha_0-\alpha_1}
|b_0-b_1|^{-1+(n-1)\tau+\beta_0+\beta_1}}
\frac{\Gamma(i\tau+\alpha_0)\Gamma(i\tau+\alpha_1)}
{\Gamma(i\tau+\beta_0)\Gamma(i\tau+\beta_1)}.$$
The $A_n$ case is similar.
Let $a_0,\dots,a_n$, $b_0,\dots,b_m$, $Z$ be points on the unit circle with $$a_0<\dots<a_n<a_{n+1}:=a_0,$$ let $\omega$ be a point in the upper half-plane, and let $\alpha^{\pm}_0,\dots,\alpha^{\pm}_n$, $\beta^{\pm}_0,\dots,\beta^{\pm}_m$ be parameters such that $$\Re(\alpha^{\pm}_0),\dots,\Re(\alpha^{\pm}_n)>0,\quad
\sum_{0\le r\le n}\alpha_r-\sum_{0\le r\le m} \beta_r,$$ where $\alpha_r=\alpha^+_r+\alpha^-_r$, $\beta_r=\beta^+_r+\beta^-_r$. Writing $q=e(v\omega)$, we have $$\begin{aligned}
\lim_{v\to 0^+}&
\frac{{\Gamma_{\!e}}(q^{\sum_r\alpha_r};p,q)}
{
\prod_{0\le r,s\le n} {\Gamma_{\!e}}(q^{\alpha^-_r+\alpha^+_s} a_s/a_r;p,q)
{\Gamma_{\!e}}(q^{\sum_r \alpha^-_r} Z/a_0\cdots a_n;p,q)
{\Gamma_{\!e}}(q^{\sum_r \alpha^+_r} a_0\cdots a_n/Z;p,q)
}\notag\\
&
I^{(m)}_{A_n}(Z|
\dots,q^{\alpha^-_r}/a_r,\dots,
\dots,p^{1/2}q^{1/2-\beta^-_r}b_r,\dots;
\dots,q^{\alpha^+_r}a_r,\dots,
\dots,p^{1/2}q^{1/2-\beta^+_r}/b_r,\dots;
p,q)\notag\\
&{}=
\prod_{0\le r<s\le n} |\theta(a_s/a_r;p)|^{1-\alpha_r-\alpha_s}
|\theta(Z/a_0\cdots a_n;p)|^{1-\sum_r \alpha_r}
\frac{\Gamma(\sum_r\alpha_r)
}
{
\prod_{0\le r\le n} \Gamma(\alpha_r)
}\notag\\
&\phantom{{}={}}
\frac{(2\pi(p;p)^2)^n}{(n+1)!}
\int_{\substack{\prod_{0\le i\le n} z_i = Z\\ z_i\in [a_i,a_{i+1}]}}
\prod_{0\le i<j\le n}
|\theta(z_i/z_j;p)|
\prod_{0\le i\le n}
\frac{\prod_{0\le r\le n} |\theta(z_i/a_r;p)|^{\alpha_r-1}}
{\prod_{0\le r\le m} |\theta(p^{1/2}b_r z_i;p)|^{\beta_r}}
\prod_{0\le i<n} \frac{dz_i}{2\pi\sqrt{-1}z_i}
\notag\end{aligned}$$
The fact that the $n+m+2$ theta functions $$\prod_{0\le i\le n} \theta(z_i/a_r;p),\
\prod_{0\le i\le n} \theta(p^{1/2} b_r z_i;p)$$ for fixed $\prod_{0\le i\le n}z_i$ span an $n+1$-dimensional space is presumably relevant to finding an appropriate change of variables to eliminate the theta functions from the integrand. Clearly, though, the resulting computations would not give a [*trivial*]{} derivation of the above integral from a more traditional multivariate beta integral.
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|
---
author:
- 'Franz Elsner, Georg Feulner'
- Ulrich Hopp
bibliography:
- 'literature.bib'
date: 'Received 24 July 2007; accepted 29 October 2007'
title: 'The impact of *Spitzer* infrared data on stellar mass estimates – and a revised galaxy stellar mass function at $0 < z < 5$'
---
[We estimate stellar masses of galaxies in the high redshift universe with the intention of determining the influence of newly available Spitzer/IRAC infrared data on the analysis. Based on the results, we probe the mass assembly history of the universe.]{} [We use the *GOODS–MUSIC* catalog, which provides multiband photometry from the U–filter to the 8 $\mu m$ Spitzer band for almost $15\,000$ galaxies with either spectroscopic (for $\approx
7~\%$ of the sample) or photometric redshifts, and apply a standard model fitting technique to estimate stellar masses. We than repeat our calculations with fixed photometric redshifts excluding Spitzer photometry and directly compare the outcomes to look for systematic deviations. Finally we use our results to compute stellar mass functions and mass densities up to redshift $z = 5$.]{} [We find that stellar masses tend to be overestimated on average if further constraining Spitzer data are not included into the analysis. Whilst this trend is small up to intermediate redshifts $z \la 2.5$ and falls within the typical error in mass, the deviation increases strongly for higher redshifts and reaches a maximum of a factor of three at redshift $z \approx 3.5$. Thus, up to intermediate redshifts, results for stellar mass density are in good agreement with values taken from literature calculated without additional Spitzer photometry. At higher redshifts, however, we find a systematic trend towards lower mass densities if Spitzer/IRAC data are included.]{}
Introduction
============
Whilst the assembly of stellar mass over cosmic history became a matter of interest decades ago, substantial research on this problem has become feasible only in the past few years. Extensive surveys had to be carried out, and techniques to infer stellar masses of galaxies from multicolor photometry had to be developed, as large complete spectroscopic samples at cosmic distances are not yet available. In general, these methods rest upon fitting a grid of stellar population models to the data to compute stellar masses by multiplying the mass–to–light ratio of the best matching template with its absolute luminosity. This procedure has been used in numerous publications to calculate stellar mass functions or stellar mass densities up to intermediate redshifts . With the increasing availability of deeper surveys, the analysis has been extended to higher redshifts, e.g. up to $z = 2$ or 3 , or even to $z = 5$ [@2005ApJ...619L.131D]. However, the derived results are typically based on observations from the U to the K–band and may be affected by extrapolation errors as the rest–frame wavelength coverage of the observed objects is shifted to the blue, and a good observational constraint to a galaxy’s optical to near–infrared rest–frame luminosity is necessary to estimate its stellar mass reliably. Results for stellar masses may therefore be impaired by systematic uncertainties in the high redshift regime if observations are not extended to longer wavelengths.
Since the Spitzer space telescope has become operational, it has been possible to complement observations up to the K filter with high–quality infrared data in the adjacent wavelength range [@2004ApJS..154....1W]. To benefit from this improvement we used the Great Observatories Origins Deep Survey, which provides deep publicly available observations from numerous facilities in different wavelength regimes [@dickinson]. Based on these data we probe the influence of Spitzer photometry on the estimated stellar masses and examine the consequences of the results on inferences about the mass assembly history of the universe. In this work we address the effects of Spitzer data to the mass calculation process only, i.e. we do not study whether additional systematic deviations of photometric redshifts emerge that, in general, can influence the result as well. Furthermore we adopt a specific set of template models for the calculations.
The paper is organized as follows. In Sect. 2 we give a short overview of the dataset, and discuss the procedure adopted to estimate stellar masses in Sect. 3. We focus on the influence of Spitzer data on the derived masses in Sect. 4 and use the outcome to calculate stellar mass functions (Sect. 5) and mass densities (Sect. 6). Finally, we summarise our results and draw our conclusions in Sect. 7.
Throughout the paper we assume $\Omega_{\mathrm{M}} = 0.3$, $\Omega_{\Lambda} = 0.7$ and $H_{0} =
70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$. Magnitudes are given in the AB system.
The dataset
===========
To study the assembly of stellar mass we focus on the Chandra Deep Field South, where numerous observations within the framework of the Great Observatories Origins Deep Survey (*GOODS*) provide data over a wide range of wavelengths. The present paper is based on *GOODS–MUSIC*, a multicolor catalog published by . We briefly discuss its main characteristics here and refer the reader to that paper for a more detailed description.
The catalog combines U–band data from the 2.2m–MPE/ESO telescope and VLT/VIMOS ($\mathrm{U_{35}}$, $\mathrm{U_{38}}$ and $\mathrm{U_{VIM}}$), Hubble/ACS images in F435W (B), F606W (V), F775W (i) and F850LP (z), VLT/ISAAC data in J, H and $\mathrm{K_S}$, and Spitzer/IRAC data at $3.6~\mu m$, $4.5~\mu m$, $5.8~\mu m$ and $8.0~\mu m$. Since observations have not yet been finished in all bands, the coverage fraction lies at around 63 % for $\mathrm{U_{VIM}}$–data and at about 54 % for the H–band (ISAAC data release 1.0). Source detection has been performed independently in both the deep z–image (14651 objects detected) and the shallower $\mathrm{K_S}$–band data (2931 sources), ending up with a z– and $\mathrm{K_S}$–complete catalog consisting of $14\,847$ objects in total. Special software was developed for photometry [@2006MSAIS...9..454D] to meet the requirements of color measurement in combined ground and space based observations with dissimilar point spread functions. Out of all objects, 13767, 12041, 6767 and 5869 sources could be detected in the IRAC data in channels 1, 2, 3 and 4, respectively. For objects undetected in a specific image, upper limits in flux were calculated on the basis of morphological information derived from the detection image. As a last step redshift information were added to complete the dataset. To do this, spectroscopic surveys available at that time were used to assign 1068 spectroscopic redshifts. For the remaining objects, a standard photometric redshift code was applied that was able to reproduce the spectroscopic redshifts with an accuracy of $\langle|\frac{\Delta z}{1
+ z}|\rangle = 0.045$. As shown in Fig. 12 of , less than 2 % of the spectroscopically observed objects reveal a photometric redshift that deviates severely ($|z_{spec} - z_{phot}| > 0.3 = 5 \cdot \langle \sigma \rangle$) from the spectroscopic one.
In summary, the catalog used here consists of $14\,847$ objects enclosing at least 72 stars and 68 AGNs over a total area of 143.2 $\mathrm{arcmin^2}$, with mean limiting magnitudes of $\mathrm{z_{lim}} \approx 26.0$ and $\mathrm{K_{S \; lim}} \approx
23.8$ at 90 % completeness level.
Deriving stellar masses
=======================
To calculate galaxy stellar masses we adopted the method described in @2004ApJ...608..742D and locally tested against spectroscopic results in @2004ApJ...616L.103D. This method is based on comparing object colors to those of a template library of stellar population synthesis models. The five-dimensional model grid used here was computed from synthetic @bruzual-2003-344 models with an underlying Salpeter initial mass function (IMF) truncated at 0.1 and 100 M$_{\sun}$. It is parameterized by a star formation history (SFH) of the form $\psi(t) \propto \mathrm{exp}(-t/\tau)$ evaluated at $\tau$ = {0.5, 1.0, 2.0, 3.0, 5.0, 8.0, 20.0} Gyr at 15 different ages $t \in [0.2, 10.0]$ Gyr, with dust extinction between $A_\mathrm{V} = 0$ and $A_\mathrm{V} = 1.5$ magnitudes using a @2000ApJ...533..682C extinction law. While covering the physical relevant parameter range in $t$ and $\tau$, the upper limit in $A_\mathrm{V}$ must be considered as a restriction. We adopted this to take into account the degeneracy in age and extinction to suppress solutions with unexpectedly high values for $A_\mathrm{V}$. However, our sample may contain a small number of heavily dust enshrouded galaxies that, in turn, will not be treated appropriately. In addition to the main component, a starburst was superimposed which was allowed to contribute at most 20 % to the z–band luminosity in rest–frame. It was modeled as a 50 Myr old episode of constant star formation with an independent extinction up to $A_\mathrm{V} (b) =
2.0$ magnitudes. To take into account polycyclic aromatic hydrocarbon (PAH) emission, which becomes important at rest–frame wavelength $\lambda \ga 6~\mu m$ and is attributed to starforming regions, the spectral energy distribution (SED) of the burst component was modified by including PAH–emission features following @2005ApJ...619..755D. We found that the object fluxes are reconstructed slightly better at low redshifts compared to a burst–SED lacking this feature, while the actual result for calculated stellar masses is not significantly affected ($|\log \,
M_\mathrm{with \; PAH}/M_\mathrm{without \; PAH}| <
0.003~\mathrm{dex}$ at $z < 1$). Because of the well known age–metallicity degeneracy, we restricted our models to solar metallicities and performed tests to ensure that we do not introduce significant systematic deviations with this constraint.
To derive stellar masses, we used photometry in the filters $\mathrm{U_{38}}$, B, V, i, z, J, H, $\mathrm{K_S}$ and the four IRAC channels. In the blue we focused on $\mathrm{U_{38}}$ only, because the $\mathrm{U_{35}}$–filter is known to be leaking and the $\mathrm{U_{VIM}}$ observations do not cover the whole field . We want to emphasize here that the Spitzer observations provide an unprecedented opportunity to include high–quality infrared photometry longward of about $3~\mu m$ in stellar mass estimates. For each object we computed the full likelihood distribution of our models shifted to the corresponding redshift. To infer the most probable mass–to–light ratio ($M/L_\mathrm{K}$) we weighted the individual $M/L_\mathrm{K}$ ratios of our templates by their likelihoods and averaged over all parameter combinations. Moreover, we were able to derive an estimate of the expected error of this quantity from the width of the distribution. By utilizing the $\mathrm{K_S}$–band $M/L$ ratio we benefit from several advantages. In general, the variation with age in $M/L_\mathrm{K}$ is small compared to its optical counterpart as shown in Fig. \[fig:ml\]. Furthermore, dust absorption is small at longer wavelengths, so potentially large uncertainties in $A_\mathrm{V}$ do not alter the result substantially. The actual mass, which was calculated by multiplying the $M/L_\mathrm{K}$ ratio with the k–corrected total $\mathrm{K_S}$–band luminosity of the best fitting SED model, turns out to be comparatively robust with a mean error of $\sigma_\mathrm{log \, M} = 0.16 \; dex$. Besides the error attributed to the template fitting process only, calculated stellar masses are affected by further important sources of uncertainty. To take them into account we performed extensive Monte Carlo simulations. We analyzed 1000 simulated catalogs where we have considered errors in the object redshifts, calculated $M/L_\mathrm{K}$–ratios and a general uncertainty in photometry. The errors were propagated to the results for stellar masses and the mean uncertainty was estimated to be about $\sigma_\mathrm{log \, M} = 0.33 \; dex$.
In Fig. \[fig:mz\] we show the distribution of all galaxies in the mass versus redshift plane. We found five galaxies at high redshifts $z > 4$ with masses $\log\,M/M_{\sun} > 11.5$. The presence of such high–mass objects less than 2 Gyr after the big bang is still controversial [see, for example, @2006astro.ph..6192D and references therein]. However, a closer analysis reveals that they are more likely to be heavily dust enshrouded galaxies at intermediate redshifts. While keeping the original upper limit in $A_\mathrm{V}$ for the remaining sample, to account for the degeneracy in age and extinction as explained above, we extended the parameter space by allowing a larger maximal dust extinction of $A_\mathrm{V} = 4$ magnitudes for these five galaxies. Refitting the objects’ redshifts with our full template library leads to much smaller redshifts $z \approx 2$ and typical masses about $\log\,M/M_{\sun} \approx 10.5$ for four of the galaxies. With this reanalysis, the fit quality of our best matching SED template increases significantly and becomes comparable to the mean for these four objects. The fifth galaxy remains at high $z$ primarily because of an unexpected low $\mathrm{K_S}$–band luminosity, which may be a result of incorrect photometry. Nevertheless, we want to emphasize that it is not possible to draw an unambiguous conclusion about the nature of these objects on the basis of currently available data. This uncertainty reflects the general character of samples with mainly photometric redshifts. While they can provide accurate data within a statistical analysis, for a dedicated study of rare object properties additional information is still required. To restrict our sample to a uniform parameter space, we excluded the five questionable galaxies from our analysis in Sect. 4.
The role of *Spitzer*
=====================
To study the impact of Spitzer/IRAC data on our analysis, we considered only those galaxies with errors in mass from the fitting process $\sigma_\mathrm{log \, M} < 0.2$ ($\approx 13\,000$). With this restriction, about half of the objects were detected in at least three IRAC channels. However, even if a source has not been identified in a specific filter, an upper limit in flux was included into the analysis, which provides a valuable additional constraint and in general affects the derived stellar mass. For this sample we repeated the calculation of stellar masses as described in Sect. 3 but reduced photometric information to the filters $\mathrm{U_{38}}$, B, V, i, z, J, H and $\mathrm{K_S}$ only, i.e. excluded the four IRAC channels. This subset of our data represents a multi–band catalog with wavelength coverage from about 3500 Å to $23\,000$ Å, which was typically available to study the high redshift universe before facilities such as the Spitzer space telescope became operational. We were then able to study the influence of additional infrared photometry by comparing the properties of the best–fitting models directly.
We found stellar masses to be overestimated without Spitzer data on average. The mean deviation between the masses calculated with all filters up to IRAC channel 4 ($M_\mathrm{U-4}$) and those with limited photometry ($M_\mathrm{U-K}$) of the whole sample is $\log \,
M_\mathrm{U-4}/M_\mathrm{U-K} = -0.18~\mathrm{dex}$ with a r.m.s. of 0.41. As shown in Fig. \[fig:dm\], this trend is relatively independent of mass itself, but has a strong dependency on redshift. Whilst the mean deviation up to a redshift of $z \approx 2$ is only moderate ($|\log \, M_\mathrm{U-4}/M_\mathrm{U-K}| <
0.2~\mathrm{dex}$) and comparable to the typical error in mass, the difference reaches a maximum of $|\log \,
M_\mathrm{U-4}/M_\mathrm{U-K}| \approx 0.5~\mathrm{dex}$ at $z \approx
3.5$. This corresponds to an overestimation of mass of more than a factor of three on average, whereas the scatter for individual sources is large in general, as indicated by the r.m.s. of the distribution. Therefore we can confirm the conjecture made by , who did the same analysis on the $\mathrm{K_S}$–selected subsample with about 3000 galaxies, although we found the effect of Spitzer/IRAC data to be somewhat more pronounced. At higher redshifts the difference in mass decreases again. We can therefore conclude directly that without further data points the extrapolation of object luminosities to longer wavelengths is not straight forward and induces systematic deviations. This result has been derived for a specific set of template SEDs with solar metallicity, a Calzetti extinction law, Salpeter IMF, and for fixed photometric redshifts. Although other choices of model parameters may result in different absolute values for stellar masses, we expect the detection of a systematic trend to be comparatively robust, as it depends on a relative deviation. This statement is supported by the results of who found a similar trend despite using a different model template library and SED fitting procedure.
The immediate reasons for the deviation can be seen in Fig. \[fig:dmzul\], where shifts in both the underlying $\mathrm{K_S}$–band luminosities and the mass–to–light ratios are evident. For a robust calculation of stellar masses with the aid of infrared luminosities, a good knowledge of the slope of the spectra beyond the $4000$ Å break is mandatory. Whereas the brightness of an old massive galaxy decreases slowly in this wavelength range, a young low–mass object has a steep spectrum and the decline is much stronger. To distinguish between early and late type galaxies solely on the basis of the blue part of their spectra is difficult, because the possible presence of a starburst can alter a galaxy’s UV luminosity substantially. Furthermore, uncertainties in extinction strongly affect this wavelength range, which makes it even more difficult to reveal the fundamental properties of the underlying SED. Therefore several data points longward of $4000$ Å are required for a reliable estimation of stellar masses. However, at a redshift of $z = 2.5$ the 4000 Å break lies at $14\,000$ Å, i.e. it has already passed the J filter, so the calculation is resting upon H and $\mathrm{K_S}$–band photometry predominantly, as the slope of the rest–frame optical/near–infrared part of a galaxy’s spectrum is characterized by the $\mathrm{H} - \mathrm{K_S}$ color only. At a redshift of $z \approx 3.5$, where the deviation is largest, the spectrum longward of the $4000$ Å break in rest–frame is covered solely by the $\mathrm{K_S}$–filter and is therefore insufficiently sampled. In general, photometry becomes increasingly uncertain at higher redshifts when objects fade, so the constraints on the slope weaken further. At this redshift, the systematic deviation in masses due to the inclusion of Spitzer data starts to dominate over possible intrinsic errors. This effect can be traced back to the assignment of model SEDs that are too old with too high an absolute infrared luminosity. A decrease in stellar mass is therefore often associated with attributing a younger model SED, i.e. there is a correlation between shift in masses and ages, as can be seen in Fig. \[fig:korr\]. Since extinction influences the shape of the spectrum in a similar way to age, the degeneracy in the two quantities reduces the correlation. We show the change of the best fitting model SED for several galaxies in the appendix.
The reasons for decreasing differences in stellar masses for even higher redshifts beyond $z = 3.5$ are twofold. First, the maximal age of a possible fit model is restricted by the age of the universe at that redshift, which acts as an upper limit. Therefore, the fit models are forced to be younger and a large decline in age with a lower resulting mass is therefore not likely. Secondly, at these extreme redshifts sources are very faint and an increasing fraction of our sample is hardly detected in the shallower J, H, and $\mathrm{K_S}$–band data. In this case, the calculation of masses without Spitzer data is mainly based on the V, i and z photometry (as galaxies are U and B–filter dropouts) and it turns out that very young low–mass model SEDs were assigned. With the inclusion of Spitzer bands the masses for this type of objects tend to increase, and the average difference of the whole sample gets smaller. We want to mention here that whereas the former effect is universal, the latter is a property of our specific catalog and may be less pronounced in other surveys.
In the next step we examined the influence of single Spitzer filters on the resulting stellar mass estimates. We repeated our calculations after adding the four Spitzer/IRAC bands to the analysis one by one. The outcome is illustrated in Fig. \[fig:stin\] where we show deviations in mass in relation to those calculated with full photometric information. Only one further data point at $3.6~\mu m$ can reduce the remaining shift in masses to less than $|\log \,
M_\mathrm{U-4}/M_\mathrm{U-1}| \le 0.25~\mathrm{dex}$ over the full redshift–range. After additionally taking into account Spitzer channel 2 at $4.5~\mu m$, the systematic deviation shrinks to $|\log
\, M_\mathrm{U-4}/M_\mathrm{U-2}| \le 0.1~\mathrm{dex}$ and thus becomes comparable to possible intrinsic errors attributed to the method used to derive stellar masses. Similarly, it is possible to reduce the deviation including just Spitzer channels 3 and 4 in the calculation. Although the remaining difference is at most $|\log \,
M_\mathrm{U-4}/M_\mathrm{U-K+3-4}| \le 0.24~\mathrm{dex}$ and therefore larger than in the case discussed above, and taking into account lower quality data, providing upper limits in flux can often significantly improve the result. Despite reduced mean deviations, the change in mass for single objects can still be large.
The stellar mass function
=========================
Because we found systematic shifts in calculated stellar masses, we wanted to examine the effects on resulting quantities such as the stellar mass function, i.e. the number of objects per comoving volume and mass interval. To do so, we subdivided our catalog into seven redshift bins from $z = 0.25$ to $z = 5$. Using this relatively coarse grid we still work with 380 objects in the highest $z$–bin and ensure that our results are based on statistically meaningful samples. Within each interval we calculated stellar mass functions after correcting our data points for incompleteness due to the flux–limited object selection utilising the $V/V_\mathrm{max}$–formalism of @1968ApJ...151..393S. In this context it is important to point out that we do not achieve a well defined completeness limit in stellar mass, as even for a sharp underlying flux limit, the dynamic range of mass–to–light–ratios results in a wide distribution in mass. Therefore the largest $M/L$–ratio at a certain redshift determines the limit in mass below which incompleteness starts to play a role. To quantify this effect we followed two different approaches. First, motivated by a more theoretical point of view, we studied the evolution of a passive evolving galaxy with subsolar metallicity and zero extinction formed at a redshift of $z = 10$ in a single burst. By assuming an underlying flux limit of $\mathrm{z_{lim}} = 26.0$ magnitudes[^1] we computed the corresponding stellar mass as a more conservative estimate for the completeness limit of the catalog. We found that galaxies with masses down to $M\mathrm{_{lim}} \approx 3 \cdot 10^9~M_{\sun}$ can be detected at $z
= 1$. The value is increasing to $M\mathrm{_{lim}} \approx 3 \cdot
10^{10}~M_{\sun}$ at $z = 2.5$ where those objects would be detected in the shallower $\mathrm{K_S}$–band with an average flux limit of $\mathrm{K_{S \; lim}} = 23.8$ magnitudes. This limit in mass is also obtained for old but moderately star–forming systems with extinctions around $A_V \approx 1$. In the second approach, which is motivated by the data actually available, we studied the evolution of the mass–to–light ratios of the catalog and calculated the 95 % quantile in $M/L_z$ as a function of redshift e.g. the limit below which 95 % of the $M/L_z$ ratios of our sample are located. Assuming a sharp flux limit such as adopted above, we computed the corresponding stellar mass as an estimate for completeness. For a redshift of $z = 1$ we found this value to be $M\mathrm{_{lim}}
\approx 2 \cdot 10^9~M_{\sun}$ increasing to $M\mathrm{_{lim}} \approx
9 \cdot 10^9~M_{\sun}$ at $z = 2.5$. Although this more aggressive method results in a lower mass limit, it should be clear that a significant fraction of heavily dust enshrouded galaxies with large $M/L$–ratios may stay undetected at intermediate $z$, while a reliable detection of typical more massive active galaxies with low to intermediate extinctions should be possible up to high redshifts. As the two methods result in different completeness limits we performed the calculations in this section for both values independently.
To estimate the errors of the stellar mass function we used 1000 realisations of randomly drawn catalogs for which we have considered errors in computed redshifts, calculated $M/L_\mathrm{K}$–ratios and a general uncertainty in photometry. Finally, to cope with incompleteness at lower masses, in view of our subsequent analysis, we fitted our datapoints from the massive end down to the completeness limit with an analytical expression suggested by @1976ApJ...203..297S of the form: $$\psi(M; \phi^*, M^*, \alpha) = log(10) \cdot \phi^* \cdot \left[ 10^{(M - M^*)} \right] ^{(1+\alpha)} \cdot exp\left[-10^{(M - M^*)}\right]$$ In this formula the number density $\psi(M)$ is parameterized via a scale factor $\phi^*$, a typical mass $10^{M^*}M_{\sun}$ and a slope–parameter $\alpha$. First we computed the values of the fit parameters through an $\chi^2$–analysis in the redshift bins up to $z
= 4$ independently. We excluded the highest redshift interval at $4
\le z < 5.01$ from the procedure, as we found the Schechter function to be insufficiently constrained due to the increasing mass limit. However, a robust estimation of the parameters is difficult as they are highly degenerated in the expression used. To deal with this problem we decided to fix the slope–parameter at its error–weighted mean value, as we found no clear evidence for an evolution with redshift as shown in the left–hand panel of Fig. \[fig:spar\]. We also adopted this procedure because an undersampling of low mass objects in the catalog may affect the determination of the slope at higher redshifts in a systematic manner. With this constant value of $\alpha$ we repeated our calculation of the two remaining parameters $M^*$ and $\phi^*$ in every $z$–bin; for both completeness limits the result is plotted in Fig. \[fig:spar\] and listed in Table \[tab:spar\]. A relatively uniform decrease in $\phi^*$ with redshift is clearly evident, which reflects the fact of a general decline in the number of detected objects in place. In contrast to the evident decrease of $\phi^*$, it is hard to say whether there is a hint at a mass dependent evolution of the number density, which would manifest itself as a shift in the parameter $M^*$. Although this trend would be expected by the downsizing scenario for galaxy evolution, where massive galaxies tend to form earlier than their low–mass counterparts, a slight increase in the typical mass with redshift followed by a decrease as indicated in the plot may also be the result of large–scale structure within the observed field.
The Schechter functions can be seen in Fig. \[fig:smf\], where we show the fit to the data with respect to the local stellar mass function of @2001MNRAS.326..255C with the parameters $M^*$ = 11.16, $\phi^*$ = 0.0031 and $\alpha$ = -1.18. For comparison we also plot the mass function of , as derived from the $\mathrm{K_S}$–selected subsample of the *GOODS–MUSIC* catalog in slightly varied redshift bins up to $z \le 4$. Besides a tendency to a smaller number of high mass objects, the datasets show a general agreement. The discrepancy at the high mass end turns out to be robust. Although results in this region rely on only a few galaxies, the error in mass is small for most of them. We discuss possible reasons for the deviations in detail in the next section. We also display the result of @2005ApJ...619L.131D calculated from the *FORS Deep Field (FDF)* and a subarea of the *GOODS–S* (*hereafter GSD*) in the same redshift intervals without additional Spitzer infrared data. While the *GSD* data can be reproduced well up to intermediate redshifts, the analysis of the *FDF* sample tends to result in larger values for stellar mass function. At high redshifts, $z > 3$, deviations are clearly visible for both the *FDF* and *GSD* datasets. They can be attributed to distinct effects; in addition to a shift in the mass scale as a result of the influence of Spitzer photometry, the total number of objects in the highest redshift intervals derived from an integral over the mass function is larger. Performing a Kolmogorov–Smirnov test on the redshift distributions of the *FDF* galaxies and the *GOODS–MUSIC* catalog used in this work, we can clearly reject the hypothesis that the two samples are drawn from the identical parent distribution at a 1 % level. However, both effects can have the same origin since the calculation of photometric redshifts becomes less reliable at high redshifts if the spectra are insufficiently constrained in the rest–frame optical, especially as spectroscopic redshifts used for comparison become very rare and are restricted to the most luminous objects.
redshift interval $M^*$ $\sigma_{M^*}$ $\phi^* \cdot 10^4$ $\sigma_{\phi^*} \cdot 10^4$ $\alpha$ $\sigma_{\alpha}$
--------------------- ------- ---------------- --------------------- ------------------------------ ---------- -------------------
$0.25 \le z < 0.75$ 11.51 0.06 8.43 0.60 -1.358 0.023
$0.75 \le z < 1.25$ 11.58 0.07 4.41 0.35 -1.358 0.023
$1.25 \le z < 1.75$ 11.59 0.07 3.20 0.31 -1.358 0.023
$1.75 \le z < 2.25$ 11.51 0.07 2.82 0.28 -1.358 0.023
$2.25 \le z < 3.01$ 11.34 0.06 2.66 0.24 -1.358 0.023
$3.01 \le z < 4.01$ 11.34 0.10 1.17 0.15 -1.358 0.023
$0.25 \le z < 0.75$ 11.50 0.06 8.77 0.62 -1.352 0.023
$0.75 \le z < 1.25$ 11.57 0.07 4.57 0.36 -1.352 0.023
$1.25 \le z < 1.75$ 11.58 0.07 3.29 0.32 -1.352 0.023
$1.75 \le z < 2.25$ 11.54 0.08 2.56 0.34 -1.352 0.023
$2.25 \le z < 3.01$ 11.48 0.08 1.89 0.28 -1.352 0.023
$3.01 \le z < 4.01$ 11.51 0.14 0.75 0.24 -1.352 0.023
{width="17cm"}
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The stellar mass density
========================
We are now able to compute stellar mass densities, i.e. the mass in stars and remnants per comoving volume at a specific redshift, on the basis of our results from Sect. 5. For this calculation we divided our stellar mass functions in each redshift bin into two mass intervals at the threshold values of the two completeness limits considered here. Above the limit, where data points and Schechter function fall together, we summed the stellar masses of our objects directly. In contrast to this procedure, we integrated the Schechter function in the low–mass range down to zero to take account of the fact that our catalog suffers from incompleteness here. The completion to lower masses contributes about 13 % (43 %) to the final mass density at the redshift interval $3.01 \le z < 4.01$ using the mass limits derived from a $M/L$–ratio analysis (passive evolution scenario). In the highest redshift bin we summed stellar masses directly, not correcting for completeness. Although we certainly underestimate the resulting outcome for stellar mass density, a comparison to the values published by @2005ApJ...619L.131D is still possible as the results were calculated without corrections there. In order to check our results for robustness, we dropped our assumption of a constant slope parameter $\alpha$ and recalculated stellar mass densities, but did not find appreciable deviations. To assign errors to the resulting data points we again used Monte Carlo simulations, and additionally considered uncertainties in the Schechter–parameters that affect the contribution of the integral over the low mass range only. However, a more careful analysis reveals that resulting errors may not include all sources of uncertainty. For example, cosmic variance can alter our result on a 20 % level when estimating the expected uncertainty in number density of observed objects [@2004ApJ...600..171S], although we were able to draw on a relatively large survey area of about 140 $\mathrm{arcmin^2}$ for our calculations. Another source of uncertainty is the proper treatment of stars in the post–AGB phase and their influence on stellar population synthesis models [see, for example, @2006ApJ...652...85M; @2007astro.ph..2091B; @2006ApJ...652...97V]. Furthermore, deviations from the assumed IMF can affect the outcome in a systematic way. In addition, it is important to point out that we can only give lower limits to the stellar mass densities, as we are not able to detect heavily dust enshrouded galaxies with large extinctions already at intermediate redshifts.
We list our results in Table \[tab:md\], where we also compare the mass densities to the local value derived by @2001MNRAS.326..255C. It turns out that at a redshift of $\langle z \rangle = 1$ at least 42 % of today’s stellar mass density is already in place. This fraction decreases to 22 % at $\langle z
\rangle = 2$ and about 6 % at $\langle z \rangle = 3.5$. A comparison with values from literature derived on the basis of a photometric catalog without additional Spitzer/IRAC data is shown in Fig. \[fig:md\]. Whilst up to intermediate redshifts the stellar mass density is reproduced well (though with much smaller scatter because of a larger observed area), we find systematic deviations at high redshifts to lower densities, which one would expect from the properties of the stellar mass functions discussed in Sect. 5.
A comparison of this work with the result of , who used the $\mathrm{K_S}$–selected subsample of an almost identical catalog[^2] with about 3000 galaxies and integrated the Schechter function using smoothed parameters, shows systematically higher values for the stellar mass density. This trend is strengthening from $\log \, \rho_\mathrm{this \;
work}/\rho_\mathrm{Fontana \; et \; al.} = 0. 11~\mathrm{dex}$ at a redshift of $\langle z \rangle = 0.5$ to $0.31~\mathrm{dex}$ at $\langle z \rangle = 3.5$. The discrepancy may have its origins in slightly different model grids used to infer mass–to–light–ratios, and in particular features of the utilized codes themselves. In contrast to the proceedings of , we restricted our template library to solar metallicity, but allowed for an independent burst component when fitting the object luminosities with model SEDs. As the difference in the inferred densities becomes more pronounced with redshift, it stands to reason that the derived masses of younger galaxies, in particular, are subject to systematic deviations. In general, they are affected by larger errors as the function of the mass–to–light–ratio steepens at low ages. Against the background of the discussed degeneracies in age, metallicity and extinction, differences in the underlying template models can affect calculated stellar masses more distinctly here. Similarly, an additional burst component can influence the derived stellar mass. If a galaxy reveals both a high UV luminosity due to a recent starburst and a large infrared luminosity i.e. substantial mass in an old population, a single component fit may not be able to reproduce the spectrum in the whole wavelength range simultaneously, as a young model is too faint in the infrared and an old model not bright enough in the blue. As a consequence, the inferred stellar mass can be lower [@2007ApJ...655...51W]. Therefore, splitting up the fit in burst and main component covers the range of mass–to–light–ratios in a more flexible fashion.
redshift interval $log \, \rho(z) / [M_{\sun} Mpc^{-3}]$ $\sigma_{log \, \rho(z)}$ $\frac{\rho(z)}{\rho(z = 0)}$
--------------------- ---------------------------------------- --------------------------- -------------------------------
$0.25 \le z < 0.75$ 8.57 0.03 68.2 %
$0.75 \le z < 1.25$ 8.37 0.02 42.2 %
$1.25 \le z < 1.75$ 8.22 0.03 30.4 %
$1.75 \le z < 2.25$ 8.10 0.04 22.9 %
$2.25 \le z < 3.01$ 7.93 0.04 15.4 %
$3.01 \le z < 4.01$ 7.59 0.05 7.0 %
$4.01 \le z < 5.01$ $>$ 6.90 0.08 1.4 %
$0.25 \le z < 0.75$ 8.57 0.03 68.1 %
$0.75 \le z < 1.25$ 8.36 0.02 42.2 %
$1.25 \le z < 1.75$ 8.22 0.03 30.4 %
$1.75 \le z < 2.25$ 8.08 0.06 21.9 %
$2.25 \le z < 3.01$ 7.89 0.06 14.1 %
$3.01 \le z < 4.01$ 7.61 0.25 7.4 %
$4.01 \le z < 5.01$ $>$ 6.90 0.08 1.4 %
: Stellar mass densities, as derived using the mass limits from a $M/L$–ratio analysis (*upper section*) and a passively evolving scenario (*lower section, see text*). The redshift interval $4.01 \le z < 5.01$ has not been corrected for completeness.[]{data-label="tab:md"}
Summary
=======
In this work we estimated the influence of newly available infrared data longward of the K–filter on stellar mass estimates. To do so we used the *GOODS–MUSIC* catalog published by , which combines photometric data in 10 filters from 0.35 to 2.3 $\mu m$ with observations from the IRAC instrument of the Spitzer space telescope at 3.6, 4.5, 5.8 and 8.0 $\mu m$. The catalog consists of 14847 objects within an area of 143.2 $\mathrm{arcmin^2}$ detected either in the z or the $\mathrm{K_S}$–band. We computed stellar masses of this sample by fitting stellar population synthesis models [@bruzual-2003-344] to the data and multiplying the k–corrected absolute $\mathrm{K_S}$–band luminosity with the $M/L_\mathrm{K}$–ratio of the best fitting model SED. To probe the influence of the IRAC data on the analysis we repeated the computation of stellar masses without Spitzer photometry, keeping the photometric redshifts fixed, and compared the outcome directly.
We found stellar masses to be overestimated on average, if further constraining infrared data from Spitzer were not included in the calculation. Whilst this trend is almost independent of mass itself, a closer analysis reveals a strong dependency on redshift. While up to $z \approx 2$ the systematic deviation in mass is only moderate ($|\log \, M_\mathrm{U-4}/M_\mathrm{U-K}| < 0.2~\mathrm{dex}$) and comparable to the intrinsic uncertainty of the method adopted to estimate stellar masses, it increases strongly for higher redshifts and reaches a maximum of a factor of three at $z \approx 3.5$. The reason for this systematic shift can be traced back to insufficient constraints on the slope of the spectra redward of the 4000 Å break at high redshifts. It turns out that, on average, models that are too old, with excessively high absolute infrared luminosities and $M/L$–ratios were assigned to the data if Spitzer photometry is not included. Thus, a shift to lower stellar masses is likely to be correlated with a decreasing age of the best fitting model SED. The inclusion of one additional data–point longward of $3~\mu m$ can already reduce the remaining error in mass significantly.
In the next step we used our results to calculate stellar mass functions in different redshift intervals utilizing the $V/V_\mathrm{max}$–formalism of @1968ApJ...151..393S to correct our sample partly for incompleteness. To assign errors we performed extensive Monte Carlo simulations where we considered uncertainties in the underlying $M/L_\mathrm{K}$–ratios, redshifts and a general error in photometry. Afterwards, the data points were fitted via three free parameters using the analytical expression suggested by @1976ApJ...203..297S. We found a pronounced general decrease in numberdensity of all objects with redshift. Beyond that, it is hard to say whether there is also a mass dependent evolution. Although the change in computed Schechter parameter may support this position, the effect can be caused by large–scale structure as well.
Finally, we computed stellar mass densities as a function of redshift. We summed the mass of our objects within each redshift interval at the high mass end and integrated the Schechter function derived on the basis of our stellar mass functions to complete the result for lower masses. To estimate errors we again used Monte Carlo simulations, but we pointed out that further effects such as biases in stellar population synthesis models may be dominating sources of uncertainty. By comparing the outcome to the local value of @2001MNRAS.326..255C we found at least 42 % of the stellar mass density to be already in place at $\langle z \rangle = 1$. This value decreases to 23 % at $\langle z \rangle = 2$ and about 7 % at $\langle z \rangle = 3.5$. Therefore, up to intermediate redshifts our results are in good agreement with values taken from literature derived without additional Spitzer/IRAC data. However, at high redshifts a systematic deviation to lower densities is present as one would expect from the effect of Spitzer photometry on the calculation.
We thank the anonymous referee for the comments that helped to improve the presentation of our results. We are grateful to Niv Drory for providing the program used here to calculate stellar masses and for valuable discussions at the final stage of this paper. We further acknowledge for making the *GOODS–MUSIC* catalog publicly available.
Examples of SED–fits
====================
We show some examples of a change in the best fitting model SED due to inclusion of Spitzer photometry. The shift in calculated stellar masses becomes manifest in a change of age, SFH and extinction (Figs. \[fig:ex\_1\], \[fig:ex\_2\]) or the overall normalization (Fig. \[fig:ex\_3\]). In general, even a slight variation of the $\mathrm{K_S}$–band luminosity can change the derived stellar mass substantially if IRAC photometry is not taken into account. On the other hand, we occasionally found the slope of the spectra at longer wavelengths insufficiently constrained because of large photometric errors (Fig. \[fig:ex\_4\]).
[^1]: The magnitude limits were calculated in using simulations. In the z–band, the limit varies little over the field as the exposure map is relatively homogeneous. The variation in the $\mathrm{K_S}$–filter is larger.
[^2]: Additional spectroscopic redshift information for about 150 objects became available in the meantime and were included in .
|
---
author:
- 'Y. Revaz'
- 'P. Jablonka'
- 'T. Sawala'
- 'V. Hill'
- 'B. Letarte'
- 'M. Irwin'
- 'G. Battaglia'
- 'A. Helmi'
- 'M. D. Shetrone'
- 'E. Tolstoy'
- 'K.A. Venn'
bibliography:
- 'bibliography.bib'
date: 'Received – – 20–/ Accepted – – 20–'
title: The Dynamical and Chemical Evolution of Dwarf Spheroidal Galaxies
---
We present a large sample of fully self-consistent hydrodynamical Nbody/Tree-SPH simulations of isolated dwarf spheroidal galaxies (dSphs). It has enabled us to identify the key physical parameters and mechanisms at the origin of the observed variety in the Local Group dSph properties. The initial total mass (gas + dark matter) of these galaxies is the main driver of their evolution. Star formation (SF) occurs in series of short bursts. In massive systems, the very short intervals between the SF peaks mimic a continuous star formation rate, while less massive systems exhibit well separated SF bursts, as identified observationally. The delay between the SF events is controlled by the gas cooling time dependence on galaxy mass. The observed global scaling relations, luminosity-mass and luminosity-metallicity, are reproduced with low scatter. We take advantage of the unprecedentedly large sample size and data homogeneity of the ESO Large Programme DART, and add to it a few independent studies, to constrain the star formation history of five Milky Way dSphs, Sextans, LeoII, Carina, Sculptor and Fornax. For the first time, \[Mg/Fe\] vs \[Fe/H\] diagrams derived from high-resolution spectroscopy of hundreds of individual stars are confronted with model predictions. We find that the diversity in dSph properties may well result from intrinsic evolution. We note, however, that the presence of gas in the final state of our simulations, of the order of what is observed in dwarf irregulars, calls for removal by external processes.
Introduction
============
Understanding the dominant physical processes at the origin of the dynamical and chemical properties of dwarf spheroidal galaxies (dSphs) is challenging. The binding energy of the interstellar medium of these low mass systems, at the faint end of the galaxy luminosity function, is weak. The injection of energy, due to violent explosions of supernovae [@dekel86; @mori97; @maclow99; @murakami99; @mori02; @hensler04; @ricotti05; @kawata06], or the cosmic UV background during reionization [@efstathiou92; @barkana99; @bullock00; @mayer06] may leave dSphs totally devoid of gas and consequently quench their star formation. In this picture, the majority of the dSphs are fossils of the reionization epoch and are characterized by an old stellar population [@ricotti05].
However, observations offer evidence for more complex star formation histories and reveal a clear variety of dwarf galactic systems [@mateo98; @dolphin02]. The spread in stellar chemical abundances, and in particular the low \[$\alpha$/Fe\] values compared to Galactic halo stars at equal metallicity, are hardly compatible with an early termination of star formation at the epoch of reionization [e.g., @harbeck01; @shetrone98; @shetrone01; @shetrone03; @tolstoy03; @geisler05; @koch08 examples taken in relation to the galaxies studied in this work]. Whilst some dSphs are indeed consistent with rather short star formation episodes, such as Sextans [@lee03] or Sculptor [@babusiaux05], others are characterized by much more extended periods, like Carina [@smecker-hane96; @hurley-keller98] or Fornax [@coleman08].
Long durations of the star formation were early advocated by purely chemical evolution models constrained by the dwarf metallicity distributions and color-magnitude diagrams [@ikuta02; @lanfranchi04]. Subsequent simulations of dwarf galaxies introduced the role of the dark matter coupled to the stellar feedback [@ferrara00], and later the full dynamical physics of the gas and dark matter, by means of N-Body+SPH treatment. Along this line, @marcolini06 [@marcolini08] concluded that a prolonged (compared to instantaneous) star formation requires an external cause for gas removal, which cannot be due to galactic winds. Intermittent episodes of star formation were at the focal point of the analysis by @stinson07. They naturally arose from the alternation between feedback and cooling of the systems. @valcke08 confirmed their self-regulated form. These authors also found a gradual shift of the star formation towards the inner galactic regions. @kawata06 had looked for evidence of spatial variation as well, in the form of metallicity gradients, but had to stop their simulations at redshift 1. Likewise, considering a cosmological box as initial conditions instead of individual halos, @read06 stopped their simulations early, and focused on the smallest and most metal-poor dwarf galaxies.
In all these works, gas remains at the end of the dSph evolutions. The resolution of this problem constitutes a challenge. @mayer06 performed simulations of gas-rich dwarf galaxy satellites orbiting within a Milky Way-sized halo and studied the combined effects of tides and ram pressure. They showed that while tidal stirring produces objects whose stellar structure and kinematics resemble that of dSphs, ram-pressure stripping is needed to entirely remove their gas. @salvadori08 proposed a semi-analytical treatment in a hierarchical galaxy formation framework and achieved the smallest final gas fraction.
Despite real limitations, such as scarce comparisons with observations, incomplete time-evolution, or ad hoc parameterizations, we are witnessing a rapid convergence toward understanding the formation and evolution of dSphs. A critical step forward must be undertaken with a large set of simulations to be confronted with an equally broad sample of data. In particular, the chemical imprints resulting from different hypotheses have not yet been fully capitalized on.
The VLT/FLAMES instrument, with fiber links to the GIRAFFE and UVES spectrographs, has enabled a revolution in spectroscopic studies of resolved stellar populations in nearby galaxies. It is now possible to measure the abundances of a wealth of chemical elements for more than 100 stars at once. Our ESO-Large Programme DART (Dwarf Abundances and Radial velocity Team) is dedicated to the measure of abundances and velocities for several hundred individual stars in a sample of three nearby dSph galaxies: Sculptor, Fornax, and Sextans. We have used the VLT/FLAMES facility in the low resolution mode to obtain CaII triplet metallicity estimates, as well as accurate radial velocities out to the galaxies’ tidal radii [@tolstoy04; @battaglia06; @helmi06; @battaglia08; @battaglia08b]. Each of the three galaxies has also been observed at high resolution for about 80 stars in their central regions, to obtain detailed abundances for a range of interesting elements such as Mg, Ca, O, Ti, Na, Eu. [@venn05; @letarte07 Hill et al in preparation; Letarte et al. in preparation].
In the following, we take advantage of the statistically significant DART sample and data homogeneity, and include some recent independent studies, to constrain the star formation history of five Milky Way dSphs, Sextans, LeoII, Carina, Sculptor, and Fornax. For the first time, populated \[$\alpha$/Fe\] vs \[Fe/H\] diagrams can be confronted with model predictions. Our first goal is to establish how well one can reproduce the apparent diversity of dSph star formation histories in a common scheme. We choose to model galaxies in isolation, as this is the only way to control the effect of all parameters at play, and to understand the dominant physical processes. We will try to see if a complex star formation history may result from intrinsic evolution or if external processes are necessary. We have performed an unprecedentedly large number of simulations. Not only do they account for the gravity of the dark matter and baryons, but they also contain a large number of additional physical mechanisms: metal-dependent gas cooling above and below $10^4\,\rm{K}$, star formation, SNIa and SNII energy feedback and chemical evolution. We focus on the luminosity, star formation history and metallicity properties of dSphs, rather than on their dynamical properties, which turn out to be less constraining.
The paper is organized as follows: The code and the implementation of physical processes are described in Section \[code\]. The initial conditions are detailed in Section \[model\]. The presentation of the results is split in three parts: Section \[models\] focuses on the global evolution of the galaxies and discusses the main driving parameters, while Section \[global\_relations\] is devoted to the scaling relations. The detailed analysis of the chemical properties of Sextans, Leo II, Carina, Sculptor, and Fornax are treated in Section \[generic\_models\]. Section \[discussion\] offers a physical interpretation of the results. Section \[summary\] summarizes our work.
The code {#code}
========
We have adapted the code *treeAsph* originally developed by @serna96, with further developments presented in @alimi03. The chemical evolution was introduced by @poirier03 [Poirier, PhD thesis] and @poirier02. For the sake of simplicity, we recall below the main features and the general philosophy of the algorithms.
Dynamics
--------
All gravitational forces are computed under the *tree* algorithm proposed by @barnes86 [see also @hernquist87]. This technique is based on a hierarchical subdivision of space into cubic cells. One approximates the forces due to a cluster of particles contained in a cubic cell and acting on a particle $i$ by a quadrupolar expansion of the cluster gravitational potential. This is done under the condition that the size of the cubic cell is small compared to its distance to the particle $i$. The ratio between size and distance must be smaller than a tolerance parameter, $\theta$, fixed to 0.7 in our simulations [@hernquist87]. Indeed, under this condition, the internal distribution of the particles within cells can be neglected. Consequently, the number of operations needed to compute the gravitational forces between $N$ particles scales as $\sim N\log N$, instead of $\sim N^2$ if one were to consider each individual pair of particles.
The hydrodynamics of the gas is followed in the Lagrangian Smooth Particle Hydrodynamics (SPH) scheme. Its allows to describe an arbitrarily shaped continuous medium with a finite number of particles [@lucy77; @gingold77], see @monaghan92 and @price05 for reviews. Each gas particle has its mass spatially smeared out by a smoothing kernel $W$ (here a spline function). Unlike the gravitational forces, which are determined from the interactions with all other particles in the system, the hydrodynamical forces result from the contributions of a modest number of neighbors. The spatial resolution is determined by the smoothing length $h_i$ associated with the particle $i$, computed through the requirement that a sphere of radius $h_i$ centered on particle $i$ contains $32$ neighbors.
The integration scheme is the symplectic leapfrog used with adaptative time-steps.
Cooling
-------
At temperatures lower than $10^4\,\rm{K}$, cooling in primordial gas is dominated by the molecule $H_2$. @galli98 have shown that the cooling efficiency of $H_2$ is determined by its mass fraction $\chi_{H_2}$. Unfortunately, an accurate computation of $\chi_{H_2}$ is difficult and requires to take into account the complex processes of the formation and destruction of $H_2$. Following @maio07, we instead fix $\chi_{H_2} $to $10^{-5}$. Once the gas is enriched with metals, these are important to the cooling properties. We consider oxygen, carbon, silicon and iron [@maio07], since they are the most-abundant heavy atoms released during stellar evolution, particularly by the SNe II and SNe Ia, that we follow in our simulations. We set the density of the free electrons over that of hydrogen , ($n_{\rm{e}^-}/n_{\rm{H}}$) to $10^{-4}\,\rm{cm^{-3}}$.
Above $10^4\,\rm{K}$, the cooling function is calculated following the metallicity dependent prescription of @sutherland93. The full normalized cooling function is shown in Figure \[cooling\_fct\], for a large range of metallicities.
A more detailed modeling of the gas cooling is not necessary, as long as simulations are limited in spatial resolution. Indeed, the cooling of the gas is directly dependent on its density. Therefore, a limited resolution smoothes the density fluctuations of the interstellar medium.
Chemical evolution and stellar feedback
---------------------------------------
The chemical enrichment of the interstellar medium (ISM) depends on the interplay between different physical processes. It requires us to follow the rate at which stars form, the amount of newly synthesized chemical elements, the mass and energy released during the different stellar phases, and, finally, the mixing of the metal-enriched stellar outflows with the ISM. The computation of this cycle is done by implementing the original equations of chemical evolution formalized by @tinsley80, as closely as possible.
Star formation
--------------
We adopt the now classical recipe of @katz92 and @katz96. A gas particle becomes eligible for star formation if it is [*i*]{}) collapsing (negative velocity divergence) and [*ii*]{}) its density is higher than a threshold of $\rho_{\rm{sfr}}=1.67\times
10^{-25}\rm{g/cm^{3}}$. However, we do not require the dynamical time to be shorter than the sound crossing time (Jeans instability). The gas particles, which satisfy the above criteria, form stars at a rate expressed by: $$\frac{d \rho_\star}{dt} = \frac{c_\star}{t_{\rm{g}}}\rho_{\rm{g}},
\label{sfr}$$ which mimics a Schmidt law [@schmidt59]. $c_\star$ is the dimensionless star formation parameter, $t_{\rm{g}}$ is taken as the maximum of the local cooling time and the free-fall time.
For a given time interval $\Delta t$, a gas particle of mass $m_{\rm{g}}$ has a probability $p_{\star}$ to form a stellar particle of mass $m_{\star}$, where $p_{\star}$ is defined by: $$p_{\star} = \frac{m_{\rm{g}}}{m_{\star}}\left[ 1-\exp\left( -\frac{c_\star}{t_{\rm{g}}}\Delta t \right) \right].
\label{pstar}$$ The new stellar particle is initially assigned the position and velocity of its gas progenitor. Subsequently, gas and stellar velocities are modified in order to conserve both energy and momentum.
Each stellar particle represent a cluster of stars, sharing the same age and metallicity, whose initial mass function (IMF) $\Phi(m)$ is described by a Salpeter law [@salpeter55]: $$\Phi(m) = \left[ \frac{x+1}{ m_u^{x+1} - m_l^{x+1}} \right] m^x,$$ with $x=-1.35$, $m_l=0.05\,\rm{M_\odot}$ and $m_u=50\,\rm{M_\odot}$.
Ejecta
------
We neglect stellar winds, since they contribute little to the evolution of the chemical elements that we consider (magnesium and iron), and because the injection power to the ISM is dominated by SN explosions [@leitherer92].
The amount of energy, mass and metals ejected by a stellar particle during a time interval $\Delta t$ is calculated by considering the mass of stars exploding between $t$ and $t+\Delta t$. The dependency of the stellar lifetimes on metallicity is taken into account following @kodama97 [private communication].
Hence, the feedback energy released by a stellar particle in the time interval $[t,t+\Delta t]$ is: $$\Delta E_{\rm{SN}} = m_{\star,0}\,\,\left[ n_{\rm{II}}(t)\,E_{\rm{II}} + n_{\rm{Ia}}(t)\,E_{\rm{Ia}} \right],$$ where $m_{\star,0}$ is the initial mass of the stellar particle, and $n_{\rm{II}}(t)$ and $n_{\rm{Ia}}(t)$ are the corresponding number of supernovae SNe II and SNe Ia per unit mass during $\Delta t$. The energy released by both SNe II ($E_{\rm{II}}$), and SNe Ia ($E_{\rm{Ia}}$), is set to $10^{51}\,\rm{erg}$.
With $m_{\rm{II},l}=8\,\rm{M_{\odot}}$ and $m_{\rm{II},u}=50\,\rm{M_{\odot}}$ being the lowest and highest masses of stars exploding as SNe II, and $m(t)$ being the mass of stars with lifetime $t$, we can express $n_{\rm{II}}(t)$ as: $$n_{\rm{II}}(t) = \int_{\max[m(t+\Delta t),m_{\rm{II},l}]}^{\min[m(t),m_{\rm{II},u}]}\frac{\Phi(m)}{m}\,dm.$$ To calculate $n_{\rm{Ia}}(t)$, we adopt the model of @kobayashi00 in which the progenitors of SNe Ia have main-sequence masses in the range of $M_{\rm{p,l}}=3\,\rm{M_\odot}$ to $M_{\rm{p,u}}=8\,\rm{M_\odot}$, and evolve into C+O white dwarfs (WDs). These white dwarfs can form two different types of binary systems (here labeled $i$), either with main sequence stars or with red giants. Hence: $$n_{\rm{Ia}}(t) = \left(\int_{M_{\rm{p,l}}}^{M_{\rm{p,u}}}\frac{\Phi(m)}{m}\,dm\right)
\sum_{i=1}^{i=2} b_i \int_{m_{i,1}}^{m_{i,2}} \frac{\Phi_d(m)}{m}\,dm,$$ with $\Phi_d(m)$ the distribution mass function of the companion stars, $m_{i,1}=\max[m(t+\Delta t),M_{i,\rm{d,l}}]$, $m_{i,2}=\min[m(t),M_{i,\rm{d,u}}]$. The lifetime of SNe Ia is $\sim
2-20$ and $0.5-1.5$ Gyr, depending on the binary system. See @kobayashi00 for the lowest $M_{i,\rm{d,l}}$ and higher mass $M_{i,\rm{d,u}}$ of the WDs progenitors and for the values of $b_i$ which weights the probability of having one or the other binary system.
The supernova feedback energy is released in the form of thermal energy only at the end of each dynamical time-step. This procedure avoids the thermal energy to be dissipated instantaneously by the strong cooling above $10^4\,\rm{K}$, and mimics the blast wave shocks of supernovae [@stinson06].
The ejected gas mass fraction due to SNe Ia is given by: $$\Delta m_{\rm{Ia}}(t) = m_{\rm{WD}} \, n_{\rm{Ia}}(t).$$ with $m_{\rm{WD}}=1.38\,\rm{M_\odot}$ being the mass of white dwarf.
The mass of each chemical element $k$ ejected by a stellar particle is: $$\Delta M_k = m_{\star,0}\,\,\left[ \Delta m_{k,\rm{II}}(t) + \Delta m_{k,\rm{Ia}}(t) \right],$$ where: $$\begin{aligned}
\Delta m_{k,\rm{II}}(t) & = & \int^{m(t)}_{\max[m(t+\Delta t),m_{\rm{II},l}]} p_{k,\rm{II}}(m) \, \Phi(m)\,dm \\\nonumber
& + & z_{k} \int^{m(t)}_{\max[m(t+\Delta t),m_{\rm{II},l}]} \left( 1- \omega(m) - p_{k,\rm{II}}(m) \right)\, \Phi(m)\,dm
\end{aligned}$$ and $$\Delta m_{k,\rm{Ia}}(t) = \Delta m_{\rm{Ia}}(t)\,p_{k,\rm{Ia}}(m).$$ $\omega(m)$ is the remnant mass fraction, which is the mass fraction of a black hole, neutron star or a white dwarf, depending on the initial mass $m$. Values of $\omega(m)$ are taken from [@kobayashi00]. $z_{k}$ is the original stellar abundance of the element $k$, $p_{k,\rm{II}}(m)$ and $p_{k,\rm{Ia}}(m)$ are the stellar yields, i.e. the mass fractions of newly produced and ejected element $k$, coming from SNe II and SNe I, respectively. They are taken from @tsujimoto95.
Since the stellar particles correspond to star clusters, we use the single stellar population mass-to-light ratios of @maraston98 [@maraston05] to calculate their luminosities in $V$-band. The effects of metallicity and age are taken into account.
Initial Conditions {#model}
==================
Mass distribution
-----------------
We consider dSphs in isolation. Gas and dark matter are initially represented by pseudo-isothermal spheres: $$\rho(r)=\frac{\rho_{\rm{c}}}{1+\left( \frac{r}{r_{\rm{c}}} \right)^2},$$ where $r$ is the radius, $r_c$ is the scale length of the mass distribution, and $\rho_c$ the central mass density. The models are truncated at $r_{\rm{max}}$. The halo and gas mass distributions differ by their central density, $\rho_{\rm{c,halo}}$ and $\rho_{\rm{c,gas}}$, respectively.
As the the total mass inside a radius $r_{\rm{max}}$ is linearly dependent on the central density, there is a proportionality relation between the fraction of baryonic matter, $f_{\rm{b}}$, and the central densities: $$f_{\rm{b}} = \frac{M_{\rm{gas}}}{M_{\rm{gas}}+M_{\rm{halo}}} =
\frac{\rho_{\rm{c,gas}}}{\rho_{\rm{c,gas}}+\rho_{\rm{c,halo}}}.$$ In the following, we will use $\rho_{\rm{c,tot}}$ = $\rho_{\rm{c,gas}}+\rho_{\rm{c,halo}}$. Using $f_{\rm{b}}$ and $\rho_{\rm{c,tot}}$, one can simply write: $$\begin{aligned}
\rho_{\rm{c,gas}} &=& f_{\rm{b}}\, \rho_{\rm{c,tot}} ,\nonumber\\
\rho_{\rm{c,halo}} &=& (1-f_{\rm{b}})\, \rho_{\rm{c,tot}} .
\end{aligned}$$ Similarly to the gas, the dark matter halo evolves under the laws of gravity.
We consider a core in the initial dark model profile. Whilst cosmological simulations predict the formation of cuspy dark halos [@navarro97b; @fukushige97; @moore98; @springel08 and the references therein], our choice is motivated by observational evidences found in normal, low brightness and dwarf galaxies [@blaisouellette01; @deblok02; @swaters03; @gentile04; @gentile05; @spekkens05; @deblock05; @deblok08; @spano08]. Measuring the inner slope of the dSph profiles is very challenging, nevertheless, @battaglia08 show that the observed velocity dispersion profiles of the Sculptor dSph are best fitted by a cored dark matter halo.
Velocities and temperature
--------------------------
The initial velocities are obtained by assuming equilibrium, free of any rotation. For a spherical distribution, we can assume that the velocity dispersion is isotropic. It can be derived from the the second moment of the Jeans equation [@binney87; @hernquist93]. In spherical coordinates, one writes: $$\sigma^2(r) = \frac{1}{\rho(r)}\int_r^\infty\! dr' \,\rho(r')\, \partial_{r'} \Phi(r').
\label{sr_sph}$$ The halo velocities are randomly generated in order to fit the velocity dispersion $\sigma^2$ at any given radius.
The temperature $T$ of the gas is deduced from the virial equation: $$\lim_{r \rightarrow \infty} \frac{G M(r)}{r} = \frac{3\,k_{\rm{B}}\,T}{\mu m_{\rm{h}}},
\label{u_gas}$$ leading, for a pseudo-isothermal sphere, to: $$T = \frac{4\pi}{3} \frac{\mu m_{\rm{h}}}{k_{\rm{B}}}G\,\rho_{\rm{c,tot}} r_{\rm{c}}^2,
\label{T_gas}$$ where $k_{\rm{B}}$ is the Boltzmann constant, $m_{\rm{h}}$ the proton mass and $\mu$ the mean molecular weight of the gas.
Initial Parameters {#parameters}
------------------
All simulations start with an initial radius of $r_{\rm{max}} =
8\,\rm{kpc}$, distance at which the gas and dark matter densities are about $1/1000$ of the central ones. We consider two different core radii, $r_{\rm{c}}$, of 0.5 and 1 kpc. The choice of the central total density $\rho_{\rm{c,tot}}$ (dark matter + gas) uniquely determines the initial total mass of the system, $M_i$, which we vary over a range of $2\times 10^8$ to $9\times 10^8\,\rm{M_\odot}$. We investigate the effect of the initial baryonic mass fraction, $f_{\rm{b}}$, by varying it from 0.1 to 0.2. Indeed, this helps in disentangling the influence of the total gravitational potential from that of the gas mass. The masses of the gas and halo particles remain constant at $1.4\times 10^4\,\rm{M_\odot}$ and $9.3\times
10^4\,\rm{M_\odot}$, respectively. The corresponding gravitational softening lengths are $0.1$ and $0.25\,\rm{kpc}$. As a consequence, the simulations start with $4000$ to $20'000$ particles. The variation in number of particles is therefore at most a factor 5, hence a factor of 1.7 in spatial resolution, which justifies the choice of fixed softening lengths. The star formation parameter $c_\star$ is varied from $0.01$ to $0.3$. The initial mass of the stellar particles is $980\,M_{\odot}$, corresponding to about one tenth of the initial mass of the gas particles.
Models
======
We performed [166]{} simulations to understand the role of each parameter at play, and to identify a series of generic models reproducing the observations. The complete list of the simulations is given in Tab. \[parameters\_1\], \[parameters\_2\] and \[parameters\_3\] of Appendix \[appendix1\]. The models have been run for $14\,\rm{Gyr}$.
Modes
-----
As mentioned in Section \[model\], the initial sphere of DM+gas is in equilibrium under adiabatic conditions. At the onset of the simulations, the energy loss due to cooling causes the gas to sinks in the potential well and contract. The total potential is deepen not only due to the central increase in gas density, but also as a consequence of the halo adiabatic contraction. Despite the large increase in density, the gas temperature is kept nearly constant due to the strong hydrogen recombination cooling above $10^4\,\rm{K}$ (see Fig. \[cooling\_fct\]). Therefore, the gravitational energy recovered from the deepening of the potential well is dissipated nearly instantaneously. For densities above $\rho_{\rm{sfr}}$, the evolution of the model depends on supernova heating, directly linked to the star formation rate.
Besides this general description common to all simulations, we identify three different major regimes. We refer to them as “full gas consumption”, “outflow” and “self-regulation”. For each of them, Fig. \[MLSfr\] presents the evolution with time of the star formation rate (SFR), the central gas density, the central gravitational potential and the mass of the gas within $3\,\rm{kpc}$ from the galaxy center.
### Full gas consumption {#collapse}
In cases where $c_\star$ is low for a given $M_{\rm{i}}$, the energy released by the supernova explosions is unable to counterbalance the radiative cooling. As a consequence, the gas keeps on sinking in the galaxy inner regions and reaches very high densities. Stars are formed continuously and at high rate. The model \#647 ($c_\star=0.05$ and $M_{\rm{i}}=6.6\times 10^8\rm{M}_\odot$) in Figure \[MLSfr\] provides a clear example of this regime: steep rise in star formation rate and central gas density. The chemical enrichment of the resulting systems is rapid, and their metallicities quickly exceed the highest ones measured in dSphs. These models were not investigated further.
### Outflow
Stars can be formed at slightly lower densities by increasing $c_\star$ at a given initial mass, or by decreasing the initial mass at fixed $c_\star$. This is sufficient to stop the drastic accumulation of gas at the center. Nevertheless, the gas density is still high, and star formation is very efficient. When SNe explode, a huge amount of energy is deposited in the gas, which in turn is expelled from the galaxie’s central regions. In parallel, the central potential increases (it is negative), primarily due to the ejection of the the gas, but also due to the ensuing DM halo expansion. The final consequence is a strong outflow. A large fraction of the total gas mass is ejected beyond a radius of $3\,\rm{kpc}$, chosen to be large enough compared to the stellar extent of the systems. For clarity, we illustrate this regime with M\#519 in Fig.\[MLSfr\], in which the outflow occurs early in the galaxy evolution: after $\sim 2\,\rm{Gyr}$, there is virtually no gas left.
### Self-regulation {#regulated}
Dwarf galaxies are formed in a regime of self-regulation, characterized by successive periods of cooling and feedback. Such intermittent star formation episodes occurring spontaneously in hydrodynamical simulations have been mentioned by @stinson07 and @valcke08.
M\#576 in Fig.\[MLSfr\] offers the example of an intermediate mass self-regulated system ($M_{\rm{i}}=4.39\times 10^8\,\rm{M_\odot}$). As usual, the first contraction of the gas leads to a peak in star formation ($t=0.6\,\rm{Gyr}$). The gas expelled by the supernova feedback is diluted at densities below $\rho_{\rm{sfr}}$, and the star formation stops. As the gas particles cool, they become eligible to star formation again, forming a new burst. Star formation occurs at high frequency in M\#576 (periods between $100$ and $200\,\rm{Myr}$). It produces a flat distribution of stellar ages, mimicking a nearly continuous star formation rate (see Fig. \[CstarvsMi2\]). We will show later that the corresponding chemical signatures are also very homogeneous.
Contrary to what has been observed by @stinson07, the fluctuation of the SF is not strictly periodic. However, we confirm the influence of the total mass $M_{\rm{i}}$ on the duration of the quiescent periods [@valcke08]. Lower mass systems ($M_{\rm{i}}
\lessapprox 3\times 10^{8}\rm{M_\odot}$) are generally characterized by star formation episodes separated by longer intervals, up to a few Gyrs. These systems exhibit inhomogeneous stellar populations.
Driving parameters
------------------
The description of the different regimes of star formation histories already points out the importance of both the initial total mass, $M_{\rm{i}}$, and the star formation parameter, $c_\star$. Figure \[CstarvsMi2\] presents the stellar age histograms for models with $f_{\rm{b}}=0.15$ and $r_{\rm{c}}=1$. From bottom to top, $c_\star$ increases by a factor 6. From left to right, $M_{\rm{i}}$ increases by a factor 3. The highest mass systems are characterized by a strong predominance of the old stellar populations. Decreasing the initial total mass extends the period of star formation, passing progressively from a continuous to a discrete distribution of stellar ages. The role of $c_\star$ appears secondary, distributing slightly differently the different peaks of star formation (position and strength).
Fig. \[CstarvsMi\] summarizes the [166]{} simulations in a diagram of $c_\star$ and $M_{\rm{i}}$, for different core radii $r_{\rm{c}}$ and baryonic fractions $f_{\rm{b}}$. Colors code the final galaxy stellar metallicity [$\langle \rm{[Fe/H]}\rangle$]{}, computed as the median of the distribution, since it best traces the position of the metallicity peaks in the observations. The size of the circles is proportional to the final stellar luminosity in the $V$-band, $L_{\rm{v}}$. The small black triangles indicate the simulations that lead to full gas consumption and have been stopped. As described earlier, the latters result from a too small $c_\star$. It can be avoided, for our purpose, by increasing $c_\star$ or decreasing $M_{\rm{i}}$. Self-regulated systems with limited outflow are found for smaller $M_{\rm{i}}$. These tendencies do not dependent on $f_{\rm{b}}$ and $r_{\rm{c}}$, which can only slightly modify the interval of mass in which a particular regime is valid. For a given $c_\star$, [$\langle \rm{[Fe/H]}\rangle$]{} increases with $M_{\rm{i}}$ and similarly, for a given $M_{\rm{i}}$, [$\langle \rm{[Fe/H]}\rangle$]{} increases with $c_\star$. At very low mass, however, [$\langle \rm{[Fe/H]}\rangle$]{} is only weakly influenced by $c_\star$. On the contrary, the larger the mass, the smaller $c_\star$ increase is needed to raise [$\langle \rm{[Fe/H]}\rangle$]{}.
The left and middle panels of Fig. \[LFeAgevsMi\] display the final galaxy stellar metallicity and stellar $V-$luminosity, respectively, as a function of $M_{\rm{i}}$. The Local Group dSphs luminosities [@mateo98; @grebel03] and mean metallicities (DART) are indicated with horizontal dotted red lines. The most outstanding result is that changing $M_{\rm{i}}$ by a factor 4 translates to a change in $L_{\rm{v}}$ by a factor 100. [$\langle \rm{[Fe/H]}\rangle$]{} is varied by a factor $\sim$ 3. By comparison, the influence of $c_\star$ on the galaxy properties appears small. In any case, increasing $c_\star$ will also help increasing both $L_{\rm{v}}$ and [$\langle \rm{[Fe/H]}\rangle$]{}. The consequence of varying $M_{\rm{i}}$ is not linear. At low initial mass, a small increase in mass is sufficient to strongly increase $L_{\rm{v}}$ and [$\langle \rm{[Fe/H]}\rangle$]{}, while at larger initial mass, the relations saturate and a larger step in mass is necessary. The mass-luminosity and metallicity-luminosity relations will be discussed in the next section.
The right panel of Fig. \[LFeAgevsMi\] presents the relation between the galaxy’s mean stellar age and $M_{\rm{i}}$. The influence of $c_\star$ looks more linear than previously on $L_{\rm{v}}$ and [$\langle \rm{[Fe/H]}\rangle$]{}. As a matter of fact, we have seen in Fig. \[CstarvsMi2\] that it plays a role in the stellar age distribution. At a given initial mass, $c_\star$ determines the length of the star formation periods as well as the interval between them.
As a conclusion, the above analyses stress the primordial impact of the initial total mass of the systems. Moreover, one can clearly identify the range of possible $M_{\rm{i}}$ leading to the formation of the Local Group dSphs as we observed them today. This range is narrow, e.g, a factor 2 centered on $M_{\rm{i}} \ge 5\times
10^{8}\rm{M_\odot}$.
Global relations {#global_relations}
================
dSph galaxies follow luminosity-mass and luminosity-metallicity relations that are considered as cornerstones to understanding their formation and evolution [@mateo98; @wilkinson06; @gilmore07; @strigari08; @geha08; @kirby08].
In the following discussion, we calculate all physical quantities (luminosities, masses, abundances) within the radius $R_{\rm{L}}$ defined as the radius containing 90% of a galaxy’s total luminosity. This choice is guided by the wish to reproduce as closely as possible the observational conditions under which these quantities are measured. The classical dSphs (as opposed to newly discovered faint ones) surrounding the Milky Way have tidal radii in the range $\sim$ $0.5\,\rm{kpc}$ to $3\,\rm{kpc}$ [@irwin95]. Fixing a constant small aperture for all dSphs would underestimate both light and mass of the largest systems. Since dark matter does not necessarily follow light, this would also bias the results. The observational estimates of the dSph total masses are based on stellar velocity dispersions measured at galactocentric radii as large as possible, thereby directly linked to the limits of the visible matter. Although the farthest measurements do not always reach the galaxies’ tidal radii, their location is determined by severe drops in stellar density, ensuring that the bulk of the galaxies’ light is enclosed. Consequently, we compare our models to the masses derived at the outermost velocity dispersion profile point [e.g, @walker07; @battaglia08; @kleyna04]. Ursa Minor is the only exception to this rule. Its mass has been derived from its central velocity dispersion [@mateo98].
Figures \[MLvsL\] and \[FevsL\] display the relations of the galaxies’ mass-to-light ratios $M/L_{\rm{v}}$ and the median of the stellar metallicity distributions [$\langle \rm{[Fe/H]}\rangle$]{}, together with the total luminosity of the model galaxies. The observations are represented in red, with squares for the Milky Way satellites and crosses for the others. In general, the values of [$\langle \rm{[Fe/H]}\rangle$]{} are taken from @mateo98 when available or from @grebel03 otherwise. The mean metallicities of Carina, Fornax, Sculptor, and Sextans are calculated from their metallicity distributions [@helmi06; @battaglia08; @battaglia06].The mean metallicity of Leo II is derived from the metallicity distribution of @bosler07. The luminosities are taken from @grebel03, with the exception of Draco [@martin08]. The masses of Carina, Fornax, Draco, Leo I and Leo II are computed by @walker07 inside $r_{\rm{max}}$. The mass of Sextans corresponds to the upper limit of @kleyna04, while the mass of Ursa Minor comes from @mateo98. Sculptor’s $M/L_{\rm{v}}$ is taken from @battaglia08.
Both $M/L_{\rm{v}}$ and [$\langle \rm{[Fe/H]}\rangle$]{} show very clear log-linear relations with $L_{\rm{v}}$: $$\log_{10} \left( M/L \right) = -0.79 \log_{10} \left( L_{\rm{v}} / L_{\odot} \right) + 1.85,
\label{MLvsL_fit}$$ and $$\langle \rm{[Fe/H]}\rangle = 0.68 \log_{10} \left( L_{\rm{v}} / L_{\odot} \right) -1.8.
\label{FeHvsL_fit}$$
Despite differences in $c_\star$, $r_{\rm{c}}$ and $f_{\rm{b}}$, all simulations nicely reproduce the observations, with a reasonably small scatter. This stresses once more that a dSph galaxy’s total initial mass drives most of its evolution.
To allow deeper insight into the building up of the $M/L_{\rm{v}}-L_{\rm{v}}$ relation, Fig. \[MvsL\] distinguishes between dark matter (DM), stars and gas. Colors encode the three different initial baryonic fractions that we have considered, $f_{\rm{b}}=0.2$ (yellow), $f_{\rm{b}}=0.15$ (green) and $f_{\rm{b}}=0.1$ (blue).
Very naturally, stellar mass scales with the luminosity. At fixed luminosity, the dispersion in stellar mass is of the order of $5
\times 10^6$M$_{\odot}$. In fact, a more appropriate way to look at this panel is to consider the dispersion in luminosity at fixed stellar mass, since the dispersion in luminosity is a direct consequence of various distributions in stellar ages and metallicities. This dispersion, of the order of $
10^6\,\rm{L_\odot}$. It increases slightly for larger masses, for which star formation can last longer, inducing a larger number of possible age/metallicity combinations.
As already discussed in Fig. \[LFeAgevsMi\], whilst $L_{\rm{v}}$ spans nearly 3 orders of magnitude, the mass of dark matter varies little. This variation is much less than one order of magnitude inside $R_{\rm{L}}$ and is mostly due to the dispersion among the models. A common mass scale (inside $R_{\rm{L}}$), around $1-5 \times
10^7\,\rm{M_\odot}$, for the dSph total masses seems also favored by the observations, although an exact value for this limit is difficult to ascertain, given the large uncertainties of the mass estimates in general [@mateo98; @gilmore07; @strigari08].
Interestingly, one can now witness the effect of varying the initial baryonic fraction. Galaxies with $L_{\rm{v}} < 3\times
10^6\,\rm{L_\odot}$ exhibit identical DM halo masses, whatever $f_{\rm{b}}$, while for larger luminosities, the models of lowest $f_{\rm{b}}$ require larger halo masses in order to generate a similar quantity of stars. For $f_{\rm{b}}=0.20$, the DM halo mass is constant over the whole luminosity range. This demonstrates that while dark matter plays a crucial role in confining the gas, the amount of the latter is also important for the most massive dSphs: it must reach a critical amount to enable their formation.
The third panel of Fig. \[MvsL\] displays the mass of residual gas after $14\,\rm{Gyr}$. It constitutes a very small fraction of the total mass, and is therefore undetectable at the level of the scaling relations. Quite interestingly, its amount is of the order of the HI mass observed in dwarf irregular galaxies (dIs) [@grebel03] and is weakly dependent on the total luminosity. Nevertheless, there is a non-intuitive tendency for the most luminous galaxies ($>5\times
10^6\,\rm{L_\odot}$) to exhibit less gas than the rest of the systems on the sequence. However, the most massive galaxies succeed in retaining most of their gas despite the supernova explosions. Less than 60% of their gas is ejected, while this fraction lies between 70% and 90% for less massive systems, in agreement with the results of @valcke08. However, the star formation efficiency is also higher in more massive galaxies. As a result, the gas consumption counterbalances the presence of the large gas reservoir. We will come back to this in Section \[discussion\].
Since the gas mass is very much constant at around $10^7\,\rm{M_\odot}$, the smallest galaxies have proportionally more gas than the massive ones (see bottom panel of Fig. \[MvsL\]). Below $ 3 \times 10^6\,\rm{L_\odot}$, galaxies have more gas left than they have formed stars. Their gas to stellar mass ratios reach 100 at the faint end of the dSph model sequence. Meanwhile, stars dominate over gas by a factor 5 for the most luminous galaxies.
As a conclusion, the lessons to be taken from Fig. \[MvsL\] are twofold: i) all dSphs are clearly dominated by dark matter. The lower the mass of the system, the lower the final baryonic fraction. ii) all model galaxies present an excess of gas at the end of their evolution, as has been found in similar studies [@marcolini06; @marcolini08; @stinson07; @valcke08]. As demonstrated earlier, dSphs cannot originate from smaller amounts of gas (for a given final luminosity, metallicity and age). It is necessary to initiate the star formation in the observed proportions. However, it is not yet clear how much of this gas must be kept in the subsequent phases of the galaxy evolution. It is clear that the excess of gas, observed in models in isolation, must in reality have been stripped some time during the galaxy evolution.
The quantity of gas falls in the range of HI mass observed in dIs, always found further away than dSphs from their parent galaxies, thereby bringing another piece of evidence for the role that interactions might play in its removal. It might also be achieved in a hierarchical formation framework, for which the gravitational potentials are initially weaker.
As we have just shown, the global scaling relations are reproduced by our model with impressive ease. In turn, this conveys the idea that the global scaling relations do not form a very precise set of constraints. They do not reflect the diversity in star formation histories that we have illustrated. In order to understand the individual histories of the Local Group dSphs, one definitely needs to go one step further and consider their chemical abundance patterns, as well as the information that color-magnitude diagrams provide on the stellar age distributions.
Generic models {#generic_models}
==============
We will now select and discuss a series of generic models reproducing the properties of four Milky Way dwarf spheroidals, Carina, Fornax, Sculptor and Sextans. The choice of these models is based on four observational constraints:
- The dSph total luminosity, which scales with the total amount of matter involved in the galaxy star formation history.
- The metallicity distribution which traces the star formation efficiency.
- The chemical abundance patterns, in particular that of the $\alpha$-elements, that determines the length and efficiency of the star formation period together with its homogeneity. We use magnesium for the $\alpha$-elements. \[Fe/H\] and \[Mg/H\] are derived from high resolution spectroscopy in the central regions of the galaxies [Hill et al in preparation; Letarte et al. in preparation; @venn05; @letarte07; @shetrone03; @koch08].
- The stellar age distributions. They complement the above constraints with information on possible series of bursts [@smecker-hane96; @hurley-keller98; @coleman08].
Table \[table2\] lists the initial parameters of our generic models. Table \[table3\] lists their final properties, namely the stellar luminosity in the $V$-band, $L_{\rm{v}}$, the mass-to-light ratio, $M/L_{\rm{V}}$, the gas mass, $M_{\rm{gas}}$, the median metallicity, [$\langle \rm{[Fe/H]}\rangle$]{}, the fraction of stars with \[Fe/H\] lower than $-3$ and, finally, the stellar age distribution divided in 3 bins: younger than $4\,\rm{Gyr}$, between $4$ and $8\,\rm{Gyr}$ and older than $8\,\rm{Gyr}$.
Fig. \[NvsFe\], \[MgFevsFe\] and \[NvsAge\] display the stellar metallicity distribution, the \[Mg/Fe\] vs \[Fe/H\] diagrams and the stellar age histograms, respectively, after $14\,\rm{Gyr}$ of evolution. Fig. \[FevsAge\] shows the stellar age-metallicity relations.
We do not try to match exactly all properties of our target dSphs. Instead, we select the four most satisfying models in our sample of [166]{}. More specifically, we allow a freedom of a factor 2 in luminosity and a shift of a few tenths of dex in peak \[Fe/H\]. It is beyond doubt that we could fine-tune $M_{\rm{i}}$, $c_\star$, $r_{\rm{c}}$ and $f_{\rm{b}}$ to exactly reproduce the observations. However, this is beyond our scope, which remains to test the hypothesis of a formation framework common to all dSphs. The dependence of the results on the model parameters is illustrated in Fig. \[L0.4b\] and \[L2.3b\]. We selected 6 models at low ($L_{\rm{v}}\cong 0.5\times 10^6\,\rm{M_\odot}$) and high ($L_{\rm{v}}\cong 3.7\times 10^6\,\rm{M_\odot}$) luminosities. The baryonic fraction is fixed each time, and only $c_\star$ and $M_{\rm{i}}$ vary. One sees that around a given fixed core of observed characteristics (e.g, luminosity, metallicity), we could run the models on a finer grid of parameters in order to match the galaxy properties in all details. This means, for example, to exactly reflect the stellar age distribution, or the spread in abundance ratios.
Global evolution
----------------
Fig. \[evolution12\] and \[evolution34\] show the stellar surface density, the gas surface density and the gas temperature between $0$ and $14\,\rm{Gyr}$ for our four selected generic models. Not only do these reproduce the properties of Carina, Fornax, Sculptor and Sextans individually, but they also depict the full spectrum of evolutions seen in our models. The size of each panel is $20\times
20\,\rm{kpc}$.
All gas particles initially share the same temperatures, corresponding to the galaxy virial temperature, as given by Eq. \[T\_gas\]. As soon as the simulations start, the gas looses energy by radiative cooling. Consequently, the gas density increases in the galaxy central regions. Red areas in the gas density maps identify densities larger than $\rho_{\rm{sfr}}$, i.e., they mark regions where gas particles may be eligible for star formation. When this is indeed the case, the newly formed stars are traced by their high brightness in the stellar density maps.
The SN explosions induce temperature inhomogeneities. Indeed, the heated central gas expands and generates a wave propagating outwards. If the SN feedback dominates the cooling, the gas is diluted and the red color vanishes from the center of the gas density maps: star formation is quenched. Such quiescent periods are characterized by a nearly constant and homogeneous central temperature $\sim
10^4\,\rm{K}$, the temperature at which the radiative cooling counterbalances the adiabatic heating. When gas has sufficiently cooled, it can condense again, and star formation is induced again. These periods of star formation and quiescence alternate at low or high frequency, depending on the mass of the galaxy. For low-mass systems, the cooling time is of the order of several Gyrs, it is much shorter for more massive ones. We will now see how each of these cases translate into dSph stellar population properties.
Sextans
-------
The stellar population of Sextans is dominated by stars older than $10\,\rm{Gyr}$ [@lee03]. With its mean metallicity $\langle
\rm{[Fe/H]}\rangle = -2.05$ and a $V$-band total luminosity of $0.5
\times 10^6\,\rm{L_\odot}$, it falls exactly on our luminosity-metallicity relation (Fig. \[FevsL\]), and its properties are reproduced by the model \#590 ($L= 0.53\times 10^6\,\rm{L_\odot}$, $\langle \rm{[Fe/H]}\rangle=-2.02$ ) that experiences an outflow. Sextans’ evolution is dominated by an early period of star formation lasting about $ 3\,\rm{Gyr}$. The gas surface density map of Fig. \[evolution34\] shows the dense central region at the origin of the star formation burst ($t=1.9\,\rm{Gyr}$). At that time, a small but bright stellar system is already formed. After this period, the gas is expelled and diluted. No star can form until the last $\rm{Gyr}$, when the gas has sufficiently cooled down to fulfill the star formation criteria. This last episode of star formation is an artifact of the gas retained by our model, as discussed in Section \[global\_relations\]. It is however negligible compared to the bulk of the population and does not influence Sextans’ properties.
Fig. \[MgFevsFe\] shows that the bulk of Sextans model stars are located at \[Mg/Fe\] $\cong 0.3$, with however a noticeable dispersion at lower values. The dispersion originates from the uneven intensities of the star formation peaks. They create regions with diverse levels of chemical enrichment. When the intensity of star formation rises again after a period of semi-quiescence, the ejecta of new SNe II are mixed with material of low $\alpha$-element abundance. Refinement of the model would require a larger sample of observed stars at high resolution, particularly to estimate the statistical significance of the dispersion in \[Mg/Fe\].
Carina
------
Carina seems to have experienced a complex evolution characterized by episodic bursts of star formation, with at least three major episodes at around $15$, $7$ and $3\,\rm{Gyr}$ , and a period of quiescence between $7$ and $3\,\rm{Gyr}$ [@smecker-hane96; @hurley-keller98]. Our set of simulations reveal that episodic bursts of star formation are intrinsic features of the self-regulated low initial masses systems ($M_{\rm{i}} \lessapprox 3\times 10^8\,\rm{M_\odot}$, see Fig. \[CstarvsMi2\]). Shortly after a star formation episode, supernovae explode, gas is heated and expands. Because its density is low, its cooling time is of the order of several Gyrs. This series of well spaced-out SF peaks translate into dispersion in \[Mg/Fe\] at fixed \[Fe/H\]. This spread is more accentuated than for Sextans, due to the extended star formation and the longer intervals between bursts.
Fig. \[FevsL\] shows that Carina, with $\langle
\rm{[Fe/H]}\rangle=-1.82$ and $L_{\rm{v}} = 0.72\times
10^6\,\rm{M_\odot}$, falls above the relation \[Fe/H\]-$L_{\rm{v}}$ defined by our models. As already mentioned, we give priority to constraints provided by the spectroscopic data, and allow some flexibility in $L_{\rm{v}}$. Under these conditions, model \#533 ($M_{\rm{i}} = 2.7\times 10^8\,\rm{M_\odot}$) provides a very reasonable fit to Carina’s properties. The three major SF episodes are reproduced: 45% of the stars have ages between $9$ and $15\,\rm{Gyr}$, 29% have $4$ to $8\,\rm{Gyr}$ and 22% are younger than $3\,\rm{Gyr}$. Whilst Carina is somewhat less luminous than Sextans, its mean metallicity is higher. This is due to a lower initial mass, locking less matter in stars, but a longer period of star formation, obtained by a slightly higher star formation parameter, which avoids outflows.
Sculptor
--------
The Sculptor dSph has formed stars early, over a period of a few Gyrs. No significant intermediate age population has been found, hence excluding star formation within the last $ \sim 5\,\rm{Gyr}$. The evidence for low $\alpha$-element enhancement agrees with an extended star formation and self-enrichment over a period of at least $2\,\rm{Gyr}$ [e.g. @babusiaux05; @shetrone03; @tolstoy03].
Sculptor’s properties are well reproduced by model \#630 ($L=
2.86\times 10^6\,\rm{L_\odot}$, $\langle \rm{[Fe/H]}\rangle=-1.82$). The outstanding feature of Sculptor, as compared to the two previous dSph, is the small dispersion in the \[Mg/Fe\] vs \[Fe/H\] diagram (Fig. \[MgFevsFe\]). This is guaranteed by a strong initial star formation, followed by subsequent episodes of lower, but smoothly decreasing intensities, ensuring the chemical homogeneity of the interstellar medium.
Fornax
------
Fornax is obviously the most challenging case of dSphs. It is the most luminous of all ($15.5\times 10^6\,\rm{L_\odot}$), it is metal-rich ($\langle \rm{[Fe/H]}\rangle=-1.07$) and has experienced multiple periods of star formation. Indeed, in addition to very old stars, Fornax reveals a dominant intermediate-age population, as well as stars of $\sim 3-4\,\rm{Gyr}$ [@coleman08].
Fornax is nicely reproduced by the self-regulated model \#575. Stars are formed in a high frequency series of short bursts, lasting between $200$ and $500\,\rm{Myr}$. The amplitude of these bursts progressively decreases with time, until $\sim 7\,\rm{Gyr}$. Thereafter, part of the gas that had been expelled by previous SN explosions, has sufficiently cooled to restore star formation, at $
\sim 3\,\rm{Gyr}$, similar to what is observed. The succession of the short bursts of similar intensities mimics a continuous star formation and leads to an efficient and homogeneous chemical enrichment ($\langle \rm{[Fe/H]}\rangle=-1.03$).
We reach a luminosity $L= 12.6 \times 10^6\,\rm{L_\odot}$, slightly below the observation of Fornax. However, just as in the case of Carina, the exact value of the $V$-band luminosity is very sensitive to the exact amount of stars formed in the last Gyrs and may easily vary. Stars form up to the end of the simulation. This is consistent with the observational evidence of a small number of $100\,\rm{Myr}$ stars [@coleman08].
Figure \[FevsAge\] show the stellar age-metallicity relations of our four generic models. In all cases, we are far from a one to one correspondence between age and metallicity, although below \[Fe/H\]=-2.5 (but more safely below \[Fe/H\]=$-3$), stars are generally older than 10 Gyr. The dispersion in metallicity is in the range 0.5 to 1 dex at ages younger than 10 Gyr.
---------- ------- -------------- -------------- ----------- ------------------------------ ------------------------
dSph \# $r_{\rm{c}}$ $f_{\rm{b}}$ $c_\star$ $\rho_{\rm{c,tot}}$ $M_{\rm{tot}}$
$[\rm{kpc}]$ $[10^8\,\rm{M_\odot/kpc^3}]$ $[10^8\,\rm{M_\odot}]$
Carina $533$ 0.5 0.20 0.100 1.20e-03 2.73
Leo II $587$ 1.0 0.15 0.050 4.00e-04 3.29
Sextans $590$ 0.5 0.15 0.050 1.60e-03 3.64
Sculptor $630$ 1.0 0.20 0.035 5.00e-04 4.12
Fornax $575$ 1.0 0.15 0.200 6.66e-04 5.48
---------- ------- -------------- -------------- ----------- ------------------------------ ------------------------
: Initial parameters of the five generic models.[]{data-label="table2"}
---------- ------- ------------------------ ---------------- ------------------------ ------------------------------ -------------------- ------------------- ------------------- -------------------
dSph \# $L_{\rm{v}}$ $M/L_{\rm{V}}$ $M_{\rm{gas}}$ $\langle \rm{[Fe/H]}\rangle$ $\rm{[Fe/H]}<-3.0$ $\le 4\,\rm{Gyr}$ $[4,8]\,\rm{Gyr}$ $\ge 8\,\rm{Gyr}$
$[10^6\,\rm{L_\odot}]$ $[10^8\,\rm{M_\odot}]$ dex % % % %
Carina $533$ 0.72 231 0.28 -1.82 2.9 26.1 29.3 44.6
Leo II $587$ 0.97 219 0.33 -1.82 2.9 24.4 26.4 49.2
Sextans $590$ 0.53 390 0.17 -2.02 5.2 1.7 0.0 98.3
Sculptor $630$ 2.86 83 0.32 -1.82 5.6 3.6 0.0 96.4
Fornax $575$ 12.76 27 0.12 -1.03 1.2 14.4 10.6 75.0
---------- ------- ------------------------ ---------------- ------------------------ ------------------------------ -------------------- ------------------- ------------------- -------------------
Leo II
------
While writing up our results, the abundance ratios in a sample of 24 stars in Leo II have been published [@shetrone09]. We keep Leo II separated from the other individual case analyses, since its abundance ratios have been obtained from lower resolution spectra and therefore have larger uncertainties.
Fig. \[leoII\] illustrates how model \#587 can reproduce the star formation history of Leo II. The observed metallicity distribution, derived from $\sim 100$ stars, is taken from @bosler07, and \[Mg/Fe\] from [@shetrone09]. Leo II has a luminosity similar to that of Sextans, but unlike Sextans, Leo II sustained a low but constant star formation rate, leading to an important intermediate age stellar population [@mighell06; @koch07]. The reason for a continuous (low) star formation in Leo II’s model is found in the slightly smaller initial total mass for the same star formation parameter as Sextans; it prevents the gas outflow. Carina and Leo II have about the same mean metallicities. They probably have probably experienced a very similar evolution, as indeed seen in Table \[table3\] for their mean properties. However, the two galaxies indeed differ: i) by the periods of quiescence which are longer in Carina, and ii) by the extent of the star formation peaks, which are between 2 to 3 times longer in Leo II, at intermediate ages. Model \#587 produced a higher final luminosity than observed, due to the presence of residual gas and therefore recent star formation, as in all the models. It is important to recall that we do not aim at reproducing Leo II properties in fine details, but instead check whether its main features exist in the series of models we ran. In this respect, model \#587 demonstrates that a low level of rather continuous star formation is indeed achievable.
The extension of the sample of stars with spectroscopic data allowing measurements of abundance ratios would greatly help the precise identification of Leo II model. This is particularly true at \[Fe/H\] $> -1.6$ which would confirm or not the apparent plateau in \[Mg/Fe\]. Even more decisive and more general, one needs to observationally confirm or disprove the dispersion in \[Mg/Fe\] at low and moderate metallicities in low mass dSphs, as it signals the discrete nature of star formation as advocated by contemporary models.
Table \[table2\] gives the fraction of very-metal poor (\[Fe/H\]$<-3$) stellar particles in the generic models. The general trend of our series of [166]{} models is that the more extended the star formation period, the smaller this fraction. This is clearly illustrated by the models of Carina and Fornax, as compared to those of Sextans and Fornax. Although the tail of very metal-poor stars is small (from$\sim$ 1 to 6%), it is still larger than suggested by the observations to date [@helmi06]. In order to properly address this particular issue, one needs i) to investigate the impact of more sophisticated IMFs, such as the one of @kroupa93, for which the number of low-mass stars is smaller than for a Salpeter IMF, and ii) to introduce the peculiar features of star formation at zero metallicity. The conditions of transition between population III and population II star formation depends on the galaxy halo masses, and so is the galaxy metallicity floor [@wise08]. This opens a independent new field of investigation. For the time being, we check our model predictions against population II properties. Interestingly, the outflows generated by these very first generation of stars offers an alternative solution to get rid of the final excess of gas.
Mass profiles and velocity dispersions
--------------------------------------
Although our primary goal is not to match the morphology of the dSphs, we have considered the shape of the final modeled dSph profiles within $R_{\rm{L}}$. Fig. \[profiles\] displays the mass density profiles of the stellar and dark components for our four main targets. Each model is successfully fitted by a Plummer model, revealing a flat inner profile, in agreement with the observations. The ratio between the stellar and dark core radii $r_{c}^{\rm{stars}}/r_{c}^{\rm{halo}}$ ranges between $0.2$ and $0.5$, indicating a spatial segregation of the stars relative to the dark halo, as discussed in [@penarrubia08]. Additionally, the line-of-sight stellar velocity dispersion profiles are flat in the enclosed within the core radii, with $6$ to $10\,\rm{km/s}$, corresponding to the least and the most massive dSph, respectively, hence well within the observed range [@walker06; @munoz06; @battaglia08; @walker06b].
Discussion
==========
We have shown that the total initial mass $M_{\rm{i}}$ and $c_{\star}$ are the two driving parameters for the evolution of dSphs. Our models which describe galaxies in isolation, i.e., not yet taking into account interactions or accretions that inevitably occur in a $\Lambda$CDM Universe, already lead to the observed variety of star formation histories. Decreasing the initial total mass of the galaxies separates the periods of star formation.
The marked differences found between high- and low mass systems is a key point in understanding the formation of dSphs. In order to illustrate our findings, we now focus on cooling time and gas temperature. The top panel of Fig. \[virial\_temperature\] displays the galaxy’s virial temperature as a function of its initial mass (Eq. \[T\_gas\]).
The green horizontal band indicates the temperature at which the cooling time of the gas is equal to $100\,\rm{Myr}$, for two threshold densities, $\rho_{\rm{sfr}}$ and $10^{-3}\,\rho_{\rm{sfr}}$ that are representative of the range in gas density observed in the course of our simulations. The value of $100\,\rm{Myr}$ is chosen to be short compared to the galaxy lifetimes. The lower border of the green band corresponds to $\rho_{\rm{sfr}}$ while the upper ones correspond to $10^{-3}\,\rho_{\rm{sfr}}$. The intersection between the green band and the curves corresponds to a total mass ranging between $3$ and $4\times 10^8\,\rm{M_\odot}$, delimiting two regimes distinguished by long and short cooling times, as can be measured in the bottom panel of Fig. \[virial\_temperature\]. The cooling times of $\rho_{\rm{sfr}}$ and $10^{-3}\,\rho_{\rm{sfr}}$ as a function of the galaxy initial masses are shown in solid and dashed lines, respectively, in the bottom panel of the figure. They are calculated by taking the virial temperature of the corresponding galaxy mass. The gas contained in galaxies of total masses larger than $4\times
10^8\,\rm{M_\odot}$, with corresponding virial temperatures larger than $10^4\,\rm{K}$ is characterized by a short cooling time ($\tau<$100[Myr]{}) at all densities. This short cooling time is due to the strong radiative cooling induced by the recombination of hydrogen above $10^4\,\rm{K}$. A galaxy which is in this cooling regime will see its gas loosing a huge amount of energy, sinking in its central regions and inducing star formation. This is precisely what happens in the model \#575 where stars are continuously formed, generating a metal-rich Fornax-like system. By decreasing the total mass from $4$ to $3\times 10^8\,\rm{M_\odot}$, the virial temperature falls below the peak of hydrogen ionization. As a consequence, the cooling function drops and the cooling time is increased by nearly three orders of magnitude. Below $3\times 10^8\,\rm{M_\odot}$, the cooling time is long, since the loss of energy by radiative cooling is strongly diminished. In this regime, star formation can only occur in episodic bursts, separated by periods corresponding to the mean cooling time. Carina gives a clear example of such a case. For even lower mass systems ($M_{\rm{tot}}<10^{8}\rm{M_\odot}$), the cooling time is longer than $10\,\rm{Gyr}$, and therefore no stars are expected to form.
From our simulations, a total mass of $\sim 4\times
10^8\,\rm{M_\odot}$ leads to a luminosity of $\sim
3\times\,10^6\,\rm{L_\odot}$. Not surprisingly, this luminosity corresponds to the critical one found in Section \[global\_relations\], below which the galaxies have more gas left than they have formed stars. The sharp cutoff of the cooling function below $4\times
10^8\,\rm{M_\odot}$ explains very nicely why a small decrease in mass induces a large drop in luminosity, seen in the middle panel of Fig. \[LFeAgevsMi\]. A direct consequence is the constancy of galaxy masses within $R_{\rm{L}}$ ($\sim 1 - 5 \times 10^7\,\rm{M_\odot}$) over the wide range of dSph luminosities, as discussed in Section \[global\_relations\] (Fig. \[MvsL\]) and suggested by @mateo98, @gilmore07 and @strigari08.
The next step in simulating the formation and evolution of dSph galaxies is to study the impact of external processes resulting from a $\Lambda$CDM complex environment, like tidal mass stripping, ram stripping, dark matter and gas accretion, as well as reionization and self-shielding. @penarrubia08 show that tidal mass stripping strongly depends on the spatial distribution of stars relative to the dark halo extent. If stars are very concentrated in large halos, dSphs will be more resilient to tidal disruption. Dwarf spheroidal galaxies may need to loose nearly $90\%$ of their mass before their star formation and therefore chemical evolution start to be affected. @mayer06 tackle all the above listed questions and show how their mechanisms can be interlaced. As an example, in their models gravitational tides help ram pressure stripping by diminishing the overall potential of the dwarf, but tides induce bar formation making subsequent stripping more difficult. Reionization prevent star formation, but self-shielding helps cooling. Clearly, understanding the impact of these physical processes on our specific modeling requires dedicated simulations.
Summary
=======
We have investigated the formation and evolution of dSph galaxies, which form a specific class among dwarf systems. Indeed, they reach the highest metallicities at fixed luminosity and are devoid of gas.
We performed [166]{} self-consistent Nbody/Tree-SPH simulations of systems initially consisting of dark matter and primordial gas. We have only considered galaxies in isolation. This has enabled us to identify the dominant physical ingredients at the origin of the observed variety in dSph properties. It has also allowed us to distinguish which of those are due to the galaxy intrinsic evolution and for which interactions are required. The diversity of star formation histories of the Milky Way dSphs, Carina, Leo II, Sextans, Sculptor and Fornax have been successfully reproduced in a single formation scheme.
$\bullet$ The crucial parameter driving the dSph evolution is the total initial mass (gas + dark matter). To a smaller extent the star formation parameter, $c_{\star}$, influences the final galaxy properties. In particular, it governs the stellar age distribution by modifying the time intervals at which star formation occurs. Since there is no physical reason for changing $c_{\star}$ from galaxy to galaxy, we understand its variation, if needed to reproduce the observations, as an indirect evidence for different interaction histories. In a hierarchical galaxy formation scheme, our initial masses correspond to the halo masses that must be reached along the merger tree before ignition of the bulk of the star formation. Besides, the chemical abundance ratios constrain the timescale of these halo mergers. For example, we have shown that Fornax could not be formed without a high initial total mass. This could in principle mean an already long accretion history. Nonetheless, its chemical homogeneity requests that this should be achieved at a very early epoch, before star formation in independent smaller halos could have widened its abundance patterns.
$\bullet$ Star formation occurs in series of short periods (a few hundreds of Myr long). A high frequency SF mode is analogous to a continuous star formation, while a low frequency SF mode produces well separated bursts, easily identified observationally as in the case of Carina. The period between the star formation events is governed by the mass of the systems. It is caused by the dependence of the gas cooling time on the galaxy mass: the more massive, the shorter it is. This mass dependency is explained by the drop of the cooling function below $10^4\,\rm{K}$. Systems less massive than $3\times
10^{8}\rm{M_\odot}$ have a virial temperature below $10^4\,\rm{K}$ and are characterized by weak cooling. Only episodic periods of star formation are expected. On the contrary, massive systems ($M_{\rm{tot}} > 4\times 10^{8}\rm{M_\odot}$) have a virial temperature above $10^4\,\rm{K}$. They are expected to form stars continuously and generate more metal-rich dSphs.
$\bullet$ The dSph scaling relations (\[Fe/H\], M/L) are reproduced and exhibit very low scatter. We have shown that there is a constant final total galaxy mass over the wide range of dSph luminosity, which appears as a direct consequence of the cooling time -mass relationship.
$\bullet$ The $\alpha$-elements were traced by magnesium. The dispersion in the \[Mg/Fe\] vs \[Fe/H\] diagrams is negligible for continuous and/or efficient star formation. It is enhanced when the star formation occurs in bursts separated by periods of quiescence, or when the star formation occurs in peaks of uneven intensities. Therefore, it is observed in galaxies as different as Sextans and Carina. Our study strengthens the need for large and homogeneous samples of stellar spectra obtained at high resolution, that are necessary to constrain and improve the models. They call for large observing programs dedicated to the chemical signatures at low metallicities in order to establish the level of homogeneity of the interstellar medium in the early phases of the galaxy evolution.
$\bullet$ The fraction of very metal-poor stars in our generic models ranges from $\sim 1$ to 6%. Although this is a rather small amount, it is still larger than suggested by the observations. In order to properly address this particular issue, one needs to carefully investigate the effect of more sophisticated IMFs and the peculiar features of star formation at zero metallicity.
$\bullet$ Gas is found in all our model galaxies after $14\,\rm{Gyr}$ ($\sim 10^7\rm{M_\odot}$). It clearly points to the need for interactions that would strip the gas in the course of the dSph evolution. The possibility that gas is gradually expelled as halos merge in a hierarchical formation scenario also deserves careful consideration.
We wish to thank the anonymous referee for his comments, which helped in improving the contents of this paper. Data reduction and galaxy maps have been performed using the parallelized Python pNbody package (see http://obswww.unige.ch/ revaz/pNbody/). This work was supported by the Swiss National Science Foundation.
Simulation list and parameters {#appendix1}
==============================
----- -------------- -------------- ----------- ------------------------------ ------------------------ ------------------------
N $r_{\rm{c}}$ $f_{\rm{b}}$ $c_\star$ $\rho_{\rm{c,tot}}$ $M_{\rm{tot}}$ $M_{\rm{gas}}$
$[\rm{kpc}]$ $[10^8\,\rm{M_\odot/kpc^3}]$ $[10^8\,\rm{M_\odot}]$ $[10^8\,\rm{M_\odot}]$
519 1.0 0.20 0.010 5.00e-04 4.12 0.82
520 1.0 0.20 0.010 4.00e-04 3.29 0.66
524 1.0 0.20 0.010 3.00e-04 2.47 0.49
630 1.0 0.20 0.035 5.00e-04 4.12 0.82
642 1.0 0.20 0.035 4.75e-04 3.91 0.78
631 1.0 0.20 0.035 4.50e-04 3.71 0.74
581 1.0 0.20 0.020 5.00e-04 4.12 0.82
582 1.0 0.20 0.020 4.00e-04 3.29 0.66
583 1.0 0.20 0.020 3.00e-04 2.47 0.49
666 1.0 0.20 0.050 1.00e-03 8.24 1.65
658 1.0 0.20 0.050 8.00e-04 6.59 1.32
662 1.0 0.20 0.050 6.60e-04 5.48 1.10
405 1.0 0.20 0.050 5.00e-04 4.12 0.82
409 1.0 0.20 0.050 4.00e-04 3.29 0.66
563 1.0 0.20 0.050 3.75e-04 3.09 0.62
555 1.0 0.20 0.050 3.50e-04 2.88 0.58
454 1.0 0.20 0.050 3.00e-04 2.47 0.49
624 1.0 0.20 0.065 5.00e-04 4.12 0.82
620 1.0 0.20 0.075 5.00e-04 4.12 0.82
667 1.0 0.20 0.100 1.00e-03 8.24 1.65
659 1.0 0.20 0.100 8.00e-04 6.59 1.32
663 1.0 0.20 0.100 6.60e-04 5.48 1.10
571 1.0 0.20 0.100 6.50e-04 5.35 1.07
570 1.0 0.20 0.100 6.00e-04 4.94 0.99
565 1.0 0.20 0.100 5.50e-04 4.53 0.91
521 1.0 0.20 0.100 5.00e-04 4.12 0.82
622 1.0 0.20 0.100 4.75e-04 3.91 0.78
618 1.0 0.20 0.100 4.50e-04 3.71 0.74
522 1.0 0.20 0.100 4.00e-04 3.29 0.66
564 1.0 0.20 0.100 3.75e-04 3.09 0.62
554 1.0 0.20 0.100 3.50e-04 2.88 0.58
523 1.0 0.20 0.100 3.00e-04 2.47 0.49
668 1.0 0.20 0.200 1.00e-03 8.24 1.65
660 1.0 0.20 0.200 8.00e-04 6.59 1.32
664 1.0 0.20 0.200 6.60e-04 5.48 1.10
562 1.0 0.20 0.200 5.00e-04 4.12 0.82
559 1.0 0.20 0.200 4.00e-04 3.29 0.66
578 1.0 0.20 0.200 3.75e-04 3.09 0.62
573 1.0 0.20 0.200 3.50e-04 2.88 0.58
572 1.0 0.20 0.200 3.00e-04 2.47 0.49
669 1.0 0.20 0.300 1.00e-03 8.24 1.65
661 1.0 0.20 0.300 8.00e-04 6.59 1.32
665 1.0 0.20 0.300 6.60e-04 5.48 1.10
632 1.0 0.20 0.300 5.00e-04 4.12 0.82
699 1.0 0.20 0.300 3.00e-04 2.47 0.49
623 1.0 0.15 0.010 6.60e-04 5.48 0.82
621 1.0 0.15 0.025 5.30e-04 4.39 0.66
649 1.0 0.15 0.050 1.00e-03 8.24 0.82
647 1.0 0.15 0.050 8.00e-04 5.60 0.99
585 1.0 0.15 0.050 6.60e-04 5.48 0.82
586 1.0 0.15 0.050 5.30e-04 4.39 0.66
641 1.0 0.15 0.050 4.65e-04 3.83 0.57
587 1.0 0.15 0.050 4.00e-04 3.29 0.49
648 1.0 0.15 0.050 2.60e-04 2.19 0.33
650 1.0 0.15 0.100 1.00e-03 8.24 0.82
569 1.0 0.15 0.100 8.00e-04 5.60 0.99
566 1.0 0.15 0.100 6.60e-04 5.48 0.82
567 1.0 0.15 0.100 5.30e-04 4.39 0.66
568 1.0 0.15 0.100 4.00e-04 3.29 0.49
579 1.0 0.15 0.100 2.60e-04 2.19 0.33
651 1.0 0.15 0.200 1.00e-03 8.24 0.82
574 1.0 0.15 0.200 8.00e-04 5.60 0.99
575 1.0 0.15 0.200 6.60e-04 5.48 0.82
576 1.0 0.15 0.200 5.30e-04 4.39 0.66
584 1.0 0.15 0.200 4.50e-04 3.71 0.56
577 1.0 0.15 0.200 4.00e-04 3.29 0.49
580 1.0 0.15 0.200 2.60e-04 2.19 0.33
652 1.0 0.15 0.300 1.00e-03 8.24 1.24
646 1.0 0.15 0.300 8.00e-04 6.59 0.99
633 1.0 0.15 0.300 6.60e-04 5.48 0.82
----- -------------- -------------- ----------- ------------------------------ ------------------------ ------------------------
----- -------------- -------------- ----------- ------------------------------ ------------------------ ------------------------
N $r_{\rm{c}}$ $f_{\rm{b}}$ $c_\star$ $\rho_{\rm{c,tot}}$ $M_{\rm{tot}}$ $M_{\rm{gas}}$
$[\rm{kpc}]$ $[10^8\,\rm{M_\odot/kpc^3}]$ $[10^8\,\rm{M_\odot}]$ $[10^8\,\rm{M_\odot}]$
634 1.0 0.15 0.300 5.30e-04 4.39 0.66
644 1.0 0.15 0.300 4.00e-04 3.29 0.49
645 1.0 0.15 0.300 2.60e-04 2.19 0.33
596 1.0 0.10 0.050 1.00e-03 8.24 0.82
597 1.0 0.10 0.050 8.00e-04 6.59 0.66
598 1.0 0.10 0.050 6.00e-04 4.94 0.49
605 1.0 0.10 0.050 4.00e-04 3.29 0.33
599 1.0 0.10 0.100 1.00e-03 8.24 0.82
600 1.0 0.10 0.100 8.00e-04 6.59 0.66
619 1.0 0.10 0.100 7.00e-04 5.76 0.58
601 1.0 0.10 0.100 6.00e-04 4.94 0.49
606 1.0 0.10 0.100 4.00e-04 3.29 0.33
602 1.0 0.10 0.200 1.00e-03 8.24 0.82
603 1.0 0.10 0.200 8.00e-04 6.59 0.66
604 1.0 0.10 0.200 6.00e-04 4.94 0.49
607 1.0 0.10 0.200 4.00e-04 3.29 0.33
670 1.0 0.10 0.300 1.00e-03 8.24 0.82
635 1.0 0.10 0.300 8.00e-04 6.59 0.66
636 1.0 0.10 0.300 6.00e-04 4.94 0.49
692 1.0 0.10 0.300 4.00e-04 3.29 0.33
528 0.5 0.20 0.010 2.00e-03 4.55 0.91
529 0.5 0.20 0.010 1.60e-03 3.64 0.73
530 0.5 0.20 0.010 1.20e-03 2.73 0.55
676 0.5 0.20 0.050 4.00e-03 9.11 1.82
673 0.5 0.20 0.050 3.20e-03 7.28 1.46
672 0.5 0.20 0.050 2.50e-03 5.69 1.14
525 0.5 0.20 0.050 2.00e-03 4.55 0.91
626 0.5 0.20 0.050 1.80e-03 4.10 0.82
526 0.5 0.20 0.050 1.60e-03 3.64 0.73
527 0.5 0.20 0.050 1.20e-03 2.73 0.55
677 0.5 0.20 0.100 4.00e-03 9.11 1.82
671 0.5 0.20 0.100 2.50e-03 5.69 1.14
531 0.5 0.20 0.100 2.00e-03 4.55 0.91
532 0.5 0.20 0.100 1.60e-03 3.64 0.73
533 0.5 0.20 0.100 1.20e-03 2.73 0.55
627 0.5 0.20 0.200 4.00e-03 9.11 1.82
628 0.5 0.20 0.200 3.00e-03 6.83 1.37
629 0.5 0.20 0.200 2.50e-03 5.69 1.14
608 0.5 0.20 0.200 2.00e-03 4.55 0.91
609 0.5 0.20 0.200 1.60e-03 3.64 0.73
610 0.5 0.20 0.200 1.20e-03 2.73 0.55
678 0.5 0.20 0.300 4.00e-03 9.11 1.82
675 0.5 0.20 0.300 3.20e-03 7.28 1.46
640 0.5 0.20 0.300 2.50e-03 5.69 1.14
637 0.5 0.20 0.300 2.00e-03 4.55 0.91
691 0.5 0.20 0.300 1.60e-03 3.64 0.73
690 0.5 0.20 0.300 1.20e-03 2.73 0.55
625 0.5 0.15 0.010 2.13e-03 4.85 0.73
643 0.5 0.15 0.025 2.13e-03 4.85 0.73
684 0.5 0.15 0.050 4.00e-03 9.11 1.37
680 0.5 0.15 0.050 3.20e-03 7.28 1.09
588 0.5 0.15 0.050 2.66e-03 6.06 0.91
589 0.5 0.15 0.050 2.13e-03 4.85 0.73
590 0.5 0.15 0.050 1.60e-03 3.64 0.55
591 0.5 0.15 0.050 1.20e-03 2.73 0.41
685 0.5 0.15 0.100 4.00e-03 9.11 1.37
681 0.5 0.15 0.100 3.20e-03 7.28 1.09
592 0.5 0.15 0.100 2.66e-03 6.06 0.91
593 0.5 0.15 0.100 2.13e-03 4.85 0.73
594 0.5 0.15 0.100 1.60e-03 3.64 0.55
595 0.5 0.15 0.100 1.20e-03 2.73 0.41
686 0.5 0.15 0.200 4.00e-03 9.11 1.37
682 0.5 0.15 0.200 3.20e-03 7.28 1.09
611 0.5 0.15 0.200 2.66e-03 6.06 0.91
612 0.5 0.15 0.200 2.13e-03 4.85 0.73
613 0.5 0.15 0.200 1.60e-03 3.64 0.55
693 0.5 0.15 0.200 1.20e-03 2.73 0.41
687 0.5 0.15 0.300 4.00e-03 9.11 1.37
683 0.5 0.15 0.300 3.20e-03 7.28 1.09
679 0.5 0.15 0.300 2.66e-03 6.06 0.91
----- -------------- -------------- ----------- ------------------------------ ------------------------ ------------------------
----- -------------- -------------- ----------- ------------------------------ ------------------------ ------------------------
N $r_{\rm{c}}$ $f_{\rm{b}}$ $c_\star$ $\rho_{\rm{c,tot}}$ $M_{\rm{tot}}$ $M_{\rm{gas}}$
$[\rm{kpc}]$ $[10^8\,\rm{M_\odot/kpc^3}]$ $[10^8\,\rm{M_\odot}]$ $[10^8\,\rm{M_\odot}]$
638 0.5 0.15 0.300 2.13e-03 4.85 0.73
695 0.5 0.15 0.300 1.60e-03 3.64 0.55
694 0.5 0.15 0.300 1.20e-03 2.73 0.41
537 0.5 0.10 0.025 4.00e-03 9.11 0.91
538 0.5 0.10 0.025 3.20e-03 7.28 0.73
539 0.5 0.10 0.025 2.40e-03 5.46 0.55
534 0.5 0.10 0.050 4.00e-03 9.11 0.91
535 0.5 0.10 0.050 3.20e-03 7.28 0.73
536 0.5 0.10 0.050 2.40e-03 5.46 0.55
547 0.5 0.10 0.050 1.80e-03 4.10 0.41
546 0.5 0.10 0.050 1.20e-03 2.73 0.27
540 0.5 0.10 0.100 4.00e-03 9.11 0.91
541 0.5 0.10 0.100 3.20e-03 7.28 0.73
542 0.5 0.10 0.100 2.40e-03 5.46 0.55
549 0.5 0.10 0.100 1.80e-03 4.10 0.41
548 0.5 0.10 0.100 1.20e-03 2.73 0.27
614 0.5 0.10 0.200 4.00e-03 9.11 0.91
615 0.5 0.10 0.200 3.20e-03 7.28 0.73
616 0.5 0.10 0.200 2.40e-03 5.46 0.55
617 0.5 0.10 0.200 1.80e-03 4.10 0.41
696 0.5 0.10 0.200 1.20e-03 2.73 0.27
689 0.5 0.10 0.300 4.00e-03 9.11 0.91
688 0.5 0.10 0.300 3.20e-03 7.28 0.73
639 0.5 0.10 0.300 2.40e-03 5.46 0.55
698 0.5 0.10 0.300 1.80e-03 4.10 0.41
697 0.5 0.10 0.300 1.20e-03 2.73 0.27
----- -------------- -------------- ----------- ------------------------------ ------------------------ ------------------------
|
---
abstract: 'The availability of new Cloud Platform offered by Google motivated us to propose nine Proof of Concepts (PoC) aiming to demonstrated and test the capabilities of the platform in the context of scientifically-driven tasks and requirements. We review the status of our initiative by illustrating 3 out of 9 successfully closed PoC that we implemented on Google Cloud Platform. In particular, we illustrate a cloud architecture for deployment of scientific software as microservice coupling Google Compute Engine with Docker and Pub/Sub to dispatch heavily parallel simulations. We detail also an experiment for HPC based simulation and workflow executions of data reduction pipelines (for the TNG-GIANO-B spectrograph) deployed on GCP. We compare and contrast our experience with on-site facilities comparing advantages and disadvantages both in terms of total cost of ownership and reached performances.'
author:
- 'Marco Landoni,$^1$, G. Taffoni$^2$, A. Bignamini$^2$ and R. Smareglia$^2$'
bibliography:
- 'P6-8.bib'
title: Application of Google Cloud Platform in Astrophysics
---
Introduction
============
Google Cloud Platform (GCP) offers a variety of services, ranging from storage to high performance computing and workflow execution, that could be exploited in the context of Computational Astrophysics. In this paper we review three Proof of Concept (PoC) out of the nine proposed to Google that have been successfully implemented on the public platform illustrating the architecture and the main results we have obtained. The paper is organized as follows: in Section 2 we illustrate the PoC for HTC oriented application while in Section 3 we comment on the implementation of HPC cluster on the Google Cloud Platform. We also tested the execution of Workflows aiming to offer instrument pipeline as a service reporting the results in Section 4.
POC 1 - HTC Workload on Google Cloud Platform. The case of DIAMONDS
===================================================================
DIAMONDS [@corsaro] is a Bayesian inference code that is design to process data from asteroseismology, a technique that study stars oscillations through photometry or spectroscopy in order to derive their internal structure and physical parameters, such as the true mass. DIAMONDS has demonstrated (on premises) to be runnable in parallel through *embarrassingly parallelism* paradigm with almost no network communication. This kind of computational approach is very suitable to test HTC workloads on GCP. We introduce in this scenario a concept of *serverless HTC scheduler* that fruitful exploit an heterogeneous set of GCP components such as Pub/Sub, Cloud Functions and Managed Groups in GCE (see Figure \[fig:diamonds\] and [@landoni] for further implementation and details). Typical resource schedulers are complex middleware that have to deal with a limited amount of resources available accordingly to a set of time-sharing policies. An advantage of this approach is to use some equivalent concepts such as the queue from the available services to manage the execution of HTC workloads while guaranteeing a general purpose approach to many possible HTC computation. In the architecture that we design for GCP, keeping in mind to be as much as general as possible, the computation starts by uploading to Cloud Storage a plain text file that contains, for each row, the data necessary to perform a single run. These rows are pushed, by a triggered Cloud Function, into a Pub/Sub topic. Then, a cluster of instances (Regular or Preemptible and configured using Google Managed Groups) dimensioned runtime is fired up accordingly to the estimated size of the whole workload. Each node of the cluster, after starting up with a pre-configured Image on Compute Engine, pulls a number of messages (proportional to the number of vCPUs available) from the PubSub queue starting the computation of various DIAMONDS simulations using Docker containers. Data produced locally by DIAMONDS on each instance are finally transferred to a bucket on Google Cloud Storage before shutting down. This method allows to deploy an HTC-based architecture, suitable for many projects that share the same kind of parallelism and requirements on the workload, that scales both vertically (number of cores per node and thus number of simulations) and horizontally through an elastic cluster fired up accordingly to the number of required simulations and CPU/hours.
![The HTC serverless architecture for DIAMONDS pipeline (POC1)[]{data-label="fig:diamonds"}](P6-8_f1.eps)
POC 2 - Exploring HPC capabilities with Google Cloud Platform
=============================================================
GADGET [@gadget] is a lagrangian code to perform numerical simulations of gravitationally interacting particles of both dark matter and baryonic matter which computes gravitational forces using a TreePM technique. A mean field approximation is used for large scales (Particle-Mesh, PM) while at smaller scales a usual Treecode is used. Hydrodynamics is solved using a so-called Smoothed Particle Hydrodynamics technique. GADGET is an HPC code based on message passing interface (MPI) libraries and OpenMP. It is written in C and requires some support libraries to run (FFW2.4, HDF5, GSL). To test the performance of the Google Cloud infrastructure for HPC applications, we executed a virtual cluster managed by Slurm scheduler composed by both reserved nodes and On-demand (non-preemptible) instances. To automate the deployment of the cluster, we use the Cloud Deployment Manager (CDM). This service aims at automate the creation of complex resources and services where various entities are described in terms of *yaml* files and deployed using *gcloud* command line interface. In this POC, we modified the Slumr official CDM files to modify the resources (we used 4 cores with 4 GB ram per core) and the software (we added MPI, FFTW, HDF5, and GSL). Moreover, we deployed a cluster where only two computing nodes are running and more resources are bootstrapped on demand using Slurm only if necessary.
![GADGET sclalability on GCloud.[]{data-label="fig:gad2"}](P6-8_f2.eps)
We tested GADGET scalability with a small cosmological BOX of 778688 particles for a $\Lambda$CDM model ($\Omega_0$ = 0.24, $\Omega_\Lambda = 0.76$, $h$ = 0.72) increasing the number of nodes and the size of nodes (up to 96 cores and 624GB Ram). We present our scalability results in Figure \[fig:gad2\]. The GCP infrastructure is based on standard ethernet connections, while for HPC applications the role of a low latency high throughput interconnect is crucial, as evident from Figure \[fig:gad2\]. On the other side, the cluster is suitable to any HTC applications where the inter-node communication is not present or limited.
POC 3 - Workflow execution. Running GIANO-B data reduction pipeline as a service
================================================================================
In this use case we report and comment about the creation of a scaled and balanced environment, whose purpose is the execution of workflows submitted by the user through the workflow environment [@yabi]. This scenario involves the user, who retrieves GIANO-B raw data from TNG archive public and private storage, and the execution of the GOFIO data reduction pipeline [@rainer] to produce reduced data that can be retrived by the user himself. The main aims of this PoC involves the simplification of the management of the infrastructure, moving from an on-premises infrastructure to PaaS/SaaS layers offered by GCP and of the deployment of software using containers to avoid incompatibility issues between packages that must coexist and work together. Finally, this PoC aims to improve software and service maintenance while optimizing and balancing the scalability of the service according to the load. The implementation of this PoC foresees these services from GCP: Google Compute Engine for virtual machine instances management, Slurm or Google Kubernetes Engine as workload manager to deploy GOFIO container and the Docker platform for the containerization of GOFIO pipeline. Since we have two workload managers, two different solutions for this PoC was implemented. In the first architecture we made use of *Yabi* and Slurm while in the second one we exploit *Yabi* coupled with Kubernetes. For both architecture *Yabi* was deployed on a Compute Engine instance that acts as frontend for final user. Slurm cluster was deployed using standard *yaml* file available through Slumr official documentation[^1]. To connect *Yabi* with Slurm, we used the native *Yabi*-Slurm backend connector, which is available in the latest version of *Yabi* (version 9). Kuberntes cluster was deployed using Kubernetes Engine following the official Google documentation[^2] deploying a Network File System (NFS) server from Cloud Launcher, configuring Persistent Volumes, POD ReplicaSet, LoadBalancer and HorizontalAutoScaler. *Yabi* does not provide a default backend connector for Kubernetes, therefore we used the default *Yabi*-SSH backend connector to connect *Yabi* to Kubernetes cluster generating SSH key in *Yabi* instance and adding it in the Kubernetes cluster. To test the performance of both architecture and to check actual scalability as function of the load, massive tests submitting simultaneously tens of jobs were performed. As a reference, for on-premises infrastructure these large workloads result in an excessive dilation of the execution times, since the total execution time of all the jobs (submitted simultaneously) is much greater than the sum of the execution times of the individual jobs performed one by one and, in most extreme cases, *Yabi* crashes. For the architecture *Yabi*-Slurm deployed on GCP the scalability is good and all jobs are completed correctly with no significant time leaks compared with the execution time of a single job. Slurm is natively supported by **Yabi** and it performs reasonably good in managing the job queue and the scaling. New Compute Engine instances are created and destroyed on demand efficiently according to the load. However, for what concerns the **Yabi**-Kubernetes the scalability is also remarkable, but some jobs (about 1 each 8) exit with error and they are not more recovered, probably due to the fact that in this configuration the job queue is completely managed by the **Yabi** SSH Backend that submit jobs to Kubernetes which seems able to manage the load, but the **Yabi** Backend fails to manage all the job queue. We evaluate a total estimated charges of about 200 EUR/month to maintain both architectures up and running.
[^1]: https://github.com/SchedMD/slurm-gcp
[^2]: https://cloud.google.com/kubernetes-engine/docs/how-to/creating-a-cluster
|
---
author:
- |
M. Backhaus\
*CERN*\
\
Now: *ETH Zürich, Otto-Stern-Weg 5, 8093 Zürich, Switzerland*\
E-mail: *backhaus@cern.ch*
title: Parametrization of the radiation induced leakage current increase of NMOS transistors
---
Introduction
============
The radiation induced leakage current of NMOS transistors and the threshold voltage shift are well known and intensively studied challenges for the design of radiation hard application specific integrated circuits (ASIC). Hardness by design (HBD) techniques [@hbd1][@hbd2] have been developed and extensively used in the circuit design in the technology node for the LHC experiments [@faccio_lhc]. The HBD techniques consist mainly of the use of enclosed gate transistors to mitigate the source to drain leakage current along the shallow trench isolation (STI) and guard ring structures surrounding the transistors to avoid leakage current between neighboring structures. The use of this technique requires an increased area as well as custom libraries. With the transition to the technology node, the amplitude of the leakage current increase decreases by three orders of magnitude, with the result that the HBD techniques are needed in sensitive nodes of the design only for radiation hard designs [@faccio_130]. However, while the chip is operational, the increase of the leakage current of NMOS transistors results in a significant increase of the supply current. A generic parametrization of the leakage current as a function of total ionizing dose (TID) of single NMOS transistors is presented in this paper. The resulting function is fitted to published data of a diversity of technologies in use or under investigation for use in extremely radiation intense environments. As demonstrated in this paper, this parametrization can be used as well to model the supply current shift with TID on full ASICs and to predict the current to be expected during operation of the ASICs as a function of the temperature and the dose rate, once the parameters of the parametrization are measured.
Parametrization of the leakage current {#sec:parametrization}
======================================
This model describes the leakage current increase of linear NMOS transistors independent of technology details. The increase of the leakage current as a function of total ionizing dose has been reported for a large number of technologies. It originates from *parasitic transistor channels* along the STI, in which the transistor is embedded as indicated in figure \[fig:TopView\].\
![Top View of a linear transistor. The source (S), gate (G), and drain (D) as well as the shallow trench isolation (STI) are indicated. The dashed line shows the place of the cross section in figure \[fig:CrossSection1\].[]{data-label="fig:TopView"}](./TopView.png){width="0.5\linewidth"}
With a positive space charge appearing in the STI due to ionizing radiation, an inversion layer is created along the STI, which results in a leakage current paths from source to drain [@faccio_130]. This is indicated in the cross section in figure \[fig:CrossSection1\].\
![Cross section of a transistor along the line indicated in figure \[fig:TopView\]. The channel of the parasitic transistor along the STI is indicated in yellow.[]{data-label="fig:CrossSection1"}](./CrossSection1.png){width="0.5\linewidth"}
This leakage current paths can be described as the channel of a parasitic transistor. As can be seen in figure \[fig:CrossSection2\], the parasitic transistor has a layout analogue to a linear transistor with the only difference, that the electric field opening the transistor channel originates from the positive space charge in the STI instead of from the potential at the gate. The space charge in the STI is always positive and therefore in PMOS transistors this transistor channel is closed by the radiation induced space charge. Thus the leakage current increase is only observed in NMOS transistors.\
![View from the top in the transistor transistor below the gate. The channel of the parasitic transistor along the STI is indicated in yellow.[]{data-label="fig:CrossSection2"}](./CrossSection2.png){width="0.5\linewidth"}
This analogue layout motivates to describe the leakage current increase due to the parasitic transistor using the *transfer characteristics* of the parasitic transistor.
Transfer characteristics of the parasitic transistor {#sec:sub:transfer}
----------------------------------------------------
The transfer characteristics (drain current as a function of the gate to source voltage) for transistors operated in saturation mode can be simplified by $$\begin{aligned}
I_D &\approx& 0
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{for} \;\;
V_G < V_{thr}
\nonumber
\\
I_D &\approx& K' \cdot (V_G - V_{thr})^2 \;\;\;\mbox{for} \;\; V_G \geq
V_{thr}\end{aligned}$$ where sub-threshold leakage is neglected. $I_D$ is the drain to source current, $V_G$ the gate to source voltage, and $V_{thr}$ the threshold voltage. $K'$ is the proportionality constant containing the widths, lengths, oxide capacitance, and the mobility of the minority charge carriers in the channel of the transistor, etc. In the parasitic transistor (which is responsible for the leakage current) the electric field originates from the effective space charge. Assuming an electric field in the silicon proportional to the effective space charge, the gate voltage of the parasitic transistor is in this model replaced by the *effective number of charges* $N_{eff}$. Similarly, the threshold voltage is expressed by the *threshold number of charges* $N_{thr}$. The transfer characteristics of the parasitic transistor are then given by $$\begin{aligned}
I_{D}^{par} &=& 0 \nonumber
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{for} \;\; N_{eff} <
N_{thr}\nonumber \\
I_{D}^{par} &=& K \cdot (N_{eff} - N_{thr})^2
\;\;\mbox{for} \;\; N_{eff} \geq N_{thr} \nonumber
\label{eqn:transferChar}\end{aligned}$$ and thus the leakage current of the regular transistor $I_{leak}$ is given by $$\begin{aligned}
I_{leak} &=& I_{leak}^0 \nonumber \\
&& \mbox{for} \;\; N_{eff} < N_{thr}\nonumber \\
\nonumber \\
I_{leak} &=& I_{leak}^0 + K \cdot (N_{eff} - N_{thr})^2 \nonumber \\
&& \mbox{for} \;\; N_{eff} \geq N_{thr}
\label{eqn:transferChar}\end{aligned}$$ with $I_{leak}^0$ the preirradiation leakage current of the transistor. $N_{thr}$ is constant in time (and therefore accumulated dose) and temperature. With these assumptions a parametrization of the number of effective charge carriers $N_{eff}$ describes the leakage current shift in NMOS transistors.
Processes of charge generation
------------------------------
A combination of four processes results in the effective space charge in the STI. First, due to ionization of the atoms, the incident radiation generates free electron hole pairs in the silicon dioxide of the STI [@ehpairs1][@ehpairs2]. Some of these pairs recombine quickly, but the mobility of the electrons is between six and twelve orders of magnitude larger than the mobility of the holes, depending on the temperature and the electrical field \[7-10\]. Therefore many electrons are quickly removed from the STI, while the left over holes move slowly in the silicon dioxide by hopping transport [@ehpairs1]. Sites with missing oxide atoms in the amorphous silicon dioxide result in energy levels above the valence band and thus electrically neutral deep hole traps [@ehpairs1]. These traps are distributed in the STI volume, and their concentration is largly influenced by the manufacturing of the STI, and thus it is technology dependent. During the movement some holes get trapped in these sites and a space charge is built. The holes have a certain probability to get detrapped by thermal energy. The lifetime $\tau_{ox}$ of the holes in the traps depends on the energy level of the traps, and of the temperature.\
Holes which are not trapped in the silicon dioxide can move to the silicon to silicon dioxide interface. At this interface are incomplete or dangling atomic bonds due to the abrupt transistion from amorphous to christalline material. The dangling bonds manifest themselves as energy levels in the band-gap, and thus they trap mainly electrons or holes, depending on the Fermi-level of the silicon [@ehpairs1]. This trapping of electrons in the case of NMOS transistors (p-type silicon) and holes in the case of PMOS transistors (n-type silicon) degrades the transistor performance, and therefore usually the dangling bonds are deactivated by the manufacturer using hydrogen. The radiation induced free holes can react with the hydrogen and the dangling bonds get re-activated [@ehpairs1]. In this case they are commonly called radiation induced *interface traps*.\
In the case of NMOS transistors the activation of the interface traps results in a negative space charge, while the trapping of holes in the STI results in a positive charge [@faccio_130]. The electric fields compensate each other, so that the effective number of charges $N_{eff}$: $$\begin{aligned}
N_{eff} &=& N_{ox} - N_{if}\end{aligned}$$ becomes relevant, with $N_{ox}$ the number of trapped holes in the STI, and $N_{if}$ the number of electrons trapped at the interface. The parametrization of these two numbers during and after ionizing radiation is explained in the following.
### Parametrization of the number of positive charges trapped in the STI
During exposure to ionizing radiation with a constant dose rate $D$ the number of holes getting trapped in the STI volume is proportional to the exposure time $t$ with a proportionality constant $k_{ox} D$, where $k_{ox}$ describes how many holes are trapped per dose unit. At the same time, the trapped holes have a life-time $\tau_{ox}$ in the traps, until they are free again and can move out of the STI. Therefore, the number of holes trapped in the STI is defined by the differential equation $$\begin{aligned}
\frac{d}{dt} N_{ox}(t) &=& k_{ox} D \; - \; \frac{1}{\tau_{ox}} N_{ox}(t)
\label{eqn:differential} \end{aligned}$$ which is solved by $$\begin{aligned}
N_{ox}(t) &=& k_{ox} D \cdot \tau_{ox} \cdot \left(1 - e^{-\frac{t}{\tau_{ox}}}\right).\end{aligned}$$ If the irradiation stops after an exposure time $t_1$, only the second term of equation (\[eqn:differential\]) stays and results in and exponential decrease of the number of holes trapped in the STI, which is usually called annealing. This decrease can be described by $$\begin{aligned}
N_{ox}(t) &=& N_{trap} \nonumber \\
&&+ \left(N_{ox}(t_1) - N_{trap}\right) \cdot
e^{-\frac{t-t_1}{\tau_{ox}}}
\label{eqn:annealing}\end{aligned}$$ where $N_{trap}$ describes the number of holes captured in traps of the oxide which are too deep for detrapping at the given temperature.
### Parametrization of the number of activated interface traps
The number of activated interface traps follows a very similar behavior, as motivated here. The holes travelling to the silicon to silicon dioxide interface and activating the radiation induced interface traps are generated again with a constant rate $k_{if}D$. $k_{if}$ describes here the number of holes available to activate interface traps per dose unit. The number of interface traps that can be activated by the radiation is technology dependent and it is limited. Therefore the probability that the holes activate new interface traps decreases with time exponentially. This can be described equivalently using the analogue equation $$\begin{aligned}
N_{if}(t) &=& k_{if} D \cdot \tau_{if} \cdot \left(1 -
e^{-\frac{t}{\tau_{if}}}\right).\end{aligned}$$
The needed temperature to anneal the interface traps is known to be well above room temperature and even more above the operational temperature of the ASICs. It is technology dependent and in the range of to . Therefore, the annealing of the interface traps is negligible for the presented parametrization.
Summary of the parametrization {#sec:sub:parasum}
------------------------------
The complete formula for the leakage current during irradiation is therefore given by $$\begin{aligned}
I_{leak} &=& I_{leak}^0
\label{eqn:IleakFullZero}\end{aligned}$$ for $$\begin{aligned}
k_{ox} D \cdot \tau_{ox} \cdot \big(1 - e^{-\frac{t}{\tau_{ox}}}\big)
\label{eqn:ileakfull0} \\
- k_{if} D \cdot \tau_{if} \cdot \big(1 - e^{-\frac{t}{\tau_{if}}}\big) &<&
N_{thr} \nonumber\end{aligned}$$ and by $$\begin{aligned}
I_{leak} &=& I_{leak}^0 \nonumber \\
&&+ K \cdot \Big[k_{ox} D \cdot \tau_{ox} \cdot \big(1 -
e^{-\frac{t}{\tau_{ox}}}\big) \nonumber \\
&&- k_{if} D \cdot \tau_{if} \cdot \big(1 - e^{-\frac{t}{\tau_{if}}}\big) \nonumber \\
&&- N_{thr}\Big]^2
\label{eqn:ileakfull1}\end{aligned}$$ for $$\begin{aligned}
k_{ox} D \cdot \tau_{ox} \cdot \big(1 - e^{-\frac{t}{\tau_{ox}}}\big)
\nonumber \\
- k_{if} D \cdot \tau_{if} \cdot \big(1 - e^{-\frac{t}{\tau_{if}}}\big) &\geq&
N_{thr}. \nonumber\end{aligned}$$
During the periods with no incident ionizing radiation (dose rate $D = 0$) $N_{ox}$ is similarly replaced by equation (\[eqn:annealing\]), and the number of activated interface traps $N_{if}$ stays constant because they do not anneal at the considered temperature range.
![Plot of the leakage current parametrization as a function of the time, including the abrupt switch-off of the ionizing radiation, resulting in the exponential decrease.[]{data-label="fig:IleakExample"}](./IleakExample.pdf){width="\linewidth"}
Figure \[fig:IleakExample\] presents the resulting leakage current increase as a function of time during an exposure to ionizing radiation with a constant dose rate. The exponential decrease step in the far right of the plot illustrates the annealing behavior when the radiation is switched-off. For many studies the leakage current is given as a function of the TID. The parametrization can be expressed during periods of constant instensity exposure using $\mbox{TID} = D \cdot t$.\
The temperature dependency of the parameters is not explicitely included here. The generation of the positive space charge depends on the temperature. This can be modelled sufficiently well with a temperature dependent de-trapping probability, and neglecting the temperature dependence of the electron and hole mobility and generation [@hmobility1][@generation_yield]. Then only $\tau_{ox}$ directly depends on the temperature. The generation of the interface traps is also expected to be a function of the temperature, but it is not well known. A dedicated measurement campaign is ongoing to provide the data that are needed for the extraction of the temperature dependence of these terms. As these data are not yet available, here the parametrization is used to describe the measured leakage current increase at a given temperature.\
In the following two sections this parametrization is first fit to published data of single NMOS transistors. Then the same function is used to describe the supply current shift of the ATLAS IBL pixel readout chip [@fe-i4] as an example of the consequence of the leakage current shift for the ASIC operation in radiation intense environments. This is an example for the power of this parametrization to predict the ASIC supply current increase once the basic parameters are known for the given technology.
Fit to single transistor data {#sec:singleTransistorFits}
=============================
Equation (\[eqn:IleakFullZero\]), (\[eqn:ileakfull0\]) and (\[eqn:ileakfull1\]) are used to describe the leakage current shift of single NMOS transistors produced in a 180nm silicon on insulator (SOI) process [@xfab_process]. The data have been published previously in [@xfab], and were obtained at a dose rate $D$ of at a temperature of . Because the proportionality factor $K$ appears in the parametrization only in products with the other parameters, $K$ was fixed to a value of per effective charge carrier. The threshold charge $N_{thr}$, the time constants $\tau_{ox}$ and $\tau_{if}$, as well as the proportionality constants $k_{ox}$ and $k_{if}$ are free fit parameters. The data and the function are shown in figure \[fig:XFAB\_n1\] to \[fig:XFAB\_n14\] and demonstrate the good agreement of data and parametrization on single transistor level.
![Fit of the leakage current parametrization as a function of the time to the leakage current measurement of a single NMOS transistor (width , length ). The data were previously published in [@xfab].[]{data-label="fig:XFAB_n1"}](./XFAB_n1_LogX.pdf){width="\linewidth"}
![Fit of the leakage current parametrization as a function of the time to the leakage current measurement of a single NMOS transistor (width , length ). The data were previously published in [@xfab].[]{data-label="fig:XFAB_n3"}](./XFAB_n3_LogX.pdf){width="\linewidth"}
![Fit of the leakage current parametrization as a function of the time to the leakage current measurement of a single NMOS transistor (width , length ). The data were previously published in [@xfab].[]{data-label="fig:XFAB_n14"}](./XFAB_n14_LogX.pdf){width="\linewidth"}
Fit to full ASIC supply current shift
=====================================
ASICs composed of a large number of linear NMOS transistors can show a significant supply current shift when operated under ionizing radiation. This supply current shift is a serious challenge for the ASIC operation and impacts the design of the system, because the services need to be able to cope with this shift. The amplitude of the increase depends on the environmental conditions, such as dose rate and temperature. An intense investigation program is currently carried out on ATLAS readout chips. The ASIC, produced in IBM technology, is composed of about 80 million transistors, and HBD techniques are not used for the large majority of the transisors. The ASIC is operated under controlled environmental conditions while being exposed to x-ray radiation. The supply current is measured as a function of the exposure time. This measurement is carried out for various dose rates and temperatures. Some preliminary results are public [@fe-i4-data] and used here to demonstrate the ability of the parametrization to describe the supply current shift of full ASICs.\
Figure \[fig:Charlotte\] shows the fit to the supply current of the ASIC as a function of the exposure time using dose rate at . The same parameters are as in section \[sec:singleTransistorFits\] free during the fit, while $K$ is fixed now to per effective charge in order to account for that the supply current is the convolution of the leakage current of about several ten millions of transistors. Additionally, the pre-irradiation supply current is added as offset. At the time $t_1 = \SI{215500}{\second}$ the irradiation was switched-off and the annealing as described in equation (\[eqn:annealing\]) is shown. The parameters are fit to the time interval 0 to $t_1$ only. For the annealing the same parameters are used, especially the same time constant $\tau_{ox}$. This reflects that the annealing is caused be the same process as the saturation of the number of positive charges trapped in the oxide during irradiation.\
![Fit of the leakage current parametrization as a function of the time to the supply current measurement of an ATLAS readout-chip. At a the time $t_1 = \SI{215500}{\second}$ the dose rate is switched-off, and the annealing starts. The data were previously published in [@fe-i4-data].[]{data-label="fig:Charlotte"}](./Charlotte.pdf){width="\linewidth"}
Figure \[fig:IceT\] shows the same fit to data taken at different temperature (). A slightly higher amplitude of the increase is observed, as expected due to the longer lifetime $\tau_{ox}$ of the positive charges in the traps.
![Fit of the leakage current parametrization as a function of the time to the supply current measurement of an ATLAS readout-chip. The data were previously published in [@fe-i4-data].[]{data-label="fig:IceT"}](./IceT.pdf){width="\linewidth"}
The data shown in figure \[fig:IBL1\] are taken again at , but using a higher dose rate of . The current maximum is significaltly higher due to the larger dose rate.\
![Fit of the leakage current parametrization as a function of the time to the supply current measurement of an ATLAS readout-chip. The data were previously published in [@fe-i4-data].[]{data-label="fig:IBL1"}](./IBL1.pdf){width="\linewidth"}
The exact dependencies of the fit parameters from the temperature and from the dose rate are currently under investigation. The good agreement between the parametrization and the data demonstrate the power of this parametrization to predict the supply current during operation in radiation environment, once the basic parameters have been measured for the ASIC in the laboratory.
Conclusions
===========
The presented parametrization of the leakage current of NMOS transistors uses simplified transfer characteristics of the *parasitic* source to drain transistor along the STI. The good agreement of single transistor measurement data and the resulting function show that the description of the gate potential and threshold voltage by the effective number of charges and threshold charge, as well as the description of their concentration as a function of the exposure time describes the observations.\
Furthermore, the parametrization can directly be used to predict the supply current increase of full ASICs in radiation environment, when HBD techniques are not used for the majority of transistors, once the parameters are known as a function of dose rate and temperature.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank S. Fernandez-Perez for the agreement to use the single transistor data for the presented study. A similar thanks belongs to A. LaRosa (MPI Munich), K. Dette (CERN), T. Obermann (University of Bonn) and D. Sultan (INFN Trento) for the collaboration in the irradiations.\
A special thanks goes to F. Faccio (CERN) for the always fruitful discussions and helpful explanations, as well as for proof-reading each draft of the work.
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McLean, F. B., H. E. Boesch, and J. M. McGarrity, *Hole transport and recovery characteristics of SiO2 gate insulators*, IEEE Transactions on Nuclear Science 23.6 (1976): 1506-1512.
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/PHYSICS/PLOTS/PIX-2015-008/*.
|
---
abstract: |
The bright, soft X-ray spectrum Seyfert 1 galaxies [Akn 564]{} and [Ton S180]{} were monitored for 35 days and 12 days respectively with [[*ASCA*]{}]{} and [[*RXTE*]{}]{} (and [[*EUVE*]{}]{} for [Ton S180]{}). These represent the most intensive X-ray monitoring of any such soft spectrum Seyfert 1 to date. Light curves were constructed for [Ton S180]{} in six bands spanning 0.1–10 keV and for [Akn 564]{} in five bands spanning 0.7–10 keV. The short time scale (hours–days) variability patterns were very similar across energy bands, with no evidence of lags between any of the energy bands studied. The fractional variability amplitude was almost independent of energy band, unlike hard spectrum Seyfert 1s, which show stronger variations in the softer bands. It is difficult to simultaneously explain soft Seyferts stronger variability, softer spectra, and weaker energy-dependence of the variability relative to hard Seyferts.
There was a trend for soft and hard band light curves of both objects to diverge on the longest time scales probed ($\sim$weeks), with the hardness ratio showing a secular change throughout the observations. This is consistent with the fluctuation power density spectra that showed relatively greater power on long time scales in the softest bands. The simplest explanation of all of these is that two continuum emission components are visible in the X-rays: a relatively hard, rapidly-variable component that dominates the total spectrum and a slowly-variable soft excess that only shows up in the lowest energy channels of [[*ASCA*]{}]{}. Although it would be natural to identify the latter component with an accretion disk and the former with a corona surrounding it, a standard thin disk could not get hot enough to radiate significantly in the [[*ASCA*]{}]{}band, and the observed variability time scales are much too short. It also appears that the hard component may have a more complex shape than a pure power-law.
The most rapid factor of 2 flares and dips occurred within $ \sim 1000
$ sec, in [Akn 564]{} and a bit more slowly in [Ton S180]{}. The speed of the luminosity changes rules out viscous or thermal processes and limits the size of the individual emission regions to ${\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}$15 Schwarzschild radii (and probably much less), that is, to either the inner disk or small regions in a corona.
author:
- 'Rick Edelson, T. J. Turner, Ken Pounds, Simon Vaughan, Alex Markowitz, Herman Marshall, Paul Dobbie, Robert Warwick'
title: 'X-ray Spectral Variability and Rapid Variability of the Soft X-ray Spectrum Seyfert 1 Galaxies [Akn 564]{} and [Ton S180]{}'
---
Introduction {#intro}
==============
Seyfert 1 galaxies and quasars are the most powerful sustained, coherent, quasi-isotropic luminosity sources known, but their distances are so large that the “central engines” in which the luminosity is actually generated are thought to be orders of magnitude too small to image from Earth. Therefore, we must rely on indirect probes such as X-ray variability to infer information about the physical conditions in Seyfert 1s. This is potentially of general interest, because the luminosity is ultimately believed to originate in the region of strong gravity ($ {\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}3
R_S $) around a supermassive ($ 10^6 - 10^9 M_\odot $) black hole, conditions that are unlikely to be reproduced in the laboratory in the foreseeable future.
Ultraviolet and optical emission-line variability “reverberation mapping" studies have yielded key information about the size and structure of the (much larger) broad-line regions of Seyfert 1s (see Netzer & Peterson 1997 for a review) that may allow estimation of the mass of the putative central black hole (Wandel, Peterson & Malkan 1999). Although Seyfert 1s are much more strongly variable in the X-rays, less spectacular results have been seen at those higher energies, quite possibly because the X-rays probe the smallest size/time scales which may still lie beyond the limit of current instrumentation. In recent [*Advanced Satellite for Cosmology and Astronomy*]{} ([[*ASCA*]{}]{}) and [*Rossi X-ray Timing Explorer*]{} ([[*RXTE*]{}]{}) surveys, Nandra [et al. ]{}(1997), Turner [et al. ]{}(1999a) and Markowitz & Edelson (2001) found evidence that variations in Seyfert 1s were the largest at softer X-ray energies. This suggests either that there are two X-ray continuum emission components, with the softer one showing stronger variability than the harder one, or that the spectrum of a single component is not constant, becoming softer as the source brightens.
Evidence for interband lags within the X-rays is less clear cut. In simultaneous [*Extreme Ultraviolet Explorer*]{} ([[*EUVE*]{}]{}), [[*ASCA*]{}]{} and [[*RXTE*]{}]{} observations of NGC 5548 and MCG–6-30-15, Chiang [et al. ]{}(1999) and Reynolds (1999) respectively reported evidence that the variations in the hard X-rays consistently lagged behind those in the soft X-rays by times shorter than or of order an single spacecraft orbit. However, Edelson [et al. ]{}(2000) found no such effect in intensive [[*RXTE*]{}]{}monitoring of NGC 3516, and called into question the reality of lag measurements on time scales shorter than or of order the orbital time scale. If such hard interband lags are confirmed, causality arguments would require rejection of “reprocessing” models in which the soft X-rays are “secondary” emission produced by passive reradiation of “primary” hard X-ray photons.
Almost all of these studies have involved what could be called “hard X-ray spectrum Seyfert 1 galaxies" (or just “hard Seyferts"): Seyfert 1s with 2–10 keV power-law slopes in the range $ \Gamma \approx 1.7 - 2.0$. These sources dominate most X-ray samples, e.g., almost all of the Piccinotti [et al. ]{}(1982) Seyfert 1s are hard Seyferts. However, it is now clear that there is a significant population of Seyfert 1s with much steeper X-ray spectra ($\Gamma \approx 2.1 - 2.6$), and particularly strong (excess) emission below $\sim$2 keV. Many of these “soft X-ray spectrum Seyfert 1 galaxies" (or “soft Seyferts") are also optically classified as “narrow-line" Seyfert 1 galaxies (Osterbrock & Pogge 1985; Boller, Brandt & Fink 1996). However, it is their strongly-variable, steep soft X-ray continua that really set these objects apart; extreme examples show giant X-ray flares (as large as a factor of 100) on time scales of days (e.g., Boller [et al. ]{}1997). This rapid X-ray variability also extends to harder X-rays (Turner [et al. ]{}1999a; Leighly 1999).
The currently favored model is that soft Seyferts are powered by black holes of relatively low mass (compared to hard Seyferts of the same luminosity), accreting at a much higher rate, closer to the Eddington limit (Pounds, Done & Osborne 1995). In this model the steep X-ray spectrum is a result of enhanced emission from the putative accretion disk, and the rapid variability results from the smaller size scales associated with a lower mass black hole (Pounds [et al. ]{}2001) and perhaps also an intrinsically less stable accretion flow.
This paper reports on the most intensive X-ray monitoring of any soft Seyferts to date: a 35 day simultaneous [[*ASCA*]{}]{} and [[*RXTE*]{}]{} observation of [Akn 564]{}, and a 12 day simultaneous [[*ASCA*]{}]{}, [[*RXTE*]{}]{} and [[*EUVE*]{}]{} observation of [Ton S180]{}. This paper focuses on the X-ray spectral variability and interband lags in both objects; other results are reported elsewhere. The observations and data reduction are reported in the next section, temporal analyses are performed and discussed in § 3, the scientific implications are discussed in § 4, and a brief summary is given in § 5.
Observations and Data Reduction {# Data }
=================================
[Akn 564]{} {# ark }
------------
[Akn 564]{} is the brightest known soft Seyfert in the hard X-ray sky ($ F_{2-10~keV} \approx 2 - 5 \times 10^{-11} $ erg cm$^{-2}$ sec$^{-1}$) with a steep X-ray spectrum both above $\sim 2$ keV ($\Gamma \approx 2.6
$) and at lower energies (Vaughan [et al. ]{}1999a; Turner, George & Netzer 1999b; Pounds [et al. ]{}2001). Unfortunately, it has rather large foreground Galactic absorption ($ N_H = 6.4 \times 10^{20} $ cm$^{-2}$; Dickey & Lockman 1990) that prevents it from being observed with [[*EUVE*]{}]{}. In the observations reported herein, [Akn 564]{} was observed simultaneously with [[*ASCA*]{}]{} over 2000 June 1 – July 5, with [[*RXTE*]{}]{} over 2000 June 1 – July 1, surrounded by a total of $\sim$2 yr of [[*RXTE*]{}]{} monitoring once every $\sim$4.3 day. Initial results on the [[*RXTE*]{}]{} fluctuation power density spectrum (PDS) and long/short term variability have been reported in Pounds [et al. ]{}(2001), and on the [[*ASCA*]{}]{} spectrum in Turner [et al. ]{}(2001b), and other results will be forthcoming.
### [[*ASCA*]{}]{} Data {# ark_asca }
[[*ASCA*]{}]{} has two solid-state imaging spectrometers (SISs; Burke [et al. ]{}1994) and two gas imaging spectrometers (GISs; Ohashi [et al. ]{}1996) yielding data over an effective bandpass $\sim$0.7–10 keV. These data were gathered in 1CCD mode. All the data were screened according to the following criteria: the source was outside the SAA, the angular offset from the nominal pointing position was $\leq$ 0.01, the RBM was $\leq$ 500, the cutoff rigidity was $\leq$ 6 GeV/c, the source was at least 10 above the Earth’s limb (5 for the GIS) and at least 20 from the bright Earth, and the observations were made $\geq$ 50 s before or after passage through the terminator. These are the same methods and screening criteria used by the [*Tartarus*]{} (Turner [et al. ]{}1999a) database. This resulted in an effective exposure of 1.245 Msec in the GISs, and 1.109 Msec in the SISs. Light curves were extracted using source events within extraction cells of radii 4.8 and 6.6 for the SIS and GIS data, respectively. In order to increase the signal-to-noise ratio in the light curves, data from the SIS pair and GIS pair of detectors were (separately) combined, requiring all time bins to be at least 99% exposed. The background was subtracted from these light curves.
### [[*RXTE*]{}]{} Data {# ark_xte }
[Akn 564]{} was observed once every $\sim$3.2 hr (= 2 orbits) during this period. The [[*RXTE*]{}]{} Proportional Counter Array (PCA) consists of five collimated Proportional Counter Units (PCUs), nominally sensitive to 2–60 keV X-rays (Jahoda [et al. ]{}1996). However, only one PCU (number 2) was in use during this campaign. The present analysis is restricted to the 2–10 keV band, where the PCA is most sensitive and the systematic errors are best understood. Data from the top (most sensitive) layer of the PCU array were extracted using the [REX]{} reduction script[^1]. Poor quality data were excluded on the basis of the following acceptance criteria: the satellite has been out of the South Atlantic Anomaly (SAA) for at least 20 min; Earth elevation angle $\geq$ 10; offset from optical position of [Akn 564]{} $\leq$ 0.02; and [ELECTRON2]{} $\leq$ 0.1. This last criterion removes data with high anti-coincidence rate in the propane layer of the PCA. These selection criteria typically yielded $\sim$1 ksec good exposure time per orbit. The background was estimated using the “L7–240” model[^2], which is currently the best available but known to exhibit anomalies that affect AGN variability studies (e.g., Edelson & Nandra 1999). Data were initially extracted with 16 sec time resolution.
[Ton S180]{} {# tons }
-------------
The X-ray spectrum of [Ton S180]{} is steep ($\Gamma \approx 2.4$) and, like [Akn 564]{} shows a strong excess at lower energies (Vaughan [et al. ]{}1999; Turner [et al. ]{}2001a). Although it is not as bright as [Akn 564]{} in the hard X-rays, it does have a much lower column ($ N_H = 1.5 \times 10^{20} $ cm$^{-2}$; Stark [et al. ]{}1992), making possible [[*EUVE*]{}]{} observations. [Ton S180]{} was observed simultaneously for 12 days with [[*EUVE*]{}]{}, [[*RXTE*]{}]{} and [[*ASCA*]{}]{}(as well as other telescopes) during 1999 December 3 – 15. (The [[*EUVE*]{}]{} and [[*RXTE*]{}]{} observations extended considerably beyond this period but, for consistency, this paper restricts itself to the 12 day period during which all three telescopes were operating.) Initial results on the [[*Chandra*]{}]{} spectrum have been reported in Turner [et al. ]{}(2001a) and the spectral energy distributions will be forthcoming (Romano [et al. ]{}2001).
### [[*ASCA*]{}]{} and [[*RXTE*]{}]{} Data {# tons_asca }
The [[*RXTE*]{}]{} observations of [Ton S180]{} utilized PCUs 0 and 2. Data were extracted from these as described in § 2.1.2., the only differences being in two of the selection criteria. The [TIME\_SINCE\_SAA]{} criterion was extended to exclude all data taken in the 30 minutes following SAA passage. (The more conservative limit was because [Ton S180]{} is fainter than [Akn 564]{} and thus more susceptible to errors in background subtraction.) An [ELECTRON0]{} $\le$ 0.1 criteria was used to eliminate periods of high background.
The [[*ASCA*]{}]{} on-source exposures were 327 ksec for the SISs and 396 ksec for the GISs. The [[*ASCA*]{}]{} data were reduced and light curves constructed using the same methods as for Ark 564, except that in this case the predominant SIS datamode was BRIGHT.
### [[*EUVE*]{}]{} Data {# tons_euve }
A light curve was extracted from the [[*EUVE*]{}]{} deep survey (DS) data using the IRAF subpackage XRAY PROS. Source counts were summed in a circular aperture of 25 pixels in radius and the background calculated from a surrounding annulus of 30 pixels in width. In some previous analyses of [[*EUVE*]{}]{} DS light curves (e.g., Marshall [et al. ]{}1996), data with a deadtime-Primbsching correction (DPC) factor $ > 1.25 $ were discarded. This correction factor accounts for the loss of events due to detector deadtime and the limited telemetry bandwidth. As the detector count rate increases, the DPC factor increases and systematic uncertainties also increase due to incomplete instrument modeling. However, during the course of reducing these data it was noted that the DS DPC factor frequently was above 1.5, significantly greater than the more typically observed values of 1.0–1.3. This is most likely due to increased geocoronal emission possibly associated with the solar maximum and/or decreasing orbital altitude of [[*EUVE*]{}]{}. Data were therefore selected between the more liberal limits of $ 1.0 < DPC < 2.0 $. The initial light curve was binned at 50 sec.
Long Time Scale Light Curve Construction {# light_curves }
------------------------------------------
The observing logs are given in Table 1. Essentially identical procedures were used to construct all light curves for both objects. Data were extracted in the following subbands: 0.1–0.2 keV (for [[*EUVE*]{}]{}), 0.7–0.95 keV (for the [[*ASCA*]{}]{} SIS), 0.95–1.3 keV (for the [[*ASCA*]{}]{} SIS and GIS), 1.3–2 keV (for the [[*ASCA*]{}]{} SIS and GIS), 2–4 keV (for the [[*ASCA*]{}]{} SIS and GIS and [[*RXTE*]{}]{}) and 4–10 keV (for the [[*ASCA*]{}]{} SIS and GIS and [[*RXTE*]{}]{}). Data were then binned by the $\sim$95 min orbit (or, in the case of the [[*RXTE*]{}]{} observations of [Akn 564]{}, every other orbit), and the mean and standard errors computed. This yielded light curves with 142–181 points in 12 days for [Ton S180]{} (some were lost due to instrument problems or scheduling conflicts) and 518–520 points in 35 days for [Akn 564]{} (231 points for [[*RXTE*]{}]{}).
Light curves taken with different instruments but in the same bands were tested for consistency. In each panel of Figure 1, the data from two different instruments were plotted in the same graph, after first dividing by the mean. As the light curves covered the same bands, they should be nearly identical, modulo the errors, sampling details, and slight mismatches in energy response. Confining the analysis first to the [[*ASCA*]{}]{} SIS and GIS data, note that the light curves show excellent agreement in both the 2–4 keV and 4–10 keV bands. (The agreement is similarly good in both sets of softer bands as well.) This gives confidence in the data and therefore the light curves were summed to produce a single [[*ASCA*]{}]{} light curve in each band where the GIS and SIS overlap, as shown in the second half of Figure 1 and in Tables 2 and 3.
Then, these summed data were compared to [[*RXTE*]{}]{} data in the same bands. Unfortunately, the [[*RXTE*]{}]{} and [[*ASCA*]{}]{} light curves do not show such good agreement. Due to its large collecting area, [[*RXTE*]{}]{} is superior to [[*ASCA*]{}]{} for monitoring the brightest 2–10 keV sources (see, e.g., Edelson [et al. ]{}2000). However, [[*RXTE*]{}]{} has a harder spectral response than [[*ASCA*]{}]{}, so the count rates are lower for soft Seyferts. Because [[*RXTE*]{}]{} is also a non-imaging instrument with a high background, the background must be modeled. As this estimated background level is larger than the mean count rate for soft Seyferts but smaller than the mean count rate for many hard Seyferts, small errors in the background model would thus cause proportionally larger problems for soft Seyferts. Indeed, Tables 2 and 3 show that the [[*RXTE*]{}]{} data for both [Akn 564]{} and [Ton S180]{}have both higher count rates and larger fractional errors in the 4–10 keV band than in the 2–4 keV band, which would not be expected if only Poisson statistics contributed to the errors. Because of this problem, it was decided that the [[*RXTE*]{}]{} data were not sufficiently reliable for this analysis, and they will not be scientifically analyzed in this paper. Instead, the summed [[*ASCA*]{}]{} SIS + GIS data are used, except where the paper specifically states otherwise (e.g., § 3.3.). The resulting light curves are shown in Figure 2.
Short Time Scale Light Curve Construction {# light_curves }
-------------------------------------------
These data were also used to study variations on the shortest accessible time scales: within a single [[*ASCA*]{}]{} orbit. These usually lasted 30–40 min without interruption, although a substantial minority of orbits were affected by SAA passage or minor telescope problems.
For this purpose, sets of eight 16 sec points were used to measure both the total 0.7–10 keV count rate and the 2–10 keV/0.7–1.3 keV hardness ratio. Standard methods were used to determine the mean and standard error for each quantity in each 128 sec bin. [Akn 564]{} showed variations of a factor of 2 or larger in 16 orbits. These data are presented in Figure 3, and will be discussed in § 3.4. The largest single-orbit variations seen in [Ton S180]{} were four orbits in which the peak-to-trough variations were 70%–85%; these will be discussed in Romano [et al. ]{}(2001).
Temporal Analysis {# temporal_analysis }
===================
In the following section the statistical properties of these light curves are examined in order to quantify any spectral variability. A complementary analysis, that of direct spectral fitting to time-resolved data, was presented in Turner [et al. ]{}(2001).
Long Time Scale Fractional Variability as a Function of Energy
---------------------------------------------------------------
The fractional variability amplitude (${F_{var}}$), a common measure of the intrinsic variability amplitude that corrects for the effects of measurement noise, is defined as $${F_{var}}= { 1 \over \langle X \rangle } \sqrt{S^2 - \langle \sigma_{err}^2
\rangle },$$ where $S^2$ is the total variance of the light curve, $\langle
\sigma_{err}^2 \rangle $ is the mean error squared and $ \langle X \rangle
$ is the mean count rate (see, e.g., Edelson, Krolik & Pike 1989). The error on ${F_{var}}$ is $$\sigma_{{F_{var}}} = {1 \over 2 {F_{var}}} \sqrt{1 \over N} {S^2 \over \langle X
\rangle^2}$$ as discussed in the Appendix.
Tables 2 and 3 the summarize the fractional variability for each band/instrument, for [Akn 564]{} and [Ton S180]{}, respectively. The fractional variability is also shown as a function of observing energy in Figure 4. Note that the variability amplitude is only weakly anticorrelated with energy. This is very different from the situation in more “normal” hard Seyfert 1s (see references in § 1), which tend to show stronger variability at softer X-ray energies. This will be discussed in § 4.
Again, note that the [[*RXTE*]{}]{} data show a behavior which is different than that seen in either of the [[*ASCA*]{}]{} instruments. The ${F_{var}}$s are significantly higher for the [[*RXTE*]{}]{} bands, and in fact for [Ton S180]{} are see to increase with energy. This apparently spurious [[*RXTE*]{}]{} result was reported (for [Akn 564]{}) by Edelson (2000a). Based on the comparison with [[*ASCA*]{}]{}, we now believe it was almost certainly due to problems with the [[*RXTE*]{}]{} background.
The [[*EUVE*]{}]{} data on [Ton S180]{} appear to show a downturn relative to extrapolation from the harder [[*ASCA*]{}]{} bands. However, the [[*EUVE*]{}]{} data are somewhat suspect because they are much noisier than, e.g., a factor of 4 worse than the [[*ASCA*]{}]{} data, as well as for reasons given in the next section.
Short Time Scale Fractional Variability as a Function of Energy {# fpp }
-----------------------------------------------------------------
${F_{var}}$ measures the variability power of the total light curve. As AGN have “red" PDS (e.g., Edelson & Nandra 1999), this quantity is dominated by variations on the longest time scales probed by a given observation (e.g., Markowitz & Edelson 2001). The short time scale variability can be probed by a related parameter, called the point-to-point fractional variability (${F_{pp}}$), defined as $${F_{pp}}= { 1 \over \langle X \rangle } \sqrt{ { 1 \over 2(N-1) }
{ \sum_{i=1}^{N-1} ( X_{i+1} - X_i )^2 } - \langle \sigma_{err}^2 \rangle
}$$ where $X_i$ is the flux for the $i$th of $N$ orbits. This measures the variations between adjacent orbits. This quantity is very similar to the “Allan Variance"[^3].
For white noise, ${F_{pp}}$ and ${F_{var}}$ give the same value, as we have confirmed by measuring these quantities for light curves in which the times have been randomized (to yield a white-noise PDS). However, for red noise, ${F_{var}}$ will be larger than ${F_{pp}}$, as the variations will be larger on longer time scales. These quantities are tabulated in Tables 2 and 3 and shown in Figure 4.
The [Ton S180]{} [[*EUVE*]{}]{} point is formally not defined, as the measured variability is slightly weaker than just that expected from the errors alone. This again suggests that the [[*EUVE*]{}]{} errors are not reliable and the [[*EUVE*]{}]{}${F_{var}}$ and ${F_{pp}}$ values should not be taken seriously.
Similarities/Differences between Long and Short Time Scale Light Curves in Different Bands {# compare }
--------------------------------------------------------------------------------------------
The complex nature of the spectral variability of these objects is concisely illustrated in Figure 5. Both objects show strong orbit-to-orbit variability in the hardness ratio ($ HR = F_{2-10keV}/F_{0.7-1.3keV}$). Furthermore, the hardness ratio shows a long-term secular trend for both objects. In [Akn 564]{}, it changed over 32 days from $ 0.527 \pm 0.005 $ at the beginning of the monitoring to $ 0.604 \pm 0.008 $ at the end, and in [Ton S180]{}, it changed over 9 days from $ 0.588 \pm 0.007 $ at the beginning to $ 0.654
\pm 0.009 $ at the end. (Mean hardness ratios and standard errors were determined by binning up hardness ratios in the first and last 3 day periods.) This is the first time such a clear difference between long and short time scale variability has been seen in different bands in a Seyfert 1 galaxy. The implications of this are discussed in § 4.
Rapid Flares and Dips {# fvar }
-----------------------
It is also interesting to examine the largest and most rapid flux and spectral flares and dips. The [Akn 564]{} data are more well-suited for this because that source showed larger variations and the duration of the observation was almost 3 times that of [Ton S180]{}. Of the 518 useful orbits in the [Akn 564]{} monitoring, 256 have 15 or more 128 sec bins (that is, $\ge$32 min of data). Of these 256 orbits, 15 (6%) show peak-to-trough variations of a factor of $\ge$2 (see Figure 3), and 143 (56%) show changes of $\ge$50%. (The fourth panel in Figure 3 has only 11 points.) That indicates that the source flux will typically change by a factor of 2 within $\sim$3000 sec, and the fastest factor of 2 variations occur on very short time scales, $\sim$1000 sec. In some flares (e.g., the third panel in Figure 3), the source appears to systematically harden as the flux increases and soften as the flux declines, in others (e.g., the eighth), it appears to harden as the flux decreases, and in yet others (e.g., the fifteenth), no clear trend is apparent.
Fluctuation Power Density Spectra {# PDS }
-----------------------------------
In order to further compare the long and short time scale variations in different energy bands, PDS were measured for each [[*ASCA*]{}]{} energy band. The [[*EUVE*]{}]{} data were not used because of the large fraction ($>20$%) of orbits without data. The [[*RXTE*]{}]{} data were also not used in this paper for reasons mentioned earlier. However, Pounds [et al. ]{}(2001) have already used the full $\sim$2 years of data on [Akn 564]{} to determine the 2–10 keV PDS over a much broader range of time scales by the technique of Edelson & Nandra (1999).
The PDS in this paper were derived using standard methods (Oppenheim & Shafer 1975, Brillinger 1981), after first creating an evenly-sampled light curve by interpolating over the few missing points (2%–3% of the data). A Welch window was applied. The zero-power and next two (very noisy) lowest-frequency points of each PDS were ignored and the remaining points binned every factor of 1.8 (0.25 in the logarithm). The PDS covered a useable frequency range of 1.94 and 1.49 decades for [Akn 564]{} and [Ton S180]{} respectively. Power-law models were then measured from an unweighted, least-squares fit to the logarithmically binned data. The PDS were not corrected for noise because the variability between different orbits was much larger than the Poisson noise (as shown in the previous section). The 0.85 keV and 5 keV PDS for [Akn 564]{} and [Ton S180]{} are shown in Figures 6a and 6b, respectively.
For [Akn 564]{}, the PDS changes monotonically from the softest (0.85 keV) band, for which the slope of the PDS was $-1.22 \pm 0.06 $, and the hardest (5 keV) band, for which the slope was $ -0.96 \pm 0.07 $. Similar behavior was seen in [Ton S180]{}, which had a PDS slope of $ -1.61 \pm 0.08 $ at 0.85 keV and $ -1.18 \pm 0.09 $ at 5 keV. In both cases the slope differences are highly significant. The sense of the difference is that the softest bands show more power on the longest time scales probed.
Fractional Variability versus Flux Level {# Fvarflux }
------------------------------------------
For [Akn 564]{}, ${F_{var}}$s were also measured for each of the 256 orbits with more than 32 min of data. These were sorted by flux levels and averaged in flux bins with 20 or more points in order to smooth out fluctuations. The result is plotted as a function of mean count rate in Figure 7. The variability amplitude is quite independent of count rate over a factor of $\sim$4 in count rate, which means that the intrinsic RMS amplitude (corrected for the measurement noise) is linearly correlated with flux. The implications of this result are discussed in detail in § 4.
Linearity of the Light Curves
------------------------------
A search for non-linear behavior (e.g., Leighly & O’Brien 1997; Green, McHardy & Done 1999) was undertaken with the [Akn 564]{} and [Ton S180]{} light curves. The surrogate data method of Theiler [et al. ]{}(1992) was used, in which a discriminating non-linear statistic is applied both to the real data and to simulated light curves. A significant difference between the values of the statistic as computed for the real and simulated data indicates a detection of non-linearity in the real light curve. Here, the Kolmogorov-Sminov (KS) D-statistic, which compares the distribution of data points above the mean with those below the mean (e.g., Press [et al. ]{}1992), is applied to all light curves. A larger value of the D-statistic implies stronger non-linearity.
For each of the two targets, 100 simulated light curves, each with a PDS slope corresponding to the PDS slope measured for the actual data, were randomly generated using the algorithm of Timmer & König (1995). Parent light curves had 4096 data points (much more than in the observation to reduce red-noise leak) and a time resolution corresponding to 1 [[*ASCA*]{}]{} orbit. A section of the light curve corresponding to the observation length was randomly chosen and sampled in the same fashion as the actual data. The KS D-statistic was calculated for each simulated light curve, and these values were ranked.
The KS D-statistic for the summed 0.7–10 keV [Akn 564]{} light curve was found to be greater than 86$\%$ of the KS D-statistic values for light curves simulated with PDS slope of $-$1.13. The KS D-statistic for the summed 0.7–10 keV [Ton S180]{} light curve was found to be greater than 63$\%$ of the KS D-statistic values for light curves simulated with PDS slope of $-$1.52. Neither of these are $ > 1.5 \sigma $ effects. Thus, this test provided no evidence for non-linear variability in either of these light curves. However, tests for non-linearity (and the related non-Gaussianity) are notoriously difficult (see Press & Rybiki 1997 for a detailed discussion) so this is perhaps not as different from previous results as it might appear.
Interband Lags {# lags }
----------------
Interband lags were searched for using the cross-correlation functions: both the discrete correlation function (Edelson & Krolik 1988) and interpolated correlation function (White & Peterson 1994). The results are shown in Figure 8, and summarized in Tables 4 and 5. They confirm that all of the [Akn 564]{} data are highly correlated, with correlation coefficients $ r = 0.85 $ to 0.99, and none of the bands appears to lead another, down to ${\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}$1 orbit, or $ |\tau| <$ 1.5 hr. (The formal errors were much smaller, but we conservatively claim no limit stronger than this; see Edelson [et al. ]{}2001 for a detailed discussion of uncertainties of interband lags and the perils of “super-resolution".) The [Ton S180]{} data are also highly correlated, although not nearly as well as for [Akn 564]{}. These data also show no lags down to limits of ${\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}$1 orbit. For CCFs that do not include the [[*EUVE*]{}]{} data, correlation coefficients are $ r = 0.48 $ to 0.92. The [[*EUVE*]{}]{} data is not as well correlated; for CCFs that do include the [[*EUVE*]{}]{} data, correlation coefficients were much lower: $ r = 0.32 $ to 0.59.
Discussion {# disco }
============
Separating Emission Components with Spectral Variability
----------------------------------------------------------
This monitoring of the soft Seyfert 1s [Akn 564]{} and [Ton S180]{} on time scales of weeks revealed a number of new and interesting results: on short time scales, the variations are similar in all bands, with no measurable interband lags down to the shortest time scales measurable and no consistent trend for the spectrum to harden or soften during flares and dips. However, especially in [Akn 564]{}, the hard and soft bands appear to diverge on longer time scales, and the soft bands had slightly larger variability amplitudes that apparently resulted from a long-term trend relative to the hard bands.
It is difficult to see how a single emission component could naturally produce spectral evolution that is so markedly different on long and short time scales. Instead, the simplest explanation is that two separate continuum emission components are visible in the X-rays: the first is a rapidly-variable hard component that dominates the emission, especially at the hardest energies, for which the shape changes only weakly, hardening slightly as the total flux changes by a factor of $>$2. The second is a much more slowly-variable “soft excess" component only seen in the lowest-energy channels of [[*ASCA*]{}]{}. Because it only contributes to the softest channels, these data alone cannot determine if its shape changes with time. There appears to be no obvious temporal connection between the two components.
The spectral and variability properties of the soft component are not consistent with the simplest models of direct thermal emission from an optically thick, geometrically thin accretion disk (e.g., Frank, King & Raine 1992). Even for the most favorable realistic parameters, the disk temperature is well below 0.1 keV, while the observed emission (from the spectral fits) extends well above 1 keV. This general problem is well known (e.g., Czerny & Elvis 1989). While gravitational focussing and Comptoniation could harden the spectrum somewhat, it is difficult to see how such a strong effect could be produced. Likewise, the relevant time scale for variations in a disk is probably the viscous time scale, which for any reasonable set of parameters is years, compared with the observed variability on time scales of ${\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}$1 week (see also Turner [et al. ]{}2001b).
The hard component is generally identified with emission from a patchy corona (e.g., Haardt, Maraschi & Ghisellini 1994). Because the cells are relatively small compared to the overall structure, and the process could proceed as quickly as the light-crossing time (e.g., in the case of magnetic reconnection), then the expected time scales are comfortably consistent with the observed variability time scales.
However, these results are not entirely consistent with the simple picture in which the spectrally-defined fit parameters fully describe the physically relevant emission components. Turner [et al. ]{}(2001b) fitted the spectrum of [Akn 564]{} with a power-law and (Gaussian) soft excess, and found that the soft excess component faded by a factor of 2.8 throughout the observation, while the harder power-law faded by only a factor of 1.68. This is consistent with the overall hardness ratio changes reported in § 3.3. However, this slowly-varying component would be nearly constant during a single orbit, and thus would provide a constant “contamination" at soft energies during any rapid flares/dips. This would yield a correlation between hardness and flux, in the sense that the source would get harder during a flare and softer during a dip. As discussed in § 3.4., this is not the case. This means that the straightforward spectral fits do not tell the full story, and that most likely the rapidly variable component contains not only the hard component (described as a power-law) but also some of the soft excess as well. That is, the hard component appears to be intrinsically more complex than the pure power-law description used in spectral fitting routines.
Implications of Rapid Variability
-----------------------------------
[Akn 564]{} shows factor of 2 flares and dips on time scales as short as 1000 sec. For its redshift of $ z = 0.0247 $ (Huchra, Vogeley & Geller 1999), this corresponds to a change in the 0.7–10 keV luminosity $ \Delta L/\Delta t
\approx 10^{41} $ erg/s$^2$. Under the assumptions of isotropic emission, the Eddington limit implies $
M_{BH} \ge 8 \times 10^5 L_{44} M_\odot $, where $L_{44}$ is the [*bolometric*]{} luminosity in units of $10^{44}$ erg sec$^{-1}$ (e.g., Peterson 1997). If we assume that [Akn 564]{}’s 0.6–10 keV luminosity is 10% of bolometric, then $ L_{44} \approx 8 $ and $ M_{BH} \ge 6 \times 10^{6} M_\odot $. We note that all of these assumptions mean that the limit is probably good to no better than an order of magnitude. Even so, for a black hole mass above this limit, both the radial drift/viscous and thermal processes, operating at distances of $ \sim 10
R_S $, give time scales that are much too long (hours to years; see Frank [et al. ]{}1992) to be compatible with the observed time scale of $\sim$1000 sec. Thus, such processes cannot be responsible for the observed X-ray emission.
The light crossing time scale yields an approximate upper limit (to within the order of magnitude uncertainties discussed above) on the size of the emitting region of $ R {\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}15 R_{S} $ for $ M_{BH} \ge 6 \times 10^{6}
M_\odot $ and $ T = 1000 $ sec. For other processes (governed, e.g., by the orbital or dynamical time scales), the upper limit on the size of the emitting region must be significantly smaller. This indicates that the bulk of the X-ray emission in [Akn 564]{} must either be produced in the inner accretion disk or else isolated clumps that are smaller than or of order a few tens of Schwarzschild radii.
Statistical Properties of the X-ray Variability
-------------------------------------------------
Decomposition of the X-ray emission into two components with very different spectral shapes and variability time scales would also significantly affect the interpretation of the PDS. Recent intensive and long-term monitoring of Seyfert 1s have begun to yield evidence that the power-law PDS measured at short time scales (e.g., Lawrence & Papadakis 1993) show a turnover at longer time scales (Edelson & Nandra 2000, Pounds [et al. ]{}2001, Uttley, McHardy & Papadakis 2001). However, the shape of this turnover is unclear, and it is consistent with a variety of shapes (Uttley [et al. ]{}2001). The PDS of Galactic X-ray binaries (XRBs), for which the shapes are much better-defined than for Seyfert 1s (due to their much shorter time scales and higher fluxes) often show a more complex structure with multiple features (Nowak 2001). These multiple features could be multiple time scales, indicating that the PDS cannot be modeled by single variability component. The spectral evidence presented in this study suggests that the same situation may be the case with Seyfert 1s.
The fact that ${F_{var}}$ is independent of flux level demonstrates that the light curve is non-stationarity but in a relatively “well-behaved" and repeatable fashion. This result confirms and expands upon the finding of Uttley & McHardy (2001) that found a similar independence of ${F_{var}}$ from flux for three other Seyfert 1s, although those were measured with only two independent flux points. These results are consistent with no zero-point offset, indicating that source does not have a large, constant flux component. More importantly, the independence of ${F_{var}}$ from flux level shows yet another remarkable parallel between Seyfert 1s and XRBs, suggesting a relationship exists between these putative accreting black hole sources independent of the mass of the central object, even though they differ by a factor of $ {\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}10^6 $ in luminosity and black hole mass.
Finally, it is interesting that neither of these two objects show strong non-linear variability, at least using the method of Theiler [et al. ]{}(1992). Although this may not be the ideal method to use, visual examination of the light curves also suggest that the variations are not wildly non-linear, as the dips are about as strong as the flares (on a logarithmic plot).
Soft and Hard Spectrum Seyfert 1 Galaxies
-------------------------------------------
With the study of these two soft Seyferts, it is now becoming feasible to systematically explore the differences between the variability in soft and hard Seyferts. It is already well known that soft Seyferts tend to have narrower optical permitted emission lines (Boller [et al. ]{}1996) and that they show much stronger X-ray variability than hard Seyferts at a similar luminosity (Leighly 1999; Turner [et al. ]{}1999a). This may be more pronounced on short time scales: Pounds [et al. ]{}(2000) finds that the PDS of [Akn 564]{} is unusually flat, meaning that there is more variability power on short time scales relative to long time scales than in hard Seyferts.
This is the first study to quantify the rapid variations in individual sources: significant variations are almost always seen within a single orbit (${\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}$40 min on source), and in the best studied case, [Akn 564]{} showed factor of two variations in $\sim$6% of all well-determined orbits. This result is consistent with the idea first put forward by Pounds [et al. ]{}(1995) that soft Seyferts are accreting at a much higher fraction of the Eddington rate than hard Seyferts.
A new clue that is emerging involves the spectral variability results in § 3.1.: both of these soft Seyferts show only a very weak dependence of variability amplitude on energy. Other observations of soft Seyferts appear to show the same behavior. A recent [[*Chandra*]{}]{} observation by Collinge [et al. ]{}(2001) found that the soft Seyfert NGC 4051 varied in 0.5–8 keV flux by a factor of $>$5 in a $\sim$4 ksec period while the 0.5–2 keV/2–8 keV flux ratio changed by less than 20%. Likewise, Gliozzi [et al. ]{}(2001) find that the soft Seyfert PKS 0558–504 actually hardens as it gets brighter. Finally, [[*XMM-Newton*]{}]{} observations of both [Ton S180]{} and the soft Seyfert 1H 0707–495 show strong variability with almost no energy dependence (Vaughan 2001). This is very different behavior than seen in hard Seyferts, which generally have X-ray spectra that appear to soften as they brighten (e.g., Markowitz & Edelson 2001).
It is difficult to construct unified phenomenological picture that can neatly explain all of these results. Soft Seyferts tend to have stronger soft excesses, stronger overall variability than hard Seyferts, yet hard Seyferts show much stronger energy dependence of the variations (in the sense that their spectra become softer as the flux increases). This is the opposite of what would be expected from mixing a soft (rapidly variable) and hard (less variable) component such that the former dominates in soft Seyferts and the latter in hard Seyferts. (It also contradicts the observation that the soft component appears to be the less variable one.) Likewise, if the harder component is the highly variable one, then one would expect hard Seyferts to show stronger variability than soft Seyferts. Of course, it may be that these objects are may be powered by completely different processes, and no unified scheme is applicable.
Summary
=========
This paper reports the most intensive X-ray monitoring ever undertaken of any soft Seyfert galaxy. These extraordinary data sets allow a deeper and more systematic quantification of soft Seyfert variability than was previously possible. Both sources show strong variability, with [Akn 564]{} showing repeated variations of a factor of 2 on time scales as short as $\sim$1000 sec. However, these relatively well-sampled light curves do not clear evidence of non-linear behavior reported for other soft Seyferts, as the number and strength of flares and dips were comparable. The hard and soft light curves track well on short time scales, with no clear trends for the hardness ratio to change in a systematic way during a flare. On longer time scales, especially for [Akn 564]{}, the hard and soft bands diverge somewhat, yielding larger long time scale variability amplitudes in the softer bands.
The rapid variations rule out thermal and viscous processes and constrain the emission to the inner ${\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}15 R_S $, most likely to the inner disk or small clumps in a corona. The spectral variability indicates the presence of two components, the dominant one (in the 0.6–10 keV [[*ASCA*]{}]{} band) being a hard, rapidly variable component that is naturally associated with a corona. However, the softer, more slowly variable component cannot be identified with the simplest optically thick, geometrically thin accretion disk models, as the emission is observed to extend well beyond 1 keV, and the observed variability time scales are much too short.
The variability amplitude was found to be almost independent of energy band for these objects, and there are indications that the same is true for other soft Seyferts as well. This indicates a possibly important difference with hard Seyferts, which generally show significantly softening of the spectrum as the flux increases. Other known differences between the sources is that soft Seyferts tend to be more rapidly variable and also to have narrower optical permitted lines. This is not easy to understand in terms of phenomenological models in which essentially identical hard and soft components are mixed together in different ratios to produce the two types of Seyferts, and instead appear to require a more complex explanation.
These observations show that intensive spectral variability monitoring has unique power to separate out emission components in a way that is complementary to single-epoch spectroscopy. As these objects are much too distant to image directly, spectral variability studies may prove our most effective tool for determining the processes responsible for the high X-ray luminosities of AGN. While the current [[*ASCA*]{}]{} and [[*RXTE*]{}]{} archives contain a great deal of relevant data, we expect that future progress will hinge on [[*XMM-Newton*]{}]{}. Its high throughput makes it the only instrument with sufficient sensitivity to obtain meaningful short term light curves for the most extreme and interesting soft Seyferts which tend to be almost an order of magnitude fainter than [Akn 564]{} and [Ton S180]{}. Its broad bandpass allows it to simultaneously study spectral variations over a much larger fraction of the X-ray spectrum than was previously possible, especially at the critical soft energies (which were not probed by [[*ASCA*]{}]{} or [[*RXTE*]{}]{}). Finally, its high-Earth orbit yields uninterrupted $\sim$40 hr light curves that can be used to study short time scale variability. The previous generation of low-Earth orbit telescopes produced light curves corrupted by interruptions that made it impossible to track the development of flares. Although the ideal parameters for such observations are not yet fully determined, it is likely that more insight will be gained from a few long observations instead of many short ones (Mushotzky 2001). It is also important that future variability studies accurately define the variability properties of both soft Seyferts (especially the most extreme examples like IRAS 13224–3908, PHL 1092 and 1H 0707-495) but also of a control of group of “standard" hard Seyferts such as NGC 5548. As such long observations are unlikely to be scheduled in great numbers in this early stage of the mission (e.g., only one Seyfert 1, MCG–6-30-15, has been scheduled for more than a single orbit in the first two years of [[*XMM-Newton*]{}]{}), patience is a necessary virtue in this area of endeavor.
The authors thank the [[*RXTE*]{}]{} and [[*ASCA*]{}]{} teams for their efforts that resulted in the data needed for this research. They also thank the referee, Niel Brandt, for helping to focus the discussion on the big picture. Edelson and Markowitz were supported by NASA grants NAG 5-7317 and NAG 5-9023, and Turner was supported by NASA grant NAG 5-7385.
The Estimation of the Fractional Excess Variability Amplitude, ${F_{var}}$, of an AGN Light Curve
===================================================================================================
Here we present a prescription for measuring the fractional excess variability parameter ${F_{var}}$ and its associated error. We also note various caveats relating to its interpretation.
Basic Equations and Derivation of ${F_{var}}$
-----------------------------------------------
Consider a light curve subdivided into $N$ time bins, where each bin is further subdivided into $n_i$ individual points ($n_i$ can be the same or different in each bin). The mean count rate in the $ith$ bin is: $$X_i = { 1 \over n_i } \sum_{j=1}^{n_i} x_{ij},$$ where $x_{ij}$ is the count rate of the $jth$ point in the $ith$ bin. The square of the standard error on $X_i$ is: $$\sigma^2_{err,i} = { 1 \over n_i(n_i-1) } \sum_{j=1}^{n_i}
(x_{ij}-X_i)^2 .$$
In considering the full light curve, the unweighted mean count rate given by: $$\langle X \rangle = { 1 \over N } \sum_{i=1}^N X_i ,$$ and the variance of the binned data comprising the light curve is: $$S^2 = { 1 \over N-1 } \sum_{i=1}^N (X_i-\langle X \rangle)^2 .$$
Both intrinsic source variability and measurement uncertainty contribute to this observed variance. Under the assumption that both components are normally distributed and combine in quadrature, the observed variance can be written as: $$S^2 = \langle X \rangle^2 {\sigma_{XS}^2}+ \langle \sigma_{err}^2 \rangle$$ The first term on the right represents the intrinsic scatter induced by the source variability. The second term is the contribution of the measurement noise. We assume that the scatter of the data points within an individual time bin is predominantly due to the statistical uncertainty of the measurements, leading to: $$\langle \sigma_{err}^2 \rangle = { 1 \over N }
\sum_{i=1}^N\sigma^2_{err,i},$$ Rearranging equation A5 yields the standard definition for the fractional excess variance $${\sigma_{XS}^2}= { S^2 - \langle \sigma_{err}^2 \rangle \over \langle X \rangle^2 }.$$ The fractional variability amplitude ${F_{var}}$ is simply the square root of the fractional excess variance: $${F_{var}}= \sqrt{ S^2 - \langle \sigma_{err}^2 \rangle \over \langle X
\rangle^2 },$$ as given in Equation 1 of the text.
Derivation of the Uncertainty on ${F_{var}}$
----------------------------------------------
We now require a measure of the uncertainties that should be assigned to $ {\sigma_{XS}^2}$ and $ {F_{var}}$. In equation A7, assume that the dominant variance will be that associated with the quantity $S^2$, and that the error term $\langle \sigma_{err}^2
\rangle$ can be neglected by comparison. The implications of this assumption are discussed at the end of this section.
This variance on $S^2$ can be estimated as $ {2 \over N-1} S^4 \approx {2
\over N} S^4 $ (e.g., Trumpler & Weaver 1962). Hence the standard deviation of ${\sigma_{XS}^2}$ is: $$\sigma_{{\sigma_{XS}^2}} = \sqrt{2 \over N} {S^2 \over \langle X \rangle^2}$$ Setting $ x = {\sigma_{XS}^2}$ and $ y = {F_{var}}$ so that $ y = \sqrt{x} $ yields $${ dy \over dx } = { 1 \over 2 \sqrt{x} } = { 1 \over 2 y }
= { 1 \over 2 {F_{var}}}$$ Transmitting the error through the equation by the standard formula $ \sigma_y = { dy \over dx } \sigma_x $ yields $$\sigma_{{F_{var}}} = { 1 \over 2 {F_{var}}} \sigma_{{\sigma_{XS}^2}}
= {1 \over 2 {F_{var}}} \sqrt{1 \over N} {S^2 \over \langle X \rangle^2}$$ as in Equation 2 of the text.
In the above analysis the assumption (made in eqn. A2) that all of the variance within a time bin is due solely to measurement errors will lead to overestimation of the latter if the source exhibits rapid variability on time scales comparable to the bin size. This is a conservative approach which in many circumstances may be a better choice than relying on the errors propagated through data extraction and data fitting algorithms (which may mix systematic and statistical errors in a manner not appropriate for variability studies). The importance of such an approach can be seen in the fact that the error estimate assumed that the variance due to systematic errors was small compared to the total variance; if they are not the derivation is incorrect.
More serious, however, is the assumption that the underlying source variability is governed by processes that are stationary and governed by Gaussian statistics. As red-noise processes are “weakly non-stationary” (e.g., Press & Rybicki 1997) the above error estimate cannot account for random fluctuations in ${F_{var}}$ as a function of time. A further point is that the weak non-stationarity and (in general) non-normal distribution of fluxes in red-noise light curves mean that the above prescription provides an increasingly poor estimate of the uncertainty on ${F_{var}}$ as the signal-to-noise in the observed light curve increases. (This will be discussed in more detail in a future work, Vaughan [et al. ]{}in prep.) A more robust approach would be to estimate the PDS, but where this is not possible ${F_{var}}$ can provide a useful measure of the degree of variability in a given light curve. In practice the value of statistics such as $F_{var}$ is as a comparative measure of the magnitude and constancy of the variability signal.
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[llcccc]{} [Ton S180]{}& [[*ASCA*]{}]{} SIS & 0.7 – 10 & 1515.55 – 1527.78 & 94.8 & 180\
& [[*ASCA*]{}]{} GIS & 0.95 – 10 & 1515.55 – 1527.78 & 94.8 & 181\
& [[*EUVE*]{}]{} & 0.1 – 0.2 & 1515.56 – 1527.79 & 94.1 & 142\
& [[*RXTE*]{}]{} & 2 – 10 & 1515.59 – 1527.72 & 95.8 & 142\
[Akn 564]{}& [[*ASCA*]{}]{} SIS & 0.7 – 10 & 1696.52 – 1731.00 & 94.0 & 518\
& [[*ASCA*]{}]{} GIS & 0.95 – 10 & 1696.52 – 1731.00 & 94.0 & 520\
& [[*RXTE*]{}]{} & 2 – 10 & 1696.63 – 1726.47 & 191.4 & 231\
[lccccccc]{} [[*ASCA*]{}]{} & 0.7 – 0.95 & 0.85& 0.57 & 31.1 & 34.2 $\pm$ 1.1% & 34.0 $\pm$ 1.1% & 14.4 $\pm$ 0.5%\
[[*ASCA*]{}]{} & 0.95 – 1.3 & 1.1 & 1.42 & 43.4 & 33.2 $\pm$ 1.0% & 33.1 $\pm$ 1.0% & 15.5 $\pm$ 0.5%\
[[*ASCA*]{}]{} & 1.3 – 2 & 1.5 & 1.57 & 43.3 & 33.4 $\pm$ 1.0% & 33.3 $\pm$ 1.0% & 15.9 $\pm$ 0.5%\
[[*ASCA*]{}]{} & 2 – 4 & 2.5 & 0.86 & 33.3 & 32.4 $\pm$ 1.0% & 32.3 $\pm$ 1.0% & 16.0 $\pm$ 0.5%\
[[*ASCA*]{}]{} & 4 – 10 & 5 & 0.27 & 19.7 & 30.3 $\pm$ 0.9% & 29.8 $\pm$ 1.0% & 16.8 $\pm$ 0.6%\
& & & & & &\
SIS & 0.7 – 0.95 & 0.85 & 0.57 & 31.1 & 34.2 $\pm$ 1.1% & 34.0 $\pm$ 1.1% & 14.4 $\pm$ 0.5%\
SIS & 0.95 – 1.3 & 1.1 & 1.00 & 39.7 & 33.2 $\pm$ 1.0% & 33.1 $\pm$ 1.0% & 15.0 $\pm$ 0.5%\
SIS & 1.3 – 2 & 1.4 & 0.95 & 37.5 & 33.3 $\pm$ 1.0% & 33.2 $\pm$ 1.0% & 15.6 $\pm$ 0.5%\
SIS & 2 – 4 & 2.5 & 0.45 & 27.0 & 32.6 $\pm$ 1.0% & 32.4 $\pm$ 1.0% & 15.8 $\pm$ 0.5%\
SIS & 4 – 10 & 5 & 0.13 & 14.8 & 31.2 $\pm$ 1.0% & 30.4 $\pm$ 1.0% & 16.4 $\pm$ 0.6%\
& & & & & &\
GIS & 0.95 – 1.3& 1.1 & 0.42 & 28.6 & 33.0 $\pm$ 1.0% & 32.8 $\pm$ 1.0% & 15.1 $\pm$ 0.5%\
GIS & 1.3 – 2 & 1.5 & 0.62 & 33.8 & 33.0 $\pm$ 1.0% & 32.8 $\pm$ 1.0% & 15.4 $\pm$ 0.5%\
GIS & 2 – 4 & 2.5 & 0.41 & 27.5 & 32.0 $\pm$ 1.0% & 31.7 $\pm$ 1.0% & 15.4 $\pm$ 0.5%\
GIS & 4 – 10 & 5 & 0.14 & 16.2 & 29.7 $\pm$ 0.9% & 29.0 $\pm$ 0.9% & 16.3 $\pm$ 0.6%\
& & & & & &\
[[*RXTE*]{}]{} & 2 – 4 & 3.3 & 0.78 & 14.0 & 34.6 $\pm$ 1.6% & 33.8 $\pm$ 1.6% & 24.2 $\pm$ 1.2%\
[[*RXTE*]{}]{} & 4 – 10 & 6 & 1.04 & 13.4 & 33.5 $\pm$ 1.6% & 32.6 $\pm$ 1.6% & 23.9 $\pm$ 1.2%\
[lccccccc]{} [[*EUVE*]{}]{} & 0.1 – 0.2 & 0.15 & 0.14 & 5.5 & 28.3 $\pm$ 1.7% & 17.8 $\pm$ 2.7% & Undefined\
& & & & & &\
[[*ASCA*]{}]{} & 0.7 – 0.95 & 0.85 & 0.19 & 17.2 & 20.3 $\pm$ 1.1% & 19.4 $\pm$ 1.1% & 7.2 $\pm$ 0.6%\
[[*ASCA*]{}]{} & 0.95 – 1.3 & 1.1 & 0.37 & 21.5 & 19.6 $\pm$ 1.0% & 19.0 $\pm$ 1.0% & 9.4 $\pm$ 0.6%\
[[*ASCA*]{}]{} & 1.3 – 2 & 1.5 & 0.42 & 23.4 & 17.8 $\pm$ 0.9% & 17.2 $\pm$ 1.0% & 9.6 $\pm$ 0.6%\
[[*ASCA*]{}]{} & 2 – 4 & 2.5 & 0.25 & 18.0 & 18.5 $\pm$ 1.0% & 17.5 $\pm$ 1.0% & 10.4 $\pm$ 0.7%\
[[*ASCA*]{}]{} & 4 – 10 & 5 & 0.08 & 9.8 & 19.4 $\pm$ 1.0% & 16.2 $\pm$ 1.2% & 9.3 $\pm$ 1.1%\
& & & & & &\
SIS & 0.7 – 0.95 & 0.85 & 0.19 & 17.2 & 20.3 $\pm$ 1.1% & 19.4 $\pm$ 1.1% & 7.2 $\pm$ 0.6%\
SIS & 0.95 – 1.3 & 1.1 & 0.28 & 20.8 & 18.6 $\pm$ 1.0% & 17.9 $\pm$ 1.0% & 8.5 $\pm$ 0.6%\
SIS & 1.3 – 2 & 1.4 & 0.24 & 19.4 & 18.3 $\pm$ 1.0% & 17.5 $\pm$ 1.0% & 9.8 $\pm$ 0.7%\
SIS & 2 – 4 & 2.5 & 0.13 & 13.9 & 19.1 $\pm$ 1.0% & 17.5 $\pm$ 1.1% & 10.9 $\pm$ 0.8%\
SIS & 4 – 10 & 5 & 0.04 & 7.1 & 21.2 $\pm$ 1.1% & 15.5 $\pm$ 1.5% & 8.6 $\pm$ 1.7%\
& & & & & &\
GIS &0.95 – 1.3 & 1.1 & 0.09 & 11.4 & 24.7 $\pm$ 1.3% & 22.9 $\pm$ 1.4% & 11.6 $\pm$ 1.0%\
GIS & 1.3 – 2 & 1.5 & 0.17 & 17.3 & 17.9 $\pm$ 0.9% & 16.8 $\pm$ 1.0% & 9.4 $\pm$ 0.7%\
GIS & 2 – 4 & 2.5 & 0.12 & 14.6 & 18.7 $\pm$ 1.0% & 17.3 $\pm$ 1.1% & 9.3 $\pm$ 0.8%\
GIS & 4 – 10 & 5 & 0.05 & 8.4 & 20.0 $\pm$ 1.1% & 15.8 $\pm$ 1.3% & 9.1 $\pm$ 1.3%\
& & & & & &\
[[*RXTE*]{}]{} & 2 – 4 & 3.3 & 0.11 & 6.1 & 26.2 $\pm$ 1.6% & 19.6 $\pm$ 2.1% & 10.5 $\pm$ 2.3%\
[[*RXTE*]{}]{} & 4 – 10 & 6 & 0.16 & 5.8 & 28.9 $\pm$ 1.7% & 22.1 $\pm$ 2.2% & 12.8 $\pm$ 2.3%\
[cccccc]{} 0.85 & 1.1 & 0.97 & 0.0 & 0.97 & 0.0 0.85 & 1.5 & 0.94 & 0.0 & 0.94 & 0.0 0.85 & 2.5 & 0.91 & 0.0 & 0.92 & 0.0 0.85 & 5 & 0.85 & 0.0 & 0.85 & 0.0 1.1 & 1.5 & 0.99 & 0.0 & 0.99 & 0.0 1.1 & 2.5 & 0.97 & 0.0 & 0.97 & 0.0 1.1 & 5 & 0.91 & 0.0 & 0.91 & 0.0 1.5 & 2.5 & 0.99 & 0.0 & 0.99 & 0.0 1.5 & 5 & 0.94 & 0.0 & 0.94 & 0.0 2.5 & 5 & 0.97 & 0.0 & 0.97 & 0.0
[cccccc]{} 0.15 & 0.85 & 0.59 & $-$1.6 & 0.63 & 0.0 0.15 & 1.1 & 0.54 & 0.0 & 0.61 & +0.8 0.15 & 1.5 & 0.46 & +1.6 & 0.54 & +0.8 0.15 & 2.5 & 0.44 & +1.6 & 0.49 & +1.6 0.15 & 5 & 0.32 & +4.8 & 0.35 & +4.0 0.85 & 1.1 & 0.92 & 0.0 & 0.92 & 0.0 0.85 & 1.5 & 0.83 & 0.0 & 0.83 & 0.0 0.85 & 2.5 & 0.77 & 0.0 & 0.77 & 0.0 0.85 & 5 & 0.48 & 0.0 & 0.49 & 0.0 1.1 & 1.5 & 0.92 & 0.0 & 0.92 & 0.0 1.1 & 2.5 & 0.84 & 0.0 & 0.84 & 0.0 1.1 & 5 & 0.61 & 0.0 & 0.62 & +0.8 1.5 & 2.5 & 0.90 & 0.0 & 0.90 & 0.0 1.5 & 5 & 0.72 & 0.0 & 0.72 & 0.0 2.5 & 5 & 0.72 & 0.0 & 0.73 & +0.8
[^1]: See [http://heasarc.gsfc.nasa.gov/docs/xte/recipes/rex.html]{}
[^2]: See [http://lheawww.gsfc.nasa.gov/$\sim$keith/dasmith/rossi2000/index.html]{}
[^3]: See [http://www.allanstime/AllanVariance/index.html]{}
|
---
abstract: 'We study the problem of learning a real-valued function that satisfies the Demographic Parity constraint. It demands the distribution of the predicted output to be independent of the sensitive attribute. We consider the case that the sensitive attribute is available for prediction. We establish a connection between fair regression and optimal transport theory, based on which we derive a close form expression for the optimal fair predictor. Specifically, we show that the distribution of this optimum is the Wasserstein barycenter of the distributions induced by the standard regression function on the sensitive groups. This result offers an intuitive interpretation of the optimal fair prediction and suggests a simple post-processing algorithm to achieve fairness. We establish risk and distribution-free fairness guarantees for this procedure. Numerical experiments indicate that our method is very effective in learning fair models, with a relative increase in error rate that is inferior to the relative gain in fairness.'
author:
- |
Evgenii Chzhen$^{1}$, Christophe Denis$^2$, Mohamed Hebiri$^2$, [**Luca Oneto$^3$, and Massimiliano Pontil$^{4,5}$**]{}\
[$^1$LMO, Université Paris-Saclay, CNRS, Inria, $^2$LAMA, Université Gustave Eiffel,]{}\
[$^3$DIBRIS, University of Genoa, $^4$Istituto Italiano di Tecnologia, $^5$University College London]{}
- |
Evgenii Chzhen$^{1}$, Christophe Denis$^2$, Mohamed Hebiri$^2$, [**Luca Oneto$^3$, and Massimiliano Pontil$^{4,5}$**]{}\
[$^1$LMO, Université Paris-Saclay, CNRS, Inria, $^2$LAMA, Université Gustave Eiffel, $^3$DIBRIS, University of Genoa,]{}\
[$^4$Istituto Italiano di Tecnologia, $^5$University College London]{}
bibliography:
- 'biblio.bib'
title: Fair Regression with Wasserstein Barycenters
---
Introduction {#sec:1}
============
A central goal of algorithmic fairness is to ensure that sensitive information does not “unfairly” influence the outcomes of learning algorithms. For example, if we wish to predict the salary of an applicant or the grade of a university student, we would like the algorithm to not unfairly use additional sensitive information such as gender or race. Since today’s real-life datasets often contain discriminatory bias, standard machine learning methods behave unfairly. Therefore, a substantial effort is being devoted in the field to designing methods that satisfy “fairness” requirements, while still optimizing prediction performance, see for example [@barocas-hardt-narayanan; @calmon2017optimized; @chierichetti2017fair; @donini2018empirical; @dwork2018decoupled; @hardt2016equality; @jabbari2016fair; @joseph2016fairness; @kilbertus2017avoiding; @kusner2017counterfactual; @lum2016statistical; @yao2017beyond; @zafar2017fairness; @zemel2013learning; @zliobaite2015relation] and references therein. In this paper we study the problem of learning a real-valued regression function which among those complying with the Demographic Parity fairness constraint, minimizes the mean squared error. Demographic Parity requires the probability distribution of the predicted output to be independent of the sensitive attribute and has been used extensively in the literature, both in the context of classification and regression [@agarwal2019fair; @chiappa2020general; @gordaliza2019obtaining; @jiang2019wasserstein; @oneto2019general]. In this paper we consider the case that the sensitive attribute is available for prediction. Our principal result is to show that the distribution of the optimal fair predictor is the solution of a Wasserstein barycenter problem between the distributions induced by the unfair regression function on the sensitive groups. This result builds a bridge between fair regression and optimal transport, [see [[*e.g., *]{}]{} @villani2003topics; @santambrogio2015optimal].
We illustrate our result with an example. Assume that $X$ represents a , $S$ is a binary attribute representing two groups of the population ([[*e.g., *]{}]{} or ), and $Y$ is the . Let $f^*(x,s)= {\mathbb{E}}[Y | X {=} x, S {=} s]$ be the regression function, that is, the optimal prediction of the salary currently in the market for candidate $(x,s)$. Due to bias present in the underlying data distribution, the induced distribution of market salary predicted by $f^*$ varies across the two groups. We show that the optimal fair prediction $g^*$ transforms the regression function $f^*$ as $$\begin{aligned}
g^*(x, s) = p_s f^*(x,s) + (1-p_s) t^*(x,s) \enspace ,\end{aligned}$$ where $p_s$ is the frequency of group $s$ and the correction $t^*(x,s)$ is determined so that the *ranking* of $f^*(x,s)$ relative to the distribution of $X|S=s$ for group $s$ ([[*e.g., *]{}]{}minority) is the same as the [ranking]{} of $t^*(x,s)$ relative to the distribution of the group $s' \neq s$ ([[*e.g., *]{}]{}majority). We elaborate on this example after Theorem \[thm:Oracle\] and in Figure \[fig:illustration\]. The above expression of the optimal fair predictor naturally suggests a simple post-processing estimation procedure, where we first estimate $f^*$ and then transform it to get an estimator of $g^*$. Importantly, the transformation step involves only unlabeled data since it requires estimation of cumulative distribution functions.
[**Contributions and organization.**]{} In summary we make the following contributions. First, in Section \[sec:2\] we derive the expression for the optimal function which minimizes the squared risk under Demographic Parity constraints (Theorem \[thm:Oracle\]). This result establishes a connection between fair regression and the problem of Wasserstein barycenters, which allows to develop an intuitive interpretation of the optimal fair predictor. Second, based on the above result, in Section \[sec:estimator\_and\_proofs\] we propose a post-processing procedure that can be applied on top of any off-the-shelf estimator for the regression function, in order to transform it into a fair one. Third, in Section \[sec:statistical\_analysis\] we show that this post-processing procedure yields a fair prediction independently from the base estimator and the underlying distribution (Proposition \[prop:distribution\_free\_fairness\]). Moreover, finite sample risk guarantees are derived under additional assumptions on the data distribution provided that the base estimator is accurate (Theorem \[thm:rate\]). Finally, Section \[sec:exp\] presents a numerical comparison of the proposed method [[*w.r.t. *]{}]{}the state-of-the-art. [**Related work.**]{} Unlike the case of fair classification, fair regression has received limited attention to date; we are only aware of few works on this topic that are supported by learning bounds or consistency results for the proposed estimator [@agarwal2019fair; @oneto2019general]. Connections between algorithmic fairness and Optimal Transport, and in particular the problem of Wasserstein barycenters, has been studied in [@chiappa2020general; @gordaliza2019obtaining; @jiang2019wasserstein; @wang2019repairing] but mainly in the context of classification. These works are distinct from ours, in that they do not show the link between the optimal fair regression function and Wasserstein barycenters. Moreover, learning bounds are not addressed therein. Our distribution-free fairness guarantees share similarities with contributions on prediction sets [@lei2013distribution; @lei2014distribution] and conformal prediction literature [@vovk2005algorithmic; @zeni2020conformal] as they also rely on results on rank statistics. Meanwhile, the risk guarantee that we derive, combines deviation results on Wasserstein distances in one dimension [@Bobkov_Ledoux16] with peeling ideas developed in [@Audibert_Tsybakov07], and classical theory of rank statistics [@van2000asymptotic].
[**Notation.**]{} For any positive integer $N \in {\mathbb{N}}$ we denote by $[N]$ the set $\{1, \ldots, N\}$. For $a, b \in {\mathbb{R}}$ we denote by $a \wedge b$ (*resp.* $a \vee b$) the minimum (*resp.* the maximum) between $a$ and $b$. For two positive real sequences $a_n, b_n$ we write $a_n \lesssim b_n$ to indicate that there exists a constant $c$ such that $a_n \leq c b_n$ for all $n$. For a finite set ${\mathcal{S}}$ we denote by $|{\mathcal{S}}|$ its cardinality. The symbols ${\mathbf{E}}$ and ${\mathbf{P}}$ stand for generic expectation and probability. For any univariate probability measure $\mu$, we denote by $F_{\mu}$ its Cumulative Distribution Function (CDF) and by $Q_{\mu}: [0, 1] \to {\mathbb{R}}$ its quantile function ([*a.k.a.*]{} generalized inverse of $F_{\mu}$) defined for all $t \in (0, 1]$ as $Q_{\mu}(t) = \inf{\left\{ y \in {\mathbb{R}}\, : \, F_{\mu}(y) {\geq} t\right\}}$ with $Q_{\mu}(0) = Q_{\mu}(0+)$. For a measurable set $A \subset {\mathbb{R}}$ we denote by $U(A)$ the uniform distribution on $A$.
The problem {#sec:2}
===========
In this section we introduce the fair regression problem and present our derivation for the optimal fair regression function alongside its connection to Wasserstein barycenter problem. We consider the general regression model $$Y = f^*(X, S) + \xi\enspace,
\label{eq:model}$$ where $\xi \in {\mathbb{R}}$ is a centered random variable, $(X, S) \sim {\mathbb{P}}_{X, S}$ on ${\mathbb{R}}^d \times {\mathcal{S}}$, with ${\left\lvert {\mathcal{S}}\right\rvert} < \infty$, and $f^*: {\mathbb{R}}^d \times {\mathcal{S}} \to {\mathbb{R}}$ is the regression function minimizing the squared risk. [Let ${\mathbb{P}}$ be the joint distribution of $(X, S, Y)$.]{} For any prediction rule $f : {\mathbb{R}}^d \times {\mathcal{S}} \to {\mathbb{R}}$, we denote by $\nu_{f | s} $ the distribution of $f(X, S) | S = s$, that is, the Cumulative Distribution Function (CDF) of $\nu_{f | s}$ is given by $$\begin{aligned}
\label{eq:push_forward}
F_{\nu_{f | s}}(t) = {\mathbb{P}}(f(X, S) \leq t | S = s)\enspace,\end{aligned}$$ to shorten the notation we will write $F_{f | s}$ and $Q_{f | s}$ instead of $F_{\nu_{f | s}}$ and $Q_{\nu_{f | s}}$ respectively.
[definition]{}[wasser]{} \[def:wass\] Let $\mu$ and $\nu$ be two univariate probability measures. The squared Wasserstein-2 distance between $\mu$ and $\nu$ is defined as $$\begin{aligned}
{\mathcal{W}}_2^2(\mu,\nu) = \inf_{\gamma \in \Gamma_{\mu, \nu}}\int {\left\lvert x - y\right\rvert}^2 d\gamma(x, y)\enspace,
\end{aligned}$$ where $\Gamma_{\mu, \nu}$ is the set of distributions (couplings) on ${\mathbb{R}}\times {\mathbb{R}}$ such that for all $\gamma \in \Gamma_{\mu, \nu}$ and all measurable [sets]{} $A, B \subset {\mathbb{R}}$ it holds that $\gamma(A \times {\mathbb{R}}) = \mu(A)$ and $\gamma({\mathbb{R}}\times B) = \nu(B)$.
In this work we use the following definition of (strong) Demographic Parity, which was previously used in the context of regression by [@agarwal2019fair; @chiappa2020general; @jiang2019wasserstein].
\[def:demographic\_parity\] A prediction (possibly randomized) $g: {\mathbb{R}}^d \times {\mathcal{S}} \to {\mathbb{R}}$ is *fair* if, for every $s,s' \in {\mathcal{S}}$ $$\begin{aligned}
\sup_{t \in {\mathbb{R}}}\Big|{\mathbf{P}}(g(X, S) \leq t | S = s) - {\mathbf{P}}(g(X, S) \leq t | S = s')\Big| = 0\enspace.
\end{aligned}$$
Demographic Parity requires the Kolmogorov-Smirnov distance between $\nu_{g | s}$ and $\nu_{g | s'}$ to vanish for all $s, s'$. Thus, if $g$ is fair, $\nu_{g | s}$ does not depend on $s$ and to simplify the notation we will write $\nu_{g}$. [Recall the model in Eq. . Since the noise has zero mean, the minimization of ${\mathbb{E}}(Y - g(X, S))^2$ over $g$ is equivalent to the minimization of ${\mathbb{E}}(f^*(X, S) - g(X, S))^2$ over $g$.]{}
![[For a new point $(x, 1)$, the value $t^*(x,1)$ is chosen such that the shaded `Green Area (//)` $= {\mathbb{P}}(f^*(X, S) \leq t^*(x, 1) | S = 2)$ equals to the shaded `Blue Area (\\)` $= {\mathbb{P}}(f^*(X, S) \leq f^*(x, 1) | S = 1)$. The final prediction $g^*(x, 1)$ is a convex combination of $f^*(x, 1)$ and $t^*(x, 1)$. The same is done for $(\bar x, 2)$]{}.[]{data-label="fig:illustration"}](illustration2.pdf){height="32.00000%"}
The next theorem shows that the optimal fair predictor for an input $(x,s)$ is obtained by a nonlinear transformation of the vector $(f^*(x, s))_{s=1}^{|{\mathcal{S}}|}$ that is linked to a Wasserstein barycenter problem [@agueh2011barycenters].
[theorem]{}[oracle]{} \[thm:Oracle\] Assume, for each $s \in {\mathcal{S}}$, that the univariate measure $\nu_{f^* | s}$ has a density and let $p_s = {\mathbb{P}}(S = s)$. Then, $$\begin{aligned}
\min_{g~\emph{is~fair}}{\mathbb{E}}(f^*(X, S) - g(X, S))^2 = \min_{\nu} \sum_{s \in {\mathcal{S}}}p_s{\mathcal{W}}_2^2(\nu_{f^*|s}, \nu)\enspace.\end{aligned}$$ Moreover, if $g^*$ and $\nu^*$ solve the l.h.s. and the r.h.s. problems respectively, then $\nu^* = \nu_{g^*}$ and $$\begin{aligned}
\label{eq:oracle_formula}
g^*(x, s)
= {\left( \sum_{s' \in {\mathcal{S}}}p_{s'}Q_{f^* | s'} \right)} \circ F_{f^* | s} \left( f^*(x, s) \right)\enspace.\end{aligned}$$
After the completion of this work, we became aware of a similar result derived independently by [@gouic2020price]. Their result applies to a general vector-valued regression problems, under assumptions that are similar to ours. They have proposed an estimator of the optimal fair prediction which leverages the connection of the fair regression and the problem of Wasserstein barycenters and derived asymptotic risk consistency. In contrast, we focus on the one dimensional case, and we provide a plug-in type estimator for which we show distribution-free fairness guarantees and finite-sample bounds for the risk. The proof of Theorem \[thm:Oracle\] relies on the classical characterization of optimal coupling in one dimension (stated in Theorem \[thm:Brenier\] in the appendix) of the Wasserstein-2 distance. We show that a minimizer $g^*$ of the $L_2$-risk can be used to construct $\nu^*$ and vice-versa, given $\nu^*$, we leverage a well-known expression for one dimensional Wasserstein barycenter (see [[*e.g., *]{}]{}[@agueh2011barycenters Section 6.1] and Lemma \[lem:barycenter\_expression\] in the appendix) and construct $g^*$.
[**The case of binary protected attribute.**]{} Let us unpack Eq. in the case that ${\mathcal{S}} = \{1, 2\}$, assuming [*w.l.o.g.*]{} that $p_2 \geq p_1$. Theorem \[thm:Oracle\] states that the fair optimal prediction $g^*$ is defined for all individuals $x \in {\mathbb{R}}^d$ in the first group as $$\begin{aligned}
g^*(x, 1) &=
p_{1}f^*(x, 1) + p_{2} t^*(x, 1),\,\, \text{with}\,\,
t^*(x, 1) = \inf{\left\{ t \in {\mathbb{R}}\, : \, F_{f^* | 2}(t) \geq F_{f^* | 1}(f^*(x, 1))\right\}}\enspace,\end{aligned}$$ and likewise for the second group. This form of the optimal fair predictor, and more generally Eq. , allows us to [understand]{} the decision made by $g^*$ at individual level. If we interpret $(x, s)$ as the candidate’s CV and candidate’s group respectively, and $f^*(x, s)$ as the current market salary (which might be discriminatory), then the fair optimal salary $g^*(x, s)$ is a convex combination of the market salary $f^*(x, s)$ and the adjusted salary $t^*(x, s)$, which is computed as follows. [ If say $s{=}1$, we first compute the fraction of individuals from the first group whose market salary is at most $f^*(x, 1)$, that is, we compute ${\mathbb{P}}(f^*(X, S) \leq f^*(x, 1) | S {=} 1)$. Then, we find a candidate $\bar x$ in group $2$, such that the fraction of individuals from the second group whose market salary is at most $f^*(\bar{x}, 2)$ is the same, that is, $\bar{x}$ is chosen to satisfy ${\mathbb{P}}(f^*(X, S) \leq f^*(\bar x, 2) | S {=} 2) = {\mathbb{P}}(f^*(X, S) \leq f^*(x, 1) | S {=} 1)$. Finally, the market salary of $\bar{x}$ is exactly the adjustment for $x$, that is, $t^*(x, 1) = f^*(\bar{x}, 2)$. ]{} This idea is illustrated in Figure \[fig:illustration\] and leads to the following philosophy: [if candidates $(x, 1)$ and $(\bar x, 2)$ share the same group-wise market salary ranking, ]{} then [they]{} should receive the same salary determined by the fair prediction $g^*(x, 1) = g^*(\bar x, 2) = p_{1}f^*(x, 1) + p_{2}f^*(\bar x, 2)$. At last, note that Eq. allows to understand the (potential) amount of extra money that we need to pay in order to satisfy fairness. While the unfair decision made with $f^*$ costs $f^*(x, 1) {+} f^*(\bar x, 2)$ for the salary of $(x, 1)$ and $(\bar x, 2)$, the fair decision $g^*$ costs $2(p_1 f^*(x, 1) {+} p_2f^*(\bar{x}, 2))$. Thus, the extra (signed) salary that we pay is $\Delta = (p_2 {-} p_1)(f^*(\bar{x}, 2) {-} f^*(\bar{x}, 1))$. Since, $p_2 {\geq} p_1$, $\Delta$ will be positive whenever the candidate ${\bar x}$ from the majority group gets higher salary according [to]{} $f^*$, and negative otherwise. [We believe that the expression Eq. could be the starting point for further more applied work on algorithmic fairness. ]{}
General form of the estimator {#sec:estimator_and_proofs}
=============================
In this section we propose an estimator of the optimal fair predictor $g^*$ that relies on the plug-in principle. The expression of $g^*$ suggests that we only need estimators for the regression function $f^*$, the proportions $p_s$, as well as the [CDF]{} $F_{f^* | s}$ and the quantile function $Q_{f^* | s}$, for all $s\in {\mathcal{S}}$. While the estimation of $f^*$ needs labeled data, all the other quantities rely only on ${\mathbb{P}}_S$, ${\mathbb{P}}_{X | S}$ and $f^*$, therefore [*unlabeled*]{} data with an estimator of $f^*$ suffices. [Thus, given a base estimator of $f^*$, our post-processing algorithm will require only unlabeled data.]{}\
For each $s \in {\mathcal{S}}$ let ${\mathcal{U}}^s {=} \{X^{s}_i\}_{i = 1}^{N_s} {\overset{\text{{{\rm i.i.d.~}}}}{\scalebox{1.1}[1]{$\sim$}}}{\mathbb{P}}_{X | S = s}$ be a group-wise unlabeled sample. In the following for simplicity we assume that $N_s$ are *even* for all $s \in {\mathcal{S}}$[^1]. Let ${\mathcal{I}}_0^{s}, {\mathcal{I}}_1^{s} \subset [N_s]$ be any fixed partition of $[N_s]$ such that $|{\mathcal{I}}_0^{s}| {=} |{\mathcal{I}}_1^{s}| {=} N_s / 2$ and ${\mathcal{I}}_0^{s} \cup {\mathcal{I}}_1^{s} {=} [N_s]$. For each $j \in \{0, 1\}$ we let ${\mathcal{U}}^s_j {=} {\left\{ X^{s}_i \in {\mathcal{U}}^s \, : \, i \in {\mathcal{I}}^s_j\right\}}$ be the restriction of ${\mathcal{U}}^s$ to ${\mathcal{I}}^s_j$. [We use ${\mathcal{U}}^s_0$ to estimate ${Q}_{f | s}$ and ${\mathcal{U}}^s_1$ to estimate ${F}_{f | s}$.]{} For each $f : {\mathbb{R}}^d {\times} {\mathcal{S}} \to {\mathbb{R}}$ and each $s \in {\mathcal{S}}$, we estimate $\nu_{f | s}$ by $$\label{eq:empirical_jittered_cdf}
\hat{\nu}^0_{f | s} = \frac{1}{|{\mathcal{I}}_0^s|}\sum_{i \in {\mathcal{I}}_0^s}\delta\big({f(X^{s}_i, s) + \varepsilon_{is}} - \cdot\big)\quad\text{and}\quad\hat{\nu}^1_{f | s} = \frac{1}{|{\mathcal{I}}_1^s|}\sum_{i \in {\mathcal{I}}_1^s}\delta\big({f(X^{s}_i, s) + \varepsilon_{is}} - \cdot\big)\enspace,$$
where $\delta$ is the Dirac measure and all $\varepsilon_{is} {\overset{\text{{{\rm i.i.d.~}}}}{\scalebox{1.1}[1]{$\sim$}}}U([-\sigma, \sigma])$, for some positive $\sigma$ set by the user. Using the estimators in Eq. , we define for all $f : {\mathbb{R}}^d {\times} {\mathcal{S}} \to {\mathbb{R}}$ estimators of $Q_{f | s}$ and of $F_{f | s}$ as $$\begin{aligned}
\label{eq:empirical_jittered_cdf1}
\hat{Q}_{f | s} \equiv Q_{\hat{\nu}^0_{f | s}}\quad\text{and}\quad\hat{F}_{f | s} \equiv {F}_{\hat{\nu}^1_{f | s}}\enspace.\end{aligned}$$ That is, $\hat{F}_{f | s}$ and $\hat{Q}_{f | s}$ are the empirical CDF and empirical quantiles of $(f(X, S) {+} \varepsilon) | S {=} s$ based on $\{f(X^{s}_i, s) {+} \varepsilon_{is}\}_{i \in {\mathcal{I}}_1^s}$ and $\{f(X^{s}_i, s) {+} \varepsilon_{is}\}_{i \in {\mathcal{I}}_0^s}$ respectively. The noise $\varepsilon_{is}$ serves as a smoothing random variable, since for all $s \in {\mathcal{S}}$ and $i \in [N_s]$ the random variables $f(X^{s}_i, s) {+} \varepsilon_{is}$ are [[i.i.d. ]{}]{}continuous for any ${\mathbb{P}}$ and $f$. In contrast, $f(X^{s}_i, s)$ might have atoms resulting in a non-zero probability to observe ties in $\{f(X^{s}_i, s)\}_{i \in {\mathcal{I}}_j^s}$. This step is also known as *jittering*, often used for data visualization [@chambers2018graphical] for tie-breaking. It plays a crucial role in the distribution-free fairness guarantees that we derive in Proposition \[prop:distribution\_free\_fairness\]; see the discussion thereafter.
()[ $s' \in {\mathcal{S}}$]{}
${\mathcal{U}}_0^{s'},{\mathcal{U}}_1^{s'} \leftarrow \mathtt{split\_in\_two}({\mathcal{U}}^{s'})$ $\texttt{ar}_0^{s'} \leftarrow \big\{{\hat f(X, {s'}) {+} U([-\sigma,
\sigma])}\big\}_{X \in {\mathcal{U}}_{0}^{s'}}$, $\texttt{ar}_1^{s'} \leftarrow \big\{\hat f(X, {s'}) {+} U([-\sigma, \sigma])\big\}_{X \in {\mathcal{U}}_{1}^{s'}}$
$\texttt{ar}_0^{s'} \leftarrow \mathtt{sort}\big(\texttt{ar}_0^{s'}\big)$,$\texttt{ar}_1^{s'} \leftarrow \mathtt{sort}\big(\texttt{ar}_1^{s'}\big)$
[$k_{s} \leftarrow \mathtt{position}{\left( \hat f( x, s) {+} U([-\sigma, \sigma]),\,\, \texttt{ar}_1^{s} \right)}$ ]{} $\hat g(x, s) \leftarrow \sum_{s' \in {\mathcal{S}}}{\hat{p}}_{s'} \times \texttt{ar}_0^{s'}\big[\ceil{N_{s'} k_{s}/{N_{ s}}}\big]$
Finally, let ${\mathcal{A}} {=} \{S_i\}_{i = 1}^N {\overset{\text{{{\rm i.i.d.~}}}}{\scalebox{1.1}[1]{$\sim$}}}{\mathbb{P}}_S$ and for each $s \in {\mathcal{S}}$ let $\hat p_s$ be the empirical frequency of $S {=} s$ evaluated on ${\mathcal{A}}$. Given a base estimator ${\hat{f}}$ of $f^*$ constructed from $n$ labeled samples ${\mathcal{L}} {=} \{(X_i, S_i, Y_i)\}_{i = 1}^{n} {\overset{\text{{{\rm i.i.d.~}}}}{\scalebox{1.1}[1]{$\sim$}}}{\mathbb{P}}$, we define the final estimator $\hat g$ of $g^*$ for all $(x, s) \in {\mathbb{R}}^d {\times} {\mathcal{S}}$ mimicking Eq. as $$\begin{aligned}
\label{eq:proposed_estimator}
\hat g(x, s) = {\left( \sum_{s' \in {\mathcal{S}}}\hat p_{s'} \hat{Q}_{{\hat{f}}| s'} \right)} \circ\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(x, s) + \varepsilon\right) \enspace,\end{aligned}$$ where $\varepsilon \sim U([-\sigma, \sigma])$ is assumed to be independent from every other random variables.
In practice one should use a very small value for $
\sigma$ ([[*e.g., *]{}]{}$\sigma {=} 10^{-5}$), which does not alter the statistical quality of the base estimator $\hat f$ as indicated in Theorem \[thm:rate\].
A pseudo-code implementation of $\hat g$ in Eq. is reported in Algorithm \[alg:preprocessing\]. It requires two primitives: $\mathtt{sort}(\texttt{ar})$ sorts the array $\texttt{ar}$ in an increasing order; $\mathtt{position}(a, \texttt{ar})$ which outputs the index $k$ such that the insertion of $a$ into $k$’th position in $\texttt{ar}$ preserves ordering ([[*i.e., *]{}]{}$\texttt{ar}[k{-}1] \leq a < \texttt{ar}[k]$). Algorithm \[alg:preprocessing\] consists of two for parts: in the for-loop we perform a preprocessing which takes $\sum_{s \in {\mathcal{S}}}O(N_s \log N_s)$ time[^2] [since it involves sorting]{}; then, the evaluation of $\hat g$ on a new point $(x, s)$ is performed in $(\max_{s\in {\mathcal{S}}}\log N_s)$ time [since it involves an element search in a sorted array]{}. Note that the for-loop of Algorithm \[alg:preprocessing\] needs to be performed only once as this step is shared for any new $(x, s)$.
Statistical analysis {#sec:statistical_analysis}
====================
In this section we provide a statistical analysis of the proposed algorithm. We first present in Proposition \[prop:distribution\_free\_fairness\] distribution-free finite sample fairness guarantees for post-processing of *any* base learner with unlabeled data and then we show in Theorem \[thm:rate\] that if the base estimator $\hat f$ is a good proxy for $f^*$, then under mild assumptions on the distribution ${\mathbb{P}}$, the processed estimator $\hat g$ in Eq. is a good estimator of $g^*$ in Eq. .
#### Distribution free post-processing fairness guarantees.
We derive two distribution-free results in Proposition \[prop:distribution\_free\_fairness\], the first in Eq. shows that the fairness definition is satisfied as long as we take the expectation over the data inside the supremum in Definition \[def:demographic\_parity\], while the second one in Eq. bounds the expected violation of Definition \[def:demographic\_parity\].
[proposition]{}[fairness]{} \[prop:distribution\_free\_fairness\] For any joint distribution ${\mathbb{P}}$ of $(X, S, Y)$, any base estimator $\hat f$ constructed on labeled data, and for all $s, s' \in {\mathcal{S}}$, the estimator $\hat g$ defined in Eq. satisfies $$\begin{aligned}
&\sup_{t\in{\mathbb{R}}}{\left\lvert {\mathbf{P}}(\hat g(X, S) \leq t | S {=} s) - {\mathbf{P}}(\hat g(X, S) \leq t | S {=} s')\right\rvert} \leq 2\left(N_s {\wedge} N_{s'} + 2\right)^{-1}{{\bf 1}_{\left\{N_s \neq N_{s'}\right\}}}\label{eq:fair1}\\\
&{\mathbf{E}}\sup_{t \in {\mathbb{R}}}{\left\lvert {\mathbf{P}}(\hat g(X, S) \leq t |S {=} s ,{{\mathcal{D}}}) - {\mathbf{P}}(\hat g(X, S) \leq t |S {=} s', {{\mathcal{D}}})\right\rvert} \leq 6 \left(N_s {\wedge} N_{s'} + 1\right)^{-1/2} \enspace.\label{eq:fair2}
\end{aligned}$$ where ${{\mathcal{D}}}= {\mathcal{L}} \cup {\mathcal{A}} \cup_{s \in {\mathcal{S}}} {\mathcal{U}}^s$ is the union of all available datasets.
[ [Let us point out that this result does not require any assumption on the distribution ${\mathbb{P}}$ as well as on the base estimator ${\hat{f}}$. This is achieved thanks to the jittering step in the definition of $\hat g$ in Eq. , which artificially introduces continuity. Continuity allows us to use results from the theory of rank statistics of exchangeable random variables to derive Eq. as well as the classical inverse transform (see [[*e.g., *]{}]{} [@van2000asymptotic Sections 13 and 21]) combined with the Dvoretzky-Kiefer-Wolfowitz inequality [@massart1990tight] to derive Eq. .]{} Since basic results on rank statistics and inverse transform are distribution-free as long as the underlying random variable is continuous, the guarantees in Eqs. – are also distribution-free and can be applied on top of *any* base estimator ${\hat{f}}$.\
[The bound in Eq. might be surprising to the reader. ]{} Yet, let us emphasize that this bound holds because the expectation [[*w.r.t. *]{}]{}the data distribution is taken inside the supremum (since ${\mathbf{P}}$ stands for the joint distribution of *all* random variables involved in $\hat g(X, S)$). Similar proof techniques are also used in randomization inference via permutations [@fisher1936design; @hoeffding1952large], conformal prediction [@vovk2005algorithmic; @lei2014distribution], knockoff estimation [@barber2019knockoff] to name a few. [However, unlike the aforementioned contributions, the problem of fairness requires a non-trivial adaptation of these techniques.]{} In contrast, Eq. might be more appealing to the machine learning community as it controls the expected (over data) violation of the fairness constraint with standard parametric rate. ]{}
#### Estimation guarantee with accurate base estimator.
In order to prove non-asymptotic risk bounds we require the following assumption on the distribution ${\mathbb{P}}$ of [$(X, S, Y) \in {\mathbb{R}}^d \times {\mathcal{S}} \times {\mathbb{R}}$.]{}
[assumption]{}[density]{} \[ass:density\_rate\] For each $s \in {\mathcal{S}}$ the univariate measure $\nu_{f^*|s}$ admits a density $q_s$, which is lower bounded by $ \underline{\lambda}_s > 0$ and upper-bounded by $\overline{\lambda}_s \geq \underline{\lambda}_s$.
[Although the lower bound on the density assumption is rather strong and might potentially be violated in practice, it is still reasonable in certain situations. We believe that it can be replaced by the assumption that $f^*(X, S)$ conditionally on $S {=} s$ for all $s {\in} {\mathcal{S}}$ admits $2 {+} \epsilon$ moments. We do not explore this relaxation in our work as it significantly complicates the proof of Theorem \[thm:rate\]. [At the same time, our empirical study suggests that the lower bound on the density is not intrinsic to the problem, since the estimator exhibits a good performance across various scenarios.]{} In contrast, the milder assumption that the density is upper bounded is crucial for our proof and [seems to be necessary.]{} ]{}
Apart from the assumption on the density of $\nu_{f^*|s}$, the actual rate of estimation depends on the quality of the base estimator $\hat f$. We require the following assumption, which states that ${\hat{f}}$ approximates $f^*$ point-wise with rate $b_n^{-{1}/{2}}$ [and a standard sub-Gaussian concentration for ${\hat{f}}$ can be derived]{}.
[assumption]{}[concentration]{} \[ass:concentration\_estimator\] There exist positive constants $c$ and $C$ independent from $n$, $N$, $N_1, \ldots, N_{|{\mathcal{S}}|}$, and a positive sequence $b_n : {\mathbb{N}}\to {\mathbb{R}}_+$ such that for all $\delta > 0$ it holds that $$\begin{aligned}
{\mathbf{P}}\left(|f^*(x, s) - \hat{f}(x, s)| \geq \delta\right) \leq c \exp\left(-Cb_{n}\delta^2\right) \text{ for almost all $(x, s)$ {{\em w.r.t.~}}${\mathbb{P}}_{X, S}$}\enspace.
\end{aligned}$$
We refer to [@Audibert_Tsybakov07; @Sara08; @Lei14; @Devroye78; @lei2014distribution] for various examples of estimators and additional assumptions such that the bound in Assumption \[ass:concentration\_estimator\] is satisfied. It includes local polynomial estimators, k-nearest neighbours, and linear regression, to name just a few.
Under these assumptions we can prove the following finite-sample estimation bound.
[theorem]{}[riskbound]{} \[thm:rate\] Let Assumptions \[ass:density\_rate\] and \[ass:concentration\_estimator\] be satisfied, and set $\sigma \lesssim \min_{s \in {\mathcal{S}}}N_s^{-1/2} {\wedge} b_{n}^{-1/2}$, then the estimator $\hat g$ defined in Eq. satisfies $$\begin{aligned}
{\mathbf{E}}{\left\lvert g^*(X, S) - \hat g(X, S)\right\rvert} \lesssim b_{n}^{-1/2} \bigvee \left(\sum_{s \in {\mathcal{S}}} p_s N_{s}^{-1/2}\right) \bigvee \sqrt{ \frac{|{\mathcal{S}}|}{N}}\enspace,
\end{aligned}$$ where the leading constant depends only on $\underline{\lambda}_s, \overline{\lambda}_s, C, c$ from Assumptions \[ass:density\_rate\] and \[ass:concentration\_estimator\].
The proof of this result combines expected deviation of empirical measure from the real measure in terms of Wasserstein distance on real line [@Bobkov_Ledoux16] with the already mentioned rank statistics and classical peeling argument of [@Audibert_Tsybakov07].\
The first term of the derived bound corresponds to the estimation error of $f^*$ by $\hat f$, the second term is the price to pay for not knowing conditional distributions $X | S = s$ while the last term correspond to the price of unknown marginal probabilities of each protected attribute. Notice that if $N_s = p_s N$, which corresponds to the standard [[i.i.d. ]{}]{}sampling from ${\mathbb{P}}_{X, S}$ of unlabeled data, the second and the third term are of the same order.
Moreover, if $N$ is sufficiently large, which in most scenarios[^3] is *w.l.o.g.*, then the rate is dominated by $b_n^{-1/2}$. [Notice that one can find a collection of joint distributions ${\mathbb{P}}$, such that $f^*$ satisfies demographic parity. Hence, if $b_n^{-1/2}$ is the minimax optimal estimation rate of $f^*$, then it is also optimal for $g^* \equiv f^*$.]{}
Empirical study {#sec:exp}
===============
In this section, we present numerical experiments[^4] with the proposed fair regression estimator defined in Section \[sec:estimator\_and\_proofs\]. In all experiments, we collect statistics on the test set ${\mathcal{T}} = \{(X_i, S_i, Y_i)\}_{i = 1}^{n_{\text{test}}}$. The empirical mean squared error (${\text{MSE}}$) is defined as $$\begin{aligned}
{\text{MSE}}\,(g) = \frac{1}{n_{\text{test}}}\sum_{(X, S, Y) \in {\mathcal{T}}} (Y - g(X, S))^2\enspace.\end{aligned}$$ We also measure the violation of fairness constraint imposed by Definition \[def:demographic\_parity\] [via]{} the empirical Kolmogorov-Smirnov (KS) distance, $$\begin{aligned}
{\text{KS}}\,(g) = \max_{s, s' \in {\mathcal{S}}}\sup_{t \in {\mathbb{R}}}\bigg|\frac{1}{|{{\mathcal{T}}}^{s}|}\sum_{(X, S, Y) \in {\mathcal{T}}^s}{{\bf 1}_{\left\{g(X, S) \leq t\right\}}} - \frac{1}{|{\mathcal{T}}^{s'}|}\sum_{(X, S, Y) \in {\mathcal{T}}^{s'}}{{\bf 1}_{\left\{g(X, S) \leq t\right\}}}\bigg| \enspace,\end{aligned}$$ where for all $s{\in}{\mathcal{S}}$ we define the set ${\mathcal{T}}^s {=} {\left\{ (X, S, Y) \in {\mathcal{T}} \, : \, S {=} s\right\}}$. For all datasets we split the data in two parts (70% train and 30% test), this procedure is repeated 30 times, and we report the average performance on the test set alongside its standard deviation. We employ the 2-steps 10-fold CV procedure considered by [@donini2018empirical] to select the best hyperparameters with the training set. In the first step, we shortlist all the hyperparameters with ${\text{MSE}}$ close to the best one (in our case, the hyperparameters which lead to 10% larger ${\text{MSE}}$ [[*w.r.t. *]{}]{}the best ${\text{MSE}}$). Then, from this list, we select the hyperparameters with the lowest ${\text{KS}}$.
[**Methods.**]{} We compare our method (see Section \[sec:estimator\_and\_proofs\]) to different fair regression approaches for both linear and non-linear regression. In the case of linear models we consider the following methods: Linear RLS plus [@berk2017convex] (RLS+Berk), Linear RLS plus [@oneto2019general] (RLS+Oneto), and Linear RLS plus Our Method (RLS+Ours), where RLS is the abbreviation of Regularized Least Squares.\
In the case of non-linear models we compare to the following methods. [i) For]{} Kernel RLS (KRLS): KRLS plus [@oneto2019general] (KRLS+Oneto), KRLS plus [@perez2017fair] (KRLS+Perez), KRLS plus Our Method (KRLS+Ours); [ii) For]{} Random Forests (RF): RF plus [@raff2017fair] (RF+Raff), RF plus [@agarwal2019fair][^5] (RF+Agar), and RF plus Our Method (RF+Ours).\
The hyperparameters of the methods are set as follows. For RLS we set the regularization hyperparameters $\lambda \in 10^{\{-4.5,-3.5,\cdots,3\}}$ and for KRLS we set $\lambda \in 10^{\{-4.5,-3.5,\cdots,3\}}$ and $\gamma \in 10^{\{-4.5,-3.5,\cdots,3\}}$. Finally, for RF we set to $1000$ the number of trees and for the number of features to select during the tree creation we search in $\{ d^{\sfrac{1}{4}}, d^{\sfrac{1}{2}}, d^{\sfrac{3}{4}} \}$.
[**Datasets.**]{} In order to analyze the performance of our methods and test it against the state-of-the-art alternatives, we consider five benchmark datasets, CRIME, LAW, NLSY, STUD, and UNIV, which are briefly described below:\
[*Communities&Crime (CRIME)*]{} contains socio-economic, law enforcement, and crime data about communities in the US [@redmond2002data] with $1994$ examples. The task is to predict the number of violent crimes per $10^5$ population (normalized to $[0, 1]$) with race as the protected attribute. Following [@calders2013controlling], we made a binary sensitive attribute $s$ as to the percentage of black population, which yielded $970$ instances of $s{=}1$ with a mean crime rate $0.35$ and $1024$ instances of $s{=}{-}1$ with a mean crime rate $0.13$.\
[*Law School (LAW)*]{} refers to the Law School Admissions Councils National Longitudinal Bar Passage Study [@wightman1998lsac] and has $20649$ examples. The task is to predict a students GPA (normalized to $[0, 1]$) with race as the protected attribute (white versus non-white).\
[*National Longitudinal Survey of Youth (NLSY)*]{} involves survey results by the U.S. Bureau of Labor Statistics that is intended to gather information on the labor market activities and other life events of several groups [@NLSY:2019]. Analogously to [@komiyama2018comparing] we model a virtual company’s hiring decision assuming that the company does not have access to the applicants’ academic scores. We set as target the person’s GPA (normalized to $[0, 1]$), with race as sensitive attribute\
[*Student Performance (STUD)*]{}, approaches $649$ students achievement (final grade) in secondary education of two Portuguese schools using $33$ attributes [@cortez2008using], with gender as the protected attribute.\
[*University *Anonymous* (UNIV)*]{} is a proprietary and highly sensitive dataset containing all the data about the past and present students enrolled at the University of *Anonymous*. In this study we take into consideration students who enrolled, in the academic year 2017-2018. The dataset contains 5000 instances, each one described by 35 attributes (both numeric and categorical) about ethnicity, gender, financial status, and previous school experience. The scope is to predict the average grades at the end of the first semester, with gender as the protected attribute. [**Comparison w.r.t. state-of-the-art.**]{} \[sec:exp:comparison\]
------------ --------------------- ------------------- --------------------- ------------------- --------------------- ------------------- -------------------- ------------------- -------------------- -------------------
Method ${\text{MSE}}$ ${\text{KS}}$ ${\text{MSE}}$ ${\text{KS}}$ ${\text{MSE}}$ ${\text{KS}}$ ${\text{MSE}}$ ${\text{KS}}$ ${\text{MSE}}$ ${\text{KS}}$
RLS $ .033 {\pm} .003 $ $ .55 {\pm} .06 $ $ .107 {\pm} .010 $ $ .15 {\pm} .02 $ $ .153 {\pm} .016 $ $ .73 {\pm} .07 $ $ 4.77 {\pm} .49 $ $ .50 {\pm} .05 $ $ 2.24 {\pm} .22 $ $ .14 {\pm} .01 $
RLS+Berk $ .037 {\pm} .004 $ $ .16 {\pm} .02 $ $ .121 {\pm} .013 $ $ .10 {\pm} .01 $ $ .189 {\pm} .019 $ $ .49 {\pm} .05 $ $ 5.28 {\pm} .57 $ $ .32 {\pm} .03 $ $ 2.43 {\pm} .23 $ $ .05 {\pm} .01 $
RLS+Oneto $ .037 {\pm} .004 $ $ .14 {\pm} .01 $ $ .112 {\pm} .012 $ $ .07 {\pm} .01 $ $ .156 {\pm} .016 $ $ .50 {\pm} .05 $ $ 5.02 {\pm} .54 $ $ .23 {\pm} .02 $ $ 2.44 {\pm} .26 $ $ .05 {\pm} .01 $
RLS+Ours $ .041 {\pm} .004 $ $ .12 {\pm} .01 $ $ .141 {\pm} .014 $ $ .02 {\pm} .01 $ $ .203 {\pm} .019 $ $ .09 {\pm} .01 $ $ 5.62 {\pm} .52 $ $ .04 {\pm} .01 $ $ 2.98 {\pm} .32 $ $ .02 {\pm} .01 $
KRLS $ .024 {\pm} .003 $ $ .52 {\pm} .05 $ $ .040 {\pm} .004 $ $ .09 {\pm} .01 $ $ .061 {\pm} .006 $ $ .58 {\pm} .06 $ $ 3.85 {\pm} .36 $ $ .47 {\pm} .05 $ $ 1.43 {\pm} .15 $ $ .10 {\pm} .01 $
KRLS+Oneto $ .028 {\pm} .003 $ $ .19 {\pm} .02 $ $ .046 {\pm} .004 $ $ .05 {\pm} .01 $ $ .066 {\pm} .007 $ $ .06 {\pm} .01 $ $ 4.07 {\pm} .39 $ $ .18 {\pm} .02 $ $ 1.46 {\pm} .13 $ $ .04 {\pm} .01 $
KRLS+Perez $ .033 {\pm} .003 $ $ .25 {\pm} .02 $ $ .048 {\pm} .005 $ $ .04 {\pm} .01 $ $ .065 {\pm} .007 $ $ .08 {\pm} .01 $ $ 3.97 {\pm} .38 $ $ .14 {\pm} .02 $ $ 1.50 {\pm} .15 $ $ .06 {\pm} .01 $
KRLS+Ours $ .034 {\pm} .004 $ $ .09 {\pm} .01 $ $ .056 {\pm} .005 $ $ .01 {\pm} .01 $ $ .081 {\pm} .008 $ $ .03 {\pm} .01 $ $ 4.46 {\pm} .43 $ $ .03 {\pm} .01 $ $ 1.71 {\pm} .16 $ $ .02 {\pm} .01 $
RF $ .020 {\pm} .002 $ $ .45 {\pm} .04 $ $ .046 {\pm} .005 $ $ .11 {\pm} .01 $ $ .055 {\pm} .006 $ $ .55 {\pm} .06 $ $ 3.59 {\pm} .39 $ $ .45 {\pm} .05 $ $ 1.31 {\pm} .13 $ $ .10 {\pm} .01 $
RF+Raff $ .030 {\pm} .003 $ $ .21 {\pm} .02 $ $ .058 {\pm} .006 $ $ .06 {\pm} .01 $ $ .066 {\pm} .006 $ $ .08 {\pm} .01 $ $ 4.28 {\pm} .40 $ $ .09 {\pm} .01 $ $ 1.38 {\pm} .12 $ $ .02 {\pm} .01 $
RF+Agar $ .029 {\pm} .003 $ $ .13 {\pm} .01 $ $ .050 {\pm} .005 $ $ .04 {\pm} .01 $ $ .065 {\pm} .006 $ $ .07 {\pm} .01 $ $ 3.87 {\pm} .41 $ $ .07 {\pm} .01 $ $ 1.40 {\pm} .13 $ $ .02 {\pm} .01 $
RF+Ours $ .033 {\pm} .003 $ $ .08 {\pm} .01 $ $ .064 {\pm} .006 $ $ .02 {\pm} .01 $ $ .070 {\pm} .007 $ $ .03 {\pm} .01 $ $ 4.18 {\pm} .38 $ $ .02 {\pm} .01 $ $ 1.49 {\pm} .14 $ $ .01 {\pm} .01 $
------------ --------------------- ------------------- --------------------- ------------------- --------------------- ------------------- -------------------- ------------------- -------------------- -------------------
In Table \[tab:results\], we present the performance of different methods on various datasets described above. [ One can notice that LAW and UNIV datasets have a least amount of disciminatory bias (quantified by KS), since the fairness *unaware* methods perform reasonably well in terms of ${\text{KS}}$. Furthermore, on these two datasets, the difference in performance between all fairness aware methods is less noticeable. In contrast, on CRIME, NLSY, and STUD, fairness unaware methods perform poorly in terms of ${\text{KS}}$. More importantly, our findings indicate that the proposed method is competitive with state-of-the-art methods and is the most effective in imposing the fairness constraint. In particular, in all except two considered scenarios (CRIME[+]{}RLS, CRIME[+]{}RF) our method improves fairness by $50\%$ (and up to $80\%$ in some cases) over the closest fairness aware method. In contrast, the accuracy of our method decreases by $1\%$ up to $30\%$ when compared to the most accurate fairness aware method. However, let us emphasize that the relative decrease in accuracy is much smaller than the relative improvement in fairness across the considered scenarios. For example, on NLSY[+]{}RLS the most accurate fairness aware method is RLS+Oneto with mean ${\text{MSE}} {=} .156$ and mean ${\text{KS}} {=} .50$, while RLS[+]{}Ours yields mean ${\text{MSE}} {=} .203$ and mean ${\text{KS}} {=} .09$. That is, compared to RLS[+]{}Oneto our method drops about $30\%$ in accuracy, while gains about $ 82 \%$ in fairness. With RF, which is a more powerful estimator, the average drop in accuracy across all datasets compared to RF[+]{}Agar is about $12\%$ while the average improvement in fairness is about $53\%$. ]{}
Conclusion and perspectives {#sec:concl}
===========================
In this work we investigated the problem of fair regression with Demographic Parity constraint assuming that the sensitive attribute is available for prediction. We derived a closed form solution for the optimal fair predictor which offers a simple and intuitive interpretation. Relying on this expression, we devised a post-processing procedure, which transforms any base estimator of the regression function into a nearly fair one, independently of the underlying distribution. Moreover, if the base estimator is accurate, our post-processing method yields an accurate estimator of the optimal fair predictor as well. Finally, we conducted an empirical study indicating the effectiveness of our method in imposing fairness in practice. In the future it would be valuable to extend our methodology to the case when we are not allowed to use the sensitive feature as well as to other notions of fairness.
Acknowledgement
===============
This work was supported by the Amazon AWS Machine Learning Research Award, SAP SE, and CISCO.
The appendix is organized as follows. In Appendix \[app:oracle\] we provide the proof of Theorem \[thm:Oracle\], in Appendix \[app:fairness\] we provide the proof of Proposition \[prop:distribution\_free\_fairness\], and in Appendix \[app:risk\] we prove Theorem \[thm:rate\]. For reader’s convenience all the results are repeated in this supplementary material and a short overview of classical results is provided.
Characterization of the optimal {#app:oracle}
===============================
Before providing the proof of Theorem \[thm:Oracle\], let us give a brief overview of classical results in the Optimal transport theory with one dimensional measures; all the results can be found in [@villani2003topics; @santambrogio2015optimal]
The coupling $\gamma$ which achieves the infimum in the definition of the Wasserstein-2 distance is called the optimal coupling.
Also let us mention that the Wasserstein-2 distance between two univariate probability measures $\nu, \mu$, defined in Definition \[def:wass\], can be expressed as $$\begin{aligned}
{\mathcal{W}}_2^2(\mu,\nu) = \inf_{\gamma}{\mathbb{E}}_{(Z_{\mu}, Z_{\nu}) \sim \gamma}(Z_{\nu} - Z_{\mu})^2\enspace,
\end{aligned}$$ where $Z_{\nu} \sim \nu$ and $Z_{\mu} \sim \mu$ and the infimum is taken over all joint distributions $\gamma$ of $(Z_{\nu}, Z_{\mu})$ which preserve marginals.
The next result establishes that as long as one of the measures in the definition of the Wasserstein-2 distance admits a density, then the optimal coupling in the infimum in Definition \[def:wass\] is deterministic (see [[*e.g., *]{}]{} [@villani2003topics Theorem 2.18] or [@santambrogio2015optimal Theorems 2.5 and 2.9]).
\[thm:Brenier\] Let $\nu, \mu$ be two univariate measures such that $\nu$ has a density and let $X \sim \nu$. Then there exists a mapping $T: {\mathbb{R}}\to {\mathbb{R}}$ such $$\begin{aligned}
{\mathcal{W}}_2^2(\mu,\nu) = {\mathbb{E}}(X - T(X))^2\enspace,
\end{aligned}$$ that is $(X, T(X)) \sim \bar{\gamma} \in \Gamma_{\mu, \nu}$ where $\bar{\gamma}$ is an optimal coupling. Moreover, the transport map is given by $T = Q_{\mu} \circ F_{\nu}$.
By the abuse of notation, for an increasing real-valued univariate function $F$ we will use $F^{\leftarrow}$ to denote its generalized inverse. For instance, if $F: {\mathbb{R}}\to [0, 1]$ is a CDF, then $F^{\leftarrow}$ is the quantile function that was defined in the introduction.
The next result is standard and can be found for instance in [@agueh2011barycenters Section 6.1] or [@santambrogio2015optimal Section 5.5.5]. It states that for one dimensional Wasserstein barycenter problem, the optimal measure admits a closed form solution.
\[lem:barycenter\_expression\] Let $\nu_1, \ldots, \nu_{|{\mathcal{S}}|}$ be $|{\mathcal{S}}|$ univariate probability measures admitting densities, for all $p_1, \ldots, p_{|{\mathcal{S}}|} \geq 0$ such that $p_1 + \ldots + p_{|{\mathcal{S}}|} = 1$ define $$\begin{aligned}
\nu^* \in \operatorname*{arg\,min}_{\nu}\sum_{s = 1}^{|{\mathcal{S}}|} p_s {\mathcal{W}}_2^2(\nu_s, \nu)\enspace.
\end{aligned}$$ Then, the cumulative distribution of $\nu^*$ is given by $$\begin{aligned}
F_{\nu^*}(\cdot) = \left(\sum_{s = 1}^{|{\mathcal{S}}|} p_s Q_{\nu_s}\right)^{\leftarrow}(\cdot)\enspace.
\end{aligned}$$
Theorem \[thm:Brenier\] and Lemma \[lem:barycenter\_expression\] are the two main ingredients that are used in the proof of Theorem \[thm:Oracle\].
We want to show that $$\begin{aligned}
\min_{g \text{ is fair}}{\mathbb{E}}(f^*(X, S) - g(X, S))^2 = \min_{\nu} \sum_{s \in {\mathcal{S}}}p_s{\mathcal{W}}_2^2(\nu_{f^*|s}, \nu)\enspace.
\end{aligned}$$ Let $\bar{g}: {\mathbb{R}}^d \times {\mathcal{S}} \to {\mathbb{R}}$ be a minimizer of the *l.h.s.* of the above equation and define by $\nu_{\bar g}$ the distribution of $\bar g$. Since $\nu_{f^* | s}$ admits density, using Theorem \[thm:Brenier\] for each $s \in {\mathcal{S}}$ there exists $T_s = Q_{\nu_{\bar g}} \circ F_{f^* | s}$ such that $$\begin{aligned}
\sum_{s \in {\mathcal{S}}}p_s{\mathcal{W}}_2^2(\nu_{f^*|s}, \nu_{\bar g})
&=
\sum_{s \in {\mathcal{S}}}p_s\int_{{\mathbb{R}}}\left(z - T_s (z)\right)^2 d\nu_{f^*|s}(z)\\
&=
\sum_{s \in {\mathcal{S}}}p_s\int_{{\mathbb{R}}^d}\left(f^*(x, s) - T_s \circ f^* (x, s)\right)^2 d{\mathbb{P}}_{X | S = s}(x)\\
&=
\sum_{s \in {\mathcal{S}}}p_s{\mathbb{E}}\left[\left(f^*(X, s) - \left(T_s \circ f^*\right) (X, s)\right)^2 | S = s\right]\\
&=
{\mathbb{E}}(f^*(X, S) - \tilde g(X, S))^2\enspace,
\end{aligned}$$ where we defined $\tilde g$ for all $(x, s) \in {\mathbb{R}}^d \times {\mathcal{S}}$ as $$\begin{aligned}
\tilde g(x, s) = \left(T_s \circ f^* \right)(x, s)
=\left(Q_{\nu_{\bar g}} \circ F_{f^* | s} \circ f^*\right)(x, s)\enspace.
\end{aligned}$$ The cumulative distribution of $\tilde g$ can be expressed as $$\begin{aligned}
{\mathbb{P}}(\tilde g (X, S) \leq t)
&= \sum_{s \in {\mathcal{S}}}p_s{\mathbb{P}}_{X | S = s}\left(Q_{\nu_{\bar g}} \circ F_{f^* | s} \circ f^*(X, s) \leq t\right)\\
&= \sum_{s \in {\mathcal{S}}}p_s{\mathbb{P}}_{X | S = s}\left(f^*(X, s) \leq Q_{f^* | s} \circ F_{\nu_{\bar g}} (t)\right) = F_{\nu_{\bar g}} (t)\enspace,
\end{aligned}$$ where the last equality is due to the fact that $\nu_{f^*|s}$ admits a density for all $s\in {\mathcal{S}}$. The above implies that $\tilde g$ is fair, thus on the one hand by optimality of $\bar g$ we have $$\begin{aligned}
{\mathbb{E}}(f^*(X, S) - \tilde g(X, S))^2 \geq {\mathbb{E}}(f^*(X, S) - \bar g(X, S))^2\enspace,
\end{aligned}$$ on the other hand we have for each $s \in {\mathcal{S}}$ $$\begin{aligned}
{\mathcal{W}}_2^2(\nu_{f^*|s}, \nu_{\bar g})
\leq {\mathbb{E}}\left[\left(f^*(X, S) - \bar g (X, S)\right)^2 | S = s\right]\enspace.
\end{aligned}$$ Thus we showed that $$\begin{aligned}
\sum_{s \in {\mathcal{S}}}p_s{\mathcal{W}}_2^2(\nu_{f^*|s}, \nu_{\bar g}) = \min_{g \text{ is fair}}{\mathbb{E}}(f^*(X, S) - g(X, S))^2\label{eq:wass00}\enspace.
\end{aligned}$$
This implies that $$\begin{aligned}
\min_{\nu} \sum_{s \in {\mathcal{S}}}p_s{\mathcal{W}}_2^2(\nu_{f^*|s}, \nu) \leq
\min_{g \text{ is fair}}{\mathbb{E}}(f^*(X, S) - g(X, S))^2\label{eq:wass0}\enspace.
\end{aligned}$$
Now we want to show that the opposite inequality also holds. To this end define $\nu^*$ as $$\begin{aligned}
\nu^* \in \operatorname*{arg\,min}_{\nu} \sum_{s \in {\mathcal{S}}}p_s{\mathcal{W}}_2^2(\nu_{f^*|s}, \nu)\enspace.
\end{aligned}$$ Set $T^*_s$ as optimal transport maps from $\nu_{f^*|s}$ to $\nu^*$ of the form $T^*_s = Q_{\nu^*} \circ F_{f^* | s}$ (provided by Theorem \[thm:Brenier\] and our assumption on the density of $\nu_{f^* | s}$) and define $g^*$ for all $(x, s) \in {\mathbb{R}}^d \times {\mathcal{S}}$ as $$\begin{aligned}
g^*(x, s) = \left(Q_{\nu^*} \circ F_{f^* | s} \circ f^* \right)(x, s)\label{eq:wass3}\enspace.
\end{aligned}$$ By the definition of $g^*$ in Eq. and Theorem \[thm:Brenier\] we clearly have $$\begin{aligned}
\label{eq:wass1}
\min_{\nu} \sum_{s \in {\mathcal{S}}}p_s{\mathcal{W}}_2^2(\nu_{f^*|s}, \nu) = {\mathbb{E}}(f^*(X, S) - g^*(X, S))^2\enspace.
\end{aligned}$$ Moreover since $\nu^*$ is independent from $S$, using similar argument as above one can show that $g^*$ satisfies the Demographic Parity constraint in Definition \[def:demographic\_parity\] and thus, Eq. yields $$\begin{aligned}
\min_{\nu} \sum_{s \in {\mathcal{S}}}p_s{\mathcal{W}}_2^2(\nu_{f^*|s}, \nu)
\geq
\min_{g \text{ is fair}}{\mathbb{E}}(f^*(X, S) - g(X, S))^2 \enspace.
\label{eq:wass2}
\end{aligned}$$ Eqs. and yield the first assertion of the result. Notice that thanks to Eq. we have also shown that $$\begin{aligned}
{\mathbb{E}}(f^*(X, S) - g^*(X, S))^2 = {\mathbb{E}}(f^*(X, S) - \bar g(X, S))^2\enspace,
\end{aligned}$$ and since $g^*$ is fair we can put $\bar g = g^*$. Finally, using Lemma \[lem:barycenter\_expression\] we derive an explicit form of $\nu^*$ and conclude using Eq. .
Proof of Proposition \[prop:distribution\_free\_fairness\] {#app:fairness}
==========================================================
Let us first recall the well-known Dvoretzky–Kiefer–Wolfowitz inequality [@massart1990tight Corollary 1].
\[thm:DKW\] Let $Z_1, \ldots, Z_n$ be [[i.i.d. ]{}]{}real valued random variables with cumulative distribution $F$. Let $\hat F$ be the empirical cumulative distribution of $Z_1, \ldots, Z_n$, then $$\begin{aligned}
{\mathbf{E}}{\lVertF - \hat{F}\rVert}_{\infty} {\vcentcolon=}{\mathbf{E}}\sup_{t \in {\mathbb{R}}}{\lvert F(t) - \hat{F}(t)\rvert} \leq \sqrt{\frac{\pi}{2n}}\enspace.
\end{aligned}$$
The proof of Eq. is based on standard results in the theory of rank statistics (see e.g. [@van2000asymptotic Sec. 13]). Meanwhile, the proof of Eq. is built upon the well-known Dvoretzky–Kiefer–Wolfowitz inequality [@massart1990tight Corollary 1].
Notice that if $X^s \sim {\mathbb{P}}_{X | S = s}$ and $X^s$ is independent from labeled, unlabeled data, and the noise variables $\varepsilon_{is}, \varepsilon$, then it holds that $$\begin{aligned}
{\mathbf{P}}(\hat g(X, S) \leq t | S = s) = {\mathbf{P}}(\hat g(X^{s}, s) \leq t),\quad \forall t\in {\mathbb{R}}\enspace.\end{aligned}$$ [**Proof of Eq. :**]{} We have for all $s, s' \in {\mathcal{S}}$ that $$\begin{aligned}
\sup_{t \in {\mathbb{R}}}&{\left\lvert {\mathbf{P}}(\hat g(X^{s}, s) \leq t) - {\mathbf{P}}(\hat g(X^{s'}, s') \leq t)\right\rvert}\\
&\leq
\sup_{t \in (0, 1)}{\left\lvert {\mathbf{P}}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\leq t\right) - {\mathbf{P}}\left(\hat{F}_{{\hat{f}}| s'}\left({\hat{f}}(X^{s'}, s') + \varepsilon\right)\leq t \right)\right\rvert}\enspace,\end{aligned}$$ where, thanks to the form of $\hat g$ in Eq. , the inequality follows from the fact that for all $s \in {\mathcal{S}}$ $$\begin{aligned}
\left(\sum_{\tilde{s} \in {\mathcal{S}}}\hat p_{\tilde{s}} \hat{Q}_{{\hat{f}}|\tilde{s}} \right) \circ \hat{F}_{{\hat{f}}| s}\left({\hat{f}}(x, s) + \varepsilon\right) \leq t \quad\Leftrightarrow\quad \hat{F}_{{\hat{f}}| s}\left({\hat{f}}(x, s) + \varepsilon\right) \leq \left(\sum_{\tilde{s} \in {\mathcal{S}}}\hat p_{\tilde{s}} \hat{Q}_{{\hat{f}}| \tilde{s}}\right)^{\leftarrow}(t)\enspace.\end{aligned}$$ Fix some $t \in (0, 1)$ and let $k_s(t) \in \{1, \ldots, |{\mathcal{I}}_1^s|\}$ be such that $\tfrac{k_s(t) - 1}{|{\mathcal{I}}_1^s|}\leq t < \tfrac{k_s(t)}{|{\mathcal{I}}_1^s|}$, then by the definition of $\hat{F}_{{\hat{f}}| s}(\cdot)$ we have $$\begin{aligned}
\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(x, s) + \varepsilon\right)\leq t \quad\Leftrightarrow\quad \sum_{i \in {\mathcal{I}}_1^s}{{\bf 1}_{\left\{{\hat{f}}(X^{s}_i, s) + \varepsilon_{is} \leq {\hat{f}}(x, s) + \varepsilon\right\}}} \leq k_s(t) - 1\enspace.\end{aligned}$$ Notice that the random variables $\{{\hat{f}}(X^{s}, s) + \varepsilon\} \cup \{{\hat{f}}(X^{s}_i, s) + \varepsilon_{is}\}_{i \in {\mathcal{I}}_1^s}$ conditionally on labeled data ${\mathcal{L}}$ are [[i.i.d. ]{}]{}and continuous. Thus, conditionally on ${\mathcal{L}}$ the random variable $\sum_{i \in {\mathcal{I}}_1^s}{{\bf 1}_{\left\{{\hat{f}}(X^{s}_i, s) {+} \varepsilon_{is} \leq {\hat{f}}(X^{s}, s) {+} \varepsilon\right\}}}$ is distributed uniformly on $\{0, \ldots, |{\mathcal{I}}_1^s|\}$ (see [[*e.g., *]{}]{} [@van2000asymptotic Lemma 13.1]), so that $$\begin{aligned}
{\mathbf{P}}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\leq t\right) = \frac{k_s(t)}{|{\mathcal{I}}_1^s| + 1}\enspace.\end{aligned}$$ Repeating the same argument for $s'$ and recalling that $|{\mathcal{I}}_1^s| = N_s / 2$ and $|{\mathcal{I}}_1^{s'}| = N_{s'} / 2$, we get $$\begin{aligned}
\sup_{t \in {\mathbb{R}}}{\left\lvert {\mathbf{P}}(\hat g(X^{s}, s) \leq t) - {\mathbf{P}}(\hat g(X^{s'}, s') \leq t)\right\rvert}
&\leq
\sup_{t \in (0, 1)}{\left\lvert \frac{k_s(t)}{N_s/2 + 1} - \frac{k_{s'}(t)}{N_{s'}/2 + 1}\right\rvert}\\
&=
2\left(N_s \wedge N_{s'} + 2\right)^{-1}{{\bf 1}_{\left\{N_s \neq N_{s'}\right\}}}\enspace.\end{aligned}$$
[**Proof of Eq. :**]{} Similarly, as in the proof of Eq. we can write $$\begin{aligned}
(*) &= \sup_{t \in {\mathbb{R}}}{\left\lvert {\mathbf{P}}(\hat g_s(X^{s}) \leq t \big| {{\mathcal{D}}}) - {\mathbf{P}}(\hat g_{s'}(X^{s'}) \leq t \big| {{\mathcal{D}}})\right\rvert}\\
&\leq
\sup_{t \in (0, 1)}{\left\lvert {\mathbf{P}}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\leq t \big| {{\mathcal{D}}}\right) - {\mathbf{P}}\left(\hat{F}_{{\hat{f}}| s'}\left({\hat{f}}(X^{s'}, s') + \varepsilon\right)\leq t \big| {{\mathcal{D}}}\right)\right\rvert}\enspace.
\end{aligned}$$ Moreover, thanks to the triangle inequality we have $$\begin{aligned}
(*) \leq
& \sup_{t \in (0, 1)}{\left\lvert {\mathbf{P}}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\leq t \big| {{\mathcal{D}}}\right) - {\mathbf{P}}\left(F_{\bar{\nu}_{{\hat{f}}| s}}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\leq t \big| {{\mathcal{D}}}\right)\right\rvert}\nonumber\\
&+
\sup_{t \in (0, 1)}{\left\lvert {\mathbf{P}}\left(\hat{F}_{{\hat{f}}| s'}\left({\hat{f}}(X^{s'}, s') + \varepsilon\right)\leq t \big| {{\mathcal{D}}}\right) {-} {\mathbf{P}}\left(F_{\bar{\nu}_{{\hat{f}}| s'}}\left({\hat{f}}(X^{s'}, s') + \varepsilon\right)\leq t \big| {{\mathcal{D}}}\right)\right\rvert}\nonumber\\
&= \sup_{t \in (0, 1)}A_s(t) + \sup_{t \in (0, 1)}A_{s'}(t)\enspace,\label{eq:DKW_fairness_intermidiate}
\end{aligned}$$ where for all $t\in {\mathbb{R}}$ and all $s \in {\mathcal{S}}$ we defined $$\begin{aligned}
F_{\bar{\nu}_{\hat f | s}}(t) = {\mathbf{P}}\left(\hat f(X^{s}, s) + \varepsilon \leq t \big| {{\mathcal{D}}}\right)\enspace,
\end{aligned}$$ and we used the fact that ${\hat{f}}(X^{s}, s) + \varepsilon$ is continuous conditionally on all the available data ${{\mathcal{D}}}$, then the random variable $F_{\bar{\nu}_{\hat f | s}}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)$ is distributed uniformly on $(0, 1)$ (see [[*e.g., *]{}]{} [@van2000asymptotic Lemma 21.1]), which means that for all $t \in (0, 1)$ and all $s, s' \in {\mathcal{S}}$ $$\begin{aligned}
t = {\mathbf{P}}\left(F_{\bar{\nu}_{\hat f | s}}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\leq t \big| {{\mathcal{D}}}\right) = {\mathbf{P}}\left(F_{\bar{\nu}_{\hat f | s'}}\left({\hat{f}}(X^{s'}, s') + \varepsilon\right)\leq t \big| {{\mathcal{D}}}\right)\enspace.
\end{aligned}$$ We bound the first term in Eq. and the bound for the second terms follows the same arguments. Fix some $t \in (0, 1)$, then we can write $$\begin{aligned}
A_s(t) &\leq {\mathbf{P}}\left({\left\lvert F_{\bar{\nu}_{\hat f | s}}\left({\hat{f}}(X^{s}, s) + \varepsilon\right) - t\right\rvert} \leq {\left\lvert F_{\bar{\nu}_{\hat f | s}}\left({\hat{f}}(X^{s}, s) + \varepsilon\right) - \hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\right\rvert} \bigg| {{\mathcal{D}}}\right)\\
&\leq
{\mathbf{P}}\left({\left\lvert F_{\bar{\nu}_{\hat f | s}}\left({\hat{f}}(X^{s}, s) + \varepsilon\right) - t\right\rvert} \leq {\left\lVertF_{\bar{\nu}_{\hat f | s}} - \hat{F}_{{\hat{f}}| s}\right\rVert}_{\infty} \bigg| {{\mathcal{D}}}\right)\leq 2 {\left\lVertF_{\bar{\nu}_{\hat f | s}} - \hat{F}_{{\hat{f}}| s}\right\rVert}_{\infty} \enspace.
\end{aligned}$$ Taking supremum on both sides and repeating the same argument for $s'$, we get $$\begin{aligned}
(*) \leq 2 {\mathbf{E}}{\left\lVertF_{\bar{\nu}_{\hat f | s}} - \hat{F}_{{\hat{f}}| s}\right\rVert}_{\infty} + 2 {\mathbf{E}}{\left\lVertF_{\bar{\nu}_{\hat f | s'}} - \hat{F}_{{\hat{f}}| s'}\right\rVert}_{\infty} \enspace,
\end{aligned}$$ we conclude applying Dvoretzky–Kiefer–Wolfowitz inequality, recalled in Theorem \[thm:DKW\], conditionally on ${\mathcal{L}}$.
Proof of Theorem \[thm:rate\] {#app:risk}
=============================
Let us first recall the assumptions that we require in order to prove Theorem \[thm:rate\].
The next simple result states that Assumption \[ass:concentration\_estimator\] yields a bound in $L_1$-norm between $f^*$ and ${\hat{f}}$.
\[lem:from\_cprob\_to\_exp\] Let Assumption \[ass:concentration\_estimator\] be satisfied, then for all $s \in {\mathcal{S}}$ it holds that $$\begin{aligned}
{\mathbf{E}}\left[|f^*(X, S) - {\hat{f}}(X, S)| | S = s\right] \leq \mathtt{A}b_n^{-1/2}\enspace,
\end{aligned}$$ with $\mathtt{A} = \tfrac{c}{2}\sqrt{\tfrac{\pi}{C}}$.
Applying Fubini’s theorem we can write $$\begin{aligned}
{\mathbf{E}}\left[|f^*(X, S) - {\hat{f}}(X, S)| | S = s\right]
&=
\int_{x \in {\mathbb{R}}^d} {\mathbf{E}}|f^*(x, s) - {\hat{f}}(x, s)| {\mathbb{P}}_{X | S = s}(dx)\\
&= \int_{x \in {\mathbb{R}}^d} {\left( \int_{0}^{+\infty} {\mathbf{P}}(|f^*(x, s) - {\hat{f}}(x, s)| > t) dt \right)} {\mathbb{P}}_{X | S = s}(dx)\\
&\stackrel{(a)}{\leq}
\int_{x \in {\mathbb{R}}^d}{\left( \int_{0}^{+\infty}c\exp{\left( -Cb_n t^2 \right)} dt \right)} {\mathbb{P}}_{X | S = s}(dx)\\
&=
c\int_{0}^{+\infty}\exp{\left( -Cb_n t^2 \right)} dt\enspace.
\end{aligned}$$ where $(a)$ follows from Assumption \[ass:concentration\_estimator\]. Making change of variables we get $$\begin{aligned}
c\int_{0}^{+\infty}\exp{\left( -Cb_n t^2 \right)} dt = c(Cb_n)^{-1/2}\int_{0}^{+\infty}\exp{\left( -t^2 \right)} dt = c(Cb_n)^{-1/2}\frac{\sqrt{\pi}}{2}\enspace.
\end{aligned}$$
We also need to define Wasserstein $1$ and $\infty$ distances.
\[def:wass1infty\] Let $\mu$ and $\nu$ be two univariate probability measures, then Wasserstein $1$ and $\infty$ distance between $\mu$ and $\nu$ are defined as $$\begin{aligned}
&{\mathcal{W}}_1(\mu, \nu) = \int_{0}^1 {\left\lvert Q_{\mu}(t) - Q_{\nu}(t)\right\rvert}dt\quad\text{and}\quad
{\mathcal{W}}_{\infty}(\mu, \nu) = \sup_{t \in [0, 1]}{\left\lvert Q_{\mu}(t) - Q_{\nu}(t)\right\rvert}\enspace,
\end{aligned}$$ respectively.
The definitions of ${\mathcal{W}}_1$ and ${\mathcal{W}}_\infty$ can be stated in terms of couplings as it is done in Definition \[def:wass\]. However, for our purposes it is more convenient to use their equivalent formulation stated in Definition \[def:wass1infty\]. We refer to [@Bobkov_Ledoux16] and in particular to their Theorem 2.10 for further details.
The final ingredient is [@Bobkov_Ledoux16 Theorem 5.11].
\[thm:W\_infty\_bound\] Let $Z_1, \ldots, Z_n$ be [[i.i.d. ]{}]{}real valued random variables from some probability measure $\mu$ and let $\hat \mu$ be the empirical measure based on $Z_1, \ldots, Z_n$. Assume that $\mu$ admits density which is lower-bounded by some constant $L > 0$, then $$\begin{aligned}
{\mathbf{E}}[{\mathcal{W}}_{\infty}(\mu, \hat\mu)] \leq L^{-1}\sqrt{\frac{2\pi}{n}}\enspace.
\end{aligned}$$
In the proof $a > 0$ is going to denote an absolute constant independent from the size of data, which can differ from line to line. First of all, define the random variable $$\begin{aligned}
\Delta(\hat g) = {\mathbb{E}}{\left\lvert \hat g(X, S) - g^*(X, S)\right\rvert} = \sum_{s \in {\mathcal{S}}}p_s {\mathbb{E}}\left[{\left\lvert \hat g(X, s) - g^*(X, s)\right\rvert} | S = s\right]\enspace,
\end{aligned}$$ where ${\mathbb{E}}$ stands for the expectation [[*w.r.t. *]{}]{}the joint distribution of $(X, S, Y)$. Recall that $$\begin{aligned}
g^*(x, s) = \sum_{s' \in {\mathcal{S}}}p_{s'}Q_{f^* | s'}\left(F_{f^* | s}\left(f^*(x, s)\right)\right) \,\,\text{and}\,\,
\hat g(x, s) = \sum_{s' \in {\mathcal{S}}}\hat p_{s'} \hat{Q}_{{\hat{f}}| s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(x, s) + \varepsilon\right) \right)\enspace.
\end{aligned}$$ Considering $g^*(x, s)$ first, we can state that $$\begin{aligned}
{\left\lvert g^*(x, s) - \sum_{s' \in {\mathcal{S}}}\hat p_{s'}Q_{f^* | s'}\left(F_{f^* | s}\left(f^*(x, s)\right)\right)\right\rvert} \leq \sum_{s' \in {\mathcal{S}}}{\left\lvert p_{s'} - \hat p_{s'}\right\rvert} \times {\left\lvert Q_{f^* | s'} \circ F_{f^* | s} \circ f^*(x, s)\right\rvert}\enspace.
\end{aligned}$$ It is clear that if we can find a bound on $|f^*(x, s')|$ which holds for almost all $x$ [[*w.r.t. *]{}]{}${\mathbb{P}}_{X | S = s'}$, it would imply an upper bound on $|Q_{f^* | s'} (t)|$ for all $t \in [0, 1]$. Fix some $a > 0$, then on the one hand for all $s' \in {\mathcal{S}}$ $$\begin{aligned}
{\mathbb{P}}(|f^*(X, S)| \leq a | S = s') \leq 1\enspace,
\end{aligned}$$ on the other hand under Assumption \[ass:density\_rate\] we can write for all $s' \in {\mathcal{S}}$ $$\begin{aligned}
{\mathbb{P}}(|f^*(X, S)| \leq a | S {=} s') = \int_{|f^*(x, s')| \leq a} {\mathbb{P}}_{X | S = s'}(dx) = \int_{|t| \leq a} q_{s'}(t) dt \geq \underline{\lambda}_{s'}\int_{|t| \leq a} dt = 2a\underline{\lambda}_{s'}\enspace,
\end{aligned}$$ which implies that $a \leq 1/(2\underline{\lambda}_{s'})$ and therefore $|f^*(x, s')| \leq 1/(2\underline{\lambda}_{s'})$ for almost all $x \in {\mathbb{R}}^d$ [[*w.r.t. *]{}]{}${\mathbb{P}}_{X | S = s'}$. Hence, we can write for all $(x, s) \in {\mathbb{R}}^d \times {\mathcal{S}}$ $$\begin{aligned}
{\left\lvert g^*(x, s) - \sum_{s' \in {\mathcal{S}}}\hat p_{s'}Q_{f^* | s'}\left(F_{f^* | s}\left(f^*(x, s)\right)\right)\right\rvert} \leq \frac{1}{2}\sum_{s' \in {\mathcal{S}}}\underline{\lambda}_{s'}^{-1}{\left\lvert p_{s'} - \hat p_{s'}\right\rvert}\enspace.
\end{aligned}$$ The above implies that $$\begin{aligned}
\Delta(\hat g) \leq &\sum_{s \in {\mathcal{S}}}p_s\sum_{s' \in {\mathcal{S}}}\hat{p}_{s'}{\mathbb{E}}\left[{\left\lvert Q_{f^* | s'}\left(F_{f^* | s}\left(f^*(X, S)\right)\right) - \hat{Q}_{{\hat{f}}| s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X, s) + \varepsilon\right) \right)\right\rvert} | S = s\right]\\
+
&\frac{1}{2}\sum_{s \in {\mathcal{S}}}\underline{\lambda}_s^{-1}{\left\lvert p_s - \hat p_s\right\rvert}\enspace.
\end{aligned}$$ Taking the total expectation we arrive at $$\begin{aligned}
{\mathbf{E}}[\Delta(\hat g)] \leq &\sum_{s, s' \in {\mathcal{S}}}p_sp_{s'}{\mathbf{E}}\left[{\left\lvert Q_{f^* | s'}\left(F_{f^* | s}\left(f^*(X, S)\right)\right) - \hat{Q}_{{\hat{f}}| s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X, S) + \varepsilon\right) \right)\right\rvert} | S = s\right]\\
+
&\frac{1}{2}\sum_{s \in {\mathcal{S}}}\underline{\lambda}_s^{-1}{\mathbf{E}}{\left\lvert p_s - \hat p_s\right\rvert}\enspace,
\end{aligned}$$ where we used the fact that $\hat p_s$ is an unbiased estimator of $p_s$. For all $s \in {\mathcal{S}}$ let $X^{s} \sim {\mathbb{P}}_{X | S = s}$ be independent from everything, for all $s', s \in {\mathcal{S}}$ set the shorthand notation $$\begin{aligned}
\mathbbm{a}_{s s'}
= {\mathbf{E}}{\left\lvert Q_{f^* | s'}\left(F_{f^* | s}\left(f^*(X^{s}, s)\right)\right) - \hat{Q}_{{\hat{f}}| s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right) \right)\right\rvert}\enspace.
\end{aligned}$$ Notice that $$\begin{aligned}
\mathbbm{a}_{s s'} = {\mathbf{E}}\left[{\left\lvert Q_{f^* | s'}\left(F_{f^* | s}\left(f^*(X, S)\right)\right) - \hat{Q}_{{\hat{f}}| s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X, S) + \varepsilon\right) \right)\right\rvert} | S = s\right]\enspace,
\end{aligned}$$ and therefore we can write $$\begin{aligned}
{\mathbf{E}}{\left\lvert \hat g(X, S) - g^*(X, S)\right\rvert}= {\mathbf{E}}[\Delta(\hat g)] \leq \sum_{s, s' \in {\mathcal{S}}}p_sp_{s'}\mathbbm{a}_{ss'} + \frac{1}{2}\sum_{s \in {\mathcal{S}}}\underline{\lambda}_s^{-1}{\mathbf{E}}{\left\lvert p_s - \hat p_s\right\rvert}\enspace.
\end{aligned}$$ Notice that the term ${\mathbf{E}}{\left\lvert p_s - \hat p_s\right\rvert} = N^{-1}{\mathbf{E}}|Np_s - V|$, where $V$ is the binomial random variable with parameters $(N, p_s)$, thus using the Cauchy–Schwarz inequality we can write ${\mathbf{E}}{\left\lvert p_s - \hat p_s\right\rvert} \leq N^{-1} \sqrt{\operatorname*{Var}(V)} = \sqrt{p_s(1 - p_s) / N}$ and the above bound reads as $$\begin{aligned}
{\mathbf{E}}{\left\lvert \hat g(X, S) - g^*(X, S)\right\rvert}
&\leq
\sum_{s, s' \in {\mathcal{S}}}p_sp_{s'}\mathbbm{a}_{ss'} + \frac{1}{2}\sum_{s \in {\mathcal{S}}}\underline{\lambda}_s^{-1}\sqrt{\frac{p_s(1 - p_s)}{N}}\nonumber\\
&\leq
\sum_{s, s' \in {\mathcal{S}}}p_sp_{s'}\mathbbm{a}_{ss'} + \frac{N^{-1/2}}{2}\max_{s \in {\mathcal{S}}}\underline{\lambda}_s^{-1}\sum_{s \in {\mathcal{S}}}\sqrt{p_s(1 - p_s)}\enspace\label{eq:norm_bound_via_a} \enspace.
\end{aligned}$$ It remains to bound $\mathbbm{a}_{s s'}$ for each $s, s' \in {\mathcal{S}}$. Fix some $s, s' \in {\mathcal{S}}$ (they can be equal), then $$\begin{aligned}
\mathbbm{a}_{s s'}
\leq
&\underbrace{{\mathbf{E}}{\left\lvert \hat{Q}_{f^* | s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\right) - \hat{Q}_{{\hat{f}}| s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right) \right)\right\rvert}}_{\mathbbm{a}_{s s'}^1}\nonumber\\
&+
\underbrace{{\mathbf{E}}{\left\lvert Q_{f^* | s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\right) - \hat{Q}_{f^* | s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\right)\right\rvert}}_{\mathbbm{a}_{s s'}^2}\label{eq:a_one_two_three_bound}\\
&+
\underbrace{{\mathbf{E}}{\left\lvert Q_{f^* | s'}\left(F_{f^* | s}\left(f^*(X^{s}, s)\right)\right) - Q_{f^* | s'}\left(\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\right)\right\rvert}}_{\mathbbm{a}_{s s'}^3}\enspace.\nonumber
\end{aligned}$$ We bound each of the three terms separately.
[**First term ($\mathbbm{a}_{s s'}^1$):**]{} Notice that $\hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)$ is distributed uniformly on $\{0, 1 / |{\mathcal{I}}_1^s|, 2 / |{\mathcal{I}}_1^s|, \ldots, 1\}$ conditionally on labeled data ${\mathcal{L}}$ (see [[*e.g., *]{}]{} [@van2000asymptotic Lemma 13.1]). Thus, we have $$\begin{aligned}
\label{eq:a_one_intermidiate}
\mathbbm{a}_{s s'}^1
&=
\frac{1}{|{\mathcal{I}}_1^s| + 1}\sum_{j = 0}^{|{\mathcal{I}}_1^s|} {\mathbf{E}}{\left\lvert \hat{Q}_{f^* | s'}\left(\frac{j}{|{\mathcal{I}}_1^s|}\right) - \hat{Q}_{{\hat{f}}| s'}\left(\frac{j}{|{\mathcal{I}}_1^s|} \right)\right\rvert}\enspace.
\end{aligned}$$ Notice that for all $j \in \{1, \ldots, |{\mathcal{I}}_1^s|\}$ and all $\alpha \in ((j - 1) / |{\mathcal{I}}_1^s| , j / |{\mathcal{I}}_1^s| ]$ it holds that $$\begin{aligned}
\hat{Q}_{f^* | s'}\left(\frac{j}{|{\mathcal{I}}_1^s|}\right) = \hat{Q}_{f^* | s'}\left(\alpha\right)\enspace.
\end{aligned}$$ The above implies that $$\begin{aligned}
\label{eq:f_hat_discrete_quantile}
\frac{1}{|{\mathcal{I}}_1^s|}\hat{Q}_{f^* | s'}\left(\frac{j}{|{\mathcal{I}}_1^s|}\right) = \int_{j / |{\mathcal{I}}_1^s|}^{(j + 1) / |{\mathcal{I}}_1^s|}\hat{Q}_{f^* | s'}\left(\alpha\right) d\alpha\enspace,
\end{aligned}$$ and the same argument repeated for $\hat{Q}_{{\hat{f}}| s'}$ implies that $$\begin{aligned}
\label{eq:f_star_discrete_quantile}
\frac{1}{|{\mathcal{I}}_1^s|}\hat{Q}_{{\hat{f}}| s'}\left(\frac{j}{|{\mathcal{I}}_1^s|}\right) = \int_{j / |{\mathcal{I}}_1^s|}^{(j + 1) / |{\mathcal{I}}_1^s|}\hat{Q}_{{\hat{f}}| s'}\left(\alpha\right) d\alpha\enspace.
\end{aligned}$$ Substituting Eqs. - in Eq. and using Definition \[def:wass1infty\] we get $$\begin{aligned}
\mathbbm{a}_{s s'}^1
\leq
2 {\mathbf{E}}\int_{0}^1{\left\lvert \hat{Q}_{f^* | s'}\left(\alpha\right) - \hat{Q}_{{\hat{f}}| s'}\left(\alpha \right)\right\rvert}d\alpha
=
2 {\mathbf{E}}{\mathcal{W}}_1({\hat{\nu}}^0_{f^* | s'}, {\hat{\nu}}^0_{{\hat{f}}| s'})\enspace,
\end{aligned}$$ where for $j = 0$ in Eq. we used the fact that $\tfrac{1}{|{\mathcal{I}}^s_1|}{\mathbf{E}}{\lvert \hat{Q}_{f^* | s'}\left(0\right) - \hat{Q}_{{\hat{f}}| s'}\left(0 \right)\rvert} \leq {\mathbf{E}}\int_{0}^1{\lvert \hat{Q}_{f^* | s'}\left(\alpha\right) - \hat{Q}_{{\hat{f}}| s'}\left(\alpha \right)\rvert}d\alpha$. Using the coupling definition of the Wasserstein distance and the way we have defined ${\hat{\nu}}^0_{f | s'}$, we can write $$\begin{aligned}
{\mathcal{W}}_1({\hat{\nu}}^0_{f^* | s'}, {\hat{\nu}}^0_{{\hat{f}}| s'}) \leq \frac{1}{|{\mathcal{I}}_0^{s'}|}\sum_{i \in {\mathcal{I}}_0^{s'}}{\left\lvert f^*(X^{s'}_i, s') + \varepsilon_{is'} - ({\hat{f}}(X^{s'}_i, s') + \varepsilon_{is'})\right\rvert}\enspace,
\end{aligned}$$ almost surely. Since $\{X_i^{s'}\}_{i \in {\mathcal{I}}_0^{s'}}$ are [[i.i.d. ]{}]{}from ${\mathbb{P}}_{X | S = s'}$, then conditionally on ${\mathcal{L}}$ the random variables $\{|{f^*(X^{s'}_i, s) - {\hat{f}}(X^{s'}_i, s')}|\}_{i \in {\mathcal{I}}_0^{s'}}$ are [[i.i.d. ]{}]{}. Furthermore, using Lemma \[lem:from\_cprob\_to\_exp\] we can write $$\begin{aligned}
\label{eq:a_one_final}
\mathbbm{a}_{s s'}^1 \leq 2 {\mathbf{E}}{\mathcal{W}}_1({\hat{\nu}}^0_{f^* | s'}, {\hat{\nu}}^0_{{\hat{f}}| s'}) \leq 2{\mathbf{E}}\left[{\left\lvert f^*(X, S) - {\hat{f}}(X, S)\right\rvert} \big| S = s'\right] \stackrel{\text{Lemma~\ref{lem:from_cprob_to_exp}}}{\leq} 2\mathtt{A}b_n^{-1/2}\enspace.
\end{aligned}$$
[**Second term ($\mathbbm{a}_{s s'}^2$):**]{} Note that under Assumption \[ass:density\_rate\], the Lipschitz constant of $Q_{f^* | s'}$ is upper bounded by $\underline{\lambda}_{s'}^{-1}$. Then, taking supremum and using Definition \[def:wass1infty\] we apply Theorem \[thm:W\_infty\_bound\] to get $$\begin{aligned}
\label{eq:a_two_final}
\mathbbm{a}_{s s'}^2 \leq {\mathbf{E}}{\mathcal{W}}_{\infty}{\left( \nu_{f^*|s'}, \hat{\nu}^0_{f^*|s'} \right)} \leq a \underline{\lambda}_{s'}^{-1} N_{s'}^{-1/2}\enspace.
\end{aligned}$$ where $a$ is an absolute positive constant ($a = 2\sqrt{2\pi}$ is sufficient).
[**Third term ($\mathbbm{a}_{s s'}^3$):**]{} We can write, using Assumption \[ass:density\_rate\] that $$\begin{aligned}
\mathbbm{a}_{s s'}^3
\leq
&\underline{\lambda}^{-1}_{s'}{\mathbf{E}}{\left\lvert F_{f^* | s}\left(f^*(X^{s}, s)\right) - \hat{F}_{{\hat{f}}| s}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\right\rvert}\nonumber\\
\leq
&\underline{\lambda}^{-1}_{s'}{\mathbf{E}}{\left\lvert F_{f^* | s}\left(f^*(X^{s}, s)\right) {-} F_{\bar\nu_{{\hat{f}}| s}}\left({\hat{f}}(X^{s}, s) {+} \varepsilon\right)\right\rvert}
+ \underline{\lambda}^{-1}_{s'}{\mathbf{E}}{\lVertF_{\bar\nu_{{\hat{f}}| s}}(t) - \hat{F}_{{\hat{f}}| s}\left(t\right)\rVert}_{\infty}\enspace,
\label{eq:a_three_intermidiate}
\end{aligned}$$ with $F_{\bar\nu_{{\hat{f}}| s}}$ defined for all $s\in {\mathcal{S}}$ and all $t \in {\mathbb{R}}$ as $$\begin{aligned}
\label{eq:cdf_noisy}
F_{\bar{\nu}_{{\hat{f}}| s}}(t) = {\mathbf{P}}\left({\hat{f}}(X^{s}, s) + \varepsilon \leq t \big| {\mathcal{L}}\right)\enspace.
\end{aligned}$$ The second term in Eq. is bounded by $\lesssim 2 \underline{\lambda}^{-1}_{s'} N_s^{-1/2}$ thanks to the Dvoretzky–Kiefer–Wolfowitz inequality recalled in Theorem \[thm:DKW\]. Thus, it remains to bound the first term in Eq. . We introduce the following shorthand notation for the first term in Eq. $$\begin{aligned}
(*) = {\mathbf{E}}{\left\lvert F_{f^* | s}\left(f^*(X^{s}, s)\right) - F_{\bar\nu_{{\hat{f}}| s}}\left({\hat{f}}(X^{s}, s) + \varepsilon\right)\right\rvert}\enspace.
\end{aligned}$$ Let $\tilde{X}^s \sim {\mathbb{P}}_{X | S = s}$ and $\tilde\varepsilon \sim U[-\sigma, \sigma]$ be independent from $\varepsilon, X^s$, ${\mathcal{L}}$ and each other. Based on this notation we can write
$$\begin{aligned}
\label{eq:term_three_intr0}
(*) {=} {\mathbf{E}}\bigg|\underbrace{{{\mathbf{P}}\left( f^*(\tilde{X}^{s}, s) {-} f^*(X^s, s){\leq} 0\big| \varepsilon, X^s, {\mathcal{L}}\right)}}_{H_0} {-} \underbrace{{\mathbf{P}}\left( {\hat{f}}(\tilde{X}^{s}, s) {+} \tilde\varepsilon {\leq} {\hat{f}}({X}^{s}, s) {+} \varepsilon\big| \varepsilon, X^s, {\mathcal{L}}\right)}_{H_1}\bigg|\enspace.
\end{aligned}$$
Furthermore, if $\Delta({X}^{s}) = f^*({X}^{s}, s) - {\hat{f}}({X}^{s}, s)$, $\Delta(\tilde{X}^{s}) = f^*(\tilde{X}^{s}, s) - {\hat{f}}(\tilde{X}^{s}, s)$, and $\Delta_{\varepsilon} = \varepsilon - \tilde\varepsilon$, then simple algebra yields $$\begin{aligned}
H_1 = {\mathbf{P}}\left( f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s) \leq \Delta_{\varepsilon} + \Delta(\tilde{X}^{s}) - \Delta(X^s)\big| \varepsilon, X^s, {\mathcal{L}}\right)\enspace.
\end{aligned}$$ For all $a, b \in {\mathbb{R}}$ it holds that $|{{\bf 1}_{\left\{a \leq 0\right\}}} - {{\bf 1}_{\left\{a \leq b\right\}}}| \leq {{\bf 1}_{\left\{0\wedge b \leq a \leq 0\vee b\right\}}} \leq {{\bf 1}_{\left\{-|b| \leq a \leq |b|\right\}}} = {{\bf 1}_{\left\{|a| \leq |b|\right\}}}$. Applying this fact to Eq. with $a = f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)$ and $b = \Delta_{\varepsilon} + \Delta(\tilde{X}^{s}) - \Delta(X^s)$ we get $$\begin{aligned}
(*) \leq
&{\mathbf{P}}\left({\left\lvert f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)\right\rvert} \leq \big|{\Delta_{\varepsilon} }\big| + \big|{\Delta(\tilde{X}^{s})}\big| + \big|{\Delta({X}^{s})}\big|\right)\\
\leq
&{\mathbf{P}}\left({\left\lvert f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)\right\rvert} \leq 3 \big|{\Delta_{\varepsilon} }\big|\right)
+
{\mathbf{P}}\left({\left\lvert f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)\right\rvert} \leq 3 \big|{\Delta(\tilde{X}^{s})}\big|\right)\\
&+
{\mathbf{P}}\left({\left\lvert f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)\right\rvert} \leq 3 \big|{\Delta({X}^{s})}\big|\right)\enspace.
\end{aligned}$$ By definition of $\tilde{X}^s$ the random variables $X^s, \tilde{X}^s$ are exchangeable, hence $$\begin{aligned}
{\mathbf{P}}(|{f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)}| \leq 3 |{\Delta(\tilde{X}^{s})}|) = {\mathbf{P}}(|{f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)}| \leq 3 |{\Delta({X}^{s})}|)\enspace.
\end{aligned}$$ Furthermore, using the fact that $|\varepsilon - \tilde\varepsilon| \leq 2 \sigma$ almost surely we get $$\begin{aligned}
\label{eq:third_temr_interm}
(*) \leq {\mathbf{P}}\left({\left\lvert f^*(\tilde{X}^{s}, s) {-} f^*(X^{s}, s)\right\rvert} \leq 6\sigma\right) + 2 {\mathbf{P}}\left({\left\lvert f^*(\tilde{X}^{s}, s) {-} f^*(X^{s}, s)\right\rvert} \leq 3 \big|{\Delta({X}^{s})}\big|\right)\enspace.
\end{aligned}$$ Thanks to Assumption \[ass:density\_rate\] we have the following bound on the first term in Eq. $$\begin{aligned}
{\mathbf{P}}\left({\lvert f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)\rvert} \leq 6\sigma\right) \leq {\mathbf{E}}\left[{\mathbf{P}}\left({\lvert f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)\rvert} \leq 6\sigma | X^s\right)\right] \leq 12\overline{\lambda}_s \sigma\enspace.
\end{aligned}$$ For the second term in Eq. , we observe that Assumption \[ass:density\_rate\] yields almost surely $$\begin{aligned}
{\mathbf{P}}\left({\lvert f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)\rvert} \leq 3 \big|{\Delta({X}^{s})}\big| \big| {\mathcal{L}}, X^s\right) \leq 6 \overline{\lambda}_s\big|{\Delta({X}^{s})}\big|\enspace.
\end{aligned}$$ Thus, taking the total expectation on both sides of this inequality we get $$\begin{aligned}
{\mathbf{P}}\left({\lvert f^*(\tilde{X}^{s}, s) - f^*(X^{s}, s)\rvert} \leq 3\big|{\Delta({X}^{s})}\big|\right) \leq 6\overline{\lambda}_s{\mathbf{E}}\big|{\Delta({X}^{s})}\big| \stackrel{\text{Lemma~\ref{lem:from_cprob_to_exp}}}{\leq} 6\overline{\lambda}_s\mathtt{A}b_n^{-1/2}\enspace.
\end{aligned}$$ Since $\sigma \lesssim b_{n}^{-1/2}$, then we have demonstrated that $(*) \lesssim \overline{\lambda}_sb_n^{-1/2}$. Substituting this bound into Eq. , we derive that $$\begin{aligned}
\label{eq:a_three_final}
\mathbbm{a}^3_{s s'} \lesssim \underline{\lambda}^{-1}_{s'}\overline{\lambda}_{s}b_{n}^{-1/2} + \underline{\lambda}^{-1}_{s'}N_s^{-1/2}\enspace.
\end{aligned}$$
[**Gathering three terms together:**]{} Finally, substituting Eqs. , , into Eq. we get $$\begin{aligned}
\mathbbm{a}_{s s'}
&\lesssim
b_{n}^{-1/2} + \underline{\lambda}^{-1}_{s'}\overline{\lambda}_{s}b_{n}^{-1/2} + \underline{\lambda}_{s'}^{-1} N_{s'}^{-1/2} + \underline{\lambda}^{-1}_{s'}N_s^{-1/2}\enspace.
\end{aligned}$$ Finally, substituting the bound above into Eq. we arrive at $$\begin{aligned}
{\mathbf{E}}{\left\lvert \hat g(X, S) - g^*(X, S)\right\rvert}
\lesssim
&b_{n}^{-1/2} + \left(\sum_{s \in {\mathcal{S}}}p_s\underline{\lambda}_{s}^{-1}\right) \left(\sum_{s \in {\mathcal{S}}}p_s \overline{ \lambda}_{s}\right)b_{n}^{-1/2}\\
&+
\sum_{s \in {\mathcal{S}}} p_s \underline{\lambda}_{s}^{-1} N_{s}^{-1/2} + \left(\sum_{s \in {\mathcal{S}}}p_s\underline{\lambda}_{s}^{-1}\right)\left(\sum_{s \in {\mathcal{S}}}p_sN_s^{-1/2}\right)\\
&+
N^{-1/2}\max_{s \in {\mathcal{S}}}\underline{\lambda}_s^{-1}\sum_{s \in {\mathcal{S}}}\sqrt{p_s(1 - p_s)}\\
\lesssim
&b_{n}^{-1/2} + \sum_{s \in {\mathcal{S}}} p_s N_{s}^{-1/2} + \sqrt{|{\mathcal{S}}|}N^{-1/2}\enspace,
\end{aligned}$$ where in the last inequality we used the fact that $$\begin{aligned}
\sum_{s \in {\mathcal{S}}}\sqrt{p_s(1 - p_s)}
\leq
\sum_{s \in {\mathcal{S}}}\sqrt{p_s} \leq \sqrt{|{\mathcal{S}}|}\sqrt{\sum_{s \in {\mathcal{S}}}p_s} = \sqrt{|{\mathcal{S}}|}\enspace.
\end{aligned}$$ This ends the proof.
Notice that the exact constant in front of the rate of convergence in Theorem \[thm:rate\] can be recovered following the proof. Furthermore, this proof can be extended to control $L_p$ norm $$\begin{aligned}
{\left( {\mathbf{E}}|g^*(X, S) - \hat g(X, S)|^p \right)}^{1/p}\enspace,
\end{aligned}$$ for all $p \in [1, \infty)$ (the current proof deals only with $p = 1$). To achieve it one only needs to extend Lemma \[lem:from\_cprob\_to\_exp\] while the rest of the proof follows line-by-line using deviation results on Wasserstein-$p$ distance on the real line [@Bobkov_Ledoux16]. Finally, it is possible to extend this result under the same assumptions to control ${\mathbf{E}}{\left\lVertg^* - \hat g\right\rVert}_{\infty}$, which induces an extra multiplicative polylogarithmic factor in $b_n^{-1/2}$.
[^1]: Since we are ready to sacrifice a factor $2$ in our bounds, this assumption is without loss of generality.
[^2]: It is assumed in this discussion that the time complexity to evaluate $\hat f$ is $O(1)$.
[^3]: One can achieve it by splitting the labeled dataset ${\mathcal{L}}$ artificially augmenting the unlabeled one, which ensures that $N > n$. In this case if $b_n^{-1/2} = O(n^{-1/2})$, then the first term is always dominant in the derived bound.
[^4]: The source of our method can be found at <https://www.link-anonymous.link>.
[^5]: We thank the authors for sharing a prototype of their code.
|
---
abstract: 'Special partial matchings (SPMs) are a generalisation of Brenti’s special matchings. Let a *pircon* be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti’s zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic $W$-sets constitute pircons. Together, these results extend a result of Can, Cherniavsky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.'
address: 'Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden'
author:
- Nancy Abdallah
- Mikael Hansson
- Axel Hultman
bibliography:
- 'Referenser\_bara\_initialer.bib'
title: Topology of posets with special partial matchings
---
Introduction {#se:intro}
============
A special matching on a poset is a complete matching of the Hasse diagram satisfying certain extra conditions. The concept was introduced by Brenti [@Brenti_intersection]. For eulerian posets, an equivalent notion was also independently introduced by du Cloux [@du_Cloux]. Their main motivation was to provide an abstract framework in which to study the Bruhat order on a Coxeter group. Namely, every non-trivial lower interval in the Bruhat order admits a special matching. Thus, Bruhat orders provide examples of *zircons*, posets in which every non-trivial principal order ideal is finite and has a special matching. Beginning with Marietti [@Marietti], zircons have been the focal point of a lot of attention; see, e.g., [@C-M; @Hultman_zircon; @Marietti_coxeter]. Notably, (the order complex of) any open interval in a zircon is a PL sphere; this is essentially a result of du Cloux [@du_Cloux [Corollary]{} 3.6], which is based on results from Dyer’s thesis [@Dyer_thesis]. Reading [@Reading] provided a different proof.[^1]
In [@A-H], two of the present authors generalised the special matching concept to special partial matchings (SPMs), which are not necessarily complete matchings satisfying similar conditions. Generalising zircons, let us say that a *pircon* is a poset in which every non-trivial principal order ideal is finite and admits an SPM. These notions, too, are originally motivated by Coxeter group theory: the dual of the Bruhat order on the fixed point free involutions in the symmetric group is a pircon [@A-H]. This is generalised considerably in Section \[se:coxeter\], where it is demonstrated that the Bruhat order on the twisted identities ${\i(\t)}$ is a pircon whenever the involution $\t$ has the so-called NOF property. Moreover, Bruhat orders on Rains and Vazirani’s [@R-V] quasiparabolic $W$-sets (under a boundedness assumption) form pircons. In particular, this applies to all parabolic quotients of Coxeter groups.
We investigate the topology of posets with SPMs. Our first main result roughly states that an SPM provides a way to “lift” the PL ball or sphere property from a subinterval; this is [Theorem]{} \[th:main\]. It follows that every open interval in a pircon is a PL ball or a PL sphere, which is our second main result. In particular, this proves a conjecture from [@Hultman3] on Bruhat orders on twisted identities, and confirms a claim from [@R-V] about quasiparabolic $W$-sets.
The overall proof strategy is inspired by that of Reading’s aforementioned proof in [@Reading]. Roughly, if $P$ is a poset with minimum $\hat0$, maximum $\hat1$, and an SPM $M$, we prove that $P$ can be obtained from the interval $[\hat0,M(\hat1)]$ using certain modifications. Investigating the effect of these modifications on the poset topology forms the technical backbone of the paper.
The remainder of the paper is structured in the following way. In the next section, we recall basic definitions and review some useful results from the literature. Then, in Section \[se:PL\_tools\], we prove a couple of elementary lemmas that later serve as the main topological tools. In Section \[se:zip\], ways to locally modify posets, including a version of Reading’s “zippings” from [@Reading], are studied. It is shown that these modifications preserve the PL ball or sphere property. After that, in Section \[se:spm\], we recall the definition of an SPM and prove that a poset which admits an SPM can be obtained from one which in some sense is easier to understand, using the modifications studied in the previous section. Combining the results of the previous two sections, the main results follow essentially at once; this is the content of Section \[se:main\]. In Section \[se:coxeter\], we explain how examples of pircons are provided by Bruhat orders, first on twisted identities and second on quasiparabolic $W$-sets in Coxeter groups. The implications of our second main result in these contexts are discussed. Finally, in the last section, we raise some open questions.
Preliminaries {#se:prel}
=============
In this section, preliminary material on posets (partially ordered sets) and topology of simplicial complexes is gathered.
Posets {#sse:posets}
------
Let $P$ be a poset. If $P$ contains an element denoted $\hat0$ or $\hat1$, it is assumed to be a minimum or a maximum, respectively, i.e., $x \geq \hat0$ and $x \leq \hat1$ for all $x \in P$. The *proper part* of $P$ is then ${\overline}{P}=P-\{\hat0,\hat1\}$.
Standard interval notation is employed for posets. Thus, if $x,y \in P$, then $$[x,y]=\{z \in P \mid x \leq z \leq y\},$$ with the induced order from $P$, and similarly for open and half-open intervals.
An *order ideal* $J \seq P$ is an induced subposet closed under going down, i.e., $x \leq y \in J \To x \in J$. The complement of an order ideal is called an *order filter*. An order ideal is *principal* if it has a maximum. For principal order ideals, the notation $P_{\leq y}=\{x \in P \mid x \leq y\}$ is convenient. Similarly, $P_{<y}$, $P_{\geq y}$, and $P_{>y}$ are defined in the obvious way.
Suppose every principal order ideal in $P$ is finite. If, for any $y \in P$, all maximal chains (totally ordered subsets) in $P_{\leq y}$ have the same number of elements, $P$ is called *graded*. In this case, there is a unique *rank function*, i.e., a function $\operatorname{rk}: P \to \{0,1,\p\}$ such that $\operatorname{rk}(x)=0$ if $x$ is minimal, and $\operatorname{rk}(y)=\operatorname{rk}(x)+1$ if $y$ covers $x$.
Suppose $\pi: P \to P'$ is an order-preserving map of posets. Then $\pi$ is called an *order projection* if for every ordered pair $x' \leq_{P'} y'$ in $P'$ there exist $x \leq_P y$ in $P$ such that $\pi(x)=x'$ and $\pi(y)=y'$. In particular, any order projection is surjective. We construct the quotient ${\mathcal{F}}_\pi$ as follows. The elements of ${\mathcal{F}}_\pi$ are the fibres $\pi^{-1}(x')=\{x \in P \mid \pi(x)=x'\}$ for $x' \in P'$. A relation on ${\mathcal{F}}_\pi$ is given by $F_1 \leq_{{\mathcal{F}}_\pi} F_2$ if $x \leq_P y$ for some $x \in F_1$ and $y \in F_2$. This is a partial order if $\pi$ is an order projection. We then call ${\mathcal{F}}_\pi$ the *fibre poset*. It is isomorphic to $P'$:
\[le:fibre\] If $\pi: P \to P'$ is an order projection, then ${\mathcal{F}}_\pi$ and $P'$ are isomorphic posets.
Simplicial complexes {#sse:complexes}
--------------------
*Throughout the present paper, all simplicial complexes are finite.* By convention, the empty set is considered to be a simplex of every non-void simplicial complex. Given an (abstract) simplicial complex $\De$, we shall denote its geometric realisation (defined up to linear homeomorphism) by $\|\De\|$, a polyhedron in some real euclidean space. The simplices of $\De$ are sometimes called its *faces*, and maximal faces are referred to as *facets*.
For a face $\si \in \De$, the subcomplex $$\operatorname{lk}_\De(\si)=\{\tau \in \De \mid \h{$\si \cap \tau=\0$ and $\si \cup \tau \in \De$}\}$$ is the *link* of $\si$.
If $V$ is a set of vertices of $\De$, the *deletion* of $V$ in $\De$ is the subcomplex $$\operatorname{del}_\De(V)=\{\si \in \De \mid \si \cap V=\0\}.$$
The *join* $\De {*}\De'$ of two simplicial complexes $\De$ and $\De'$ is a new simplicial complex defined (up to isomorphism) as follows. Suppose the vertex sets of $\De$ and $\De'$ are disjoint (otherwise, first replace $\De'$, say, by a suitable isomorphic copy), and let $$\De {*}\De'=\{\si \cup \tau \mid \h{$\si \in \De$ and $\tau \in \De'$}\}.$$
If $\mathcal{F}$ is a finite family of finite sets, $\operatorname{cl}(\mathcal{F})$ denotes the simplicial complex generated by $\mathcal{F}$, i.e., $$\operatorname{cl}(\mathcal{F})=\{\si \mid \h{$\si \seq F$ for some $F \in \mathcal{F}$}\};$$ it is called the *closure* of $\mathcal{F}$.
Let $\si \prec \tau$ indicate that $\si \subset \tau$ and $\dim\si=\dim\tau-1$. If $\si \prec \tau$ and $\tau$ is the unique face (necessarily a facet) of $\De$ which properly contains $\si$, then the modification $\De \searrow \De-\{\si,\tau\}$ is an *elementary collapse*. A simplicial complex $\De$ is *collapsible* if $\De \searrow \cd \searrow \0$. Forman’s discrete Morse theory [@Forman] provides a convenient method to establish collapsibility. The formulation in terms of matchings which we use here is due to Chari [@Chari]; see also Forman [@Forman2].
A *complete matching* on $\De$ is a function $\mu: \De \to \De$ which satisfies $\mu^2={\mathrm{id}}$ and either $\si \prec \mu(\si)$ or $\mu(\si) \prec \si$ for all $\si \in \De$. Then $\mu$ is *acyclic* if $$\si_0 \prec \mu(\si_0) \succ \si_1 \prec \mu(\si_1) \succ \cd \prec \mu(\si_{t-1}) \succ \si_t$$ with $\si_0 \neq \si_1$ implies that $\si_t \neq \si_0$.
\[le:forman\] A simplicial complex is collapsible if it has an acyclic complete matching.
Given a finite poset $P$, its *order complex* $\De(P)$ is the simplicial complex whose faces are the chains in $P$. In order to prevent proliferation of brackets when taking order complexes of poset intervals, we shall write ${\De(x,y)}$ instead of $\De((x,y))$, $\De[x,y)$ instead of $\De([x,y))$, and so on.
PL topology {#sse:topology}
-----------
Next, some notions from PL topology are reviewed. We refer to, e.g., [@Hudson] or [@Rourke-Sanderson] for this and much more information.
Suppose $\De$ and $\De'$ are simplicial complexes. A continuous map $f: \|\De\| \to \|\De'\|$ is *piecewise linear*, or *PL*, if its graph is a euclidean polyhedron. This is equivalent to there being simplicial subdivisions $\tilde{\De}$ and $\tilde{\De}'$ of $\De$ and $\De'$, respectively, with respect to which $f$ is a simplicial map of the corresponding triangulations of $\|\De\|$ and $\|\De'\|$.
Say that $\De$ and $\De'$ are *PL homeomorphic* if there exists a PL homeomorphism $f: \|\De\| \to \|\De'\|$ (it follows that $f^{-1}$ is also PL).
A *PL $d$-ball* is a simplicial complex which is PL homeomorphic to the simplicial complex $\De^d$ whose only facet is the $d$-dimensional simplex. A *PL $(d-1)$-sphere* is a simplicial complex which is PL homeomorphic to the simplicial complex obtained by removing the facet from $\De^d$. In the PL category, balls and spheres behave as expected with respect to joins:
\[le:join\] Let $B^d$ denote a PL $d$-ball and $S^d$ a PL $d$-sphere. Then $B^k {*}B^l \cong B^k {*}S^l \cong B^{k+l+1}$ and $S^k {*}S^l \cong S^{k+l+1}$, where $\cong$ means PL homeomorphic.
In particular, the cone over a PL $d$-ball or a PL $d$-sphere is a PL $(d+1)$-ball, since a cone is a join with the $0$-ball.
A *PL $d$-manifold* is a simplicial complex satisfying that, for all $k \geq 0$, the link of every $k$-dimensional face is a PL $(d-1-k)$-ball or sphere. If $\De$ is a PL $d$-manifold, its *boundary* $\pa\De$ is the simplicial complex whose facets are the $(d-1)$-dimensional faces of $\De$ that are contained in only one facet of $\De$. PL $d$-balls are PL $d$-manifolds with PL $(d-1)$-spheres as boundaries. PL $d$-spheres are PL $d$-manifolds without boundaries.
If $P$ is a finite poset with $\hat0$ and $\hat1$, every link in the order complex $\De({\overline}{P})$ is a join of order complexes of open intervals in $P$. Hence, by Lemma \[le:join\], $\De({\overline}{P})$ is a PL manifold if and only if $P$ is graded and ${\De(x,y)}$ is a PL ball or sphere for every interval $(x,y) \neq (\hat0,\hat1)$ in $P$.
As we shall see, the next lemma opens up for inductive arguments. However plausible it seems, the first statement would be false without the PL condition.
\[le:PL\] ${}$
- If $\De_1$ and $\De_2$ are PL $d$-balls and $\De_1 \cap \De_2$ is a PL $(d-1)$-ball contained in $\pa\De_1 \cap \pa\De_2$, then $\De_1 \cup \De_2$ is a PL $d$-ball.
- If $\De_1$ and $\De_2$ are PL $d$-balls with $\De_1 \cap \De_2=\pa\De_1=\pa\De_2$, then $\De_1 \cup \De_2$ is a PL $d$-sphere.
For a proof of (i), see [@Hudson [Corollary]{} 1.28]. A proof of (ii) can be found in [@Mandel_thesis].
Although the second sentence of the following result is rarely stated explicitly, it follows from, e.g., the first part of Hudson’s proof; see [@Hudson [Theorem]{} 1.26].
\[le:newman\] The closure of the complement of a PL $d$-ball embedded in a PL $d$-sphere is a PL $d$-ball. Moreover, the two balls have the same boundary.
In particular, the deletion of a single vertex $v$ in a PL $d$-sphere is a PL $d$-ball, since it is the closure of the complement of a cone over the link of $\{v\}$.
\[Hudson\] If $A$ is a PL $d$-ball and $F$ is a PL $(d-1)$-ball contained in $\pa{A}$, then any PL homeomorphism $\|F\| \to \|\De^{d-1}\|$ extends to a PL homeomorphism $\|A\| \to \|\De^d\|$.
\[le:whitehead\] A collapsible PL manifold is a PL ball.
PL topological tools {#se:PL_tools}
====================
In this section, we develop elementary PL topological machinery that will serve as our toolbox in the proofs of the main results.
Let ${\mathbf{2}}$ denote the totally ordered, two-element poset $\{\al,\be\}$ where $\al<\be$.
\[le:collapsible\] If $P$ is a finite poset with $\hat0$ and $\hat1$, then $\De({\overline}{P \times {\mathbf{2}}}-\{(\hat0,\be)\})$ is collapsible.
We shall apply Lemma \[le:forman\]. For brevity, let $Q={\overline}{P \times {\mathbf{2}}}-\{(\hat0,\be)\}$. Given a chain $$C=\{(x_1,\ga_1) < (x_2,\ga_2) <\cd <(x_m,\ga_m)\} \seq Q,$$ put $(x_{m+1},\ga_{m+1})=(\hat1,\be)$, and let $j$ be the smallest index such that $\ga_j=\be$. Define $p(C)=(x_j,\al)$. Observe that $C \cup \{p(C)\}$ is a chain in $Q$, and that $p(C \cup \{p(C)\})=p(C)=p(C-\{p(C)\})$. Therefore, $$\mu(C)=\begin{cases}C \cup \{p(C)\} & \h{if $p(C) \notin C$,} \\ C-\{p(C)\} & \h{otherwise}\end{cases}$$ defines a complete matching $\mu$ on $\De(Q)$. Now, if $C_0 \prec \mu(C_0) \succ C_1 \prec \mu(C_1)$ for chains $C_0 \neq C_1$, then $C_1$ has fewer elements than $C_0$ with $\be$ as the second component. Hence $\mu$ is acyclic.
\[le:extension\] Suppose $P$ is a finite poset with $\hat0$ and $\hat1$. If $\De({\overline}{P})$ is a PL $d$-ball (a PL $d$-sphere), then $\De({\overline}{P \times {\mathbf{2}}})$ is a PL $(d+1)$-ball (a PL $(d+1)$-sphere). In either case, $\De({\overline}{P \times {\mathbf{2}}}-\{(\hat0,\be)\})$ is a PL $(d+1)$-ball.
Let $R={\overline}{P \times {\mathbf{2}}}$ and $Q=R-\{(\hat0,\be)\}$. We induct on $d$, all assertions being clear when $d=0$.
For $p \in {\overline}{P}$, we have the following two poset isomorphisms: $$Q_{<(p,\ga)} \cong \begin{cases}{\overline}{P_{\leq p}} & \h{if $\ga=\al$,} \\ {\overline}{P_{\leq p} \times {\mathbf{2}}}-\{(\hat0,\be)\} & \h{if $\ga=\be$,}\end{cases}$$ and $$Q_{>(p,\ga)} \cong \begin{cases}{\overline}{P_{\geq p} \times {\mathbf{2}}} & \h{if $\ga=\al$,} \\ {\overline}{P_{\geq p}} & \h{if $\ga=\be$.}\end{cases}$$ Moreover, $Q_{<(1,\al)} \cong {\overline}{P}$. The induction assumption therefore implies that all links of non-empty faces in $\De(Q)$ are PL balls or spheres. Hence $\De(Q)$ is a PL $(d+1)$-manifold. Now Lemmas \[le:whitehead\] and \[le:collapsible\] imply that $\De(Q)$ is a PL $(d+1)$-ball.
Next observe that $$\De(R)=\De(Q) \cup \De\Big(R_{\geq (\hat0,\be)}\Big).$$ Both complexes in the union are PL $(d+1)$-balls; the latter is isomorphic to a cone over $\De({\overline}{P})$. Furthermore, we have $$\De(Q) \cap \De\Big(R_{\geq (\hat0,\be)}\Big)=\De\Big(R_{> (\hat0,\be)}\Big),$$ which is contained in the boundary of both balls. On the other hand, this intersection is isomorphic to $\De({\overline}{P})$. The desired conclusions about $\De(R)$ now follow from Lemma \[le:PL\].
We shall frequently find the need to modify simplicial complexes by replacing balls with other balls. The following two statements describe circumstances under which the topology is left unchanged.
\[ball\] Suppose $\De$, $A$, and $A'$ are PL $d$-balls such that $A \seq \De$ and $A' \cap \De=\pa{A'}=\pa{A}$. Then $(\De-A) \cup A'$ is a PL $d$-ball.
Let $C$ be a cone over $\pa\De$ whose apex $v$ is disjoint from $A'$ and $\De$. By Lemma \[le:PL\](ii), $S=\De \cup C$ is a PL $d$-sphere. Put $a=\operatorname{cl}(S-A)$, which is a PL $d$-ball with $\pa{a}=\pa{A}$ by Lemma \[le:newman\]. Since $A' \cap \De=\pa{A'}=\pa{A}$, $\Si=a \cup A'$ is a PL $d$-sphere by Lemma \[le:PL\](ii). Hence, $$\operatorname{del}_\Si(\{v\})=(\De-A) \cup A'$$ is a PL $d$-ball.
\[homeo\] Let $\De$ be a simplicial complex. Suppose $A$ and $A'$ are PL $d$-balls and $F$ is a PL $(d-1)$-ball such that $A \seq \De$ and $F \seq \pa{A} \cap \pa{A'}$. If both $\operatorname{cl}(\De-A) \cap A$ and $\operatorname{cl}(\De-A) \cap A'$ are contained in $F$, then $\De$ and $(\De-A) \cup A'$ are PL homeomorphic.
There is a PL homeomorphism $\f: \|F\| \to \|\De^{d-1}\|$. By Lemma \[Hudson\], it extends to PL homeomorphisms $\f_1: \|A\| \to \|\De^d\|$ and $\f_2: \|A'\| \to \|\De^d\|$. Let $\psi=\f_2^{-1} \circ \f_1$. Then $\psi: \|A\| \to \|A'\|$ is a PL homeomorphism whose restriction to $\|F\|$ is the identity map. Obviously, $\De=\operatorname{cl}(\De-A)\cup A$. Moreover, $(\De-A) \cup A'=\operatorname{cl}(\De-A) \cup A'$ because $\operatorname{cl}(\De-A) \seq (\De-A) \cup F$. Now define $f: \|\De\| \to \|(\De-A) \cup A'\|$ by $$f(x)=\begin{cases}\psi(x) & \h{if $x \in \|A\|$,} \\ x & \h{if $x \in \|\operatorname{cl}(\De-A)\|$.}\end{cases}$$ Then $f$ is a well-defined PL map because $\operatorname{cl}(\De-A)\cap A \seq F$, and the same holds for $f^{-1}$ since $\operatorname{cl}(\De-A)\cap A' \seq F$.
Zippings and removals {#se:zip}
=====================
In [@Reading], Reading introduced the concept of a zipper in a poset. We restrict his definition somewhat.
Let $P$ be a finite poset with $\hat0$ and $\hat1$, and distinct elements $x,y,z \in P$. Call $(x,y,z)$ a *zipper* if
- $z$ covers only $x$ and $y$,
- $z=x \vee y$, where $\vee$ denotes join (supremum), and
- $[\hat0,x)=[\hat0,y)$.
The zipper is *proper* if $z \neq \hat1$.
Given $P$ with a partial order $\leq$ and a proper zipper $(x,y,z)$, let $P'=(P-\{x,y,z\}) \biguplus \{{x'}\}$, and define a partial order $\leq'$ on $P'$ by
- $a \leq' b$ if $a \leq b$,
- ${x'}\leq' a$ if $x \leq a$ or $y \leq a$,
- $a \leq' {x'}$ if $a \leq x$ (or, equivalently, $a \leq y$), and
- ${x'}\leq' {x'}$.
The fact that $\leq'$ is a partial order on $P'$ is [@Reading [Proposition]{} 4.1]. We say that $P'$ is the result of a *zipping* in $P$. The effect is that $P'$ is obtained from $P$ by identifying the elements $x$, $y$, and $z$; they become the element ${x'}$. Reading proved that this preserves PL spheres:
\[4.7\] If $P'$ is obtained from $P$ by zipping a proper zipper and $\De({\overline}{P})$ is a PL $d$-sphere, then so is $\De({\overline}{P'})$.
We shall prove a similar result for PL balls. In contrast to spheres, balls have boundaries. This causes complications that can be overcome by imposing additional restrictions on zippers. A version which suffices for our needs is the content of the next definition.
Recall that a *coatom* in a poset with $\hat1$ is an element covered by $\hat1$.
\[de:clean\] A zipper $(x,y,z)$ is *clean* if it is proper, and for some coatom $c$ there exists a poset isomorphism $\f: [x,\hat1] \to [x,c] \times {\mathbf{2}}$ such that $\f(z)=(x,\be)$.
\[MT1\] If $P'$ is obtained from $P$ by zipping a clean zipper and $\De({\overline}{P})$ is a PL $d$-ball, then so is $\De({\overline}{P'})$.
Suppose $\De({\overline}{P})$ is a PL $d$-ball and $(x,y,z)$ is a clean zipper in $P$. Let ${\De_{xyz}}$ be the simplicial complex whose facets are the maximal chains in ${\overline}{P}$ containing $x$ or $y$ (note that this includes all that contain $z$), and let ${\De_{{x'}}'}$ be the simplicial complex whose facets are the maximal chains in ${\overline}{P'}$ containing ${x'}$. By the definition of a zipping, $\De({\overline}{P}-\{x,y,z\})=\De({\overline}{P'}-\{{x'}\})$ and ${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}={\operatorname{del}_{{\De_{{x'}}'}}(\{{x'}\})}$. Hence, $$\begin{aligned}
\label{eq:ball_exchange}
\De({\overline}{P'}) &= \De({\overline}{P'}-\{{x'}\}) \cup {\De_{{x'}}'}\\
&= \De({\overline}{P}-\{x,y,z\}) \cup {\De_{{x'}}'}\nonumber \\
&= (\De({\overline}{P})-{\De_{xyz}})\cup {\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}\cup {\De_{{x'}}'}\nonumber \\
&= (\De({\overline}{P})-{\De_{xyz}})\cup {\operatorname{del}_{{\De_{{x'}}'}}(\{{x'}\})}\cup {\De_{{x'}}'}\nonumber \\
&= (\De({\overline}{P})-{\De_{xyz}})\cup {\De_{{x'}}'}. \nonumber\end{aligned}$$ That is, $\De({\overline}{P'})$ is obtained from $\De({\overline}{P})$ by removing ${\De_{xyz}}$ and inserting ${\De_{{x'}}'}$. Our goal is to apply either Lemma \[ball\] or Lemma \[homeo\] with $\De=\De({\overline}{P})$, $A={\De_{xyz}}$, $A'={\De_{{x'}}'}$, and (if needed) $F={\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}={\operatorname{del}_{{\De_{{x'}}'}}(\{{x'}\})}$. The hypotheses must be verified.
Even though it originally concerns the situation when $\De({\overline}{P})$ is a sphere, the appropriate part of Reading’s proof of [@Reading [Theorem]{} 4.7] shows that ${\De_{xyz}}$ is a PL $d$-ball also in our situation.[^2]
Next we observe that ${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}\seq \pa{\De_{xyz}}$. Indeed, since $z=x \vee y$, the cleanness of $(x,y,z)$ implies that every facet $C$ in ${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}$ contains some $w$ which covers exactly one of $x$ and $y$, say $x$. Hence, $C$ extends uniquely to a facet in ${\De_{xyz}}$, namely by adding $x$.
If ${\De{(y,\hat1)}}$ and $\De(\hat0,x)$ are PL spheres, ${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}$ is a PL $(d-1)$-sphere. Otherwise, ${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}$ is a PL $(d-1)$-ball.
Let us assume this claim for now and turn to its proof later.
Suppose first that ${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}$ is a sphere. Since it cannot be a proper subcomplex of another $(d-1)$-sphere, ${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}=\pa{\De_{xyz}}$. Since ${\De_{{x'}}'}$ is a cone over the PL sphere ${\operatorname{del}_{{\De_{{x'}}'}}(\{{x'}\})}={\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}$ with apex ${x'}$, ${\De_{{x'}}'}$ is a PL $d$-ball and ${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}=\pa{\De_{{x'}}'}$. By Lemma \[ball\] and , $\De({\overline}{P'})$ is a PL $d$-ball.
Now suppose ${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}$ is a ball. Since ${\De_{{x'}}'}$ is a cone over this ball with apex ${x'}$, ${\De_{{x'}}'}$ is a PL $d$-ball with the PL $(d-1)$-ball ${\operatorname{del}_{{\De_{{x'}}'}}(\{{x'}\})}={\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}$ contained in its boundary. Observe that $$\operatorname{cl}(\De({\overline}{P})-{\De_{xyz}}) \cap {\De_{xyz}}\seq \De({\overline}{P}-\{x,y,z\}) \cap {\De_{xyz}}={\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}$$ and $$\operatorname{cl}(\De({\overline}{P})-{\De_{xyz}}) \cap {\De_{{x'}}'}\seq \De({\overline}{P'}-\{{x'}\}) \cap {\De_{{x'}}'}={\operatorname{del}_{{\De_{{x'}}'}}(\{{x'}\})}.$$ Lemma \[homeo\] now shows that $\De({\overline}{P})$ and $(\De({\overline}{P})-{\De_{xyz}}) \cup {\De_{{x'}}'}$ are PL homeomorphic. By , $\De({\overline}{P'})$ is a PL $d$-ball.
It remains to verify the claim. Define $\De={\De{(y,\hat1)}}$, $A=\De[z,\hat1)$, and $A'=\De({(x,\hat1)}-\{z\})$. Observe that $${\operatorname{del}_{{\De_{xyz}}}(\{x,y,z\})}=\De(\hat0,x) {*}((\De-A)\cup A').$$ By Lemma \[le:join\], the claim follows if $\De$ and $(\De-A)\cup A'$ are PL homeomorphic. There are two cases:
**Case 1: $\De$ is a PL $k$-sphere.** In this case, ${\De{(z,\hat1)}}$ and ${\De{(x,\hat1)}}$ are spheres, the former because it is a link in ${\De{(y,\hat1)}}$, the latter by Lemma \[le:extension\] because $[x,c] \cong [z,\hat1]$ with $c$ being the coatom of [Definition]{} \[de:clean\]. Hence, $A$ and $A'$ are PL $k$-balls, and $\pa{A}=\pa{A'}={\De{(z,\hat1)}}=\De \cap A'$. By Lemma \[le:newman\], $\operatorname{cl}(\De-A)$ is a PL $k$-ball, and thus Lemma \[le:PL\](ii) implies that $\operatorname{cl}(\De-A) \cup A'=(\De-A) \cup A'$ is a PL $k$-sphere, as desired.
**Case 2: $\De$ is a PL $k$-ball.** We shall apply Lemma \[ball\] or Lemma \[homeo\], the latter with $F={\De{(z,\hat1)}}$. Again, there is a coatom $c$ such that $[x,c] \cong [z,\hat1]$. By Lemma \[le:extension\], $A'$ is a PL $k$-ball, as are $A$ and $\De$, whereas $F$ is either a PL $(k-1)$-ball or a PL $(k-1)$-sphere. Since $A$ is a cone over $F$, $F \seq \pa{A}$. Consider a maximal chain $C$ in ${(z,\hat1)}$ with minimum $w$ (let $w=\hat1$ if ${(z,\hat1)}$ is empty). Then $\f(w)=(v,\be)$ for some $v \leq c$ which covers $x$, where $\f: [x,\hat1] \to [x,c] \times {\mathbf{2}}$ is the poset isomorphism provided by [Definition]{} \[de:clean\]. The only way to extend $C$ to a maximal chain in ${(x,\hat1)}-\{z\}$ is to add $v$. Hence $F \seq \pa{A'}$.
If $F$ is a sphere, we have $F=\pa{A}=\pa{A'}$ since a sphere cannot be a proper subcomplex of another sphere of the same dimension. Lemma \[ball\] then shows that $(\De-A)\cup A'$ is a PL $k$-ball.
If, instead, $F$ is a ball, we observe that $$\operatorname{cl}(\De-A) \cap A \seq \De({(y,\hat1)}-\{z\}) \cap \De[z,\hat1)={\De{(z,\hat1)}}=F$$ and $$\operatorname{cl}(\De-A) \cap A' \seq \De({(y,\hat1)}-\{z\}) \cap \De({(x,\hat1)}-\{z\})={\De{(z,\hat1)}}=F.$$ Thus, Lemma \[homeo\] implies that $(\De-A) \cup A'$ is a PL $k$-ball. The claim is established.
In addition to zippings, we shall find the need for another way to modify posets which also preserves PL balls.
\[de:removable\] Let $P$ be a finite poset with $\hat0$ and $\hat1$. An element $z \neq \hat1$ is called *removable* if $z$ covers exactly one element $x$, and for some coatom $c$ there exists a poset isomorphism $\f: [x,\hat1] \to [x,c] \times {\mathbf{2}}$ such that $\f(z)=(x,\be)$.
If $z \in P$ is removable, we shall refer to $P-\{z\}$ as obtained by a *removal*. Alternatively, in analogy with zippings, we may consider $P-\{z\}$ as being obtained by identifying $x$ and $z$. Removals produce balls from PL balls or spheres:
\[MT2\] Suppose $z \in P$ is removable. If $\De({\overline}{P})$ is a PL $d$-ball or a PL $d$-sphere, then $\De({\overline}{P}-\{z\})$ is a PL $d$-ball.
Let $x$ and $c$ be as in [Definition]{} \[de:removable\]. Since $\De(x,c)$ is a PL ball or sphere, $\De({(x,\hat1)}-\{z\})$ is a PL ball by Lemma \[le:extension\]. If $x=\hat0$ we are done, so suppose $x>\hat0$. Then $\De({\overline}{P})$ is a ball since the link of $\{z\}$ is a cone with apex $x$ and therefore not a sphere. Let $\De_x$ be the simplicial complex whose facets are maximal chains in ${\overline}{P}$ containing $x$. We shall apply Lemma \[homeo\] with $\De=\De({\overline}{P})$, $A=\De_x$, $A'=\operatorname{del}_{\De_x}(\{z\})$, and $F=\operatorname{del}_{\De_x}(\{x,z\})$. Since $\De_x$ is a cone over , $A$ is a PL $d$-ball satisfying $F \seq \pa{A}$. Furthermore, $F=\De(\hat0,x) {*}\De({(x,\hat1)}-\{z\})$, which is a PL $(d-1)$-ball by Lemma \[le:join\], and $A'$ is a cone over $F$, hence a PL $d$-ball with $F$ in its boundary. Finally, $\operatorname{cl}(\De-A) \cap A' \seq \operatorname{cl}(\De-A) \cap A \seq F$ because every chain which contains $z$ or $x$ is contained in $\De_x$. By Lemma \[homeo\], $$(\De-A) \cup A'=(\De({\overline}{P})-\De_x) \cup \operatorname{del}_{\De_x}(\{z\})=\De({\overline}{P}-\{z\})$$ is PL homeomorphic to $\De=\De({\overline}{P})$.
Special partial matchings {#se:spm}
=========================
The following definition is taken from [@A-H].
\[de:spm\] Suppose $P$ is a finite poset with $\hat1$, and let $\cov$ denote its cover relation. A *special partial matching*, or *SPM*, on $P$ is a function $M: P \to P$ such that
- $M^2={\mathrm{id}}$,
- $M(\hat1) \cov \hat1$,
- for all $x\in P$, we have $M(x) \cov x$, $M(x)=x$, or $x\cov M(x)$, and
- if $x\cov y$ and $M(x) \neq y$, then $M(x)<M(y)$.
The terminology comes from the fact that an SPM without fixed points is precisely a *special matching* as defined by Brenti [@Brenti_intersection].
For special matchings, the following important lemma is essentially due to Brenti; see [@Brenti_intersection Lemma 4.2], which is, however, stated under a gradedness assumption. A proof without this assumption appears in [@Hultman_zircon]. We provide here a different proof which is valid also for SPMs.
\[le:lifting\] Suppose that $P$ is a finite poset with $\hat1$, and $M$ is an SPM on $P$. If $x,y \in P$ with $x<y$ and $M(y) \leq y$, then
- $M(x) \leq y$,
- $M(x) \leq x \To M(x)<M(y)$, and
- $M(x) \geq x \To x \leq M(y)$.
It suffices to prove (i) and (ii) because together they imply (iii).
Consider a saturated chain $x=x_0 \cov x_1 \cov \cd \cov x_k=y$. By the definition of an SPM, for each $i<k$, either $M(x_i)<M(x_{i+1})$ or $M(x_i)=x_{i+1}$.
\(i) We either have $M(x_0)<M(x_1)<\cd<M(y) \leq y$ or $M(x_0)<M(x_1)<\cd<M(x_i)=x_{i+1} \leq y$ for some $i<k$.
\(ii) We either have $M(x_0)<M(x_1)<\cd<M(y)$ or $M(y)>M(x_{k-1})>\cd>M(x_{i+1})=x_i \geq x \geq M(x)$ for some $i<k$.
Next, a fundamental construction is described. It presents a poset with an SPM as the image of an order projection of a poset which in an appropriate sense is easier to understand. This extends Reading’s corresponding construction for Bruhat intervals [@Reading Section 5].
Let $P$ be a finite poset with $\hat0$ and $\hat1$. Assume $M$ is an SPM on $P$, and define $\pi: [\hat0,M(\hat1)] \times {\mathbf{2}}\to P$ by $$(p,\ga) \mapsto \begin{cases}M(p) & \h{if $\ga=\be$ and $p \cov M(p)$,} \\ p & \h{otherwise.}\end{cases}$$ It is readily checked that the fibres of $\pi$ are as follows: $$\label{eq:fibres}
\pi^{-1}(p)=\begin{cases}\{(M(p),\be)\} & \h{if $p \not\leq M(\hat1)$,} \\
\{(p,\al)\} & \h{if $p<M(p)$,} \\
\{(p,\al),(p,\be)\} & \h{if $p=M(p)$,} \\
\{(p,\al),(M(p),\be),(p,\be)\} & \h{if $M(p)<p \leq M(\hat1)$.}\end{cases}$$
The map $\pi$ is an order projection. In particular, $P$ is isomorphic to the fibre poset ${\mathcal{F}}_\pi$.
For brevity, define $Q=[\hat0,M(\hat1)] \times {\mathbf{2}}$. First we show that $\pi:Q \to P$ is order-preserving. Suppose $(p',\ga') \leq (p,\ga)$ in $Q$. The only non-obvious case to consider is when $\pi((p',\ga'))=M(p')$. Then, if $\pi((p,\ga))=M(p)$, $M(p') \leq M(p)$ follows from the lifting property since $p<M(p)$ in this case. If, instead, $\pi((p,\ga))=p$ we have $M(p)\leq p$ because $\ga=\be$. Hence, lifting yields $M(p') \leq p$, as desired. Thus $\pi$ is order-preserving.
Now assume $p' \leq p$ in $P$. We have to produce $q' \in \pi^{-1}(p')$ and $q \in \pi^{-1}(p)$ such that $q' \leq q$ in $Q$.
- If $p \leq M(\hat1)$, we may use $q'=(p',\al)$ and $q=(p,\al)$.
- If $p \not\leq M(\hat1)$ and $M(p') \geq p'$, use $q'=(p',\al)$ and $q=(M(p),\be)$; lifting first implies $M(p)<p$ and then $p' \leq M(p)$.
- Finally, if $p \not\leq M(\hat1)$ and $M(p')<p'$, we may take $q'=(M(p'),\be)$ and $q=(M(p),\be)$; here lifting first yields $M(p)<p$ and then $M(p') \leq M(p)$.
Thus $\pi$ is an order projection. By Lemma \[le:fibre\], $P$ and ${\mathcal{F}}_\pi$ are isomorphic.
The previous lemma describes a poset with an SPM as a fibre poset. Next, we show that the fibre poset can be constructed from the domain of the order projection using modifications that change the topology in a controlled manner. This is analogous to Reading’s [@Reading [Theorem]{} 5.5].
\[th:convert\] Let $P$ be a finite poset with $\hat0$ and $\hat1$. If $M$ is an SPM on $P$, then $P$ can be obtained from $[\hat0,M(\hat1)] \times {\mathbf{2}}$ by a sequence of clean zippings and removals.
Again, let $Q=[\hat0,M(\hat1)] \times {\mathbf{2}}$. Suppose $F_1,\p,F_t$ is a linear extension of the fibre poset ${\mathcal{F}}_\pi$ of the order projection $\pi: Q \to P$. This means that $$\label{eq:classes}
F_k \ni x \leq y \in F_l \To k \leq l.$$
Consider the sequence of posets $Q=P_0,P_1,\p,P_t={\mathcal{F}}_\pi \cong P$, where $P_i$ is obtained from $P_{i-1}$ by identifying the elements of $F_i$. More precisely, as sets, $$P_i=\Bigg(Q-\bigcup_{j=1}^i{F_j}\Bigg) \cup \{F_1,\p,F_i\},$$ and the order on $P_i$ is given by $a \leq_{P_i}b$ if and only if (i) $a,b \in Q$ and $a \leq_Q b$, (ii) $a=F_k$, $b \in Q$, and $x \leq_Q b$ for some $x \in F_k$, or (iii) $a=F_k$, $b=F_l$, and $x \leq_Q y$ for some $x \in F_k$ and $y\in F_l$.
Clearly, $P_i \cong P_{i+1}$ if $|F_{i+1}|=1$. It suffices to prove that $P_{i+1}$ is obtained from $P_i$ by a removal if $|F_{i+1}|=2$ and by a clean zipping if $|F_{i+1}|=3$.
Suppose first $|F_{i+1}|=2$, so that $F_{i+1}=\{(p,\al),(p,\be)\}$ for some $p=M(p)$. We must show that $(p,\be)$ only covers $(p,\al)$ in $P_i$. By , all other elements below $(p,\be)$ in $P_i$ are of the form $F_k$, $k \leq i$. Suppose $(p,\be)$ covers $F_k$. Then there exists $(p',\be) \in F_k$ such that $p$ covers $p'$ in $P$. Since $M$ is an SPM, $M(p') \leq p'$, which, by , implies $(p',\al) \in F_k$. Hence $(p,\al)>_{P_i} F_k$, which is the desired contradiction.
The conditions on removable elements that are left to check involve only the structure of the order filter generated by $(p,\al)$. By , this order filter in $P_i$ is equal to the same order filter in $Q$. In $Q$, however, the conditions are obvious (as the coatom $c$, take $(M(\hat1),\al)$).
Second, assume $|F_{i+1}|=3$ with $F_{i+1}=\{(p,\al),(M(p),\be),(p,\be)\}$; in particular, this means that $M(p)<p \leq M(\hat1)$. We have to show that $((p,\al),(M(p),\be),(p,\be))$ is a clean zipper in $P_i$. That $(p,\be)$ only covers $(p,\al)$ and $(M(p),\be)$ in $P_i$ is shown in the same way as when $|F_{i+1}|=2$. Next, let us verify that $(p,\al)$ and $(M(p),\be)$ are above the same elements. By , only fibres $F_k$, $k \leq i$, need to be considered. So, suppose $F_k <_{P_i} (p,\al)$. That is, there exists $(p',\al) \in F_k$ with $p'<p$.
- If $M(p')<p'$, $(M(p'),\be) \in F_k$. The lifting property asserts that $M(p')<M(p)$. Therefore, $F_k <_{P_i} (M(p),\be)$.
- If $M(p') \geq p'$, lifting implies $p' \leq M(p)$. Hence, $(p',\al) <_Q (M(p),\be)$ so that, again, $F_k <_{P_i} (M(p),\be)$.
Now, suppose instead $F_k <_{P_i} (M(p),\be)$.
- If $(p',\al) \in F_k$ for some $p' \leq M(p)$, then $(p',\al) <_Q (p,\al)$ and $F_k <_{P_i} (p,\al)$.
- Otherwise, shows that $\{(M(p'),\be),(p',\al)\} \seq F_k$ holds for some $M(p') \leq p'$ and $M(p')<M(p)$. Then the lifting property yields $p'<p$. Thus, $(p',\al) <_Q (p,\al)$ and $F_k <_{P_i} (p,\al)$.
The conditions on clean zippers that remain to be verified involve only the structure of the order filter generated by $(p,\al)$ and $(M(p),\be)$. As before, the conditions hold in $Q$, hence in $P_i$ by .
Main results {#se:main}
============
Combining the material of the previous two sections, we obtain strong topological statements about posets with special partial matchings. These assertions, which are recorded in this section, form our main results.
\[th:main\] Let $P$ be a finite poset with $\hat0$ and $\hat1$, and suppose $M$ is an SPM on $P$. If $\De(\hat0,M(\hat1))$ is a PL $d$-ball, then $\De({\overline}{P})$ is a PL $(d+1)$-ball. If $\De(\hat0,M(\hat1))$ is a PL $d$-sphere, then $\De({\overline}{P})$ is a PL $(d+1)$-ball or a PL $(d+1)$-sphere; the latter holds if and only if $M$ is actually a special matching.
It follows from Lemma \[le:extension\] that is a PL $(d+1)$-ball (sphere) if $\De(\hat0,M(\hat1))$ is a PL $d$-ball (sphere). According to [Theorem]{} \[th:convert\], a sequence of clean zippings and removals converts $[\hat0,M(\hat1)] \times {\mathbf{2}}$ into $P$. Moreover, removals are used precisely when $M$ has fixed points; this follows from the proof of [Theorem]{} \[th:convert\].
By [Theorems]{} \[4.7\], \[MT1\], and \[MT2\], $\De({\overline}{P})$ is a PL $(d+1)$-ball or sphere, the latter occurring precisely when $\De(\hat0,M(\hat1))$ is a sphere and $M$ has no fixed points, i.e., is a special matching.
Let us now formally define the notions of zircons and pircons, which were discussed in the introduction. Given a poset $P$, recall that $P_{\leq x}=\{y \in P \mid y \leq x\}$.
\[de:zircon\] A poset $P$ is a *zircon* if, for every non-minimal element $x \in P$, the order ideal $P_{\leq x}$ is finite and admits a special matching.
Actually, Marietti [@Marietti] originally defined zircons in a slightly different way. His definition and [Definition]{} \[de:zircon\] are, however, equivalent; see [@Hultman_zircon [Proposition]{} 2.3]. It is obvious how to generalise this to the SPM setting:
A poset $P$ is a *pircon* if, for every non-minimal element $x \in P$, the order ideal $P_{\leq x}$ is finite and admits an SPM.
Clearly, zircons are pircons. Recall from the introduction that all open intervals in zircons are topological spheres. This characterises zircons among pircons:
\[th:pircon\] Suppose $P$ is a pircon and $x<y$ in $P$. Then ${\De(x,y)}$ is a PL ball or a PL sphere. Moreover, there exist $x<y$ in $P$ such that ${\De(x,y)}$ is a ball if and only if $P$ is not a zircon.
First, observe that every principal order ideal $P_{\leq y}$ has a unique minimum. Indeed, the lifting property shows that every minimal element in $P_{\leq y}$ also belongs to $P_{\leq M(y)}$, where $M$ is an SPM on $P_{\leq y}$. The observation now follows by induction on the cardinality of a longest chain in the ideal.
Let $\hat0$ be the minimum of $P_{\leq y}$. Using similar induction, we may assume $\De(\hat0,M(y))$ is a PL ball or sphere. By [Theorem]{} \[th:main\], $\De(\hat0,y)$ is a PL ball or sphere, too. The same holds for ${\De(x,y)}$ since it is a link in $\De(\hat0,y)$.
For the final statement, we know that open intervals in zircons are spheres. On the other hand, if $P$ is not a zircon, some $P_{\leq y}$ admits an SPM with fixed points. [Theorem]{} \[th:main\] then shows that $\De(\hat0,y)$ is a ball, where again $\hat0$ is the minimum of $P_{\leq y}$.
Pircons in Coxeter group theory {#se:coxeter}
===============================
In this section, we demonstrate how [Theorem]{} \[th:pircon\] can be applied to certain posets appearing in Coxeter group theory. Acquaintance with the basics of this theory, as explained for example in [@B-B] or [@Humphreys], is assumed.
Twisted identities {#sse:twist}
------------------
As a first application, we shall prove [@Hultman3 [Conjecture]{} 6.3]. The reader may consult [@Hultman3] for context. Here we only describe the necessary ingredients for the statement and its proof.
Let $(W,S)$ be a Coxeter system with an involutive automorphism $\t$. Define two subsets of $W$ as follows. The set of *twisted involutions* is $${{\mathfrak{I}}(\t)}=\{w \in W \mid \t(w)=w^{-1}\},$$ and the set of *twisted identities* is $${\i(\t)}=\{\t(w)w^{-1} \mid w \in W\}.$$ It is clear that ${\i(\t)}\seq {{\mathfrak{I}}(\t)}$.
Say that $\t$ has the *no odd flip*, or *NOF*, *property* if $s\t(s)$ has even or infinite order for every $s \in S$ with $s \neq \t(s)$.[^3] For any $X \seq W$, let ${\mathrm{Br}}(X)$ denote the subposet of the Bruhat order on $W$ which is induced by $X$. The identity element $e \in W$ is the minimum in ${\mathrm{Br}}(W)$, hence in ${\mathrm{Br}}({\i(\t)})$.
The poset ${\mathrm{Br}}({{\mathfrak{I}}(\t)})$ is always graded; denote its rank function by $\rho$. Whenever ${\mathrm{Br}}({\i(\t)})$ is graded, its rank function is the restriction of $\rho$. Furthermore, ${\mathrm{Br}}({\i(\t)})$ is graded if $\t$ satisfies the NOF property [@Hultman3].
When $W$ is of type $A_{2n+1}$ and $\t$ is the unique non-trivial involution, [@A-H [Theorem]{} 4.3] shows that ${\mathrm{Br}}({\i(\t)})$ is a pircon. This is generalised substantially in the next result. The main proof ideas are, however, the same.
\[th:nof\] If $\t$ has the NOF property, then ${\mathrm{Br}}({\i(\t)})$ is a pircon.
Choose $w \in {\i(\t)}$ and $s \in S$ such that $ws<w$ in the Bruhat order. For $x \in {\mathrm{Br}}({\i(\t)})_{\leq w}$, put $M(x)=\t(s)xs$. We shall prove that $M$ is an SPM on this (finite) order ideal.
Observe that $$M(x)=\begin{cases}\f(x) & \h{if $\f(x) \in {\i(\t)}$,} \\ x & \h{otherwise,}\end{cases}$$ where the map $$\f(x)=\begin{cases}xs & \h{if $M(x)=x$,} \\ M(x) & \h{otherwise}\end{cases}$$ is a special matching on ${\mathrm{Br}}({{\mathfrak{I}}(\t)})_{\leq w}$ by [@Hultman2 [Theorem]{} 4.5]. Hence, $M$ preserves ${\mathrm{Br}}({\i(\t)})_{\leq w}$ by the lifting property applied to $\f$.
It follows from [@Hultman3] that for $x \in {\mathrm{Br}}({\i(\t)})$, $$\label{eq:fact}
M(x)=x \To \f(x)>x.$$ Therefore, the second property of an SPM (see [Definition]{} \[de:spm\]) holds, and the first and third properties are readily checked. It remains to verify the fourth.
Suppose $x \cov y$ in ${\mathrm{Br}}({\i(\t)})_{\leq w}$ and $M(x) \neq y$. Since ${\mathrm{Br}}({\i(\t)})$ has the induced rank function of ${\mathrm{Br}}({{\mathfrak{I}}(\t)})$, $x \cov y$ in ${\mathrm{Br}}({{\mathfrak{I}}(\t)})_{\leq w}$, too. We have to show that $M(x)<M(y)$. Since $\f$ is a special matching, this is obvious if $M(x) \neq x$ and $M(y) \neq y$. Apart from some trivial cases, we thus have to consider (1) $M(x)=x$ and $M(y)<y$, and (2) $M(x)>x$ and $M(y)=y$. However, we shall see that both cases are impossible.
In the former case, by we have $\f(x)>x \neq \f(y)<y$, which contradicts the lifting property. In the latter case, implies $\f(y) \voc y$. Since $\f(y) \voc \f(x)$, too, we have a contradiction because according to [@Hultman3 Lemma 4.5], under the NOF assumption, an element in ${{\mathfrak{I}}(\t)}-{\i(\t)}$ can cover at most one twisted identity in ${\mathrm{Br}}({{\mathfrak{I}}(\t)})$.
In general, [Theorem]{} \[th:nof\] is false without the NOF assumption. For example, suppose $W$ is of type $A_4$ with generating set $S=\{s_1,s_2,s_3,s_4\}$ such that $s_1s_2$, $s_2s_3$, and $s_3s_4$ have order $3$, and all other generator pairs commute. Let $\t$ be the unique non-trivial involution of $(W,S)$, mapping $s_i$ to $s_{5-i}$. Define $w=s_2s_1s_3s_2s_4s_3$. One readily checks that ${\mathrm{Br}}({{\mathfrak{I}}(\t)})_{\leq w}$ is isomorphic to the rank $3$ boolean lattice, and that ${\mathrm{Br}}({\i(\t)})_{\leq w}$ is obtained from ${\mathrm{Br}}({{\mathfrak{I}}(\t)})_{\leq w}$ by removing the rank $2$ element $s_2s_3s_2$. The resulting poset does not admit an SPM, hence ${\mathrm{Br}}({\i(\t)})$ cannot be a pircon.
In light of [Theorem]{} \[th:pircon\], [Theorem]{} \[th:nof\] immediately implies the following result, which is the previously mentioned conjecture.
\[co:main\] Suppose $\t$ has the NOF property and let $I$ be an open interval in ${\mathrm{Br}}({\i(\t)})$. Then $\De(I)$ is a PL ball or a PL sphere.
${}$
1\. Can, Cherniavsky, and Twelbeck [@C-C-T] established [Corollary]{} \[co:main\] for $W$ of type $A_{2n+1}$ using shellability methods.
2\. It follows from [@Hultman3 [Theorem]{} 4.12] that $\De(I)$ is a sphere precisely when $I$ is *full*, meaning that it coincides with an interval in ${\mathrm{Br}}({{\mathfrak{I}}(\t)})$, i.e., $I=\{x \in {\i(\t)}\mid u<x<w\}=\{x \in {{\mathfrak{I}}(\t)}\mid u<x<w\}$ for some $u,w \in {\i(\t)}$.
3\. The remark after [Theorem]{} \[th:nof\] shows that ${\mathrm{Br}}({\i(\t)})$ is not a pircon if $W$ is of type $A_{2m}$, $m \geq 2$, with $\t \neq {\mathrm{id}}$. It is, however, an open question whether the open intervals are PL balls or spheres. This is not true for arbitrary $W$ and $\t$. For example, as shown in [@Hultman3 [Example]{} 4.7], if $W$ is of type $\widetilde{A}_2$ with $\t \neq {\mathrm{id}}$, there are intervals in ${\mathrm{Br}}({\i(\t)})$ which are not even graded.
Quasiparabolic -sets {#sse:quasi}
--------------------
Our second application concerns quasiparabolic $W$-sets as introduced by Rains and Vazirani [@R-V] as a context to which many nice properties of parabolic quotients extend. Let us recall some crucial definitions and results from [@R-V]. The reader should consult the original source for much more background and motivation.
Again $(W,S)$ denotes a Coxeter system. Say that $X$ is a *scaled $W$-set* if $X$ is a (left) $W$-set equipped with a function ${\mathrm{ht}}: X \to {\mathbf{Z}}$ such that $|{\mathrm{ht}}(sx)-{\mathrm{ht}}(x)| \leq 1$ for all $x \in X$ and all $s \in S$. An element $x \in X$ is called *$W$-minimal* if ${\mathrm{ht}}(x) \leq {\mathrm{ht}}(sx)$ for all $s \in S$. Say that $X$ is *bounded from below* if the function ${\mathrm{ht}}$ is bounded from below.
Let $T=\{wsw^{-1} \mid w \in W, \: s \in S\}$ denote the set of reflections.
\[de:QB\] A scaled $W$-set $X$ is called *quasiparabolic* if it satisfies the following two properties.
1. For all $t \in T$ and $x \in X$, if ${\mathrm{ht}}(tx)={\mathrm{ht}}(x)$, then $tx=x$.
2. For all $t \in T$, $x \in X$, and $s \in S$, if ${\mathrm{ht}}(tx)>{\mathrm{ht}}(x)$ and ${\mathrm{ht}}(stx)<{\mathrm{ht}}(sx)$, then $tx=sx$.
\[le:minmax\] Each orbit of a quasiparabolic $W$-set contains at most one $W$-minimal element.
Suppose now that $X$ is quasiparabolic with a $W$-minimal element $x_0$. Assume, [without loss of generality]{}, that ${\mathrm{ht}}(x_0)=0$. If $y \in X$ with ${\mathrm{ht}}(y)=k$, then $s_1 \cd s_kx_0$ is a *reduced expression* for $y$ if $y=s_1 \cd s_kx_0$ for some $s_i \in S$. All elements in the orbit of $x_0$ have reduced expressions [@R-V]. Define the *Bruhat order* $\leq$ on $X$ as follows.
\[de:Bruhat\] Let $y=s_1 \cd s_k x_0$ be a reduced expression. Then $$\h{$x \leq y \iff x=s_{i_1} \cd s_{i_j}x_0$ for some $1 \leq i_1<\cd<i_j \leq k$.}$$
In particular, elements in different $W$-orbits are incomparable. Although not obvious from [Definition]{} \[de:Bruhat\], the Bruhat order is a partial order on $X$, which we denote by ${\mathrm{Br}}(X)$; it is graded with rank function ${\mathrm{ht}}$ [@R-V]. In particular, $W$-minimal elements are minimal in the Bruhat order.
Again there is a “lifting property”:
\[le:lift\] Suppose $x,y \in X$ and $s \in S$. If $x \leq y$ and $sx \not\leq sy$, then $sx \leq y$ and $x \leq sy$.
\[th:quasipircon\] If $X$ is a quasiparabolic $W$-set bounded from below, then ${\mathrm{Br}}(X)$ is a pircon. In particular, the order complex of every open interval in ${\mathrm{Br}}(X)$ is a PL ball or a PL sphere.
Suppose $z\in X$ is a non-minimal element. Since $X$ is bounded from below, there is a minimal element $x_0<z$. By Lemma \[le:minmax\], $x_0$ is in fact unique since elements in different $W$-orbits are incomparable. Hence ${\mathrm{Br}}(X)_{\leq z}=[x_0,z]$. By [Definition]{} \[de:Bruhat\], $[x_0,z]$ is finite. Choose a reduced expression $s_1 \cd s_kx_0$ for $z$. For $x \in [x_0,z]$, let $M(x)=s_1x$. We shall prove that $M$ is an SPM on $[x_0,z]$.
- For all $x \leq z$, $s_1s_1x=x$. Thus $M^2={\mathrm{id}}$.
- Since ${\mathrm{ht}}(s_1z)={\mathrm{ht}}(s_2 \cd s_kx_0)=k-1$, $M(z) \cov z$. Lemma \[le:lift\] thus shows that $M(x) \leq z$ for all $x \leq z$.
- For all $x \leq z$, $s_1x$ and $x$ are comparable by [@R-V Remark 5.2], and $|{\mathrm{ht}}(s_1x)-{\mathrm{ht}}(x)| \leq 1$. Hence, $M(x) \cov x$, $M(x)=x$, or $x \cov M(x)$.
- Suppose $x \cov y \leq z$ and $M(x) \neq y$. Then $s_1x \neq y$, $x \neq s_1y$, and $s_1x \neq s_1y$. By Lemma \[le:lift\], we either have $s_1x<s_1y$, or else $s_1x<y$ and $x<s_1y$. In the latter case, $s_1x \not> x$, so $s_1x \leq x<s_1y$. Hence, in either case, $M(x)<M(y)$.
The topological conclusion of [Theorem]{} \[th:quasipircon\] is implied by [@R-V [Theorem]{} 6.4], which claims CL-shellability of the intervals. Unfortunately, the proof of that result has turned out to be flawed; see the discussion in [@C-C-T].
A familiar example of a quasiparabolic $W$-set is the parabolic quotient $W^J$, $J \seq S$, which consists of the minimal length representatives of the left cosets of the parabolic subgroup $W_J$ in $W$. In this setting, the topological conclusion of [Theorem]{} \[th:quasipircon\] was established by Björner and Wachs [@B-W_coxeter] using shellability techniques.
Other examples include several instances of ${\i(\t)}$ (with $W$ acting by twisted conjugation, i.e., the action of $w$ on $x$ is given by $wx\t(w^{-1})$), including the odd rank type $A$ case [@R-V]. In fact, it seems possible that ${\i(\t)}$ is always a quasiparabolic $W$-set with this action whenever $\t$ has the NOF property; if so, [Theorem]{} \[th:nof\] would be a special case of [Theorem]{} \[th:quasipircon\]. We neither know of a proof nor of a counterexample.
Open questions {#se:questions}
==============
We conclude the paper with a couple of questions that suggest themselves naturally.
Clearly, all zircons and pircons have rank functions.[^4] Indeed, the rank of an element $x$ equals the dimension of the ball or sphere $\De(\hat0,x)$ plus two, where $\hat0$ is the unique minimal element below $x$; the uniqueness was shown in the proof of [Theorem]{} \[th:pircon\].
Let $Z$ be a zircon with rank function $\operatorname{rk}$. For a non-minimal element $z \in Z$, let $M_z$ denote a fixed special matching on $Z_{\leq z}$. Given an induced subposet $P \seq Z$ and $p \in P$, define $$M_p'(x)=\begin{cases}M_p(x) & \h{if $M_p(x) \in P$,} \\ x & \h{otherwise.}\end{cases}$$ Suppose $M_p'$ is an SPM on $P_{\leq p}$ for every non-minimal element $p \in P$. If, moreover, the restriction of $\operatorname{rk}$ to $P$ is a rank function of $P$, call $P$ an *induced pircon* of $Z$.
It follows from the proof of [Theorem]{} \[th:nof\] that every pircon of the form ${\mathrm{Br}}({\i(\t)})$ is an induced pircon of the corresponding zircon ${\mathrm{Br}}({{\mathfrak{I}}(\t)})$. Similarly, ${\mathrm{Br}}(W^J)$ is an induced pircon of ${\mathrm{Br}}(W)$ for any $J \seq S$.
Is every pircon an induced pircon of some zircon?
A common way to establish topological consequences such as those of [Theorem]{} \[th:pircon\] is to prove shellability. Beginning with Björner [@Bjorner], there are several variations of lexicographic shellability; see, e.g., Wachs’ survey [@Wachs]. Under this umbrella are gathered several similarly flavoured combinatorial methods that can be used to establish shellability of order complexes by means of certain labellings of the posets.
Concerning zircons, the following question is known to have an affirmative answer for ${\mathrm{Br}}(W)$ in arbitrary type [@B-W_coxeter], as well as for ${\mathrm{Br}}({{\mathfrak{I}}(\t)})$ in types $A$, $B$, and $D$ [@Incitti_A; @Incitti_B; @Incitti_D]. For other pircons, it has been established for ${\mathrm{Br}}(W^J)$ [@B-W_coxeter] and for ${\mathrm{Br}}({\i(\t)})$ in type $A$ of odd rank [@C-C-T].
Is every interval in every pircon lexicographically shellable?
In case both the previous questions turn out to have affirmative answers, one may speculate that even more could be true. The aforementioned result from [@C-C-T] can be interpreted in the following way. For $W$ of type $A_n$, Incitti [@Incitti_A] established lexicographic shellability of ${\mathrm{Br}}({{\mathfrak{I}}(\t)})$ by producing an EL-labelling of this poset. When $n$ is odd and $\t \neq {\mathrm{id}}$, Can, Cherniavsky, and Twelbeck proved that the restriction of this labelling to the induced pircon ${\mathrm{Br}}({\i(\t)})$ is an EL-labelling, too.
Is it true that every induced pircon has an EL-labelling which is induced from an EL-labelling of the corresponding zircon?
Acknowledgements {#acknowledgements .unnumbered}
================
N. Abdallah was funded by a stipend from the Wenner-Gren Foundations.
[^1]: Although Reading worked in the context of Bruhat orders, his proof is valid in the more general zircon setting.
[^2]: One invokes Lemma \[le:PL\](i) using that ${\De_{xyz}}$ is the union of the PL $d$-balls $\De(\hat0,x] {*}{\De{(x,\hat1)}}$ and $\De(\hat0,y] {*}{\De{(y,\hat1)}}$ whose intersection is the PL $(d-1)$-ball $\De(\hat0,x) {*}\De[z,\hat1)$ which is contained in the boundary of both.
[^3]: This means that $\t$ does not flip any edges with odd labels in the Coxeter graph.
[^4]: For zircons, the existence of a rank function is part of Marietti’s [@Marietti] original definition; see the discussion after [Definition]{} \[de:zircon\].
|
---
abstract: |
[*Energy*]{} is often the most constrained resource for battery-powered wireless devices and the lion’s share of energy is often spent on transceiver usage (sending/receiving packets), not on computation. In this paper we study the [*energy complexity*]{} of and in several models of wireless radio networks. It turns out that energy complexity is very sensitive to whether the devices can generate random bits and their ability to [*detect collisions*]{}. We consider four collision-detection models: (in which transmitters and listeners detect collisions), and (in which only transmitters or only listeners detect collisions), and (in which no one detects collisions.)
The take-away message of our results is quite surprising. For randomized algorithms, there is an exponential gap between the energy complexity of and : $$\mbox{Randomized:} \; \mbox{{\textsf{No-CD}}} \;=\; \mbox{{\textsf{Sender-CD}}} \;\gg\; \mbox{{\textsf{Receiver-CD}}} = \mbox{{\textsf{Strong-CD}}}$$ and for deterministic algorithms, there is another exponential gap in energy complexity, *but in the reverse direction*: $$\mbox{Deterministic:} \; \mbox{{\textsf{No-CD}}} \;=\; \mbox{{\textsf{Receiver-CD}}} \;\gg\; \mbox{{\textsf{Sender-CD}}} = \mbox{{\textsf{Strong-CD}}}$$ In particular, the randomized energy complexity of is $\Theta(\log^* n)$ in but $\Theta(\log(\log^* n))$ in , where $n$ is the (unknown) number of devices. Its deterministic complexity is $\Theta(\log N)$ in but $\Theta(\log\log N)$ in , where $N$ is the (known) size of the devices’ ID space.
There is a tradeoff between time and energy. We give a new upper bound on the time-energy tradeoff curve for randomized and . A critical component of this algorithm is a new deterministic algorithm for [*dense*]{} instances, when $n=\Theta(N)$, with inverse-Ackermann-type ($O(\alpha(N))$) energy complexity.
author:
- |
Yi-Jun Chang\
University of Michigan\
`cyijun@umich.edu`
- |
Tsvi Kopelowitz\
University of Michigan\
`kopelot@gmail.com`
- |
Seth Pettie\
University of Michigan\
`pettie@umich.edu`
- |
Ruosong Wang\
Tsinghua University\
`wrs13@mails.tsinghua.edu.cn`
- |
Wei Zhan\
Tsinghua University\
`jollwish@gmail.com`
nocite: '[@FinemanGKN16; @FinemanNW16]'
title: |
Exponential Separations in the Energy\
Complexity of Leader Election[^1]
---
#### Acknowledgement.
We would like to thank Tomasz Jurdziński for discussing the details of [@JurdzinskiKZ02; @JurdzinskiKZ02b; @JurdzinskiKZ02c; @JurdziskiKZ03] and to Calvin Newport for assisting us with the radio network literature.
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[^1]: Supported by NSF grants CNS-1318294, CCF-1514383, and CCF-1637546.
|
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abstract: 'We have now completed detailed abundance analyses of more than 100 stars selected as candidate extremely metal-poor stars with \[Fe/H\] $< -3.0$ dex. Of these 18 are below $-3.3$ dex on the scale of the First Stars VLT project led by Cayrel, and 57 are below $-3.0$ dex on that scale. Ignoring enhancement of carbon which ranges up to very large values, and two C-rich stars with very high N as well, there are 0 to 3 high or low [*[strong]{}*]{} outliers for each abundance ratio tested from Mg to Ni. The outliers have been checked and they are real. Ignoring the outliers, the dispersions are in most cases approximately consistent with the uncertainties, except those for \[Sr/Fe\] and \[Ba/Fe\], which are much larger. Approximately 6% of the sample are strong outliers in one or more elements between Mg and Ni. This rises to $\sim$15% if minor outliers for these elements and strong outliers for Sr and Ba are included. There are 6 stars with extremely low \[Sr/Fe and \[Ba/Fe\], including one which has lower \[Ba/H\] than Draco 119, the star found by Fulbright, Rich & Castro to have the lowest such ratio known previously. There is one extreme $r$-process star.'
author:
- 'Judith G. Cohen, Norbert Christlieb, Ian Thompson, Andrew McWilliam & Stephen Shectman'
title: Outliers in the 0Z Survey
---
Outliers in Abundance Ratio Trends
==================================
Extremely metal poor (EMP) stars were presumably among the first stars formed in the Galaxy, and hence represent in effect a local high-redshift population. Such stars provide important clues to the chemical history of our Galaxy, the role and type of early SN, the mode of star formation in the proto-Milky Way, and the formation of the Galactic halo. [@beers05] compiled the small sample of EMP stars known as of 2005. The goal of our 0Z Project is to increase this sample substantially.
Our sample selection is based on mining the database of the Hamburg/ESO Survey [@wis00] for candidate EMP stars with \[Fe/H\] $< -3.0$ dex [@christlieb03]. Our abundance determination procedures are described in [@cohen04]. The determination of stellar parameters, measurement of equivalent widths, and detailed abundance analyses were all carried out by J. Cohen.
Our data in general follow the well established trends from numerous studies of Galactic halo stars between abundance ratios \[X/Fe\] and overall metallicity as measured by \[Fe/H\] [see e.g. @cayrel_04; @cohen04]. The interesting question is whether in the low metallicity regime studied here we can detect the effect of only a small number of SN contributing to a star’s chemical inventory or inhomogeneous mixing within the ISM at these early stages of formation of the Galaxy. Thus the size of the scatter around these trends and whether there are major outliers is of great interest.
After all the abundance analyses were completed, we looked for [*[strong]{}*]{} outliers, either high or low, in plots of \[X/Fe\] vs \[Fe/H\]. We checked these in detail. The Ca abundance turned out to be problematical in those very C-rich stars whose spectra were obtained prior to HIRES detector upgrade in mid-2004 and thus included only a limited wavelength range. In an effort to derive Ca abundances from these early HIRES spectra, we ended up using lines which were crowded/blended, presumably by molecular features. This was only realized fairly recently when we obtained additional C-star HIRES spectra extending out to 8000 Å which covered key isolated Ca I lines in the 6160 Å region. We found much lower Ca abundances from the additional Ca lines in these carbon stars. Our earlier claims in [@cohen06] of high Ca/Fe for some C-rich stars are not correct.
These abundance analyses were carried out over a period of a decade, and some updates were made in J. Cohen’s master list of adopted $gf$ values during that period. The next step, completed in Dec 2011 after the conference, was to homogenize the $gf$ values.
Linear Fits to Abundance Ratios
===============================
Fig. 1 shows \[Ca/Fe\] vs \[Fe/H\] for our sample. Linear fits to the abundance ratios vs \[Fe/H\] where there is adequate data for the species X were calculated, excluding C-rich stars and a small number of [*[strong]{}*]{} outliers. An example of these fits is shown for Ca in Fig 2, where in the lower panel the included stars are shown together with the fit (thick solid line) and the fit $\pm$0.15 dex (dashed lines). Note that the fit for \[Ca/Fe\] vs \[Fe/H\] is constant, \[Ca/Fe\] = 0.26 dex. In the upper panel the histogram of deviations from the linear fit is shown. Although a number of very deviant low outliers were excluded, an assymetric distribution of $\delta$(\[Ca/Fe\]) still remains, suggesting the presence of a small tail of stars with low \[Ca/Fe\], though not so extreme that the stars were rejected as strong outliers. This is also apparent in the lower panel, where there are four stars with \[Ca/Fe\] $< 0$; these values were not low enough for them to be rejected as strong outliers. Current work focuses on determining whether the dispersion about these fits is larger than the expected uncertainties.
There are six stars which are very deviant low outliers in \[Ba/Fe\]. One of these has \[Ba/H\] below that of Draco 119, the star with the lowest Ba abundance previously known [@draco119].
We are very grateful to the Palomar, Las Campanas, and Keck time allocation committees for their long-term support of this effort. J. Cohen acknowledges partial support from NSF grants AST–0507219 and AST–0908139. I. Thompson acknowledges partial support from NSF AST-0507325. We are grateful to the many people who have worked to make the Keck Telescopes and their instruments, and the Magellan Telescopes and their instruments, a reality and to operate and maintain these observatories.
Beers, T. C. & Christlieb, N., 2005, , 43, 531
Cayrel, R. et al, 2004, , 416, 1117
Christlieb, N., 2003, Rev. Mod. Astron., 16, 191
Cohen, J. G.,et al, 2004, , 612, 1107
Cohen, J., McWilliam, A., Shectman, S., Thompson, I., Christlieb, N., Ramírez, S., Swenson, A. & Zickgraf, F. J., 2006, , 132, 137
Fulbright, J., Rich, R. M. & Castro, S., 2004, , 612, 447
Wisotzki, L., Christlieb, N., Bade, N., Beckmann, V., Köhler, T., Vanelle, C. & Reimers, D., 2000, , 358, 77
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abstract: 'In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, without assuming the convexity of the prescribed solution and using the theory of fully nonlinear elliptic equations. If the given data admits a suitable radial graph as a subsolution, then we prove that there exists a radial graph with constant curvature and realizing the prescribed boundary. As an application, it is proved that if $\Omega\subset\mathbb{S}^n$ is a mean convex domain whose closure is contained in an open hemisphere of $\mathbb{S}^n$ then, for $0<R<n(n-1),$ there exists a radial graph of constant scalar curvature $R$ and boundary $\partial\Omega.$'
address: |
Departamento de Matemática\
Universidade Regional do Cariri\
Campus Crajubar\
Juazeiro do Norte, Ceará\
Brazil\
63041-141
author:
- 'Flávio F. Cruz'
title: Radial graphs of constant curvature and prescribed boundary
---
[^1]
Introduction {#section1}
============
The aim of this work is to study the following Plateau type problem: given a smooth symmetric function $f$ of $n$ ($n\geq 2$) variables and a $(n-1)$-dimensional compact embedded submanifold $\Lambda$ of $\mathbb{R}^{n+1},$ decide whether there exists a hypersurface $\Sigma$ of constant curvature $$\label{intro1}
f(\kappa[\Sigma])=c$$ with boundary $$\label{intro2}
\partial \Sigma = \Lambda,$$ where $\kappa[\Sigma]=(\kappa_1, \ldots, \kappa_n)$ denotes the principal curvatures of $\Sigma$ and $c$ is a constant. The classical Plateau problem for minimal or constant mean curvature surfaces, as well as the corresponding problem for Gauss or scalar curvature, are important particular cases of the problem.
Although the solvability of the problem in this generality still remains open, there are various existence results for some important particular cases. These results have shown that the theory of nonlinear elliptic PDEs is a powerful tool in order to understand the solvability of the problem. In order to apply the PDE techniques a successful strategy is describe the hypersurface $\Sigma$ as the graph of a solution of the Dirichlet problem associated to a certain PDE. After the works of Bernstein, Leray, Jenkins, Finn and others, Serrin applied this approach in [@SERRIN] and proved the existence of hypersurfaces of constant mean curvature and prescribed boundary $\Lambda$ in two geometric settings: Firstly, when the boundary $\Lambda$ is a (vertical) graph over the boundary of a domain in a hyperplane and, secondly, when $\Lambda$ is a radial graph over the boundary of a domain in a hypersphere. For more general curvature functions, the first breakthroughs about the solvability of the problem were due to Caffarelli, Nirenber and Spruck [@CNSV]. Applying the techniques developed in [@CNSI] and [@CNSIII], they proved the existence of solutions to - for a large class of curvature functions, which includes the scalar and Gaussian curvature. Importantly, however, they only treat the cases where the boundary date is constant, $\Lambda$ is the boundary of a strictly convex domain in a hyperplane and the solution is a graph over this domain. ,For the specific case of high order mean curvature functions, Ivochkina [@IVO2] was able to extend the existence for general boundary values and nonconvex domains. Much subsequent work aimed to improve and extend they results, as we can see in [@GUAN-LI], [@IVO1], [@JORGE-FLAVIO], [@SHENG-URBAS-WANG] and [@TRU1]. Later, Guan and Spruck [@GUAN-SPRUCK-1] established existence results for constant Gaussian curvature hypersurfaces which are radial graphs over a domain in a hypersphere and whose boundary is a radial graph over the boundary of the domain. Their results were extended in [@GUAN-SPRUCK-2] and [@TRU-WANG] to convex curvature functions. However, the existing results to date leave open the case of radial graphs with constant non-convex curvature functions. In particular, there is no result for the fundamental case of the scalar curvature in this context. The main purpose of this paper is to establish the existence of radial graphs with constant higher order curvature $f=H_r$, when the prescribed hypersurface is not assumed convex. In particular, our results embrace the scalar curvature case.
Let us now explain more precisely the framework we are considering. Let $\Omega$ be a smooth domain in $\mathbb{S}^n\subset \mathbb{R}^{n+1}$ with boundary $\partial \Omega.$ In order to solve the problem - we seek for a smooth hypersurface $\Sigma$ that can be represented as a radial graph $$X(x)=\rho(x) x, \quad \rho>0,\, x\in \bar{\Omega}$$ with prescribed curvature and boundary $$\begin{aligned}
\label{equation}
\begin{split}
f(\kappa_{\Sigma}[X]) &=\psi(x), \quad x\in\Omega\\
X(x) & =\phi(x)x, \quad x\in\partial\Omega
\end{split}\end{aligned}$$ where $\kappa_{\Sigma}[X]=(\kappa_1, \ldots, \kappa_n)$ denotes the principal curvatures of $\Sigma$ at $X(x)$ with respect to the inward unit normal, $\psi\in C^{\infty}(\overline\Omega), \phi\in C^{\infty}
(\partial\Omega),$ $\psi,\phi>0$ and $f$ is a high order curvature function $$\label{Hr}
f(\kappa)=H_r(\kappa)=\frac{S_r(\kappa)}{S_r(1,\ldots, 1)}$$ where $1< r\leq n$ and $S_r$ is the $r-$th order elementary symmetric function, $$\label{Sr}
S_r(\kappa)=\sum \kappa_{i_1}\kappa_{i_2}\ldots \kappa_{i_r}$$ the sum being taken over all increasing $k-$tuples $i_1, i_2, \ldots, i_k\subset\{ 1, \ldots, n\}.$
In this context, a function $\rho\in C^2(\overline\Omega)$ is called [*admissible*]{} if $\kappa_\Sigma[X]\in \Gamma_r$ at each point $X$ of its radial graph $\Sigma,$ where $\Gamma_r$ is the open convex cone in $\mathbb{R}^n$ with vertex at the origin and given by $$\label{gamma-r}
\Gamma_r=\{\kappa\in \mathbb{R}^n\,:\, S_j(\kappa)>0, \, j=1, \ldots, r\}.$$
We shall assume the existence of a suitable admissible subsolution: there exists a smooth admissible radial graph $\bar\Sigma$: $\bar{X}(x)=\bar{\rho}(x)x$ over $\bar{\Omega}$ that is locally strictly convex (up to the boundary) in a neighbourhood of $\partial\Omega$ and satisfies $$\begin{aligned}
\label{subsolution}
\begin{split}
f(\kappa_{\bar{\Sigma}}[\bar{X}])>&\psi(x)\quad\textrm{in } \Omega,\\
\bar{\rho}=&\phi\quad \textrm{on } \partial\Omega.
\end{split}\end{aligned}$$
Our main result may be state as follows:
\[teorema1\] Let $\Omega$ be a smooth domain whose closure is contained in an open hemisphere of $ \mathbb{S}^n.$ Suppose the mean curvature of $\partial\Omega$ as a submanifold of $\Omega,$ computed with respect to the unit normal pointing to the interior of $\Omega,$ is nonnegative. Then, under condition , there exists a smooth radial graph $\Sigma$ satisfying .
In general, solutions to equation are not unique. See for instance, Example 8.5.2 in [@LOPEZ]. It follows from the Gauss equation that the case of scalar curvature $R$ of $\Sigma$ is given by $R=n(n-1) H_2$, therefore the scalar curvature case is included in Theorem \[teorema1\]. Moreover, using the function $\bar\rho=1$ as a subsolution, we obtain the following result:
\[corolario\] Let $\Omega\subset \mathbb{S}^n$ be as in Theorem \[teorema1\]. Then, for $0<R<n(n-1),$ there exists a radial graph $\Sigma$ of constant scalar curvature $R$ and boundary $\partial\Sigma=\partial\Omega.$
A central issue in solving is to derive [*a priori*]{} $C^2$ estimates for admissible solutions. The height and boundary gradient bounds follows from the existence of a subsolution and the assumption on the geometry of $\Omega.$ Hessian and gradient interior estimates are obtained applying the results of [@CNSIV] to a suitable auxiliary equation. Our main contribution here is the establishment of the second derivatives estimates on the boundary without imposing any condition on the geometry of $\Omega.$ As this estimate is of independent we describe it separately:
\[teorema1-2\] Let $\rho\in C^3(\Omega)\cap C^2(\bar\Omega)$ be an admissible solution of . Suppose that there exists a smooth admissible subsolution $\bar{\rho}$ of , i.e., the radial graph $\bar\Sigma$: $\bar{X}(x)=\bar{\rho}(x)x$ satisfy $$\begin{aligned}
\begin{split}
f(\kappa_{\bar{\Sigma}}[\bar{X}])>&\psi(x)\quad\textrm{in } \Omega\\
\bar{\rho}=&\phi\quad \textrm{on } \partial\Omega,
\end{split}\end{aligned}$$ and $\bar\Sigma$ is locally strictly convex (up to the boundary) in a neighbourhood of $\partial\Omega.$ Then there exists a constant $C$ depending on $\sup_{\Omega}\bar\rho,
\|\bar \rho \|_{C^2(\bar\Omega)}, $ the convexity of $\bar \Sigma$ in a neighbourhood of $\partial\Omega$ and other known data, that satisfies $$|\nabla^2 \rho|< C \quad \textrm{on } \partial\Omega$$ where $\nabla^2 \rho$ denotes the Hessian of $\rho.$
An outline of the paper is as follows. In Section \[section2\] we list some basic formulae which are needed later and define two elliptic operators to express . In Section \[section3\] we deal with the [*a priori*]{} estimates for prospective solutions and prove Theorem \[teorema1-2\]. Finally in Section \[section4\] we complete the proof of Theorem \[teorema1\] using the continuity method and a degree theory argument with the aid of the established estimates.
Preliminaries {#section2}
=============
In this section we recall the expressions of the second fundamental form and other relevant geometric quantities of a smooth radial graph $\Sigma$ given by $X(x)=\rho(x)x,$ where $\rho$ is a smooth function defined in a domain $\Omega$ of the unit sphere $\mathbb{S}^n\subset\mathbb{R}^{n+1}.$
Let $e_1, \ldots, e_n$ be a smooth local orthonormal frame field on $\mathbb{S}^n$ and let $\nabla$ denote the covariant differentiation on $\mathbb{S}^n.$ The metric of $\Sigma$ is then given in terms of $\rho$ by $$g_{ij}=\langle\nabla_i X,\nabla_j X\rangle= \rho^2\delta_{ij}+\nabla_i\rho\nabla_j\rho,$$ where $\nabla_i=\nabla_{e_{i}}$ and $\langle \cdot , \cdot\rangle$ denotes the standard inner product in $\mathbb{R}^{n+1}.$ The interior unit normal to $\Sigma$ is $$N=\frac{1}{(\rho^2+|\nabla \rho|^2)^{1/2}}(\nabla \rho-\rho x),$$ where $\nabla\rho=\textrm{grad} \rho,$ and the second fundamental form of $\Sigma$ is $$h_{ij}=\langle \nabla_{ij} X, N\rangle = \frac{1}{(\rho^2+|\nabla \rho|^2)^{1/2}}
(\rho^2\delta_{ij} +2\nabla_i\rho\nabla_j\rho-\rho\nabla_{ij}\rho),$$ where $\nabla_{ij}=\nabla_i\nabla_j.$
Setting $u=1/\rho$ we can rewrite the expressions of the metric, its inverse and second fundamental form of $\Sigma$ at $X(x)= \frac{1}{u(x)}x$ in terms of $u$ by $$g_{ij}=\frac{1}{u^2}(\delta_{ij}+\frac{1}{u^2}\nabla_iu\nabla_j u),
\quad g^{ij}=u^2\left(\delta_{ij}-\frac{\nabla_i u\nabla_j u}{w^2}\right)$$ and $$\label{segunda-forma}
h_{ij}=\frac{1}{uw}(u\delta_{ij}+\nabla_{ij}u),$$ respectively, where $w=\sqrt{u^2+|\nabla u|^2}.$ The principal curvatures of $\Sigma$ are the eigenvalues of the Weingarten matrix $[h_i^j]=[g^{jk}h_{ki}].$ However, as in [@CNSIV], here we will work with its similar symmetric matrix $A[u]=[a_{ij}]=[\gamma^{ik}h_{kl}\gamma^{jl}],$ where $[\gamma^{ij}]$ and its inverse matrix $[\gamma_{ij}]$ are given, respectively, by $$\label{raiz-gij}
\gamma^{ij}= u\delta_{ij}-u\frac{\nabla_i u\nabla_j u}{w(u+w)}$$ and $$\label{raiz-gij2}
\gamma_{ij}= \frac{1}{u}\delta_{ij}+\frac{\nabla_i u\nabla_j u}{u^2(u+w)}.$$ Geometrically, $[\gamma_{ij}]$ is the square root of the metric, i.e., $\gamma_{ik}\gamma_{kj}=g_{ij}.$
Now we present a reformulation of equation in the form $$\label{equation2}
G(\nabla^2 u,\nabla u, u)=\tilde{\psi}(x),$$ where $\tilde{\psi}=\psi^{1/r}.$ Let $\mathcal{S}$ be the space of all symmetric matrices and $\mathcal{S}_{r}$ the open subset of those symmetric matrices $A\in\mathcal S$ for which the eigenvalues are contained in $\Gamma_r.$ We define the function $F$ by $$F(A)=f\big(\lambda(A)\big)=H_r^{1/r}\big(\lambda(A)\big), \quad A\in\mathcal{S}_{r}$$ where $\lambda(A)=(\lambda_1, \cdots, \lambda_n)$ are the eigenvalues of $A.$ In the sequel we use $f$ to denote both $H_r$ and $H_r^{1/r}$. Thus equation can be written in the form $$F(A[u])=\tilde{\psi}\big(X(x)\big).$$ Therefore, the function $G$ in is defined by $$G(\nabla^2 u,\nabla u, u)=F(A[u])$$ and equation can be rewritten as $$\begin{aligned}
\begin{split}
\label{equation3}
G(\nabla^2 u,\nabla u, u)&=\tilde{\psi} (x) \quad\textrm{in}\, \Omega,\\
u&=\varphi \quad\textrm{on}\, \partial\Omega,
\end{split}\end{aligned}$$ where $\varphi=1/\phi.$
Next we will describe some properties of the functions $F$ and $G.$ We denote the first derivatives of $F$ by $$F^{ij}(A)=\frac{\partial F}{\partial a_{ij}}(A).$$ Since $H_r$ is positively monotonous on $\Gamma_r ,$ the symmetric matrix $[F^{ij}(A)]$ is positive definite for any $A\in\mathcal{S}_{r}$ and it follows from the concavity of $H_r^{1/r}$ that $F$ is a concave function in $\mathcal{S}_{r}.$ $[F^{ij}(A)]$ and $A$ can be orthogonally diagonalized simultaneously. Thereafter, we have $$\label{euler}
F^{ij}(A)a_{ij}=\sum f_i\lambda_i\leq f(\lambda)$$ where the last inequality follows from the concavity of $f.$ Also we point out that $$\begin{aligned}
\label{new} \sum f_i(\lambda)\lambda_i^2\leq C_0 (\lambda_j \mathbf{1}_{\lambda_j>0}&+
\sum_{k\neq j} f_k(\lambda)\lambda_k^2), \quad\textrm{for all } \lambda\in \Gamma_\psi,\end{aligned}$$ where $\Gamma_\psi=\{\lambda\in\Gamma : \psi_0\leq f(\lambda)\leq\psi_1\}$ and $C_0$ is a positive constant depending on $\psi_0$ and $\psi_1.$ This inequality was first proved by Ivochkina in [@IVO2]. Using the expression for $A[u]$ we compute $$\label{G-I-J}
G^{ij}=\frac{\partial G}{\partial\nabla_{ij}u}=\frac{1}{uw}\sum_{k,l}F^{kl} \gamma^{ik}\gamma^{jl}.$$ Then equation is elliptic for $A[u]\in\mathcal{S}_{r}.$ The concavity of $F$ implies that $G$ is concave with respect to $\nabla_{ij}u.$ By assumption , the function $\underline{u}=1/\bar\rho$ is a subsolution of equation , i.e., $$\begin{aligned}
\begin{split}
\label{subsolution2}
G(\nabla^2 \underline u,\nabla \underline u, \underline u)&=\underline \psi(x)>\tilde{\psi}(x)
\quad\textrm{in}\, \Omega,\\
\underline u&=\varphi \quad\textrm{on}\, \partial\Omega.
\end{split}\end{aligned}$$
In order to establish the existence of solution for we will apply the continuity method and a degree theory argument on two auxiliary forms of . Consider, for each fixed $t\in [ 0,1],$ the functions $\Psi^t$ and $\Xi^t$ defined in $\Delta= \{ X\in \mathbb{R}^{n+1} \, : \, \frac{X}{\|X\|}\in\overline\Omega\}$ by $$\label{psi-t}
\Psi^t(\rho x)= \left(\frac{\overline \rho(x)}{\rho}\right)^3 \big(t\psi(x)+(1-t)
\underline{\psi}(x)\big)$$ and $$\label{phi-t}
\Xi^t(\rho x)= t\psi(x)+(1-t)\left(\frac{\overline \rho(x)}{\rho}\right)^3\psi(x).$$ We shall work on the two corresponding auxiliary forms of . In sections \[section3\] and \[section4\] we will represent generically these equations by $$\begin{aligned}
\begin{split}
\label{equation4}
G(\nabla^2 u,\nabla u, u)&=\Upsilon (X(x)) \quad\textrm{in}\, \Omega,\\
u&=\varphi \quad\textrm{on}\, \partial\Omega,
\end{split}\end{aligned}$$ where $X(x)= \frac{1}{u(x)} x$ and $\Upsilon$ denotes a general positive smooth function on $\Delta$. We finalize this section observing that the concavity of $f$ implies $$\label{H-positiva}
\sum \lambda_i >0,$$ for any $\lambda=(\lambda_1, \ldots, \lambda_n)\in\Gamma_{\psi}$ (e.g., [@CNSIII]).
A priori estimates {#section3}
==================
In this section we obtain the [*a priori*]{} $C^2$ estimates for admissible solutions $u$ of satisfying $u\geq \underline{u}.$
In order to derive an upper bound for $u,$ we note that as the closure of $\Omega$ is contained in an open hemisphere and the mean curvature of $\partial\Omega$ is nonnegative, there exist [@SERRIN] a minimal radial graph $\hat{\Sigma}: \hat{X}(x)=\underline{\rho}(x)x$ over $\Omega$ with boundary value $\underline{\rho}=\varphi.$ On the other hand, as implies that the mean curvature of $\Sigma$ is positive, we can apply the comparison principle to obtain $u\leq \overline{u},$ where $\overline{u}=1/\underline{\rho}.$ Then $\underline{u}\leq u\leq \overline{u}$ in $\Omega$ and $\underline{u}=u=\overline{u}$ on $\partial\Omega,$ which yields the height and the boundary gradient bounds. For the interior gradient estimate we first observe that $$\label{derivada-ro}
\frac{\partial}{\partial\rho}(\rho\Upsilon(\rho x))\leq 0\quad \textrm{if} \quad \rho\leq \bar\rho.$$ Therefore, the interior gradient bounds can be established as in [@CNSIV] (see also [@ABL]). Then we have the following result.
\[teorema2\] Let $u\geq \underline{u}$ an admissible solution of . Then we have the estimates $$\label{S3-22}
L^{-1}\leq u\leq L, \quad |\nabla u|\leq C \quad \textrm{in}\,\, \bar\Omega,$$ where $L$ and $C$ depend on $\inf_{\Omega}\underline u, \|\underline u\|_{C^1(\bar\Omega)}$ and other known data.
The only places we need assumptions on the geometry of $\Omega$ is in getting an upper bound for $u$ and the gradient boundary estimate. In what follows, when we use $L,$ it always means the same constant and we will denote $$\Delta_L=\{X\in \Delta
\, : \, L^{-1}\leq \|X\|\leq L\}.$$
Now we shall to establish the second derivatives estimates. First we will obtain bounds for $|\nabla^2 u|$ on $\partial \Omega.$
Consider an arbitrary point $x_0\in\partial\Omega,$ let $e_1, \ldots, e_n$ be a local orthonormal frame field on $\mathbb{S}^n$ around $x_0,$ obtained by parallel translation of a local orthonormal frame field on $\partial\Omega$ and the interior, unit, normal vector field to $\partial\Omega,$ along the geodesic perpendicular to $\partial\Omega$ on $\mathbb{S}^n.$ We assume that $e_n$ is the parallel translation of the unit normal vector field on $\partial\Omega.$
As $u=\varphi$ on $\partial\Omega$ we have $$\label{U-2-bordo}
\nabla_{ij}(u-\varphi)=-\nabla_{n}(u-\varphi)
B_{ij}\quad\textrm{for} \, i,j<n,$$ where $B_{ij}=\langle \nabla_{e_{i}} e_j, e_n\rangle$ is the second fundamental form of $\partial\Omega.$ It follows that $$\label{EST-TANG}
|\nabla_{ij} u(x_0)|\leq C, \quad i,j<n,$$ for a uniform constant $C.$
We now proceed to estimate the mixed tangential-normal derivatives $\nabla_{k n} u(x_0), \, k<n.$ By a straightforward computation, we get $$\begin{aligned}
\label{Gs}
\begin{split}
G^{s}&=\frac{\partial G}{\partial \nabla_s u}=-\frac{1}{w^2}F^{ij}a_{ij}\nabla_s u
-\frac{2}{u^2w} F^{ij}a_{ik}\gamma^{ks}\nabla_j u
\end{split}\end{aligned}$$ and $$\begin{aligned}
\label{Gu}
\begin{split}
G_u=&\frac{\partial G}{\partial u}= \frac{|\nabla u|^2}{uw^2}F^{ij}a_{ij}+\frac{2}{uw^2}\sum_kF^{ij}a_{ik}\nabla_j u
\nabla_ku \\ &+\frac{u}{w}F^{ij}\Big(\delta_{ij}-\frac{\nabla_i u\nabla_j u}{w^2}\Big).
\end{split}\end{aligned}$$ In particular, as $[F^{ij}]$ and $[a_{ij}]$ can be diagonalized simultaneously, it follows from Theorem \[teorema2\] that $$\begin{aligned}
\begin{split}
\label{bound-gs}
|G^s|\leq & C\big(1+\sum f_i |\kappa_i|\big) \\
|G_u|\leq C&\big(1+\sum (f_i |\kappa_i| + f_i)\big),
\end{split}\end{aligned}$$ for a uniform constant $C$ depending on $\|u\|_{C^{1}(\bar\Omega)}$ and $\sup_{\Delta_L}\psi^t.$
Now we present some key preliminary lemmas. Let $\varrho(x)$ denote the distance from $x\in\Omega$ to $x_0,$ $\varrho(x)=\textrm{dist}_{\mathbb{S}^n}(x,x_0),$ and set $$\Omega_{\delta}=\{x\in\Omega\, :\, \varrho(x)<\delta\}.$$ Since $\nabla_{ij}\varrho^2(x_0)= 2\delta_{ij}$, by choosing $\delta>0$ sufficiently small we can assume that $\varrho$ is smooth in $\Omega_{\delta},$ $$\label{RHO-2}
\delta_{ij}\leq \nabla_{ij}\varrho^2\leq 3\delta_{ij} \quad\textrm{in}\,\, \Omega_\delta,$$ and the distance function $d(x)=\textrm{dist}_{\mathbb{S}^n}(x,\partial\Omega)$ to the boundary $\partial\Omega$ is smooth in $\Omega_\delta.$
\[DEF-W\] For some positive constants $K$ and $M$ sufficiently large depending on $
\|u\|_{C^{1}(\bar\Omega)}, \|\Upsilon\|_{C^1(\Delta_L)}$ and other known data, the function $$\label{CARA-W}
\Phi=\nabla_k (u-\varphi) -\frac{K}{2}\sum_{l<n}\big(\nabla_l (u-\varphi)\big)^2$$ satisfies $$\label{LW}
G^{ij}\nabla_{ij} \Phi\leq M(1+|\nabla \Phi|+G^{ij}\delta_{ij}+G^{ij}\nabla_i \Phi\nabla_j \Phi)
\quad \textrm{in} \quad \Omega_\delta.$$
A straightforward computation yields $$\begin{aligned}
G^{ij}&\nabla_{ij} \Phi = G^{ij} \nabla_{ijk}u
-K \sum_{l<n}\nabla_l (u-\varphi)G^{ij}\nabla_{ijl}u-G^{ij}\nabla_{ijk}\varphi
\\&-K \sum_{l<n}G^{ij}\nabla_{li} (u-\varphi)\nabla_{lj} (u-\varphi)
+K \sum_{l<n}\nabla_{l}(u-\varphi)G^{ij}\nabla_{ijl}\varphi.\end{aligned}$$ Then, using Theorem \[teorema2\] we easily get the bound $$\begin{aligned}
\label{conta-v3-2}
\begin{split}
G^{ij}&\nabla_{ij} \Phi \leq G^{ij} \nabla_{ijk}u
-K \sum_{l<n}\nabla_l (u-\varphi)G^{ij}\nabla_{ijl}u
\\ & -K \sum_{l<n}G^{ij}\nabla_{li} (u-\varphi)\nabla_{lj} (u-\varphi)+ C\sum G^{ii},
\end{split}\end{aligned}$$ for a uniform constant $C$ depending on $\|u\|_{C^{1}(\bar\Omega)},
\|\varphi\|_{C^{3}(\partial\Omega)}$ and $K.$ Differentiating equation we get $$\begin{aligned}
\label{CONTA-PSI-K-1}
G^{ij}\nabla_{pij} u+G^s\nabla_{ps}u+G_u\nabla_p u=\nabla_p\Upsilon.\end{aligned}$$ Hence, applying the standard formula for commuting the order of covariant derivatives on $\mathbb{S}^n$ we obtain $$\begin{aligned}
\label{v3-psik-2}
\begin{split}
G^{ij}\nabla_{ijp} u &= G^{ij}(\nabla_{pij}u+\delta_{ij}\nabla_pu-\delta_{pj}\nabla_i u)\\
&= -G^s\nabla_{sp}u+G_u\nabla_p u
+G^{ij}(\delta_{ij}\nabla_pu-\delta_{pj}\nabla_i u)+\nabla_p\Upsilon.
\end{split}\end{aligned}$$ Thus, as $$\begin{aligned}
\begin{split}
G^{s}\nabla_{sk} u - &K \sum_{l<n}\nabla_l (u-\varphi)G^s\nabla_{sl}u=
G^s\nabla_s \Phi+G^s\nabla_{ks}\varphi \\ &-K \sum_{l<n}\nabla_l (u-\varphi)G^s\nabla_{sl}\varphi,
\end{split}\end{aligned}$$ we get $$\begin{aligned}
\begin{split}
&G^{ij}\nabla_{ijk}u-K \sum_{l<n}\nabla_l (u-\varphi)G^{ij}\nabla_{ijl}u = -G^s\nabla_s \Phi
-G^s\nabla_{ks}\varphi \\ &+K \sum_{l<n}\nabla_l (u-\varphi)\big(G^s\nabla_{sl}\varphi
-G_u\nabla_l u-G^{ij}(\delta_{ij}\nabla_l u-\delta_{lj}\nabla_i u )-\nabla_l\Upsilon\big)
\\ &+G_u\nabla_ku+G^{ij}(\delta_{ij}\nabla_k u-\delta_{kj}\nabla_i u)+\nabla_k\Upsilon.
\end{split}\end{aligned}$$ Therefore, replacing this expression into and using and we find $$\begin{aligned}
\label{conta-v3-4}
\begin{split}
G^{ij}&\nabla_{ij} \Phi \leq -G^s\nabla_s \Phi -K \sum_{l<n}G^{ij}\nabla_{li} (u-\varphi)\nabla_{lj} (u-\varphi)
\\& + C\big(1+ \sum (f_i |\kappa_i| + f_i)\big).
\end{split}\end{aligned}$$
Let $P=[\eta_{ij}]$ be an orthogonal matrix that simultaneously diagonalizes $[F^{ij}]$ and $[a_{ij}],$ and let $\{\tau_1, \ldots, \tau_n\}$ be a basis of vectors that induce by the parametrization $X$ a basis of principal vectors of $\Sigma,$ that is, a basis of eigenvectors of the Weingarten operator of $\Sigma.$ Henceforth, we will use the greek letters for derivatives in the basis $\tau_1, \ldots, \tau_n$ and latin letters for derivatives in the frame $e_1, \ldots, e_n.$ For instance, $\nabla_{\alpha\beta}u$ and $\nabla_{s\alpha}u$ will denote respectively $\nabla^2u(\tau_\alpha, \tau_\beta)$ and $\nabla^2 u(e_s, \tau_\alpha).$ In particular, as $\gamma_{\alpha\beta}$ is the unique positive square root of $g_{\alpha\beta},$ we have $$g_{\alpha\beta}=\langle \nabla_\alpha X,\nabla_\beta X\rangle=\gamma_{\alpha\beta}=\delta_{\alpha\beta}.$$ Thus, inequality can be written as $$\begin{aligned}
\begin{split}
\label{LW-2}
\sum f_\alpha \nabla_{\alpha\alpha}\Phi \leq M\left(1+|\nabla \Phi|+
\sum (f_\alpha(\nabla_\alpha \Phi)^2+ f_\alpha)\right).
\end{split}\end{aligned}$$ In the sequel, we will often denote by $C$ a uniform constant under control. As $\nabla_{ij} u=uw\sum_{k,l}\gamma_{ik}a_{kl}\gamma_{jl}-u\delta_{ij},$ it follows from and that $$\begin{aligned}
\label{S2-1}
G^{ij}\nabla_{li} (u-\varphi)\nabla_{lj} (u-\varphi)\geq \sum_\alpha \big( \theta_{0} f_\alpha
\kappa_\alpha^2\eta_{l\alpha}^2-C (f_\alpha|\kappa_\alpha|+ f_\alpha)\big),\end{aligned}$$ for a positive uniform constant $\theta_0.$ Similarly, applying Theorem \[teorema2\] and inequality we get the bound $$\begin{aligned}
\label{S2-2}
|G^s\nabla_s \Phi|\leq C \big( |\nabla\Phi|+\sum f_\alpha|\kappa_\alpha \nabla_\alpha \Phi|\big).\end{aligned}$$ Then we replace - into to obtain $$\begin{aligned}
\label{S2-3}
\begin{split}
\sum f_\alpha \nabla_{\alpha\alpha} \Phi &\leq\sum\big( Cf_\alpha|\kappa_\alpha \nabla_\alpha \Phi|
-K\theta_0\sum_{l<n} f_\alpha\kappa_\alpha^2\eta_{l\alpha}^2\big)
\\& + C\big(1+ |\nabla \Phi|+\sum (f_\alpha |\kappa_\alpha|+ f_\alpha)\big).
\end{split}\end{aligned}$$
Now let us consider two cases. First we assume that, for all $\alpha\in\{1,\ldots, n\},$ it holds $$\label{I}
\sum_{l<n}\eta_{l\alpha}^2\geq K^{-2}.$$ The second case occurs when (\[I\]) does not hold. In the first case $$\label{I1}
-\sum_\alpha\sum_{l<n} f_\alpha\kappa_\alpha^2 \eta_{l\alpha}^2
\leq -\sum_\alpha f_\alpha\kappa_\alpha^2.$$ Then follows from and the inequalities $$\begin{aligned}
\begin{split}
\label{I2}
\sum f_\alpha|\kappa_\alpha \nabla_\alpha \Phi | &\leq \sum \big( \epsilon f_\alpha\kappa_{\alpha}^2
+\epsilon^{-1} f_\alpha(\nabla_\alpha \Phi)^2\big)
\\ \sum f_\alpha |\kappa_\alpha| &\leq \sum \big(\epsilon f_\alpha \kappa_{\alpha}^2+\epsilon^{-1} f_\alpha\big),
\end{split}\end{aligned}$$ for an appropriate constant $\epsilon>0.$
In the second case, there exists some $1\leq \gamma\leq n$ such that $$\label{delta}
\sum_{l<n}\eta_{l\gamma}^2< K^{-2}.$$ As in [@IVO1], we can prove (see the appendix) that implies $$\label{delta2}
\sum_{l<n}\eta_{l\alpha}^2\geq\epsilon_0$$ for all $\alpha\neq \gamma,$ where $\epsilon_0>0$ is a uniform positive constant that does not depends on $K.$ For simplicity, let us assume that $\gamma=1.$ To proceed we consider two subcases: $\kappa_1\leq 0$ and $\kappa_1>0$.\
If $\kappa_1\leq 0$ then inequality yields $$f_1\kappa_1^2\leq C_0\sum_{\alpha>1}f_\alpha\kappa_\alpha^2.$$ Hence $$\label{C3}
\sum f_\alpha\kappa_\alpha^2 \leq C \sum_{\alpha>1} f_\alpha \kappa_\alpha^2$$ and we can estimate $$\begin{aligned}
\begin{split}
\label{II2}
\sum f_\alpha| \kappa_\alpha\nabla_\alpha \Phi | &\leq \sum f_\alpha\big(\nabla_\alpha \Phi\big)^2+
C\sum_{\alpha>1} f_\alpha\kappa_{\alpha}^2 \\ \sum f_\alpha |\kappa_\alpha|
&\leq \sum f_\alpha+ C \sum_{\alpha> 1} f_\alpha \kappa_{\alpha}^2.
\end{split}\end{aligned}$$ On the other hand, it follows from that $$\label{C4}
\sum_\alpha \sum_{l<n} f_\alpha\eta_{l\alpha}^2\kappa_\alpha^2
\geq \epsilon_0\sum_{\alpha>1} f_\alpha\kappa_\alpha^2.$$ Applying and into we then obtain by choosing $K$ sufficiently large.
Now suppose that $\kappa_1>0.$ A straightforward computation and the expression $\nabla_{\alpha\beta} u=u(w\kappa_\alpha\delta_{\alpha\beta}-\sigma_{\alpha\beta}),$ where $\sigma_{\alpha\beta}=\langle \tau_\alpha, \tau_\beta\rangle, $ yield $$\begin{aligned}
\begin{split}
f_1|\kappa_1 \nabla_1\Phi| &=
f_1\kappa_1\big|\sum_{\alpha}\eta_{k\alpha}\nabla_{1\alpha }u-\nabla_{1k}\varphi
-K\sum_{\alpha}\sum_{l<n}\eta_{l\alpha}\nabla_l
(u-\varphi)\nabla_{1\alpha}(u-\varphi)\big| \\ &
\leq uw\big|\eta_{k1}-K\sum_{l<n}\eta_{l1}\nabla_l (u-\varphi)\big| f_1\kappa_1^2
+ Cf_1\kappa_1.
\end{split}\end{aligned}$$ Moreover, $$\begin{aligned}
\begin{split}
|\nabla\Phi| & \geq |\nabla_1\Phi|=\big|\sum_{\alpha}\eta_{k\alpha}\nabla_{1\alpha }u
-\nabla_{1k}\varphi-K\sum_{\alpha}\sum_{l<n}\eta_{l\alpha}\nabla_l
(u-\varphi)\nabla_{1\alpha}(u-\varphi)\big|
\\ & \geq \big|\sum_{\alpha}\eta_{k\alpha}u(w\kappa_\alpha -1)\delta_{\alpha 1}
-K\sum_{\alpha} \sum_{l<n}\eta_{l\alpha}\nabla_l (u-\varphi)u(w\kappa_\alpha -1)\delta_{\alpha 1}\big|-C\\
&\geq uw\big|\eta_{k1}-K\sum_{l<n}\eta_{l1}\nabla_l (u-\varphi)\big|\kappa_1-C.
\end{split}\end{aligned}$$ Thus, applying we get the bound $$\begin{aligned}
\label{S2-6}
\begin{split}
f_1|\kappa_1 &\nabla_1\Phi| \leq C(1+|\nabla \Phi|)+Cf_1\kappa_1
\\ &+C\Big|\eta_{k1}-K\sum_{l<n}\eta_{l1}\nabla_l (u-\varphi)\Big|
\sum_{\alpha>1}f_\alpha\kappa_\alpha^2.
\end{split}\end{aligned}$$ On the other hand, inequality gives $$\Big|\eta_{k1}-K\sum_{l<n}\eta_{l1}\nabla_l (u-\varphi)\Big| \leq C_1,$$ for a uniform positive constant $C_1$ that does not depend on $K.$ Therefore $$\label{S2-7}
f_1|\kappa_1 \nabla_1\Phi|\leq C(1+|\nabla \Phi|)+Cf_1\kappa_1
+C_1\sum_{\alpha>1} f_\alpha \kappa_\alpha^2.$$ To control the term $f_1\kappa_1$ we use to obtain $$\begin{aligned}
\label{S2-8}
\begin{split}
f_1\kappa_1&= f_\alpha\kappa_\alpha -\sum_{\alpha>1}f_\alpha\kappa_\alpha
\\ &\leq C\big(1+\sum f_\alpha\big)+\sum_{\alpha>1}f_\alpha\kappa_\alpha^2.
\end{split}\end{aligned}$$ Then $$f_1|\kappa_1 \nabla_1\Phi|\leq C(1+|\nabla \Phi|+\sum f_\alpha)
+C_1\sum_{\alpha>1} f_\alpha \kappa_\alpha^2$$ and we get the bound $$\begin{aligned}
\label{S2-9}
\begin{split}
\sum f_\alpha|\kappa_\alpha \nabla_\alpha \Phi | & \leq C(1+|\nabla \Phi|+\sum f_\alpha)
+C_1\sum_{\alpha>1} f_\alpha \kappa_\alpha^2\\ &+
\sum_{\alpha>1} f_\alpha\kappa_{\alpha}^2+
\sum f_\alpha(\nabla_\alpha \Phi)^2.
\end{split}\end{aligned}$$ Finally, as $\kappa_1>0$ we can use to get $$\begin{aligned}
\label{S2-10}
\begin{split}
f_\alpha |\kappa_\alpha|=\big(f_1\kappa_1+\sum_{\alpha>1}f_\alpha |\kappa_\alpha |\big)\leq
C\big(1+\sum f_\alpha\big)+\sum_{\alpha>1}f_\alpha \kappa_\alpha^2.
\end{split}\end{aligned}$$ Hence, using and replacing and into , we obatin by choosing $K$ sufficiently large.
Setting $$\label{w-til}
\tilde{\Phi}= 1-e^{-a_0 \Phi}$$ for a positive constant $a_0$ large such that $a_0\geq M,$ where $M$ is the constant given in , we get $$\begin{aligned}
\label{LW-TIL}
G^{ij}\nabla_{ij}\tilde{\Phi} \leq M(1+|\nabla \tilde{\Phi}|+G^{ij}\delta_{ij}).\end{aligned}$$
Now we present the following improved version of Lemma 3.3 in [@SU]. In what follows, we denote by $d(x)=\textrm{dist}(x,\partial\Omega)$ the distance function to the boundary.
\[DEF-V\] There exist some uniform positive constants $t, \delta , \varepsilon$ sufficiently small and $N$ sufficiently large depending on $\inf_{\bar\Omega}\underline u,
\|\underline u\|_{C^2(\bar\Omega)}, \sup_{\Delta_L}\Upsilon,$ the convexity of $\bar\Sigma$ in a neighbourhood of $\partial\Omega$ and other known data, such that the function $$\label{V}
\Theta=u-\underline{u} +td-Nd^2$$ satisfies $$\label{LV<0}
G^{ij}\nabla_{ij}\Theta\leq -(1+|\nabla \Theta|+ G^{ij}\delta_{ij}) \quad \textrm{in } \Omega_\delta$$ and $$\label{theta-bordo}
\Theta\geq 0 \quad\textrm{on } \partial\Omega_\delta.$$
As the surface $\bar{X}(x)=\frac{1}{\underline u}x$ is convex in a neighbourhood of $\partial\Omega,$ we can find $\beta>0$ and $\delta>0$ such that $$\label{convex1}
[\underline{u} I+\nabla^2\underline{u}]\geq 4\beta I \quad \textrm{in } \,\Omega_{\delta}.$$ In particular, $\lambda(\underline{u}I+\nabla^2\underline{u}-3\beta I)$ lies in a compact set of $\Gamma_n^+\subset\Gamma_r.$ Since $|\nabla d|=1$ and $-CI\leq \nabla^2d\leq C I,$ for a constant $C$ depending only on the geometry of $\Omega,$ we have $$\label{d1}
G^{ij}\nabla_{ij} d\leq C G^{ij}\delta_{ij}$$ and $$\begin{aligned}
\begin{split}
\label{convex3}
\lambda(\underline{u}I+\nabla^2\underline{u}+N \nabla^2d^2-2\beta I)
\geq \lambda(\underline{u}I+\nabla^2\underline{u}+2N \nabla d\otimes\nabla d-3\beta I)
\end{split}\end{aligned}$$ in $\Omega_\delta,$ when $\delta$ is sufficiently small (so that $2N\delta<\beta/C).$ Using the concavity of $f$ we get $$\begin{aligned}
F\Big( & \big[\frac{1}{uw}\gamma^{ik}(\underline{u}\delta_{kl}
+\nabla_{kl}\underline{u}+2N\nabla_l d\nabla_k d
-3\beta \delta_{lk})\gamma^{jl}\big]\Big)-\psi\big(X\big)\\
& \leq G^{ij}\Big(\nabla_{ij}\underline{u}+\underline{u}\delta_{ij}
+N\nabla_{ij}d^2-2\beta\delta_{ij}-(u\delta_{ij}+\nabla_{ij}u)\Big)
\\ & = G^{ij} \nabla_{ij}(\underline{u}-u+Nd^2)+(\underline{u}-u)G^{ij}\delta_{ij}
-2\beta G^{ij}\delta_{ij}.\end{aligned}$$ Then, using , and that $u\geq \underline u,$ we get $$\begin{aligned}
\begin{split}
\label{conta-convex1}
G^{ij}&\nabla_{ij}(u-\underline{u}+td-Nd^2)\leq \Upsilon\big(X\big)-2\beta G^{ij}\delta_{ij}
+tCG^{ij}\delta_{ij}\\ &- F\Big( \big[\frac{1}{uw}\gamma^{ik}
(\underline{u}\delta_{kl}+\nabla_{kl}\underline{u}+2N\nabla_l d\nabla_k d
-3\beta \delta_{lk})\gamma^{jl}\big]\Big) \\
= & - f\Big( \lambda\big(\frac{1}{uw}\gamma^{ik}
(\underline{u}\delta_{kl}+\nabla_{kl}\underline{u}-3\beta \delta_{kl})\gamma^{jl}
+\frac{2N}{uw}\gamma^{ik}\nabla_l d\nabla_k d\gamma^{jl}\big)\Big)\\
&+(tC-2\beta)G^{ij}\delta_{ij}+\Upsilon\big(X\big) .
\end{split}\end{aligned}$$ By the choice of $\beta$ and Theorem \[teorema2\], there exists a uniform positive constant $\lambda_0$ satisfying $$\label{convex4}
\Big[\frac{1}{uw}\gamma^{jk}(\underline{u}\delta_{kl}
+\nabla_{kl}\underline{u}-3\beta \delta_{lk})\gamma^{jl}\Big]\geq \lambda_0 I.$$ Then we can find a uniform positive constant $\mu_0$ such that $$\begin{aligned}
P^T[\frac{1}{uw}\gamma^{ik}
(\underline{u}\delta_{kl}+&\nabla_{kl}\underline{u}-3\beta \delta_{kl})\gamma^{jl}
+\frac{2N}{uw}\gamma^{ik}\nabla_l d\nabla_k d\gamma^{jl}\big]P \\ & \geq
\textrm{diag}\{\lambda_0,\lambda_0, \ldots, \lambda_0+N\mu_0\},\end{aligned}$$ where $P$ is an orthogonal matrix that diagonalizes $\big[\frac{2N}{uw}\gamma^{kl}\nabla_l d\nabla_k d\gamma^{jl}\big].$ Then, by the ellipticity and concavity of $f$ we get $$\begin{aligned}
\begin{split}
f&\Big( \lambda\big(\frac{1}{uw}\gamma^{ik}
(\underline{u}\delta_{kl}+\nabla_{kl}\underline{u}-3\beta \delta_{lk})\gamma^{jl}
+\frac{2N}{uw}\gamma^{ik}\nabla_l d\nabla_k d\gamma^{jl}\big)\Big)\\
&=f\Big( \lambda\big(P^T\big[\frac{1}{uw}\gamma^{ik}(\underline{u}\delta_{kl}+
\nabla_{kl}\underline{u}-3\beta \delta_{lk})\gamma^{jl}
+\frac{2N}{uw}\gamma^{ik}\nabla_l d\nabla_k d\gamma^{jl}\big]P\big)\Big) \\ &
\geq f(\lambda_0,\lambda_0, \ldots, \lambda_0+N\mu_0).
\end{split}\end{aligned}$$ Since $$f(\lambda_0,\lambda_0, \ldots, \lambda_0+N\mu_0)
\rightarrow +\infty \quad \textrm{as} \quad N\rightarrow+\infty,$$ it follows from that, for $t$ small enough such that $Ct\leq \beta$ and $N$ sufficient large, we have $$G^{ij}\nabla_{ij}\Theta\leq -C-3t-\beta G^{ij}\delta_{ij}$$ where $C$ is a uniform constant that satisfies $|\nabla(u-\underline u)|\leq C.$ Finally, choosing $\delta$ even smaller, such that $\delta N<t,$ we get $|\nabla \Theta|\leq C+3t$ and $\Theta\geq 0$ on $\partial(\Omega\cap \Omega_\delta).$
We are now in position to derive the mixed second derivatives boundary estimate. Consider the functions $$\label{cara-w-barra}
\bar{\Phi}=\tilde{\Phi}+b_0(u-\underline{u})$$ and $$\label{cara-theta}
\bar{\Theta}=-c_0\varrho^2-d_0 \Theta,$$ where $b_0, c_0$ and $d_0$ are positive constants to be chosen. Following the reasoning in the proof of Lemma \[DEF-V\], we can easily prove that $$\begin{aligned}
\label{V-1}
G^{ij}\nabla_{ij}(u-\underline{u})\leq C-\beta G^{ij}\delta_{ij}\end{aligned}$$ in $\Omega_\delta,$ for sufficiently small $\delta,\beta>0.$ Then we conclude from that $$G^{ij}\nabla_{ij}\bar{\Phi} \leq M(1+b_0+|\nabla \tilde{\Phi}|)+b_0\big(C-\beta G^{ij}\delta_{ij}\big).$$ Hence, choosing $b_0$ sufficiently large we get $$\label{INEL-PHI}
G^{ij}\nabla_{ij}\bar{\Phi}\leq M_0(1+|\nabla \bar{\Phi}|)\quad \textrm{in } \Omega_\delta,$$ for a uniform positive constant $M_0=M_0(a_0,b_0, M).$
On the other hand, using Lemma \[DEF-V\] and inequality we can estimate $$\begin{aligned}
G^{ij}\nabla_{ij}\bar{\Theta} =& -c_0G^{ij}\nabla_{ij}\varrho^2-d_0G^{ij}\nabla_{ij} d \\
& \geq -3c_0 G^{ij}\delta_{ij}+d_0\big(1+|\nabla \Theta|+\beta G^{ij}\delta_{ij}\big).\end{aligned}$$ As $|\nabla \bar{\Theta}|\leq 2\delta c_0 + d_0|\nabla \Theta|,$ choosing $d_0>>c_0$ sufficiently large, we get $$\label{INEL-THETA}
G^{ij}\nabla_{ij}\bar{\Theta}\geq M_0(1+|\nabla \bar{\Theta}|) \quad \textrm{in } \Omega_\delta.$$
Now we compare $\bar{\Phi}$ and $\bar{\Theta}$ on $\partial\Omega_\delta.$ At this point we need to assume that the index $k$ fixed in where $\Phi$ is defined, is chosen so that $1\leq k\leq n-1.$ In particular, $e_k$ is tangent along $\partial\Omega$ and we have $\bar{\Phi}=0$ on $\partial\Omega_\delta\cap \partial\Omega.$ By we have $\bar{\Theta}\leq -c_0\varrho^2$ on $\partial\Omega_\delta,$ then $\bar{\Phi}=0\geq -c_0\varrho^2\geq \bar{\Theta}$ on $\partial\Omega_\delta\cap \partial\Omega.$ For $\partial\Omega_\delta\cap\Omega,$ notice that $|\bar{\Phi}|\leq C$ on $\partial\Omega_\delta\cap\Omega$ for a uniform constant $C.$ Hence, choosing $c_0$ sufficiently large we get $$\bar{\Theta}=-c_0\varrho^2-d_0 \Theta\leq -c_0 \delta^2\leq \bar{\Phi}
\quad\textrm{on} \quad \partial\Omega_\delta\cap \Omega.$$ Therefore $$\label{PHI-THETA}
\bar{\Theta}\leq \bar{\Phi} \quad \textrm{on} \, \partial\Omega_\delta.$$
Finally, it follows from , and the Comparison Principle (see e.g. [@GT]) that $\bar{\Phi}\geq \bar{\Theta}$ in $\Omega_\delta.$ As $\bar{\Phi}(x_0)=\bar{\Theta}(x_0)$ we get $\nabla_n \bar{\Theta}(x_0)\leq \nabla_n \bar{\Phi}(x_0),$ which give us $$\label{mista-explicita}
\nabla_{kn}u(x_0) \geq \nabla_{kn}\varphi(x_0)
-\frac{d_0}{a_0}\big(\nabla_n(u-\underline u)(x_0)+t\big).$$ for $1\leq k\leq n-1.$ Then, as $x_0\in\partial\Omega$ is arbitrary, we get $$\label{TANGENTE-NORMAL}
|\nabla_{kn} u|<C \quad \textrm{on}\, \partial\Omega.$$ The mixed second derivatives boundary estimate is established.
Now we consider the pure normal second derivative bound. Since $\Sigma$ has positive mean curvature, we only need to derive an upper bound $$\label{NORMAL-NORMAL}
\nabla_{nn}u < C \quad \textrm{on}\,\, \partial\Omega.$$
Let $\kappa'=(\kappa'_1\ldots,\kappa'_{n-1})$ the roots of $\det(h_{\alpha\beta}-t g_{\alpha\beta})
=0\, (1\leq \alpha,\beta\leq n-1).$ Notice that $\kappa'$ do not denotes the first $n-1$ principal curvatures of $\Sigma.$ For an arbitrary fixed $x\in\partial\Omega,$ let $\tau_1, \ldots, \tau_{n-1}\in T_x\partial\Omega$ be a basis of vectors that diagonalize $h_{\alpha\beta}$ with respect to the inner product defined by $g_{\alpha\beta}.$ Then the function curvature $S_r$ of the hypersurface $\Sigma$ at $X(x)$ is given by (see, for instance, [@ALM]) $$\begin{aligned}
\label{Sr-boundary}
\begin{split}
S_r=S_{r-1}(\kappa')\nabla_{nn} u+ D
\end{split}\end{aligned}$$ where $D$ depends only on $u, \nabla u$ and the tangential and mixed second derivatives of $u.$ Therefore $$\begin{aligned}
S_{r-1}(\kappa')=\frac{\partial S_r}{\partial \nabla_{nn}u}=\frac{\partial F}{\partial a_{ij}}\frac{\partial a_{ij}}{\partial \nabla_{nn}u}
=g^{nn}\frac{\partial F}{\partial a_{nn}}>0,\end{aligned}$$ by ellipticity. In particular, $(\kappa',0)\in \Gamma_{r-1}\subset\mathbb{R}^n.$ Now we adapt the techniques used in [@CNSIII] and [@GUAN-2], which are based on a brilliant idea introduced by Trudinger in [@TRU-2]. First we show that an upper bound on $\nabla_{nn} u$ on $\partial \Omega$ amounts to a lower bound on $S_{r-1}(\kappa')$ on $\partial \Omega$ by a uniform positive quantity.
Let $\Gamma'_{r-1}$ be the projection of $\Gamma_{r-1}$ into $\mathbb{R}^{n-1}$ and denote by $\tilde d(x)$ the distance from $\kappa'(x)$ to $\partial\Gamma'_{r-1}.$ In what follows, we estimate $\nabla_{nn} u$ at a point $x_0$ of $\partial\Omega$ where $\tilde d$ is minimum. So, let $x_0\in\partial\Omega$ be a point where $\tilde d$ attains its minimum. As above, choose a local frame field $\tau_1, \ldots, \tau_{n-1}$ on $\partial\Omega$ around $x_0$ which is orthogonal with respect to the inner product given by $g_{\alpha\beta}$ and that diagonalizes $h_{\alpha\beta}$ at $x_0.$ Let $\tau_1, \ldots, \tau_{n-1}, e_n$ be the frame field on $\mathbb{S}^n$ obtained by parallel translation of the local frame field $\tau_1, \ldots, \tau_{n-1}$ along the geodesic perpendicular to $\partial\Omega$ and $e_n$ denotes the parallel translation of the unit normal field on $\partial\Omega.$ Choose the first $n-1$ indices so that $$\kappa'_1\leq \ldots\leq \kappa'_{n-1}.$$ Using Lemma 6.1 of [@CNSIII], we can find a vector $\gamma'=(\gamma_1,\ldots , \gamma_{n-1})
\in\mathbb{R}^{n-1}$ such that $$\begin{aligned}
\gamma_1\geq \cdots \geq\gamma_{n-1}\geq 0,\quad\sum_{\alpha<n}\gamma_\alpha=1\end{aligned}$$ and $$\label{D-X-0}
\tilde d(x_0)=uw\sum_{\alpha<n}\gamma_{\alpha}\kappa'_\alpha(x_0) =
\sum_{\alpha<n}\gamma_{\alpha}\big(u\sigma_{\alpha\alpha}+\nabla_{\alpha\alpha}u\big) (x_0),$$ where $\sigma_{\alpha\beta}=\langle \tau_\alpha, \tau_\beta\rangle$ and we have used that $g_{\alpha\beta}=\delta_{\alpha\beta}.$ Here we are also using that the distance of $\kappa'$ and $uw\kappa'$ to $\partial\Gamma'_{r-1}$ is equal. Furthermore, $$\label{CASA-GAMA-LINHA}
\Gamma'_{r-1}\subset \left\{\lambda'\in\mathbb{R}^{n-1}\, :\, \gamma'\cdot \lambda'>0 \right\}.$$ It follows by Lemma 6.2 of [@CNSIII], with $\gamma_n=0,$ that for all $x\in\partial
\Omega$ sufficiently near $x_0$ we have $$\begin{aligned}
\label{DES-T}
\sum_{\alpha<n}\gamma_{\alpha}\big(u\sigma_{\alpha\alpha} +\nabla_{\alpha\alpha} u\big)(x)
\geq uw\sum_{\alpha<n}\gamma_\alpha\kappa'(x)\geq \tilde d(x)\geq\tilde d(x_0),\end{aligned}$$ where we have used and $|\gamma'|\leq 1$ in the second inequality. Then $$\begin{aligned}
\label{DES-B-D}
\begin{split}
\nabla_n u (x)\sum_{\alpha<n}&\gamma_\alpha B_{\alpha\alpha}(x)
= \sum_{\alpha<n}\gamma_\alpha \nabla_{\alpha\alpha}\varphi(x)
-\sum_{\alpha<n}\gamma_\alpha \nabla_{\alpha\alpha}u(x)
\\ &\leq \sum_{\alpha<n}\gamma_\alpha\big(\varphi\sigma_{\alpha\alpha}
+\nabla_{\alpha\alpha}\varphi\big)(x)-\tilde d(x_0),
\end{split}\end{aligned}$$ where we have used in the last inequality.
Since the matrix $\{\underline u\sigma_{\alpha\beta} +\nabla^2_{\alpha\beta}\underline u\}$ is positive definite in a neighbourhood of $\partial\Omega,$ it follows that $\kappa'[\underline u]:=(\underline u\sigma_{11} +\nabla_{11}\underline u, \ldots,
\underline u\sigma_{(n-1)(n-1)} +\nabla_{(n-1)(n-1)}\underline u)(x_0)$ belongs to $\Gamma'_{r-1}.$ We may assume $$\tilde d(x_0)<\frac{1}{2} \textrm{dist}\big(\kappa'[\underline u], \partial\Gamma'_{r-1}\big),$$ otherwise we have a uniform positive lower bound for $S_{r-1}(\kappa')(x_0)$ and follows directly from . Thus, we conclude from and Lemma 6.2 of [@CNSIII] that $$\begin{aligned}
\nabla_n(u-\underline u) (x_0)\sum_{\alpha<n}&\gamma_\alpha B_{\alpha\alpha}(x_0)=
\sum_{\alpha<n}\gamma_\alpha\nabla_{\alpha\alpha}\underline u(x_0)
-\sum_{\alpha<n}\gamma_\alpha \nabla_{\alpha\alpha}u(x_0)
\\ & \geq\tilde d\big(\kappa'[\underline{u}]\big)-\tilde d(x_0)
> \frac{1}{2} \tilde d\big(\kappa'[\underline{u}]\big)>0.\end{aligned}$$ As $\nabla_n(u-\underline u)\geq 0$ on $\partial\Omega,$ then $\sum_{\alpha<n}\gamma_\alpha B_{\alpha\alpha}(x_0)>0$ and we conclude that there exist uniform positive constants $ c,\delta>0,$ such that $$\sum_{\alpha<n}\gamma_\alpha B_{\alpha\alpha}(x)\geq c>0,$$ for every $x\in \Omega$ satisfying $\textrm{dist}_{\mathbb{S}^n}(x, x_0)<\delta.$ Hence we may define the function $$\begin{aligned}
\label{CARA-MU}
\mu(x) =\frac{1}{\sum_{\alpha<n}\gamma_\alpha B_{\alpha\alpha}(x)}
\left(\sum_{\alpha<n}\gamma_\alpha\big(\varphi\sigma_{\alpha\alpha}
+\nabla_{\alpha\alpha}\varphi\big)(x)-\tilde d(x_0)\right),\end{aligned}$$ for $x\in \Omega_{\delta}=\{ x\in\Omega\, :\, \textrm{dist}_{\mathbb{S}^n}(x,x_0)<\delta\}.$ It follows from that $ \nabla_n u \leq \mu$ on $\partial\Omega\cap\partial\Omega_{\delta}$ for a uniform constant $\delta>0.$ On the other hand, implies that $\nabla_n u (x_0) = \mu(x_0).$ Then we may proceed as it was done for the mixed normal-tangential derivatives to get the estimate $\nabla_{nn}u(x_0)
\leq C,$ for a uniform constant $C.$ In fact, redefining the function $\Phi$ given in by replacing $\nabla_{k}(u-\varphi)$ for $\mu-\nabla_n u,$ i.e., defining $$\label{CARA-W-2}
\Phi=\mu-\nabla_n u-\frac{K}{2}\sum_{l<n}\big(\nabla_l (u-\varphi)\big)^2,$$ we conclude from the uniform bound $|\nabla^2\mu|\leq C$ that inequality remain valid for this new function $\Phi.$ Defining $\bar\Theta$ as in , clearly inequality remains true. Finally, as $ \nabla_n u \leq \mu$ on $\partial\Omega\cap\partial\Omega_{\delta}$ the function $\bar{\Phi}$ defined in satisfies $\bar{\Phi}\geq 0$ on $\partial\Omega\cap\partial\Omega_{\delta}.$ Therefore, proceeding as above we get the uniform bound $$\label{U-NUNU-X-0}
\nabla_{nn} u(x_0)\leq C.$$
Hence, it follows from the previous estimates that the principal curvatures $\kappa_{\Sigma}[X(x_0)]=(\kappa_1, \ldots ,\kappa_n)(x_0)$ of $\Sigma$ at $X(x_0)$ is contained in an [*a priori*]{} bounded subset of $\Gamma_{r}\subset\Gamma_{r-1}.$ Therefore, as $$H_r^{1/r}(\kappa[u])=\Upsilon\geq \inf_{\Delta_L}\Upsilon>0$$ and $H_r=0$ on $\partial\Gamma_r,$ it follows that $\textrm{dist}\big((\kappa_1,\ldots, \kappa_{n-1})(x_0),
\partial\Gamma'_{r-1}\big)\geq \bar{c}_0>0$ for a uniform constant $\bar{c}_0>0.$ On the other hand, by Lemma 1.2 of [@CNSIII], the principal curvatures $\kappa_{\Sigma}=(\kappa_1, \ldots ,\kappa_n)$ of $\Sigma$ behave like $$\begin{aligned}
\label{ki}\kappa_\alpha &=\kappa'_\alpha+o(1), \quad 1\leq \alpha\leq n-1,\\
\label{kn}\kappa_n &=\frac{h_{nn}}{g_{nn}}\left(1+O\left(\frac{1}{h_{nn}}\right)\right),\end{aligned}$$ as $|h_{nn}|\rightarrow\infty,$ where $o(1)$ and $O(1/h_{nn})$ are uniform, depending only on $\kappa'_1, \ldots, \kappa'_{n-1}$ and the bounds on $|u|,$ $|\nabla u|$ and $|\nabla_{\alpha n} u|,$ $(1\leq \alpha\leq n-1).$ Then there exists a uniform constant $N_0$ such that, if $\nabla_{nn} u(x_0)\geq N_0$ the distance of $\big(\kappa_1,\ldots,\kappa_{n-1}\big)(x_0)$ to $\kappa'(x_0)$ is less then $\bar{c}_0/2,$ where $\bar{c}_0$ is the constant given above. In particular, if $\nabla_{nn} u(x_0)\geq N_0$ then $\tilde{d}(x_0)\geq c_0$ for a uniform constant $c_0>0,$ which implies that $S_{r-1}(\kappa')$ admits itself a uniform positive bound on $\partial\Omega$ and, in this case, follows from . This establish and completes the proof of Theorem \[teorema1-2\].
In [@CNSIV] it is also shown how to derive the global estimates for $|\nabla^2 u|$ on $\bar\Omega$ from its bound on the boundary $\partial\Omega,$ if $\Upsilon$ satisfy . Then we have the following result.
\[teorema3\] Let $u\geq \underline{u}$ be an admissible solution of and suppose that $\Upsilon$ satisfy . Then we have the estimate $$\label{S3-2}
\| u\|_{C^{2}(\bar{\Omega})}\leq C \quad \textrm{in } \bar\Omega,$$ where $C$ depends on $ \inf_{\bar\Omega}\underline u, \|\underline{u}\|_{C^2(\bar\Omega)},
\|\psi^t\|_{C^2(\Delta_L)},$ the convexity of $\bar\Sigma$ in a neighbourhood of $\partial\Omega$ and other known data.
Proof of Theorem \[teorema1\] {#section4}
=============================
In this section we complete the proof of Theorem \[teorema1\] by applying the method of continuity and a degree theory argument with the aid of the [*a priori*]{} estimates we have established. Our proof is inspired in [@CNSI] and [@SU], where Monge-Ampère type equations are treated.
Here we will not deal with equation because $G_u$ is positive and can not be bounded easily. Then we need to express in a different form. Setting $v=-\ln \rho=\ln u,$ the matrix $A[u]=[a_{ij}]$ can be written in terms of $v$ by $$\label{g-v}
a_{ij}=\frac{e^v}{w}\big(\delta_{ij}+\gamma^{ik}\nabla_{kl}v\gamma^{jl}\big)$$ where $$\label{h-v}
w=\sqrt{1+|\nabla v|^2}, \quad \gamma^{ij}=\delta_{ij}-\frac{\nabla_i v\nabla_j v}{w(1+w)}.$$ Hence, for $\Upsilon=\Psi^t,$ takes the form $$\begin{aligned}
\begin{split}
\label{equation5}
H(\nabla^2 v,\nabla v, v)&=\Psi^t(X(x))=e^{3(v-\underline v)} \big(t\psi(x)+(1-t)
\underline{\psi}(x)\big)\quad\textrm{in}\, \Omega,\\
v&=-\ln \phi \quad\textrm{on}\, \partial\Omega,
\end{split}\end{aligned}$$ Notice that $\underline v=-\ln\bar\rho=\ln \underline{u}$ is a strictly subsolution of fot $t>0$ and it is a solution for $t=0.$ Moreover, as $$\begin{aligned}
\begin{split}
H_v=F^{ij} a_{ij}=\sum f_i\kappa_i \leq f(\kappa) =\Psi^t
\end{split}\end{aligned}$$ and $$\begin{aligned}
\Psi^t_v= \frac{\partial \Psi^t }{\partial v}=3\Psi^t,\end{aligned}$$ it follows that $H_v-\Psi^t_v\leq 0.$ Then we can apply the comparison principle to equation to conclude that any solution $v^t$ of for $t>0$ satisfy $v^t> \underline{v}.$ Hence Theorem \[teorema3\] can be applied and we get the $C^2$ estimates for any solution $v^t$ of . Therefore the holder estimates follows from the Evans-Krylov Theorem and we can apply the continuity method to conclude that there exist a unique solution $v^0$ of for $t=1.$ Now we consider the family of equations ($s\in [0,1]$) $$\begin{aligned}
\begin{split}
\label{equation6}
H(\nabla^2 v,\nabla v, v)&=\Xi^s(X(x))=s\psi(x)+(1-s)e^{3(v-\underline v)}
\underline{\psi}(x)\quad\textrm{in}\, \Omega,\\
v&=-\ln\phi \quad\textrm{on}\, \partial\Omega.
\end{split}\end{aligned}$$ From Theorem \[teorema3\], the Evans-Krylov Theorem and by the standard regularity theory for second order uniformly elliptic equations we can get higher order estimate $$\begin{aligned}
\label{c4forvt}
\| v^s\|_{C^{4,\alpha} (\bar\Omega)}< \bar C\quad\textrm {independent of } s,\end{aligned}$$ for any solution $v^s$ of equation satisfying $v^s\geq \underline v.$ We also point out that, if $s>0$ and $v^s\geq \underline{v}$ is a solution of then $v^s$ is a supersolution of for $t=s.$ In particular, we have the strictly inequality $v^s>\underline v$ in this case.
Let $C_0^{4,\alpha}(\bar\Omega)$ be the subspace of $C^{4,\alpha}(\bar\Omega)$ consisting of functions vanishing on the boundary. Consider the cone $$\begin{aligned}
\mathcal{O} = \{ & z\in C_0^{4,\alpha}(\bar\Omega) : z>0 \textrm{ in } \Omega, \,
\nabla_n z >0 \textrm{ on } \partial\Omega, \\ & z+\underline{v} \textrm{ is admissible} \textrm{ and }
\|z\|_{C^{4,\alpha}(\bar\Omega)}\leq \bar C+\|\underline{v}\|_{C^{4,\alpha}(\bar\Omega)}\}, \end{aligned}$$ where $\bar C$ is the constant given in . Now we construct a map from $\mathcal{O}\times [0,1]$ to $C^{2,\alpha}(\bar\Omega)$ given by $$M_s[w]=H(\nabla^2(z+\underline{v}), \nabla (z+\underline{v}), z+\underline{v})-
\Xi^s(z+\underline{v}), \quad z\in\mathcal{O},$$ where $\Xi^s$ is the function given in : $$\Xi^s(z+\underline{v})= t\psi(x)+(1-t)e^{3z}\psi(x).$$ Clearly, $z$ is a solution of $M_s[z]=0$ iff $v^s=z+\underline{v}$ is a solution of . In particular, $z^0=v^0-\underline{v}$ is the unique solution of $M_0[z]=0$ and $z^0\in\mathcal{O}.$ Moreover, there is no solution of $M_s[z]=0$ on $\mathcal{O}$ for any $s.$ Therefore, the degree of $M_s$ on $\mathcal{O}$ at $0$ $\deg(M_s,\mathcal{O},0)$ is well defined and independent of $s.$ For more details, we refer the reader to [@LI1] and [@LI2].
Now we compute $\deg(M_0,\mathcal{O},0).$ We know that $M_0[z]=0$ has a unique solution $z^0$ in $\mathcal{O}.$ The Fréchet derivative of $M_0$ at $z^0$ is a linear elliptic operator from $C^{4,\alpha}_0(\bar\Omega)$ to $C^{2,\alpha}(\bar\Omega)$, $$M_{0,z^0}(h)=H^{ij}|_{v^0}\nabla_{ij}h+H^i|_{v^0}\nabla_i h+(H_{v}|_{v^0}-\Xi^0_v|{v^0})h.$$ By , $H_{v}|_{v^0}-\Xi^0_v|{v^0}<0.$ So $M_{0,z^0}$ is invertible. By the theory in [@LI1], we can see $$\deg(M_0,\mathcal{O},0)=\deg(M_{0, z^0},B_1,0)=\pm 1\neq 0,$$ where $B_1$is the unit ball of $C_0^{4,\alpha}(\bar\Omega).$ Therefore $$\deg(M_s,\mathcal{O},0)\neq 0 \quad \textrm{for all}\, s\in [0,1].$$ Then equation $M_s[z]=0$ has at least one solution for any $s\in[0,1].$ In particular, the function $v^1=z^1-\underline{v}$ is then a solution of . Therefore $\rho = e^{-v^1}$ is a solution of .
Appendix
========
For completeness, we present here the prove that inequality implies inequality , i.e., the existence of some $1\leq \gamma\leq n$ such that $$\label{S5-1}
\sum_{l<n}\eta_{l\gamma}^2< K^{-2}.$$ implies that $$\label{S5-2}
\sum_{l<n}^{n-1}\eta_{l\alpha}^2\geq\epsilon_0$$ for all $\alpha\neq \gamma$ and for a uniform positive constant $\epsilon_0>0$ that does not depend on $K.$ For simplicity, let us assume $\gamma=1.$ As $P=[\eta_{ij}]$ is an orthogonal matrix, $\eta_{ij}$ is the cofactor of index $(j,i)$ of $P.$ Developing the determinant of $P$ with respect to the first line we get $$\begin{aligned}
\label{lema-c1}
\begin{split}
1&= |\det(\eta_{ij})|
\leq |\eta_{1n}\eta_{n1}|+\sum_{l<n}|\eta_{l1}\eta_{1l}|\\ &\leq |\eta_{1n}\|\eta_{n1}|+
\Big(\sum_{l<n}(\eta_{l1})^2\Big)^{1/2}\Big(\sum_{l<n}(\eta_{1l})^2\Big)^{1/2}
\\ &\leq |\eta_{n1}|+K^{-1}.
\end{split}\end{aligned}$$ Now we develop the cofactor $\eta_{n1}$ with respect to the $\alpha^{th}$ line, $2\leq\alpha\leq n,$ to obtain $$\begin{aligned}
|\eta_{n1}|& = \Big| \sum_{l<n} \eta_{l\alpha} \zeta_{\alpha l}\Big|
\leq (n-2)!\sum_{l<n} | \eta_s^\alpha| (n-2)!
\\ & \leq (n-2)!(n-1)^{1/2}\Big(\sum_{s=1}^{n-1} (\eta_s^\alpha)^2\Big)^{1/2}.\end{aligned}$$ where $\zeta_{l\alpha}$ denotes the cofactor of index $(\alpha, l)$ of the $[n-1]$ matrix $(\eta_{ij}),$ where $2\leq i\leq n$ and $1\leq j\leq n-1.$ Thus, replacing this inequality into we obtain $$\begin{aligned}
1\leq (n-2)!(n-1)^{1/2}\Big(\sum_{l<n} (\eta_{l\alpha})^2\Big)^{1/2}+K^{-1}.\end{aligned}$$ Therefore, for $K\geq 2,$ $$\begin{aligned}
\Big(\sum_{l<n} (\eta_{l\alpha})^2\Big)^{1/2}
\geq \frac{1}{2(n-1)^{1/2}(n-2)!},\end{aligned}$$ which proves .
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work has been partially supported by CAPES. Part of this work was written while the author was visiting the Universidad de Murcia at Murcia, Espain. He thanks that institution for its hospitality.
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[^1]: Work partially supported by CAPES-Brazil
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---
abstract: 'A nonnegative integer is called a *fertility number* if it is equal to the number of preimages of a permutation under West’s stack-sorting map. We prove structural results concerning permutations, allowing us to deduce information about the set of fertility numbers. In particular, the set of fertility numbers is closed under multiplication and contains every nonnegative integer that is not congruent to $3$ modulo $4$. We show that the lower asymptotic density of the set of fertility numbers is at least $1954/2565\approx 0.7618$. We also exhibit some positive integers that are not fertility numbers and conjecture that there are infinitely many such numbers.'
address: $^1$Princeton University
author:
- Colin Defant$^1$
title: Fertility Numbers
---
Introduction
============
Throughout this article, the word “permutation" refers to a permutation of a finite set of positive integers. We write permutations as words in one-line notation. Let $S_n$ denote the set of permutations of $\{1,\ldots,n\}$. We say a permutation is *normalized* if it an element of $S_n$ for some $n$ (e.g., the permutation $12547$ is not normalized).
The study of permutation patterns, which has now developed into a vast area of research, began with Knuth’s investigation of stack-sorting in [@Knuth]. In his 1990 Ph.D. thesis, Julian West [@West] explored a deterministic variant of Knuth’s stack-sorting algorithm, which we call the *stack-sorting map*. This map, denoted $s$, is defined as follows.
Assume we are given an input permutation $\pi=\pi_1\cdots\pi_n$. Throughout this algorithm, if the next entry in the input permutation is smaller than the entry at the top of the stack or if the stack is empty, the next entry in the input permutation is placed at the top of the stack. Otherwise, the entry at the top of the stack is annexed to the end of the growing output permutation. This procedure stops when the output permutation has length $n$. We then define $s(\pi)$ to be this output permutation. Figure \[Fig1\] illustrates this procedure and shows that $s(4162)=1426$.
{width="1\linewidth"}
\[Fig1\]
There is an alternative recursive description of the stack-sorting map. Specifically, if $m$ is the largest entry appearing in the permutation $\pi$, we can write $\pi=LmR$, where $L$ and $R$ are the substrings of $\pi$ appearing to the left and right of $m$, respectively. Then $s(\pi)=s(L)s(R)m$. For example, $s(4162)=s(41)s(2)6=s(41)26=s(1)426=1426$. It is also possible to describe the stack-sorting algorithm in terms of in-order readings and postorder readings of decreasing binary plane trees [@Bona; @Defant].
West defined the *fertility* of a permutation $\pi$ to be $|s^{-1}(\pi)|$, the number of preimages of $\pi$ under the stack-sorting map [@West]. He proceeded to compute the fertilities of the permutations of the forms $$23\cdots k1(k+1)\cdots n,\quad 12\cdots(k-2)k(k-1)(k+1)\cdots n,\quad\text{and}\quad k12\cdots(k-1)(k+1)\cdots n.$$ Bousquet-Mélou then defined a *sorted* permutation to be a permutation that has positive fertility [@Bousquet]; she provided an algorithm for determining whether or not a given permutation is sorted. She also mentioned that it would be interesting to find a method for computing the fertility of any given permutation. The current author found such a method in [@Defant]. In fact, the results in that paper are even more general; they allow one to enumerate certain types of decreasing plane trees that have a given permutation as their postorder readings. The current author has since used this method to improve the best-known upper bounds for the enumeration of so-called $3$-stack-sortable and $4$-stack-sortable permutations in [@Defant2]. See [@Bona; @BonaSurvey; @Defant2; @Zeilberger] for more information about $t$-stack-sortable permutations.
The method developed in [@Defant] and [@Defant2] for computing fertilities makes use of new combinatorial objects called *valid hook configurations*. The authors of [@Defant3] gave a concise description of valid hook configurations and exhibited a bijection between these objects and certain ordered pairs of set partitions and acyclic orientations. They then exploited this bijection to study permutations with fertility $1$, showing that these permutations[^1] are counted by an interesting sequence known as Lassalle’s sequence (which Lassalle introduced in [@Lassalle]). This bijection also allowed the authors to connect cumulants arising in free probability theory with valid hook configurations and the stack-sorting map (building upon results from [@Josuat]). For completeness, we repeat the short description of valid hook configurations from [@Defant3] in Section 2. See also [@DefantCatalan; @DefantMotzkin; @Defant3; @Hanna; @Maya] for further investigation of the combinatorics of valid hook configurations.
\[Def1\] Say a nonnegative integer $f$ is a *fertility number* if there exists a permutation with fertility $f$. Say a nonnegative integer is an *infertility number* if it is not a fertility number.
For example, $0,1$, and $2$ are fertility numbers because $|s^{-1}(21)|=0$, $|s^{-1}(1)|=1$, and $|s^{-1}(12)|=2$. In Section 3, we prove the following statements about fertility numbers. These are Theorems \[Thm1\]–\[Thm7\] below.
- The set of fertility numbers is closed under multiplication.
- If $f$ is a fertility number, then there are arbitrarily long permutations with fertility $f$.
- Every nonnegative integer that is not congruent to $3$ modulo $4$ is a fertility number. The lower asymptotic density of the set of fertility numbers is at least $1954/2565\approx 0.7618$.
- The smallest fertility number that is congruent to $3$ modulo $4$ is $27$.
- If $f$ is a positive fertility number, then there exist a positive integer $n\leq f+1$ and a permutation $\pi\in S_n$ such that $f=|s^{-1}(\pi)|$.
The fourth bullet point above shows, in particular, that the notion of a fertility number is not pointless because infertility numbers exist. The fifth bullet shows that determining whether or not a given number is a fertility number can be reduced to a finite search. This finite search can be very long, but we will see in our proof of the fourth bullet point that we can often cut corners to reduce the computations. In Section 4, we give suggestions for future work, including three conjectures.
Valid Hook Configurations
=========================
In this section, we review some of the theory of valid hook configurations. Our presentation is virtually the same as that given in [@Defant3], but we include it here for completeness. It is important to note that the valid hook configurations defined below are, strictly speaking, different from those defined in [@Defant] and [@Defant2]. For a lengthier discussion of this distinction, see [@Defant3].
The construction of a valid hook configuration commences with the choice of a permutation $\pi=\pi_1\cdots\pi_n$. A *descent* of $\pi$ is an index $i$ such that $\pi_i>\pi_{i+1}$. Let $d_1<\cdots<d_k$ be the descents of $\pi$. We use the example permutation $3142567$ to illustrate the construction. The *plot* of $\pi$ is the graph displaying the points $(i,\pi_i)$ for $1\leq i\leq n$. The left image in Figure \[Fig2\] shows the plot of our example permutation. A point $(i,\pi_i)$ is a *descent top* if $i$ is a descent. The descent tops in our example are $(1,3)$ and $(3,4)$.
A *hook* of $\pi$ is drawn by starting at a point $(i,\pi_i)$ in the plot of $\pi$, moving vertically upward, and then moving to the right until reaching another point $(j,\pi_j)$. We must necessarily have $i<j$ and $\pi_i<\pi_j$. The point $(i,\pi_i)$ is called the *southwest endpoint* of the hook, while $(j,\pi_j)$ is called the *northeast endpoint*. The right image in Figure \[Fig2\] shows our example permutation with a hook that has southwest endpoint $(3,4)$ and northeast endpoint $(6,6)$.
A *valid hook configuration* of $\pi$ is a configuration of hooks drawn on the plot of $\pi$ subject to the following constraints:
1. The southwest endpoints of the hooks are precisely the descent tops of the permutation.
2. A point in the plot cannot lie directly above a hook.
3. Hooks cannot intersect each other except in the case that the northeast endpoint of one hook is the southwest endpoint of the other.
![Four configurations of hooks that are forbidden in a valid hook configuration.[]{data-label="Fig3"}](FertilityPIC6){width=".7\linewidth"}
![All of the valid hook configurations of $3142567$.[]{data-label="Fig4"}](FertilityPIC4){width=".7\linewidth"}
Figure \[Fig3\] shows four placements of hooks that are forbidden by conditions 2 and 3. Figure \[Fig4\] shows all of the valid hook configurations of $3142567$. Note that the total number of hooks in a valid hook configuration of $\pi$ is exactly $k$, the number of descents of $\pi$. Because the southwest endpoints of the hooks are the points $(d_i,\pi_{d_i})$, we have a natural ordering of the hooks. Namely, the $i^\text{th}$ hook is the hook whose southwest endpoint is $(d_i,\pi_{d_i})$. We can write a valid hook configuration of $\pi$ concisely as a $k$-tuple $\mathcal H=(H_1,\ldots,H_k)$, where $H_i$ is the $i^\text{th}$ hook.
A valid hook configuration of $\pi$ induces a coloring of the plot of $\pi$. To begin the process of coloring the plot, draw a “sky" over the entire diagram. As one might expect, we color the sky blue. Assign arbitrary distinct colors other than blue to the $k$ hooks in the valid hook configuration.
There are $k$ northeast endpoints of hooks, and these points remain uncolored. However, all of the other $n-k$ points will be colored. In order to decide how to color a point $(i,\pi_i)$ that is not a northeast endpoint, imagine that this point looks directly upward. If this point sees a hook when looking upward, it receives the same color as the hook that it sees. If the point does not see a hook, it must see the sky, so it receives the color blue. However, if $(i,\pi_i)$ is the southwest endpoint of a hook, then it must look around (on the left side of) the vertical part of that hook. See Figure \[Fig5\] for the colorings induced by the valid hook configurations in Figure \[Fig4\]. Note that the leftmost point $(1,3)$ is blue in each of these colorings because this point looks around the first (red) hook and sees the sky.
To summarize, we started with a permutation $\pi$ with exactly $k$ descents. We chose a valid hook configuration of $\pi$ by drawing $k$ hooks according to the rules 1, 2, and 3 above. This valid hook configuration then induced a coloring of the plot of $\pi$. Specifically, $n-k$ points were colored, and $k+1$ colors were used (one for each hook and one for the sky). Let $q_i$ be the number of points colored the same color as the $i^\text{th}$ hook, and let $q_0$ be the number of points colored blue (sky color). Then $(q_0,q_1,\ldots,q_k)$ is a composition of $n-k$ into $k+1$ parts.[^2] We call a composition obtained in this way a *valid composition of* $\pi$. Let $\operatorname{\mathsf{VHC}}(\pi)$ be the set of valid hook configurations of $\pi$. Let $\mathcal V(\pi)$ be the set of valid compositions of $\pi$.
![The different colorings induced by the valid hook configurations of $3142567$.[]{data-label="Fig5"}](FertilityPIC7){width=".7\linewidth"}
The following theorem is the main reason why valid hook configurations are so useful when studying the stack-sorting map. Let $C_j=\frac{1}{j+1}{2j\choose j}$ denote the $j^\text{th}$ Catalan number. We will find it convenient to introduce the notation $$C_{(q_0,\ldots,q_k)}=\prod_{t=0}^kC_{q_t}$$ for any composition $(q_0,\ldots,q_k)$.
\[Thm5\] If $\pi$ has exactly $k$ descents, then the fertility of $\pi$ is given by the formula $$|s^{-1}(\pi)|=\sum_{(q_0,\ldots,q_k)\in\mathcal V(\pi)}C_{(q_0,\ldots,q_k)}.$$
Note in particular that a permutation is sorted if and only if it has a valid hook configuration. See [@Defant; @Defant2; @Defant3] for extensions and refinements of Theorem \[Thm5\].
\[Exam1\] The permutation $\pi=3142567$ has six valid hook configurations, which are shown in Figure \[Fig4\]. The colorings induced by these valid hook configurations are portrayed in Figure \[Fig5\]. The valid compositions of these valid hook configurations are (reading the first row before the second row, each from left to right) $$(3,1,1),\quad (2,2,1),\quad(1,3,1),\quad(2,1,2),\quad(1,2,2),\quad(1,1,3).$$It follows from Theorem \[Thm5\] that $$|s^{-1}(\pi)|=C_{(3,1,1)}+C_{(2,2,1)}+C_{(1,3,1)}+C_{(2,1,2)}+C_{(1,2,2)}+C_{(1,1,3)}=27.$$ Consequently, $27$ is a fertility number.
Throughout this paper, we implicitly make use of the following result, which is Lemma 3.1 in [@Defant2].
\[Thm6\] Let $\pi$ be a permutation. The map $\operatorname{\mathsf{VHC}}(\pi)\to\mathcal V(\pi)$ sending each valid hook configuration of $\pi$ to its induced valid composition is injective.
Proofs of the Main Theorems
===========================
We now exploit the valid hook configurations discussed in the previous section to prove our main theorems concerning fertility numbers. Let us begin with some useful definitions.
Let $\pi=\pi_1\cdots\pi_n$ be a permutation. Let $H$ be a hook in a valid hook configuration of $\pi$ with southwest endpoint $(i,\pi_i)$ and northeast endpoint $(j,\pi_j)$. When referring to a point “below" $H$, we mean a point $(x,y)$ with $i<x<j$ and $y<\pi_j$. In particular, the endpoints of a hook do not lie below that hook.
\[Def2\] Let $\pi=\pi_1\cdots\pi_n$ be a permutation, and let $H$ be a hook drawn on the plot of $\pi$. We say $H$ is a *stationary hook* if it appears in every valid hook configuration of $\pi$.
For example, suppose $\pi\in S_n$, $\pi_n=n$ and $\pi_i=n-1$, where $i\leq n-2$. Let $H$ be the hook with southwest endpoint $(i,n-1)$ and northeast endpoint $(n,n)$. The point $(i,n-1)$ is a descent top of $\pi$, so every valid hook configuration of $\pi$ must have a hook whose southwest endpoint is $(i,n-1)$. The northeast endpoint of such a hook must be $(n,n)$, so it follows that $H$ is a stationary hook of $\pi$. One can check that the hook drawn in Figure \[Fig6\] is another example of a stationary hook.
![A stationary hook of the permutation $1\,\,8\,\,11\,\,4\,\,3\,\,5\,\,7\,\,6\,\,13\,\,14\,\,2\,\,12\,\,15\,\,9\,\,10\,\,16$.[]{data-label="Fig6"}](FertilityPIC9){width=".4\linewidth"}
\[Prop1\] Let $\pi=\pi_1\cdots\pi_n$ be a permutation with a stationary hook $H$. Let $(i,\pi_i)$ and $(j,\pi_j)$ be the southwest and northeast endpoints of $H$, respectively. Let $\sigma=\pi_1\cdots\pi_{i+1}\pi_j\cdots\pi_n$ and $\tau=\pi_{i+1}\cdots\pi_{j-1}$. We have $$|s^{-1}(\pi)|=|s^{-1}(\sigma)||s^{-1}(\tau)|.$$
There is a natural bijection $$\operatorname{\mathsf{VHC}}(\sigma)\times \operatorname{\mathsf{VHC}}(\tau)\to\operatorname{\mathsf{VHC}}(\pi)$$ obtained by combining a valid hook configuration of $\sigma$ and a valid hook configuration of $\tau$ into a valid hook configuration of $\pi$. Furthermore, the colorings of the plots of $\sigma$ and $\tau$ combine into one coloring of $\pi$. Note that the non-blue colors used to color $\sigma$ must be different from those used to color $\tau$. The blue points in the plot of $\tau$ must change to the color of $H$ in the plot of $\pi$. See Figure \[Fig10\] for a depiction of this combination of valid hook configurations and induced colorings. In that figure, $H$ is the hook with southwest endpoint $(3,11)$ and northeast endpoint $(10,14)$.
Let $k_\sigma=\text{des}(\sigma)$ and $k_\tau=\text{des}(\tau)$ be the number of descents of $\sigma$ and the number of descents of $\tau$, respectively. Note that $H$ is a stationary hook of $\sigma$. If $i$ is the $r^\text{th}$ descent of $\sigma$, then every valid composition of $\sigma$ is of the form $(q_0,\ldots,q_{r-1},1,q_{r+1},\ldots,q_{k_\sigma})$. It follows from the above paragraph that the map $\mathcal V(\sigma)\times\mathcal V(\tau)\to\mathcal V(\pi)$ given by $$((q_0,\ldots,q_{r-1},1,q_{r+1},\ldots,q_{k_\sigma}),(q_0',\ldots,q_{k_\tau}'))\mapsto(q_0,\ldots,q_{r-1},q_0',\ldots,q_{k_\tau}',q_{r+1},\ldots,q_{k_\sigma})$$ is a bijection. Invoking Theorem \[Thm5\], we find that $$|s^{-1}(\pi)|=\sum_{(q_0,\ldots,q_{r-1},1,q_{r+1},\ldots,q_{k_\sigma})\in\mathcal V(\sigma)}\:\sum_{(q_0',\ldots,q_{k_\tau}')\in\mathcal V(\tau)}C_{(q_0,\ldots,q_{r-1},q_0',\ldots,q_{k_\tau}',q_{r+1},\ldots,q_{k_\sigma})}$$
$$=\sum_{(q_0,\ldots,q_{r-1},1,q_{r+1},\ldots,q_{k_\sigma})\in\mathcal V(\sigma)}\:\sum_{(q_0',\ldots,q_{k_\tau}')\in\mathcal V(\tau)}C_{(q_0,\ldots,q_{r-1},1,q_{r+1},\ldots,q_{k_\sigma})}C_{(q_0',\ldots,q_{k_\tau}')}$$ $$=\left[\sum_{(q_0,\ldots,q_{r-1},1,q_{r+1},\ldots,q_{k_\sigma})\in\mathcal V(\sigma)}C_{(q_0,\ldots,q_{r-1},1,q_{r+1},\ldots,q_{k_\sigma})}\right]\left[\sum_{(q_0',\ldots,q_{k_\tau}')\in\mathcal V(\tau)}C_{(q_0',\ldots,q_{k_\tau}')}\right]$$ $$=|s^{-1}(\sigma)||s^{-1}(\tau)|.\qedhere$$
![Valid hook configurations of $\sigma=1\,\,8\,\,11\,\,4\,\,14\,\,2\,\,12\,\,15\,\,9\,\,10\,\,16$ and $\tau=4\,\,3\,\,5\,\,7\,\,6\,\,13$ combine to form a valid hook configuration of $\pi=1\,\,8\,\,11\,\,4\,\,3\,\,5\,\,7\,\,6\,\,13\,\,14\,\,2\,\,12\,\,15\,\,9\,\,10\,\,16$. In Proposition \[Prop1\], we consider a stationary hook $H$ of $\pi$. In this example, $H$ is the (red) hook with southwest endpoint $(3,11)$ and northeast endpoint $(10,14)$.[]{data-label="Fig10"}](FertilityPIC10){width="1\linewidth"}
The following corollary allows us to explicitly construct permutations with certain fertilities by positioning stationary hooks appropriately. Given $\pi=\pi_1\cdots\pi_n\in S_n$, let $\widetilde\pi=(n+1)\pi(n+2)$. If $\pi_n=n$, put $\pi^*=\pi_1\cdots\pi_{n-1}\in S_{n-1}$. If $\lambda=\lambda_1\cdots\lambda_\ell\in S_\ell$ and $\mu=\mu_1\ldots\mu_m\in S_m$, then the *sum* of $\lambda$ and $\mu$, denoted $\lambda\oplus\mu$, is obtained by placing the plot of $\mu$ above and to the right of the plot of $\lambda$. More formally, the $i^\text{th}$ entry of $\lambda\oplus\mu$ is $$(\lambda\oplus\mu)_i=\begin{cases} \lambda_i & \mbox{if } 1\leq i\leq \ell; \\ \mu_{i-\ell}+\ell, & \mbox{if } \ell+1\leq i\leq \ell+m. \end{cases}$$
\[Cor1\] Let $\ell$ and $m$ be positive integers. Let $\lambda=\lambda_1\cdots\lambda_\ell\in S_\ell$ and $\mu=\mu_1\ldots\mu_m\in S_m$, and assume $\lambda_\ell=\ell$. Letting $\pi=\lambda^*\oplus\widetilde\mu\in S_{\ell+m+1}$, we have $$|s^{-1}(\pi)|=|s^{-1}(\lambda)||s^{-1}(\mu)|.$$
Note that $\pi_\ell=\ell+m$ and $\pi_{\ell+m+1}=\ell+m+1$. The hook with southwest endpoint $(\ell,\ell+m)$ and northeast endpoint $(\ell+m+1,\ell+m+1)$ is a stationary hook of $\pi$. Following Proposition \[Prop1\], let $\sigma=\pi_1\cdots\pi_{\ell+1}\pi_{\ell+m+1}$ and $\tau=\pi_{\ell+1}\cdots\pi_{\ell+m}$. That proposition tells us that $|s^{-1}(\pi)|=|s^{-1}(\sigma)||s^{-1}(\tau)|$. We have $\tau_i=\mu_i+(\ell-1)$ for all $i\in\{1,\ldots,m\}$, so $\tau$ and $\mu$ are order isomorphic. It is immediate from the definition of the stack-sorting map that two permutations that are order isomorphic have the same fertility. Thus, $|s^{-1}(\tau)|=|s^{-1}(\mu)|$. Also, $\sigma$ is order isomorphic to the permutation $\lambda'=\lambda_1\cdots\lambda_{\ell-1}(\ell+1)\ell(\ell+2)$. We have $$\mathcal V(\lambda')=\{(q_0,\ldots,q_r,1):(q_0,\ldots,q_r)\in\mathcal V(\lambda)\}.$$ According to Theorem \[Thm5\], $$|s^{-1}(\sigma)|=|s^{-1}(\lambda')|=\sum_{(q_0,\ldots,q_r,1)\in\mathcal V(\lambda')}C_{(q_0,\ldots,q_r,1)}=\sum_{(q_0,\ldots,q_r)\in\mathcal V(\lambda)}C_{(q_0,\ldots,q_r)}=|s^{-1}(\lambda)|. \qedhere$$
The following theorem is now an immediate consequence of Corollary \[Cor1\].
\[Thm1\] The set of fertility numbers is closed under multiplication.
The next theorem also follows easily from the above corollary.
\[Thm2\] If $f$ is a fertility number, then there are arbitrarily long permutations with fertility $f$.
If $f$ is a fertility number, then there is a permutation $\lambda$ such that $|s^{-1}(\lambda)|=f$. We may assume that $\lambda$ is normalized. That is, $\lambda\in S_\ell$ for some $\ell\geq 1$. Now let $\mu=1\in S_1$. The permutation $\pi$ constructed in Corollary \[Cor1\] has length $\ell+2$ and has fertility $f$. Repeating this procedure yields arbitrarily long permutations with fertility $f$.
Given a set $S$ of nonnegative integers, the quantity $$\liminf_{N\to\infty}\frac{|S\cap\{0,1,\ldots,N-1\}|}{N}$$ is called the *lower asymptotic density* of $S$. We next construct explicit permutations with certain fertilities in order to prove the following theorem.
\[Thm3\] Every nonnegative integer that is not congruent to $3$ modulo $4$ is a fertility number. The lower asymptotic density of the set of fertility numbers is at least $1954/2565\approx 0.7618$.
We begin by showing that the permutation $$\xi_m=m(m-1)\cdots 321(m+1)(m+2)\cdots (2m)$$ has fertility $2m$. The descent tops of this permutation are precisely the points of the form $(i,m+1-i)$ for $i\in\{1,\ldots,m-1\}$. In a valid hook configuration of $\xi_m$, the southwest endpoints of the hooks are precisely these descent tops. The northeast endpoints of hooks form an $(m-1)$-element subset of $\{(m+1,m+1),\ldots,(2m,2m)\}$. Of course, this subset is determined by choosing the number $j\in\{1,\ldots,m\}$ such that $(m+j,m+j)$ is not in the subset. Once this number is chosen, the hooks themselves are determined by the fact that hooks cannot intersect in a valid hook configuration. The valid composition induced from this valid hook configuration is $(1,\ldots,1,2,1,\ldots,1)$, where the $2$ is in the $(m+1-j)^\text{th}$ position. Since $C_{(1,1,\ldots,1,2,1,\ldots,1)}=2$, it follows from Theorem \[Thm5\] that $|s^{-1}(\xi_m)|=2m$. Thus, every even positive integer is a fertility number. This computation is illustrated in Figure \[Fig7\] in the case $m=4$.
![The valid hook configurations of $\xi_4=43215678$ along with their induced colorings.[]{data-label="Fig7"}](FertilityPIC11){width="1\linewidth"}
Suppose we have a permutation $\pi\in S_n$. Every valid hook configuration of $1\oplus\pi$ is obtained by placing a valid hook configuration of $\pi$ above and to the right of the point $(1,1)$. In the induced coloring of the plot of $1\oplus\pi$, the point $(1,1)$ must be blue. Every other point is given the same color as in the coloring of the plot of $\pi$ induced from the original valid hook configuration. It follows that $$\mathcal V(1\oplus\pi)=\{(q_0+1,q_1,\ldots,q_r):(q_0,\ldots,q_r)\in\mathcal V(\pi)\}.$$
We have seen that the valid compositions of $\xi_m$ are precisely the compositions consisting of $m-1$ parts that are equal to $1$ and one part that is equal to $2$. Therefore, the valid compositions of $1\oplus\xi_m$ are $$(3,1,1,1,\ldots,1),\hspace{.2cm}(2,2,1,1,\ldots,1),\hspace{.2cm}(2,1,2,1,\ldots,1),\hspace{.2cm}\ldots,\hspace{.2cm}(2,1,1,\ldots,1,2).$$ Invoking Theorem \[Thm5\], we find that $$|s^{-1}(1\oplus\xi_m)|=5+4(m-1)=4m+1.$$ It follows that every positive integer that is congruent to $1$ modulo $4$ is a fertility number.
We saw in Example \[Exam1\] that $27$ is a fertility number. The valid compositions of $1243567$ are $(5,1)$, $(4,2)$, and $(3,3)$, so $$|s^{-1}(1243567)|=C_{(5,1)}+C_{(4,2)}+C_{(3,3)}=42+28+25=95.$$ This shows that $95$ is also a fertility number. If we combine Theorem \[Thm1\] with the fact that all positive integers congruent to $1$ modulo $4$ are fertility numbers, then we find that all positive integers congruent to $3$ modulo $4$ that are multiples of $27$ or $95$ are also fertility numbers. In summary, every nonnegative integer $f$ satisfying one of the following conditions is a fertility number:
- $f\not\equiv 3\pmod 4$;
- $f\equiv 3\pmod 4$ and $27\mid f$;
- $f\equiv 3\pmod 4$ and $95\mid f$.
The natural density of the set of nonnegative integers satisfying one of these conditions is $$\frac 34+\frac{1}{4\cdot 27}+\frac{1}{4\cdot 95}-\frac{1}{4\cdot 27\cdot 95}=\frac{1954}{2565}. \qedhere$$
The constant $1954/2565$ in Theorem \[Thm3\] is not optimal. Indeed, we can increase the constant by simply exhibiting a fertility number that is congruent to $3$ modulo $4$ and is not already counted. Let us briefly describe one method for doing this. Let $$\zeta_m=(m+1)1(m+2)2(m+3)3\cdots(2m)m(2m+1)(2m+2)(2m+3).$$ The valid compositions of $\zeta_m$ are precisely the compositions consisting of either one $3$ and $m$ $1$’s or two $2$’s and $m-1$ $1$’s. This is not difficult to see, but one can also give a rigorous proof using Theorem 2.4 from [@Defant5]. For example, $\zeta_2$ is the permutation $3142567$ from Example \[Exam1\]. It follows from Theorem \[Thm5\] that $$|s^{-1}(\zeta_m)|=5(m+1)+4{m+1\choose 2},$$ and this is congruent to $3$ modulo $4$ whenever $m\equiv 2\pmod 4$.
Proving that a given positive integer $f$ is a fertility number amounts to constructing a permutation with fertility $f$, as we did in the proof of Theorem \[Thm3\]. Showing that a number is an infertility number is more subtle and requires additional tools. Bousquet-Mélou introduced the notion of the *canonical tree* of a permutation and showed that the shape of a permutation’s canonical tree determines that permutation’s fertility [@Bousquet]. She then asked for an explicit method for computing the fertility of a permutation from its canonical tree. The current author reformulated the notion of a canonical tree in the language of valid hook configurations, defining the *canonical hook configuration* of a permutation [@Defant2]. He then described a theorem that yields an explicit method for computing a permutation’s fertility from its canonical hook configuration. This result appears as Theorem 2.4 in the more recent article [@Defant5]. The following lemma is a consequence of this theorem; we omit the discussion describing how to compute the numbers $e_j$, $\mu_j$, and $\alpha_j$ because our present applications do not require it.
\[Lem2\] Let $\pi\in S_n$ be a permutation, and let $d_1<\cdots<d_k$ be the descents of $\pi$. There exist integers $e_0,\ldots,e_k,\mu_0,\ldots,\mu_k,\alpha_1,\ldots,\alpha_{k+1}$ (depending on $\pi$) with the following property. A composition $(q_0,\ldots,q_k)$ of $n-k$ into $k+1$ parts is a valid composition of $\pi$ if and only if the following two conditions hold:
(a) For every $m\in\{0,1,\ldots,k\}$, $$\sum_{j=m}^{e_m-1}q_j\geq\sum_{j=m}^{e_m-1}\mu_j.$$
(b) If $m,p\in\{0,1,\ldots,k\}$ are such that $m\leq p\leq e_m-2$, then $$\sum_{j=m}^pq_j\geq d_{p+1}-d_m-\sum_{j=m+1}^{p+1}\alpha_j.$$
Suppose $q=(q_0,\ldots,q_k)$, $q'=(q_0',\ldots,q_k')$, and $q''=(q_0'',\ldots,q_k'')$ are compositions of $n-k$ into $k+1$ parts (where $n$ and $k$ are as in Lemma \[Lem2\]). We say $q$ *interval dominates* $q'$ and $q''$ if $$\sum_{j=m_1}^{m_2}q_j\geq\min\left\{\sum_{j=m_1}^{m_2}q_j',\sum_{j=m_1}^{m_2}q_j''\right\}\quad\text{whenever }0\leq m_1\leq m_2\leq k.$$ If $q',q''\in\mathcal V(\pi)$ and $q$ interval dominates $q'$ and $q''$, then it follows immediately from Lemma \[Lem2\] that $q\in\mathcal V(\pi)$. In fact, this is the only reason why we need Lemma \[Lem2\].
\[Thm4\] The smallest fertility number that is congruent to $3$ modulo $4$ is $27$.
We saw in Example \[Exam1\] that $27$ is a fertility number. Assume by way of contradiction that there exists a fertility number $f\in\{3,7,11,15,19,23\}$. Let $n$ be the smallest positive integer such that there exists a permutation in $S_n$ with fertility $f$. Let $\pi\in S_n$ be one such permutation, and let $k$ be the number of descents of $\pi$. We say a composition $c$ *has type* $\lambda$ if $\lambda$ is the partition formed by rearranging the parts of $c$ into nonincreasing order. For example, $(1,2,1,2)$ has type $(2,2,1,1)$.
Because $|s^{-1}(\pi)|=f$ is odd, Theorem \[Thm5\] tells us that $\pi$ must have a valid composition $q$ such that $C_q$ is odd. If any of the parts in $q$ were greater than $4$, the sum representing $|s^{-1}(\pi)|$ in Theorem \[Thm5\] would be at least $42$, which is larger than $f$. If any of the parts were $2$ or $4$, $C_q$ would be even. This shows that all of the parts of $q$ are equal to $1$ or $3$. Furthermore, there is at most one part equal to $3$ (otherwise, the sum in Theorem \[Thm5\] would be at least $25$).
We know from Section 2 that every valid composition of $\pi$ is a composition of $n-k$ into $k+1$ parts. If $q=(1,1,\ldots,1)$, then $n=2k+1$. In this case, $(1,1,\ldots,1)$ is the only valid composition of $\pi$ (it is the only composition of $n-k$ into $k+1$ parts), so it follows from Theorem \[Thm5\] that $|s^{-1}(\pi)|=1$. This is a contradiction, so $q$ must have type $(3,1,\ldots,1)$. Since $q$ is a composition of $n-k$, we must have $n=2k+3$. This implies that every composition of $n-k$ into $k+1$ parts is of type $(3,1,\ldots,1)$ or of type $(2,2,1,\ldots,1)$. Thus, every valid composition of $\pi$ is of one of these types.
Let $Q_1,\ldots,Q_a$ be the valid compositions of $\pi$ of type $(3,1,\ldots,1)$, and let $b$ be the number of valid compositions of $\pi$ of type $(2,2,1,\ldots,1)$. By Theorem \[Thm5\], $5a+4b=f$. Reading this equation modulo $4$ shows that $a\equiv 3\pmod 4$. Since $f\leq 23$, we must have $a=3$. For $1\leq u<v\leq 3$, let $Q_{u,v}$ be the composition whose $i^\text{th}$ part is the arithmetic mean of the $i^\text{th}$ part of $Q_u$ and the $i^\text{th}$ part of $Q_v$. It is straightforward to see that $Q_{u,v}$ is a composition of $n-k$ into $k+1$ parts that has type $(2,2,1,\ldots,1)$ and that interval dominates $Q_u$ and $Q_v$. According to the discussion preceding this theorem, $Q_{1,2}$, $Q_{1,3}$, and $Q_{2,3}$ are valid compositions of $\pi$. Consequently, $b\geq 3$. It follows that $f=5a+4b\geq 27$, which is our desired contradiction.
Among the bulleted statements in the introduction, only the last one remains to be proven. The proof requires us to use Proposition \[Prop2\], which is stated below. The proof of this proposition relies on the following lemma, which is interesting in its own right.
\[Lem1\] Let $\pi$ be a sorted permutation with descents $d_1<\cdots<d_k$. Suppose there is an index $i\in\{1,\ldots,k\}$ such that $q_i=1$ for all $(q_0,\ldots,q_k)\in\mathcal V(\pi)$. If $H$ is a hook in a valid hook configuration of $\pi$ with southwest endpoint $(d_i,\pi_{d_i})$, then $H$ is a stationary hook of $\pi$.
Recall from the previous section that we write valid hook configurations as tuples of hooks. Let $\mathcal H=(H_1,\ldots,H_k)$ be a valid hook configuration containing the hook $H$. Necessarily, we have $H=H_i$ (this is simply due to the conventions we chose in Section 2 concerning how to order hooks). Suppose by way of contradiction that there is a valid hook configuration $\mathcal H'=(H_1',\ldots,H_k')$ with $H_i'\neq H$. The southwest endpoint of $H_i'$ must be $(d_i,\pi_{d_i})$. Let $(j,\pi_j)$ and $(j',\pi_{j'})$ be the northeast endpoints of $H_i$ and $H_i'$, respectively. Without loss of generality, we may assume $j<j'$.
There exists $r\in\{i,\ldots,k\}$ such that $(d_{i+1},\pi_{d_{i+1}}),\ldots,(d_r,\pi_{d_r})$ are the descent tops of $\pi$ lying below $H$. Let $$\mathcal H''=(H_1',\ldots,H_i',H_{i+1},\ldots,H_r,H_{r+1}',\ldots,H_k').$$ One can check that $\mathcal H''$ is a valid hook configuration of $\pi$. In the coloring of the plot of $\pi$ induced by $\mathcal H''$, both $(d_i+1,\pi_{d_i+1})$ and $(j,\pi_j)$ are given the same color as the hook $H_i'$. Letting $(q_0'',\ldots,q_k'')$ denote the valid composition of $\pi$ induced by $\mathcal H''$, we have $q_i''\geq 2$. This contradicts our hypothesis.
\[Prop2\] Assume $n\geq 3$. Let $\pi=\pi_1\cdots\pi_n$ be a sorted permutation with descents $d_1<\cdots<d_k$. Suppose there is an index $i\in\{1,\ldots,k\}$ such that $q_i=1$ for all $(q_0,\ldots,q_k)\in\mathcal V(\pi)$. Let $\mathcal X=\{(q_0,\ldots,q_{i-1},q_{i+1},\ldots,q_k):(q_0,\ldots,q_k)\in\mathcal V(\pi)\}$. There exists a permutation $\zeta\in S_{n-2}$ such that $\mathcal V(\zeta)=\mathcal X$.
According to Lemma \[Lem1\], $\pi$ has a stationary hook $H$ with southwest endpoint $(d_i,\pi_{d_i})$. Let $\lambda_{\text I},\lambda_{\text{II}},\lambda_{\text{III}},\lambda_{\text{IV}},\mu$ be the parts of the plot of $\pi$ as indicated in Figure \[Fig8\]. Let us slide all of the points of $\lambda_{\text I}\cup\lambda_{\text{II}}\cup\mu$ up by some integral distance so that the lowest point of $\lambda_{\text I}\cup\lambda_{\text{II}}\cup\mu$ is now higher than the highest point of $\lambda_{\text{III}}\cup\lambda_{\text{IV}}$. We can then slide the points in $\lambda_{\text I}\cup\lambda_{\text{II}}$ up by another integral distance so that the lowest point in $\lambda_{\text I}\cup\lambda_{\text{II}}$ is now higher than the highest point in $\mu$. These two operations, illustrated in Figure \[Fig8\], produce a new permutation $\pi'$.
Given a valid hook configuration of $\pi$, we obtain a valid hook configuration of $\pi'$ by keeping the hooks attached to their endpoints throughout these two sliding operations. Every valid hook configuration of $\pi'$ is obtained in this way because we can easily undo these sliding operations. Each valid hook configuration of $\pi$ induces a valid composition of $\pi$, and the corresponding valid hook configuration of $\pi'$ induces a valid composition of $\pi'$. These two valid compositions are identical because no points or hooks were ever moved horizontally and no hooks could have moved through each other during the sliding. Therefore, $\mathcal V(\pi)=\mathcal V(\pi')$. To ease notation, let us replace $\pi$ with this new permutation $\pi'$. In other words, we have shown that, without loss of generality, we may assume the plot of $\pi$ has the shape depicted in the rightmost part of Figure \[Fig8\].
![The two sliding operations described in the proof of Proposition \[Prop2\].[]{data-label="Fig8"}](FertilityPIC12){width=".75\linewidth"}
Let us now remove the hook $H$ and its endpoints from the plot of $\pi$. After shifting the remaining points in $\mu$ to the left by $1$ and shifting the points in $\lambda_{\text{I}}\cup\lambda_{\text{IV}}$ left by $2$, we obtain the plot of a permutation $\xi$. We claim that $\mathcal V(\xi)=\mathcal X$. Indeed, there is a natural bijection $\varphi:\operatorname{\mathsf{VHC}}(\pi)\to\operatorname{\mathsf{VHC}}(\xi)$. To apply $\varphi$ to a valid hook configuration of $\pi$, we first leave unchanged every hook whose endpoints were not deleted (i.e., those hooks whose endpoints were not also endpoints of $H$). If there was a hook whose southwest endpoint was the northeast endpoint of $H$, replace its southwest endpoint with the rightmost remaining point from $\mu$. This is allowed because the rightmost remaining point in $\mu$ is a descent top of $\pi$ ($\lambda_{\text{IV}}$ lies below $\mu$). If there was a hook whose northeast endpoint was the southwest endpoint of $H$, replace its northeast endpoint with the leftmost remaining point from $\mu$. See Figure \[Fig9\] for two examples of applications of $\varphi$.
If $\mathcal H\in\operatorname{\mathsf{VHC}}(\pi)$ induces a valid composition $(q_0,\ldots,q_k)\in\mathcal V(\pi)$, then $\varphi(\mathcal H)$ induces the valid composition $(q_0,\ldots,q_{i-1},q_{i+1},\ldots,q_k)\in\mathcal V(\xi)$. It follows that $\mathcal V(\xi)=\mathcal X$, as desired. Finally, we can normalize the permutation $\xi$ to obtain a permutation $\zeta\in S_{n-2}$ with $\mathcal V(\zeta)=\mathcal X$.
![Two example applications of the map $\varphi$ from the proof of Proposition \[Prop2\][]{data-label="Fig9"}](FertilityPIC13){width=".75\linewidth"}
The following corollary is now an immediate consequence of Theorem \[Thm5\].
\[Cor2\] In the notation of Proposition \[Prop2\], the permutation $\zeta\in S_{n-2}$ has the same fertility as $\pi$.
We can finally prove the last of our main theorems. As mentioned in the introduction, this theorem reduces the problem of determining whether a given positive integer is a fertility number to a finite problem.
\[Thm7\] If $f$ is a positive fertility number, then there exist a positive integer $n\leq f+1$ and a permutation $\pi\in S_n$ such that $f=|s^{-1}(\pi)|$.
We know that there exist a positive integer $n$ and a permutation $\pi\in S_n$ such that $f=|s^{-1}(\pi)|$. Let us choose $n$ minimally. We will show that $n\leq f+1$. The theorem is easy when $f\in\{1,2\}$, so we may assume $f\geq 3$. This forces $n\geq 3$.
Let $(q_{10},\ldots,q_{1k}),\ldots,(q_{m0},\ldots,q_{mk})$ be the valid compositions of $\pi$. Form the $m\times (k+1)$ matrix $M=(q_{i(j-1)})$ so that the rows of $M$ are precisely the valid compositions of $\pi$. If there is a column of $M$ whose entries are all $1$’s, then we can use Corollary \[Cor2\] to see that there is a permutation in $S_{n-2}$ with fertility $f$, contradicting the minimality of $n$. Hence, every column of $M$ contains at least one number that is not $1$.
Given an $a\times b$ matrix $D=(d_{ij})$ with positive integer entries, define $$N_D=b-1+\frac{1}{a}\sum_{i=1}^a\sum_{j=1}^b d_{ij}$$ and $$F_D=\sum_{i=1}^aC_{(d_{i1},\ldots,d_{ib})}.$$ From the fact that every valid composition of $\pi$ is a composition of $n-k$ into $k+1$ parts, we find that $N_M=n$. We know from Theorem \[Thm5\] that $F_M=f$. Consequently, it suffices to prove the following claim.
[**Claim:**]{} If $D$ is a matrix with positive integer entries and every column of $D$ contains at least one number that is not $1$, then $N_D\leq F_D+1$.
To prove this claim, we first describe a useful reduction. We can choose an entry $d_{ij}\geq 2$ of $D$ and replace it with $d_{ij}-1$ to produce a new matrix $D'$. Note that $F_{D'}\leq F_D-1$ and $N_{D'}=N_D-1/a\geq N_D-1$. We can repeat this operation repeatedly until we are left with a matrix $D^*$ such that every entry of $D^*$ is either a $1$ or a $2$ and such that every column of $D^*$ contains exactly one $2$. If we performed the above operation $\ell$ times to obtain $D^*$ from $D$, then $F_{D^*}\leq F_D-\ell$ and $N_{D^*}=N_D-\ell/a\geq N_D-\ell$. It suffices to show that $N_{D^*}\leq F_{D^*}+1$.
Let $u_i$ be the number of $2$’s in the $i^\text{th}$ row of $D^*$. Note that $u_1+\cdots+u_a=b$ because every column of $D^*$ has exactly one $2$. We have $$N_{D^*}=b-1+\frac{1}{a}(ab+u_1+\cdots+u_a)=\left(2+\frac 1a\right)(u_1+\cdots+u_a)-1$$ and $$F_{D^*}+1=2^{u_1}+\cdots+2^{u_a}+1.$$ We will show that $$\label{Eq1}
\left(2+\frac 1a\right)(u_1+\cdots+u_a)-1\leq 2^{u_1}+\cdots+2^{u_a}+1$$ for every choice of nonnegative integers $u_1,\ldots,u_a$.
If one of the integers $u_i$ is at least $3$, we can replace it by $u_i-1$. This has the effect of decreasing the expression on the left-hand side of by $2+1/a$ and decreasing the expression on the right-hand side by at least $4$. Therefore, it suffices to prove the inequality in after decreasing $u_i$ by $1$. We can repeatedly decrease the integers that are at least $3$ until every integer in the list $u_1,\ldots,u_a$ is at most $2$. In other words, it suffices to prove the inequality in under the assumption that $u_i\in\{0,1,2\}$ for all $i\in\{1,\ldots,a\}$. In this case, let $X_j=|\{i\in\{1,\ldots,a\}:u_i=j\}|$. With this notation, becomes $$\left(2+\frac{1}{X_0+X_1+X_2}\right)(X_1+2X_2)-1\leq X_0+2X_1+4X_2+1.$$ This simplifies to $$\frac{-X_0+X_2}{X_0+X_1+X_2}\leq X_0+1,$$ which obviously holds.
Future Directions
=================
The primary objective of this article has been to gain an understanding of fertility numbers. Of course, the ultimate goal here is to obtain a complete description of all fertility numbers. This appears to be difficult, but there are less formidable problems whose solutions would still interest us greatly. For example, Theorem \[Thm3\] leads us to ask the following question.
\[Quest1\] Does the set of fertility numbers have a natural density? If so, what is this natural density?
We also have some conjectures spawning from our main theorems.
\[Conj1\] There are infinitely many infertility numbers.
The proof of Theorem \[Thm3\] made use of the fact that $27$ and $95$ are fertility numbers. We saw in Theorem \[Thm4\] that $27$ is the smallest fertility number that is congruent to $3$ modulo $4$, so we are led to make the following conjecture.
\[Conj3\] The smallest fertility number that is congruent to $3$ modulo $4$ and is greater than $27$ is $95$.
It is desirable to have more efficient methods for determining whether or not a given positive integer is a fertility number. It is possible that such a method could arise by extending the techniques used in the proof of Theorem \[Thm4\]. Such methods could certainly be useful for answering the above conjectures. This also leads to the problem of improving Theorem \[Thm7\]. Given a fertility number $f$, let $\mathcal N(f)$ denote the smallest positive integer $n$ such that there exists a permutation in $S_n$ with fertility $f$. Theorem \[Thm7\] states that $\mathcal N(f)\leq f+1$ for every fertility number $f$. We would like to have better estimates for $\mathcal N(f)$. In particular, we have the following conjecture.
\[Conj4\] We have $$\lim_{f\to\infty}\mathcal N(f)/f=0,$$ where the limit is taken along the sequence of positive fertility numbers.
Finally, recall that Theorem \[Thm1\] tells us that the product of two fertility numbers is again a fertility number. We would like to have additional methods for combining fertility numbers in order to produce new ones.
Acknowledgments
===============
The author was supported by a Fannie and John Hertz Foundation Fellowship and an NSF Graduate Research Fellowship.
[1]{}
M. Bóna, Combinatorics of permutations. CRC Press, 2012.
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[^1]: These permutations are called *uniquely sorted*. They are studied further in [@DefantCatalan] and [@Hanna]
[^2]: Throughout this article, a *composition of $b$ into $a$ parts* is an $a$-tuple of positive integers that sum to $b$. For $i\in\{1,\ldots,k\}$, the number $q_i$ is positive because the point immediately to the right of the southwest endpoint of the $i^\text{th}$ hook is given the same color as the $i^\text{th}$ hook. The number $q_0$ is positive because $(1,\pi_1)$ is colored blue.
|
---
abstract: 'We study the ground-state configurations and spin textures of rotating two-component Bose-Einstein condensates (BECs) with Rashba-Dresselhaus spin-orbit coupling (RD-SOC), which are confined in a two-dimensional (2D) optical lattice plus a 2D harmonic trap. In the absence of rotation, a relatively small isotropic 2D RD-SOC leads to the generation of ghost vortices for initially miscible BECs, while it gives rise to the creation of rectangular vortex-antivortex lattices for initially immiscible BECs. As the strength of the 2D RD-SOC enhances, the visible vortices or the 2D vortex-antivortex chains are created for the former case, whereas the rectangular vortex-antivortex lattices are transformed into vortex-antivortex rings for the later case. For the initially immiscible BECs with fixed 2D RD-SOC strength, the increase of rotation frequency can result in the structural phase transition from square vortex lattice to irregular triangular vortex lattice and the system transition from initial phase separation to phase mixing. In addition, we analyze the combined effects of 1D RD-SOC and rotation on the vortex configurations of the ground states for the case of initial phase separation. The increase of 1D SOC strength, rotation frequency or both of them may result in the formation of vortex chain and phase mixing. Furthermore, the typical spin textures for both the cases of 2D RD-SOC and 1D RD-SOC are discussed. It is shown that the system favors novel spin textures and skyrmion configurations including an exotic skyrmion-half-skyrmion lattice (skyrmion-meron lattice), a complicated meron lattice, a skyrmion chain, and a Bloch domain wall.'
author:
- Hui Yang
- Qingbo Wang
- Ning Su
- Linghua Wen
title: 'Topological excitations in rotating Bose-Einstein condensates with Rashba-Dresselhaus spin-orbit coupling in a two-dimensional optical lattice'
---
Introduction
============
The realization of Bose-Einstein condensates (BECs) is a milestone in the study of ultracold atomic gases [@Dalfovo]. Owing to the unprecedented level of control and precision, ultracold atomic gases provide an ideal test ground to emulate various quantum phenomena in condensed matter physics [Bloch,Zapf]{}. Recent experimental realization of artificial spin-orbit coupling (SOC) [@Lin; @Cheuk; @ZWu; @Huang; @JRLi] which couples the internal states and the orbit motion of the atoms not only offers a platform to simulate the response of charged particles to external electromagnetic field, but also give opportunities to search of novel quantum states [Zhai,Ho,Sinha,YZhang,YLi,Kawakami,Stringari,Kartashov,Ruokokoski,XLi]{}. Relevant investigations show that the SOC can lead to many new quantum phases such as plane-wave phase [@CWang], stripe phase [XQXu,XZhou,HHu]{}, bright soliton [@Sakaguchi; @Gautam], dark soliton [YXu]{}, half-quantum vortex configuration [@HHu; @Ramach; @XQLi], and topological superfluid phase [@ZWu], which enrich the phase diagram and physics of BEC system. In particular, the combined effects of SOC and rotation on the BECs are predicted to generate various novel features. Recently, Radić *et al.* [@Radic] has proposed an experimental scheme for rotating spin-orbit-coupled BECs by using a suitable control of the BECs. On the other hand, some groups have studied the properties of BECs in various external potentials including harmonic trap [@Aftalion; @Fetter; @Xu], toroidal trap [@ACWhite; @XFZhang], concentrically coupled annular traps [@XFZhang2], double-well potential [@LWen; @Javanainen; @Kartashov2], one-dimensional (1D) optical lattice [JGWang]{}, and so on. It is demonstrated that the shape and dimension of the external potential plays an important role in the stationary states and dynamic properties of the BECs.
In this work, we investigate the topological excitations of rotating two-component BECs with Rashba-Dresselhaus SOC (RD-SOC) in a two-dimensional (2D) optical lattice plus a 2D harmonic trap. Actually, ultracold bosonic gases loaded in a 2D optical lattice have attracted considerable interest. By using two pairs of counterpropagating laser beams with orthogonal polarization, a 2D optical lattice can be created [Grynberg,Greiner,Clark]{}. Early investigations showed that the BECs in a rotating optical lattice support interesting properties [@Pu; @Tung], such as structural phase transition, domain formation, and vortex pinning, due to the dynamically tunable periodicity and depth of the optical lattice. Recent studies [@Radic2; @DWZhang] demonstrated that SOC can significantly affect the quantum phase transition of a spin-orbit-coupled bosonic gas in an optical lattice from a superfluid to a Mott insulator, and may lead to some novel magnetic phases, such as spiral phase and skyrmion crystals. Here we show how the 2D isotropic RD-SOC, the 1D anisotropic RD-SOC and the rotation frequency affect the ground-state structures and spin textures of BECs in a 2D optical lattice and a harmonic trap. In the absence of rotation, a small 2D RD-SOC can yield ghost vortices for the initially miscible BECs, while it can induce the formation of vortex-antivortex lattices for the initially immiscible BECs. For strong 2D RD-SOC, the visible vortices are generated in the case of initial component mixing, while the vortex-antivortex rings are formed in the case of initial component separation. When there exists rotation driving, with the increasing rotation frequency a structural phase transition from a square vortex lattice to a triangular vortex lattice occurs for the initially immiscible BECs. In addition, the combined effect of 1D RD-SOC and rotation on the ground state of the system for initially immiscible BECs are discussed. It is found that the system supports novel spin textures and topological defects including a peculiar skyrmion-half-skyrmion lattice (skyrmion-meron lattice), a complicated meron lattice, a skyrmion chain, and a Bloch domain wall.
This paper is organized as follows. In Sec. II, we present the theoretical model of a rotating pseudospin-1/2 BEC with RD-SOC in a 2D optical lattice and a harmonic trap. The topological structures and relevant spin textures of the system are described and analyzed in Sec. III. Our findings are summarized in Sec. IV.
Model
=====
We consider a rotating quasi-2D pseudospin-$1/2$ BEC with RD-SOC in a 2D optical lattice and a 2D harmonic trap. The Hamiltonian of the system can be written as$$\hat{H}=\int dxdy\hat{\psi}^{\dagger }\left[ -\frac{\hbar ^{2}\triangledown
^{2}}{2m}+V(x,y)-\Omega L_{z}+v_{so}+\frac{g_{1}}{2}\hat{n}_{1}^{2}+\frac{%
g_{2}}{2}\hat{n}_{2}^{2}+g_{12}\hat{n}_{1}\hat{n}_{2}\right] \hat{\psi},
\label{Hamiltonian}$$where $\hat{\psi}$ $=[\hat{\psi}_{1}(r),\hat{\psi}_{2}(r)]^{T}$ represents collectively the spinor Bose field operators with 1 and 2 denoting spin-up and spin-down, respectively. $\hat{n}_{1}=\hat{\psi}_{1}^{\dagger }\hat{\psi}%
_{1}$ and $\hat{n}_{2}=\hat{\psi}_{2}^{\dagger }\hat{\psi}_{2}$ are the density operators of spin-up and spin-down atoms, respectively. Here we assume that the two component atoms have the same mass $m$. The coefficients $g_{j}$ $=\sqrt{8\pi }\hbar ^{2}a_{j}/ma_{z}$ $(j=1,2)$ and $g_{12}$ $=\sqrt{%
8\pi }\hbar ^{2}a_{12}/ma_{z}$ denote the intra- and interspecies interaction strengths characterized by the $s$-wave scattering lengths $%
a_{j} $ and $a_{12}$ between intra- and intercomponent atoms, and $a_{z}=%
\sqrt{\hbar /m\omega _{z}}$ is the oscillation length in the $z$ direction. For simplicity, we assume $g_{1}$ $=g_{2}$ $=$ $g$ throughout this paper. The RD-SOC is given by $v_{so}=-i\hbar (k_{x}\hat{\sigma}_{x}\partial
_{x}+k_{y}\hat{\sigma}_{y}\partial _{y})$ [@Goldman], where $\hat{\sigma}%
_{x}$ and $\hat{\sigma}_{y}$ are Pauli matrices, and $k_{x}$ and $k_{y}$ denote the SOC strength in the $x$ and $y$ directions. $\Omega $ is the rotation frequency along the $z$ direction, and $L_{z}=-i\hbar (x\partial
_{y}-y\partial _{x})$ is the $z$ component of the angular-momentum operator. The combined potential of a 2D optical lattice and a 2D harmonic trap is expressed as [@Pu; @Kartashov3]$$V\left( x,y\right) =V_{0}\left[ \sin ^{2}\left( \frac{2\pi x}{\lambda }%
\right) +\sin ^{2}\left( \frac{2\pi y}{\lambda }\right) \right] +\frac{1}{2}%
m\omega _{\perp }^{2}\left( x^{2}+y^{2}\right) , \label{Potential}$$where $V_{0}$ denotes the depth of the optical lattice, which can be controlled by the intensity of retroreflected laser beams, $\lambda $ is the wavelength of the retroreflected laser beams, and $\omega _{\perp }$ is the radial trap frequency. Based on mean-field theory, the Gross-Pitaevskii (GP) energy functional of the system is given by $$\begin{aligned}
E &=&\int dxdy[{\psi _{1}^{\ast }\left( -\frac{\hbar ^{2}}{2m}\nabla
^{2}+V\right) \psi _{1}+\psi _{2}^{\ast }\left( -\frac{\hbar ^{2}}{2m}\nabla
^{2}+V\right) \psi _{2}-\Omega \psi _{1}^{\ast }L_{z}\psi _{1}-\Omega \psi
_{2}^{\ast }L_{z}\psi _{2}} \notag \\
{\newline
} &&{+\psi _{1}^{\ast }\hbar \left( -ik_{x}\partial _{x}-k_{y}\partial
_{y}\right) \psi _{2}+\psi _{2}^{\ast }\hbar \left( -ik_{x}\partial
_{x}+k_{y}\partial _{y}\right) \psi _{1}+\frac{g}{2}\left( \left\vert \psi
_{1}\right\vert ^{4}+\left\vert \psi _{2}\right\vert ^{4}\right)
+g_{12}\left\vert \psi _{1}\right\vert ^{2}\left\vert \psi _{2}\right\vert
^{2}].} \label{EnergyFunctional}\end{aligned}$$In our calculation, it is convenient to make the following parameter transformations $\tilde{x}=x/a_{0}$, $\tilde{y}=y/a_{0}$, $\tilde{t}=\omega
_{\perp }t$, $\tilde{V}\left( x,y\right) =V\left( x,y\right) /\hbar \omega
_{\perp }$, $\tilde{\Omega}=\Omega /\omega _{\perp }$, $\beta =gN/\hbar
\omega _{\perp }a_{0}^{2}$, $\beta _{12}=g_{12}N/\hbar \omega _{\perp
}a_{0}^{2}$, $\tilde{k}_{q}=k_{q}/\omega _{\perp }a_{0}$ $(q=x,y)$, and $%
\tilde{\psi}_{j}=\psi _{j}a_{0}/\sqrt{N}$ $(j=1,2)$, where $a_{0}=\sqrt{%
\hbar /m\omega _{\perp }}$ is the characteristic length of the harmonic trap, and $N=\int (\left\vert \psi _{1}\right\vert ^{2}+\left\vert \psi
_{2}\right\vert ^{2})dxdy$ is the number of atoms. In terms of the variational method, we obtain the dimensionless coupled 2D GP equations,$$\begin{aligned}
i\partial _{t}\psi _{1} &=&\left[ -\frac{1}{2}\nabla ^{2}+V+\beta \left\vert
\psi _{1}\right\vert ^{2}+\beta _{12}\left\vert \psi _{2}\right\vert
^{2}-i\Omega \left( y\partial _{x}-x\partial _{y}\right) \right] \psi _{1}
\notag \\
&&+\left( -ik_{x}\partial _{x}-k_{y}\partial _{y}\right) \psi _{2},
\label{GPE1} \\
i\partial _{t}\psi _{2} &=&\left[ -\frac{1}{2}\nabla ^{2}+V+\beta \left\vert
\psi _{2}\right\vert ^{2}+\beta _{12}\left\vert \psi _{1}\right\vert
^{2}-i\Omega \left( y\partial _{x}-x\partial _{y}\right) \right] \psi _{2}
\notag \\
&&+\left( -ik_{x}\partial _{x}+k_{y}\partial _{y}\right) \psi _{1},
\label{GPE2}\end{aligned}$$where the tilde is omitted for simplicity, and the dimensionless external potential with $a=2\pi a_{0}/\lambda $ reads$$V=V_{0}\left[ \sin ^{2}\left( ax\right) +\sin ^{2}\left( ay\right) \right] +%
\frac{1}{2}\left( x^{2}+y^{2}\right) . \label{DimensionlessPotential}$$
In order to better understand the physical properties of this system, we use the nonlinear Sigma model [@Mizushima; @Kasamatsu; @Kasamatsu1] and introduce a normalized complex-valued spinor $\mathbf{\chi }=\left[ \chi
_{1},\chi _{2}\right] ^{T}$ with $\left\vert \chi _{1}\right\vert
^{2}+\left\vert \chi _{2}\right\vert ^{2}=1$. The main idea of the nonlinear Sigma model is that pseudospin representation of the order parameter of a system with internal degrees of freedom is userful to obtain a physical understanding by mapping the system to a magnetic system. In this context, two-component BECs can be treated as a spin-$1/2$ BEC. An exact mathematical correspondence can be established between the two systems, where $\psi _{1}$($\psi _{2}$) corresponds to the up (down) component of the spin-$1/2$ spinor. The detailed discussion can be referred to Refs. [Mizushima,Kasamatsu,Kasamatsu1]{}. The two-component wave functions can be expressed as $\psi _{1}=\sqrt{\rho }\chi _{1}$ and $\psi _{2}=\sqrt{\rho }%
\chi _{2}$, where $\rho =\left\vert \psi _{1}\right\vert ^{2}+\left\vert
\psi _{2}\right\vert ^{2}$ is the total density of the system. In the pseudospin representation, the spin density is given by $\mathbf{S=\bar{\chi}%
\sigma \chi \ }$in which $\mathbf{\sigma }=(\sigma _{x},\sigma _{y},\sigma
_{z})$ are the pauli matrices. The components of $\mathbf{S}$ can be written as [@Aftalion; @WHan2; @CFLiu]$$\begin{aligned}
S_{x} &=&2\left\vert \chi _{1}\right\vert \left\vert \chi _{2}\right\vert
\cos \left( \theta _{1}-\theta _{2}\right) , \label{SpindensityX} \\
S_{y} &=&-2\left\vert \chi _{1}\right\vert \left\vert \chi _{2}\right\vert
\sin \left( \theta _{1}-\theta _{2}\right) , \label{SpindensityY} \\
S_{z} &=&\left\vert \chi _{1}\right\vert ^{2}-\left\vert \chi
_{2}\right\vert ^{2}, \label{SpindensityZ}\end{aligned}$$where $\theta _{j}$ $(j=1,2)$ is the phase of component wave function $\psi
_{j}$.
Ground-state structures and spin textures for the case of 2D RD-SOC
===================================================================
In our calculations, we numerically solve the GP Eqs. (\[GPE1\]) and ([GPE2]{}) and obtain the ground-state structure of the system by using the imaginary-time propagation algorithm based on the Peaceman-Rachford method [@Peaceman; @LWen2]. We consider the isotropic 2D RD-SOC and 1D RD-SOC effects on the ground states of rotating BECs in an optical lattice plus a harmonic potential. In the present work, the parameters of the optical lattice are chosen as $V_{0}=70$ and $a=4$, and the intra- and interspecies interactions are both assumed to be repulsive. For the convenience of discussion, when $\beta ^{2}<\beta _{12}^{2}$ $(\beta ^{2}>\beta _{12}^{2})$ we call the regime briefly initial component separation (initial component mixing).
![(Color online) Ground-state phase diagram of nonrotating spin-orbit-coupled spin-$1/2$ BEC in an optical lattice plus a harmonic trap with respect to $k$ ($k_{x}=k_{y}=k$) and $\protect\beta _{12}$ for $\protect%
\beta =200$. There are six different phases marked by A-F.[]{data-label="Figure1"}](Figure1.eps){width="7.2cm"}
Firstly, we discuss the effect of isotropic 2D RD-SOC on the ground-state structure of spin-$1/2$ BEC without rotation in an optical lattice and a harmonic trap. Relevant studies [@Zhai; @Ho; @Sinha; @YZhang; @YLi] showed that the mean-field ground state for a nonrotating spin-orbit-coupled spin-$1/2$ BEC in a harmonic trap has two typical phases, plane-wave phase (i.e. Thomas-Fermi (TF) phase) and stripe phase, depending on the nonlinear interactions. In Fig. 1, we give the ground-state phase diagram of nonrotating spin-orbit-coupled spin-1/2 BEC loaded in an optical lattice plus a harmonic trap with respect to the isotropic SOC strength $k$ ($%
k_{x}=k_{y}=k$) and the interspecies interaction $\beta _{12}$. There are six different phases marked by A-F, which differ in terms of their density and phase distributions. In the following discussion, we will give a detailed description of each phase. The density and phase profiles of the B-F in Fig.1 are shown in Figs. 2(a)-2(e), respectively, where the interaction parameters are $\beta =$ $200$, $\beta _{12}$ $=50$ for the first three columns, and $\beta _{12}$ $=300$ for the last two columns. The isotropic 2D RD-SOC strengths are $k_{x}=k_{y}=0.5$ (a, d), $k_{x}=k_{y}=2$ (b), $k_{x}=k_{y}=5$ (c), and $k_{x}=k_{y}=2.5$ (e). Note that the odd and even rows present component 1 and component 2, respectively. We start from the case where SOC is sufficiently weak, which is indicated by the region A in Fig. 1. The A phase is a periodically modulated TF phase in which no vortex exists due to the very small isotropic RD-SOC (the density and phase profiles are not shown here for the sake of simplicity). In the case of relatively weak SOC, the system supports the B phase for $\beta _{12}<200$ and the E phase for $\beta _{12}>200$, whose ground states are shown in Figs. 2(a) and 2(d), respectively. The ground state of the system exhibits obvious phase mixing in Fig. 2(a), where several ghost vortices are generated in the outskirts of each component (see the bottom two rows in Fig. 2(a)) and they carry no angular momentum [@LWen; @Kasamatsu2; @LHWen]. It is known that there are three fundamental types of vortices in cold atom physics: visible vortex, ghost vortex, and hidden vortex. The visible vortex is the ordinary quantized vortex which is visible in both the density distribution and the phase distribution and carries angular momentum [Fetter2]{}. For the ghost vortex, it shows up in the phase distribution as a phase singularity and has no visible vortex core in the density distribution and carries no angular momentum [@Kasamatsu2]. Ghost vortices can be detected by the interference between two BECs, at least one of which contains ghost vortices. Like ghost vortex, the hidden vortex as a phase defect is invisible in the density profile but it carries angular momentum. Only after including the hidden vortices can the well-known Feynman rule be satisfied [@LWen; @LHWen], and the hidden vortices can be observed by using the scheme of free expansion. In Fig. 2(d) the two component densities display evident phase separation, where the topological defects in individual components composed of alternately arranged vortices (clockwise rotation) and antivortices (anticlockwise rotation) form rectangular vortex-antivortex lattices.
![(Color online) Ground-state density distributions (the top two rows) and phase distributions (the bottom two rows) for nonrotating spin-1/2 BEC with isotropic 2D RD-SOC in an optical lattice plus a harmonic potential. (a) $k_{x}=k_{y}=0.5$, (b) $k_{x}=k_{y}=2$, (c) $k_{x}=k_{y}=5$, (d) $k_{x}=k_{y}=0.5$, and (e) $k_{x}=k_{y}=2.5$. The interaction parameters are $\protect\beta =200$, $\protect\beta _{12}=50$ for (a)-(c), and $\protect%
\beta _{12}=300$ for (d)-(e). The odd and even rows correspond to component 1 and component 2, respectively. The unit length is $a_{0}$.[]{data-label="Figure2"}](Figure2.eps){width="12cm"}
With the increase of isotropic RD-SOC strength, for the case of initial component mixing the C phase emerges as the ground state of the system, as shown by the region C in Fig. 1. The typical density and phase distributions of this phase are given in Fig. 2(b), where the ground-state density distributions of the system are similar to those in Fig. 2(a), but the ghost vortices disappear and ordinary visible vortices occur. The main reason is that the increased SOC offers more energy and angular momentum to the system. As the SOC is further increased, the D phase emerges, as displayed by the region D in Fig. 1. The density and phase distributions are shown in Fig. 2(c). The phase consists of 2D vortex-antivortex chain where vortices and antivortices form alternately arranged 2D chain structure in space. For the case of initial phase separation, with the increase of isotropic RD-SOC strength, the E phase transforms to the F phase in Fig. 1. Typical density and phase distributions of such a phase are shown in Fig. 2(e). The component densities keep separated (see the upper two rows in Fig. 2(e)) but the topological defects evolve into vortex-antivortex rings in which the vortices and antivortices are arranged alternately to form ring structures (see the lower two rows in Fig. 2(e)).
![(Color online) Ground-state phase diagram of rotating spin-orbit coupled spin-1/2 BEC in an optical lattice plus a harmonic trap with respect to $\Omega $ and $k$ ($k_{x}=k_{y}=k$) for $\protect\beta =300$ and $\protect%
\beta _{12}=200$. There are seven different phases marked by A-G.[]{data-label="Figure3"}](Figure3.eps){width="7.2cm"}
Secondly, we give the ground-state phase diagram spanned by the rotation frequency $\Omega $ and the isotropic SOC strength $k$ in Fig. 3. For the case of rotating spin-orbit-coupled spin-$1/2$ BECs in a simple harmonic trap only, previous investigations [@XQXu; @Aftalion] showed that the interplay between rotation frequency, SOC strength, and interparticle interactions can lead to various ground-state phases, such as half-quantum vortex, giant vortex, ringlike structures with triangular vortex lattices. For the present system, there are seven different phases marked by A-G, which differ in terms of their different density and phase distributions. In the following discussion, we will give a description for each phase. We start from the case where both the rotation and SOC are sufficiently weak, which is indicated by the yellow region A in Fig. 3. This phase is the periodically modulated TF phase without vortex in each component, which is the same with the A Phase in Fig.1. The typical density and phase profiles of the B phase, C phase and D phase in Fig. 3 are similar to those in Figs. 2(a), 2(b) and 2(c), respectively. At the same time, the density and phase profiles of the phases E-G in Fig. 3 are shown in Fig. 7(a) and Figs. 4(a)-4(b), respectively.
![(Color online) Ground-state density distributions (the top two rows) and phase distributions (the bottom two rows) for spin-1/2 BEC with isotropic 2D RD-SOC and fixed rotation frequency $\Omega =0.5$ in an optical lattice and a harmonic trap. (a) $k_{x}=k_{y}=0$, (b) $k_{x}=k_{y}=1$, (c) $%
k_{x}=k_{y}=0$, and (d) $k_{x}=k_{y}=1$. The interaction parameters are $%
\protect\beta =300$ and $\protect\beta _{12}=200$ for (a)-(b), and $\protect%
\beta =200$ and $\protect\beta _{12}=300$ for (c)-(d). The odd and even rows correspond to component 1 and component 2, respectively. The unit length is $%
a_{0}$.[]{data-label="Figure4"}](Figure4.eps){width="8cm"}
For relatively large rotation frequency but very weak SOC, the system exhibits a vortex ring where the vortices form a ring structure, indicated by red region F in Fig. 3. The main results are illustrated in Fig. 4(a). In Fig. 4, we consider the effect of isotropic 2D RD-SOC on the ground-state structure of the system with fixed rotation frequency $\Omega =0.5$. For initially miscible two-component BECs without SOC, i.e., $k_{x}=k_{y}=0$, a vortex ring forms in each component and the density distributions are almost the same (see Fig. 4(a)), i.e., the F phase emerges. When the SOC strength further increases, the F phase transforms to the G phase, as shown in Fig. 3. The typical ground state is that more vortices occur in any of individual components and these vortices tend to form a triangular vortex lattice, where some of them enter the central region of the external potential (Fig. 4(b)). This G phase occupies the largest region of the ground-state phase diagram in Fig. 3. By comparison, for the case of initially immiscible BECs without SOC, the component densities are fully separated and the vortices form an irregular vortex cluster, which is displayed in Fig. 4(c). With the increase of SOC strength, e.g., $k_{x}=k_{y}=1$, the system exhibits partial phase mixing in spite of the two components being separated initially, and the vortices and the vortex-antivortex pairs in each component constitute complex topological structure (see Fig. 4(d)), which is resulted from the competition among the repulsive interatomic interaction, SOC, rotation and the optical lattice.
Next, we move to the case of relatively small rotation frequency and weak SOC. In this regime, the system sustains E phase, which is denoted by region E in Fig. 3. Typical density and phase distributions of the E phase are displayed in Fig. 7(a). Obviously, the ground state is the known half-quantum vortex state, which is characterized by one vortex in one component and no vortex in the other component (see the third and fourth columns of Fig. 7(a)).
We find that the present system supports not only the line-like vortex excitation with respect to the spatial degrees of freedom of the BECs but also the point-like topological excitation (skyrmion excitation) concerning the spin degrees of freedom. The skyrmion is a type of topological soliton, which was originally suggested in nuclear physics by Skyrme to elucidate baryons as a quasiparticle excitation with spin pointing in all directions to wrap a sphere [@Skyrme]. It can be viewed as the reverse of the local spin, which has been observed in many condensed-matter systems, such as quantum Hall system, liquid crystals, helical ferromagnets, liquid $^{3}$He-A, and BECs [@Anderson2; @Wright; @XZYu; @CFLiu]. The nonsingular skyrmion in two-component BECs is related to the Mermion-Ho coreless vortices [Mermin]{}, and the combination of SOC and rotation can cause various topological defects including circular-hyperbolic skyrmion [@CFLiu], giant skyrmion [@Aftalion], and so on. In Fig. 5 we display the topological charge density and the local enlargements of the spin texture for the ground state in Fig. 4(a). Here the topological charge is expressed by $Q=\int q(\mathbf{r})dxdy$ with the topological charge density $q(\mathbf{%
r})=\frac{1}{4\pi }\mathbf{S\bullet (}\frac{\partial \mathbf{S}}{\partial x}%
\times \frac{\partial \mathbf{S}}{\partial y}\mathbf{)}$**.** Our numerical calculation shows that the local topological charges in Figs. 5(b) and 5(c) approach $Q=0.5$, which indicates that the local topological defects in Figs. 5(b) and 5(c) are circular half-skyrmion (meron) and hyperbolic half-skyrmion, respectively. At the same time, the local topological charges in Figs. 5(d)-5(h) approach $Q=1$, which means that the local topological defects are circular-hyperbolic skyrmion \[see Figs. 5(d)-5(f)\], hyperbolic-radial(out) skyrmion \[Fig. 5(g)\] and hyperbolic-radial(in) skyrmion \[Fig. 5(h)\], respectively. Therefore the spin texture in Fig. 5(a) forms an exotic skyrmion-half-skyrmion lattice (skyrmion-meron lattice) composed of circular-hyperbolic skyrmions, hyperbolic-radial(out) skyrmion, circular half-skyrmions, and hyperbolic half-skyrmions. Essentially, the interesting topological structure is resulted from the interplay among the optical lattice, rotation and the interatomic interactions.
![(Color online) Topological charge density (a) and local enlargements of the spin texture (b-h) for rotating spin-1/2 BEC in an optical lattice plus a harmonic potential, where $\protect\beta =300$, $%
\protect\beta _{12}=200$, $k_{x}=k_{y}=0$,$\ $and $\Omega =0.5$.The corresponding ground state is shown in Fig. 4(a). The unit length is $a_{0}$. []{data-label="Figure5"}](Figure5.eps){width="15cm"}
Figure 6(a) shows the topological charge density in the case of $\beta =300$, $\beta _{12}=200$, $k_{x}=k_{y}=1\ $and $\Omega =0.5$, where the corresponding ground state is displayed in Fig. 4(b). The local spin texture are given in Figs. 6(b)-6(d). Our computation results demonstrate that the two local topological defects in Fig. 6(b) are a meron (half-skyrmion) pair composed of two circular merons (half-skyrmions) with each local topological charge being $Q=0.5$. In the mean time, the central meron pair is surrounded by some other spin defects (the full spin texture is not shown here in view of the limited resolution ratio and the brevity of the article). Our simulation shows that the local topological charge for each of the outer spin defects \[e.g., see Figs. 6(c) and 6(d)\] is $Q=0.5$, which implies that these outer spin defects are merons (half-skyrmions). Thus the circular meron pair and the half-skyrmions (merons) jointly form a complex meron lattice (half-skyrmion lattice), which has not been reported in previous literature. Physically, the asymmetry of the complicated topological structure (i.e., the composite meron lattice) is caused by the destruction of the SU(2) symmetry of the system in the presence of RD-SOC. The interesting and exotic spin textures as mentioned in Figs. 5 and 6 allow to be tested and observed in the future cold atom experiments.
![(Color online) Topological charge density (a) and local enlargements of the spin texture (b-d) for rotating spin-1/2 BEC in an optical lattice plus a harmonic potential, where $\protect\beta =300$, $%
\protect\beta _{12}=200$, $k_{x}=k_{y}=1$,$\ $and $\Omega =0.5$.The corresponding ground state is given in Fig. 4(b). The unit length is $a_{0}$. []{data-label="Figure6"}](Figure6.eps){width="8cm"}
Finally, we investigate the combined effects of RD-SOC, rotation and interatomic interactions on the ground state of the system. Figure 7 shows the density distributions and phase distributions for the ground states of rotating two-component BECs with RD-SOC in an optical lattice and a harmonic trap, where $k_{x}=k_{y}=0.5$. The rotation frequencies for the case of initial phase mixing with $\beta =300$ and $\beta _{12}=200$ in Figs. 7(a) and 7(b) are $\Omega =0.1$ and $\Omega =0.8$, and those for the case of initial phase separation with $\beta =200$ and $\beta _{12}=300$ in Figs. 7(c) and 7(d) are $\Omega =0.3$ and $\Omega =0.8$, respectively. The columns from left to right denote $\left\vert \psi _{1}\right\vert ^{2}$, $%
\left\vert \psi _{2}\right\vert ^{2}$, $\arg \psi _{1}$, $\arg \psi _{2}$, and $\left\vert \psi _{1}\right\vert ^{2}-\left\vert \psi _{2}\right\vert
^{2}$, respectively.
![(Color online) Ground states of rotating 2D spin–1/2 BEC with RD-SOC in an optical lattice and a harmonic trap, where $k_{x}=k_{y}=0.5.$ (a) $\Omega =0.1$, (b) $\Omega =0.8$, (c) $\Omega =0.3$, and (d) $\Omega
=0.8 $. The effective interaction parameters are $\protect\beta =300$ and $%
\protect\beta _{12}=200$ for (a)-(b), and $\protect\beta =200$ and $\protect%
\beta _{12}=300$ for (c)-(d). The columns from left to right represent $%
\left\vert \protect\psi _{1}\right\vert ^{2}$, $\left\vert \protect\psi %
_{2}\right\vert ^{2}$, $\arg \protect\psi _{1},$ $\arg \protect\psi _{2}$, and $\left\vert \protect\psi _{1}\right\vert ^{2}-\left\vert \protect\psi %
_{2}\right\vert ^{2}$, respectively. The unit length is $a_{0}$.[]{data-label="Figure7"}](Figure7.eps){width="12cm"}
For the case of initial component mixing, as the rotation frequency increases from zero, the ghost vortices on the outskirts of the atom cloud enter the condensates and become visible vortices, where the phase defects tend to form a triangular vortex lattice for large rotation frequency \[see Fig. 2(a), Fig. 7(a) and Fig. 7(b)\] and the energy of the system reaches the minimum in the rotating frame. For the case of initial component separation, with the increase of rotation frequency, the topological structure of the system gradually evolves from a square vortex lattice composed of vortex-antivortex pairs into a complex triangular vortex lattice made of pure vortices, and the two component densities transform from full phase separation into partial phase mixing \[see Fig. 2(d), Fig. 7(c) and Fig. 7(d)\], which is quite different from the usual prediction results in rotating two-component BECs with or without SOC [XQXu,XZhou,HHu,Aftalion,Fetter,Kasamatsu]{}. In addition, we find that the higher the rotation frequency is, the more the component densities of the BECs expand. This point can be understood. Physically, when the rotation frequency (with fixed RD-SOC and other parameters) increases, more angular momentum contributes to the system and leads to the creation of more phase defects and the expansion of the atom cloud, regardless of the initial state of the system being mixed or separated.
Ground-state structures and spin textures for the case of 1D RD-SOC
===================================================================
Now, we consider the ground-state structures of rotating two-component BECs loaded in an optical lattice and a harmonic trap in the presence of 1D RD-SOC. From Fig. 8, we see that the larger the 1D SOC strength or the rotation frequency is, the stronger the 1D SOC effect becomes. Taking the case of fixed 1D RD-SOC strength with $k_{x}$ $=0$ and $k_{y}=2$ \[see Figs. 8(c) and 8(d)\] as an example, we first aim to discuss the influence of the rotation frequency on the ground-state structure of the system. Figs. 8(c) and 8(d) display the ground states of the system with $\Omega =0.3$ and $%
\Omega =0.8$, respectively. For the low rotation frequency $\Omega =0.3$, there is an obvious visible vortex chain along $x=0$ axis in each component due to the 1D RD-SOC along the $y$ direction \[see columns 3 and 4 from left to right in Fig. 8(c)\], where the two component densities exhibit partial mixing and partial separation. As the rotation frequency increases to $%
\Omega =0.8$, more vortices are generated along the $x=0$ axis and the both sides of the central vortex chain, where the two component densities shows good phase mixing except that the densities along the $x=0$ axis display obvious phase separation \[see columns 3 and 4 from left to right in Fig. 8(d)\]. The physical mechanism is that large rotation frequency generates more energy and more angular momentum. Thus the $x$-direction vortex chain caused by the combined effect of 1D RD-SOC and rotation can only carry finite energy and angular momentum, and the remaining energy and angular momentum are inevitably carried by the transverse vortices beside the $x=0$ axis.
![(Color online) Ground states of rotating spin-1/2 BEC with 1D RD-SOC in an optical lattice and a harmonic trap, where $\protect\beta =200$, and $\protect\beta _{12}=300$. (a) $k_{x}=0$, $k_{y}=1$, $\Omega =0.3$, (b) $k_{x}=0$, $k_{y}=1$, $\Omega =0.8$, (c) $k_{x}=0$, $k_{y}=2$, $\Omega
=0.3$, and (d) $k_{x}=0$, $k_{y}=2$, $\Omega =0.8$. The columns from left to right represent $\left\vert \protect\psi _{1}\right\vert ^{2}$, $\left\vert
\protect\psi _{2}\right\vert ^{2}$, $\arg \protect\psi _{1},$ $\arg \protect%
\psi _{2}$, and $\left\vert \protect\psi _{1}\right\vert ^{2}-\left\vert
\protect\psi _{2}\right\vert ^{2}$, respectively. The unit length is $a_{0}$. []{data-label="Figure8"}](Figure8.eps){width="12cm"}
Then we consider the influence of 1D RD-SOC on the ground-state structure of the system. For instance, the rotation frequency is fixed as $\Omega =0.3$. From Figs. 8(a) and 8(c), it is shown that the stronger 1D RD-SOC enhances the creation of vortex chain and the formation of phase mixing. Similarly, for the BECs with 1D RD-SOC along the $x$ direction ($k_{y}$ $=0$), our simulation demonstrates that the density modulation occurs only along the $x$ direction. The above phenomena can be obtained and understood if one performs an unitary transformation, $\sigma _{x}\rightarrow \sigma _{y}$ and $\sigma _{y}\rightarrow -\sigma _{x}$, and sets $k_{x}$ or $k_{y}=0$ for the RD-SOC term.
The topological defects can be observed in a phase profile, but a better visualization is to use the pseudospin representation based on Eqs. ([SpindensityX]{})-(\[SpindensityZ\]). We can plot the functions $S_{x},$ $%
S_{y}$ and $S_{z}$ which reveal the presence of all the spin defects. The corresponding spin-density distributions of Figs. 8(a)-8(d) are shown in Figs. 9(a)-9(d), respectively. In the pseudo-spin representation, the red region denotes $1$ (spin-up) and the blue region denotes $-1$ (spin-down). According to Eq. (\[SpindensityZ\]), the spin-density component $S_{z}$ is related to the density difference of the two components, therefore the variation tendency of the last row of Fig. 9 is consistent with that of the last column of Fig. 8. $S_{x}$ and $S_{y}$ obey neither even parity distribution nor odd parity distribution along the $x$ direction or the $y$ direction.
![(Color online) Spin densities of rotating spin-1/2 BECs with 1D RD-SOC in an optical lattice and a harmonic trap, where $\protect\beta =200$ and $\protect\beta _{12}=300$. (a) $k_{y}=1$, $\Omega =0.3$, (b) $k_{y}=1$, $\Omega =0.8$, (c) $k_{y}=2$, $\Omega =0.3$, and (d) $k_{y}=2$,$\ \Omega
=0.8$. The rows from top to bottom denote $S_{x},$ $S_{y}$ and $S_{z}$ components of the spin density vector, respectively. The corresponding ground states for (a)-(d) are shown in Figs. 8(a)-8(d), respectively. The unit length is $a_{0}$. []{data-label="Figure9"}](Figure9.eps){width="9cm"}
For further comparison, we choose Fig. 9(a) ($\Omega =0.3\ $and $k_{y}$ $=1$) and Fig. 9(c) ($\Omega =0.3$ and $k_{y}$ $=2$) as an example, and we can see that for the region of $x>0$ ($x<0$) the value of $S_{y}$ gets larger (smaller) with the increasing $k_{y}$. Similarly, from Figs. 9(c) and 9(d), we can observe that for the region of $x>0$ ($x<0$) $S_{y}$ approaches $1$ ($%
-1$) when the rotation frequency $\Omega $ increases (see the middle row of Fig. 9). Thus we conclude that the spin component $S_{y}$ develops into two remarkable spin domains due to the increase of $k_{y}$ or $\Omega $. At the same time, an obvious spin domain wall forms on the boundary region between the two spin domains, which can be seen in the middle row of Fig. 9. In general, the spin domain wall for nonrotating two-component BECs is a classical Néel wall, where the spin flips only along the vertical direction of the wall. Whereas, our numerical simulation of the spin texture demonstrates that the spin in the region of spin domain wall flips not only along the vertical direction of domain wall (the $x$ direction) but also along the domain-wall direction (the $y$ direction), which indicates that here the domain wall is an unique Bloch wall instead of the conventional Néel wall. This domain wall is the product of the phase-separated two-component BECs in response to external rotation or RD-SOC, and reflects the influence of rotation or RD-SOC on the magnetism of BECs.
![(Color online) Topological charge densities and spin textures of rotating spin-1/2 BECs with RD-SOC in an optical lattice and a harmonic trap. (a) topological charge density, and (b)-(c) local amplifications of the spin texture, where the ground state is given in Fig. 8(b). (d) topological charge density, (e) the corresponding spin texture, and (f) local enlargement of the spin texture in (e), where the ground state is given in Fig. 8(d). The unit length is $a_{0}$.[]{data-label="Figure10"}](Figure10.eps){width="12cm"}
Displayed in Fig. 10(a) is the topological charge density for the parameters in Fig. 8(b). The typical local spin textures are shown in Figs. 10(b)-10(f). Our computation results show that the local topological charges in Figs. 10(b) and 10(c) are both $Q=1$ while those in Figs. 10(d)-10(f) are all $Q=0.5$. Thus the spin defects in Figs. 10(b) and 10(c) denote an irregular circular skyrmion and an irregular hyperbolic skyrmion [Kasamatsu,CFLiu]{}, respectively. At the same time, the topological defects in Figs. 10(d)-10(f) represent circular half-skyrmion (meron) [@Mermin], hyperbolic half-skyrmion and circular half-skyrmion, respectively. These topological defects alternately appear in the spin representation of Fig. 8(b) and constitute a complicated skyrmion-half-skyrmion (skyrmion-meron) lattice. In addition, we find that for strong 1D RD-SOC the system favors an exotic skyrmion chain (i.e., the skyrmions form a chain-like structure) which traverse the BECs. Figure 10(g) shows the topological charge density, where the ground state is given in Fig. 8(d). Figure 10(h) gives the corresponding spin texture, and the typical local amplification is exhibited in Fig. 10(i). Our numerical calculation shows that the local topological charge in Fig. 10(i) is $Q=1$ and the total topological charge in Fig. 10(h) is $Q=5$. Therefore the spin structure of the system is a skyrmion chain that is composed of a string of elliptic skyrmions with unit topological charge in spin space as shown in Fig. 10(h). Obviously, the skyrmion configurations observed in the present system are remarkably different fromthe previously reported results in rotating two-component BECs with or without SOC [@XQXu; @XZhou; @HHu; @Aftalion; @Fetter; @Kasamatsu; @CFLiu].
Conclusion
==========
In summary, we have investigated the topological excitations of rotating two-component BECs with RD-SOC in a 2D optical lattice and a 2D harmonic trap. The effects of 2D RD-SOC, 1D RD-SOC, rotation, interatomic interactions and optical lattice on the topological structures of the ground states of the system are systematically discussed. Two ground-state phase diagrams for the nonrotating case and the rotating case are given with respect to the SOC strength and the interspecies interaction, and with respect to the rotation frequency and the SOC strength, respectively. Without rotation, a relatively weak isotropic 2D RD-SOC induces the formation of rectangular vortex-antivortex lattice for initially separated BECs, but strong 2D RD-SOC leads to generation of vortex-antivortex rings. For fixed isotropic 2D RD-SOC strength, the depth of optical lattice and the interaction parameters, the increase of rotation frequency can trigger the structural phase transition from square vortex lattice to irregular triangular vortex lattice and the system evolves from initial phase separation into phase mixing. For the case of 1D RD-SOC, the increase of SOC strength or rotation frequency may result in the creation of vortex chain and Bloch domain wall. In addition, the system sustains novel spin texture and skyrmion structures including an exotic skyrmion-half-skyrmion lattice (skyrmion-meron lattice), a complicated meron lattice (half-skyrmion lattice) and a skyrmion chain. These new topological excitations are quite from the predictions in the previous literature of rotating two-component BECs with or without SOC. Theoretically, the rotating spin-orbit-coupled BEC in an optical lattice plus a harmonic trap is feasible and can be achieved in principle. The experimental realization of the model Hamiltonian Eq. ([Hamiltonian]{}) is still a challenge within the current experimental conditions. However, considering that the present system has various novel physical properties, with the continuous development of experimental techniques, the system may be implemented in the future and its novel topological excitations are expected to be observed in experiments.
L.W. thanks Professor Zhaoxin Liang and Professor Chaofei Liu for helpful discussions, and acknowledges the research group of Professor W. Vincent Liu at The University of Pittsburgh, where part of the work was carried out. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11475144 and 11047033), the Natural Science Foundation of Hebei Province of China (Grant Nos. A2019203049 and A2015203037), and Research Foundation of Yanshan University (Grant No. B846).
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abstract: 'We investigate the properties of galaxies between the blue and the red sequence (i.e., the transition region, 4.5$<NUV-H<$6 mag) by combining ultraviolet (UV) and near-infrared imaging to 21 cm [H[i]{}]{} line observations for a volume-limited sample of nearby galaxies. We confirm the existence of a tight relation between colour and [H[i]{}]{}-fraction across all the range of colours, although outside the blue cloud this trend becomes gradually weaker. Transition galaxies are divided into two different families, according to their atomic hydrogen content. ‘[H[i]{}]{}-deficient’ galaxies are the majority of transition galaxies in our sample. They are found in high density environments and all their properties are consistent with a quenching of the star formation via gas stripping. However, while the migration from the blue cloud is relatively quick (i.e., [[$ \stackrel{<}{\sim}$]{}]{}1 Gyr), a longer amount of time (a few Gyr at least) seems required to completely suppress the star formation and reach the red sequence. At all masses, migrating ‘[H[i]{}]{}-deficient’ galaxies are mainly disks, implying that the mechanism responsible for today’s migration in clusters cannot have played a significant role in the creation of the red sequence at high-redshift. Conversely, ‘[H[i]{}]{}-normal’ transition galaxies are a more heterogeneous population. A significant fraction of these objects show star formation in ring-like structures and evidence for accretion/minor-merging events suggesting that at least part of the [H[i]{}]{} reservoir has an external origin. The detailed evolution of such systems is still unclear, but our analysis suggests that, in at least two cases, galaxies might have migrated back from the red sequence after accretion events. Interestingly, the [H[i]{}]{} available may be sufficient to sustain star formation at the current rate for several billion years. Our study clearly shows the variety of evolutionary paths leading to the transition region and suggests that the transition galaxies may not be always associated with systems quickly migrating from the blue to the red sequence.'
author:
- |
L. Cortese[^1] and T.M. Hughes\
School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK\
date: 'Accepted 2009 August 14. Received 2009 August 13; in original form 2009 June 22'
title: |
Evolutionary paths to and from the red sequence:\
Star formation and properties of transition galaxies at $z\sim$0
---
\[firstpage\]
galaxies:evolution–galaxies: fundamental parameters–galaxies: clusters:individual: Virgo–ultraviolet: galaxies– radio lines:galaxies
Introduction
============
The last decades have seen the rise and success of the hierarchical paradigm for galaxy formation in a cold dark-matter dominated universe. Although very powerful, the concordance model is still far from providing us with a complete and coherent view of how galaxies form and evolve. This is mainly because we still do not understand the physics involving the baryonic component. The current challenge for galaxy formation and evolution studies is thus to improve our knowledge of the astrophysical processes responsible for [*transforming*]{} simple dark matter halos into the bimodal population of galaxies inhabiting today’s universe.
It is in fact well established that, when we look at their integrated optical colours, galaxies constitute a bimodal population (e.g., [@tully82; @baldry04]) composed of a ‘red sequence’, dominated by old stellar populations, and a ‘blue cloud’ where the vast majority of new stars in the universe are formed. However, the dichotomy in the colour distribution does not automatically reflect a difference in morphological type (e.g., light distribution) and we now know that the red sequence is not only composed of quiescent early-type galaxies (e.g., [@scodeggio02; @franzetti07]). For example, while the red sequence accounts for $\sim$60-85% (depending on the colour cut used to define star-forming galaxies) of the total stellar-mass density in the local universe (e.g., [@baldry04; @bell03; @borch06; @perez08]), stars in late-type galaxies contribute to at least half the local stellar mass budget (e.g., [@driver06; @driver07b; @driver07a; @kochanek01]). This automatically implies that a significant fraction of massive late-type galaxies lie in the red sequence, whereas high-mass blue ellipticals are extremely rare. Moreover, not all red galaxies have stopped forming stars, as revealed by recent ultraviolet (UV) investigations (e.g., [@kaviraj07]). It thus emerges that, at least at optical wavelength, the red sequence is a heterogeneous family of objects which have likely followed different evolutionary paths.
How galaxies end-up in the red sequence is still partly a mystery, but recently high-redshift surveys have made it possible to start tracing the growth of the star-forming and quiescent galaxy population up to $z\sim$1 and beyond. Despite the observational and theoretical uncertainties (e.g., [@conroy08]), it seems now commonly accepted that the stellar mass of the blue cloud shows very little growth since $z\sim$1, while the red sequence has grown by at least a factor $\sim$2 (e.g. [@cimatti06; @arnouts07; @bell07; @brown07; @faber07]). The most popular scenario invoked to explain the growth of red galaxies is a migration of a significant fraction of star-forming systems from the blue cloud. Although not always supported [@blanton06], the possibility of an exchange of galaxies between the two sequences is exciting, and several theoretical and observational studies have started to look for the possible astrophysical processes responsible for such transition. Several mechanisms have been proposed so far, among the most popular are different modes of gas accretion [@keres05; @dekel06], feedback from active galactic nuclei (AGN, [@schawinsky09]), and environmental effects (e.g., [@hughes09], hereafter HC09). However, whether a population of migrating galaxies does really exist and what causes the quenching of their star formation is still not clear.
In this context, the advent of the [*Galaxy Evolution Explorer*]{} (GALEX) large-area UV surveys is allowing us to tackle this problem from a different angle. Thanks to its high sensitivity to low-level star formation activity, UV magnitudes can be used to better discriminate between quiescent and still active (although optically-red) galaxies. In fact, contrary to what is observed at optical wavelengths, the UV-optical colour distribution at a given mass is not well fitted by two gaussian distributions [@wyder07], but it shows a significant excess of objects in the region between the blue and red sequence (i.e., the ‘so-called’ transition region or ‘green-valley’, [@martin07]). Transition galaxies may thus represent the missing link to understand if and how galaxies move from one population to the other. However, it is worth reminding that, despite its potential, UV emission is significantly affected by dust and, only after accurate dust corrections, can the UV-optical colour be used to identify transition galaxies. A significant fraction of galaxies found between the two sequences may in fact be composed of reddened systems [@cowie08].
Once transition galaxies are properly identified, a reconstruction of their past evolution is not straightforward. The correct discrimination between various physical mechanisms able to suppress star formation requires, in theory, a detailed investigation of [*all*]{} the galactic components, i.e., stars, gas and dust. Of particular importance is the atomic gas content ([H[i]{}]{}), which represents the fuel for the future star formation activity. The mechanism responsible for the migration from the blue cloud must in fact inhibit the condensation of atomic into molecular hydrogen and the onset of star formation. Unfortunately, not only is [H[i]{}]{} astronomy still technically limited to the nearby universe (e.g., [@catinella08b]), but also our knowledge of [H[i]{}]{} properties of local galaxies is generally restricted to the blue cloud (e.g., [@ages1367]). The very local universe (e.g., up to the distance of the Virgo cluster) is currently the only place where it is possible to investigate the link between [H[i]{}]{}-content and quenching of star formation in transition galaxies.
For all these reasons, we have collected UV to near-infrared imaging and [H[i]{}]{} 21cm line data for a volume-limited sample of nearby galaxies covering different environments. In our previous paper (HC09), we have highlighted the power of a combination of UV and [H[i]{}]{} observations to understand the properties of transition galaxies. Our analysis suggested a strong relationship between UV-near-infrared colour and [H[i]{}]{} content showing that migrating spirals are mainly [H[i]{}]{}-deficient objects found in high density environments. This result apparently rules out AGN-feedback as the main mechanism responsible for the quenching of the star formation in nearby spirals. However, a number of important questions still remain to be answered. Are transition galaxies really migrating from the blue to the red sequence? What are the time-scales of such migration? Is the quenching followed by a change in morphology? While in HC09 we have shown that a large fraction of spirals outside the blue cloud is [H[i]{}]{}-deficient, this is not true for all transition spirals. How have [H[i]{}]{}-rich systems reached the transition region? The aim of this paper is thus to extend the analysis presented in HC09, in order to provide important constraints on the recent mass growth of the red sequence.
The paper is arranged as follows. In § 2 we briefly describe the sample and discuss possible biases related to the dust extinction correction. In § 3 we define the transition region and in § 4 discuss the relation between colour and gas content. The properties of transition galaxies are presented in § 5 and their evolutionary histories and implications for galaxy evolution studies are discussed in § 6. Finally, our main results are briefly summarized in § 7.
Throughout the paper we use H$_{0} =\ $70 km s$^{-1}$ Mpc$^{-1}$. In the Virgo Cluster, where peculiar motions are dominant, we use distances as determined in [@gav99]. Star formation rates (SFRs) are computed from the NUV luminosities, following the conversions by [@buat08].
The sample
==========
The analysis presented in this paper is based on the K-band selected sample described in HC09. Briefly, it consists of a volume-limited sample of galaxies having 2MASS [@2massall] K-band magnitude K$_{Stot} \le$ 12 mag and distance between 15 and 25 Mpc. Additional selection criteria are a high galactic latitude (b $>$ +55$^{\circ}$) and low galactic extinction, A$_{B}$ $<$ 0.2 [@schlegel98], to minimize galactic cirrus contamination. The total sample contains 454 galaxies. Observations from the GALEX [@martin05] GR2 to GR4 data releases in the near- (NUV; $\lambda$=2316 Å: $\Delta \lambda$=1069 Å) and far-ultraviolet (FUV; $\lambda$=1539 Å: $\Delta \lambda$=442 Å) band were available for 394 and 325 objects, respectively. In the rest of the paper we focus our attention mainly on the $NUV-H$ colour, given the larger number statistics available. UV magnitudes were obtained by integrating the flux over the galaxy optical size, determined at the surface brightness of $\mu$(B) = 25 mag arcsec$^{-2}$. The typical uncertainty in the UV photometry is $\sim$0.1 mag and $\sim$0.15-0.20 mag in NUV and FUV respectively. Stellar masses are determined from H-band luminosities using the $B-V$ colour-dependent stellar mass-to-light ratio relation from [@bell03], assuming a [@kroupa93] initial mass function. Single-dish [H[i]{}]{} 21 cm line emission data, necessary for quantifying the [H[i]{}]{} content of galaxies, was mainly taken from [@spring05hi], [@goldmine] and the [*NASA/IPAC Extragalactic Database*]{} (NED). Given the variety of sources from which the [H[i]{}]{} fluxes are taken, it is impossible to define a 21 cm sensitiveness limit for our sample. However, as discussed in § 4, non detections start to be significant ($\sim$20%) at $log(M(HI)/M_{star})\sim$-1.3 and dominate ($>$50%) for $log(M(HI)/M_{star})$[[$ \stackrel{<}{\sim}$]{}]{}-2.1. Estimates of atomic hydrogen mass or upper limits are available for $\sim$83% (326/394) of the galaxies with NUV photometry. We estimate the [H[i]{}]{} deficiency parameter ($DEF(HI)$) as defined by [@haynes]: i.e., the difference, in logarithmic units, between the observed [H[i]{}]{} mass and the value expected from an isolated galaxy with the same morphological type $T$ and optical linear diameter $D$: $DEF(HI) = <\log M_{HI}(T^{obs},D^{obs}_{opt})> - log M^{obs}_{HI}$. We used the equation in [@haynes] for early-type galaxies (E/S0 and earlier types) and the four revised values presented by [@solanes96] for late-types (Sa-Sab, Sb, Sbc, Sc and later) to calculate the expected [H[i]{}]{} mass from the optical diameter. The regression line coefficients are almost identical from Sa to Sbc types, whereas they both significantly vary going to E/S0 or to Sc and later types [@haynes; @solanes96]. The typical uncertainty in the estimate of $DEF(HI)$ is $\sim$0.3 (e.g. [@haynes; @fumagalli09]), but it might slightly increase for dwarf galaxies and early-type systems. Given its large uncertainty, in the following, we will mainly use the [H[i]{}]{} deficiency to select those galaxies which have likely lost a significant amount of atomic hydrogen. In detail, we use a threshold of $DEF(HI)=$0.5 to discriminate between ‘[H[i]{}]{}-deficient’ and ‘[H[i]{}]{}-normal’ galaxies. [H[i]{}]{}-deficient systems are thus objects with $\geq$70% less atomic hydrogen than expected for isolated objects of the same optical size and morphological type.
{width="12.5cm"}
Extinction correction
---------------------
The estimate of the internal UV dust attenuation $A(UV)$ is a crucial step for a correct interpretation of the colour-mass diagram. This is particularly true outside the blue cloud where a) evolved stellar populations can contribute to the dust heating [@afuv_luca], b) a significant fraction of the UV emission may come from evolved stars and not from young stellar populations (in particular in the red sequence, e.g. [@bosell05]). Previous works have shown how dramatic the effect of inaccurate dust corrections can be on the transition region and red sequence [@wyder07; @schiminovich07; @afuv_luca]. [@afuv_luca] have recently calibrated new recipes to estimate the dust attenuation, taking into account the contribution of evolved stellar population to the dust heating. Although this method provides more realistic dust corrections with an average error of $\sim$0.5 mag in $A(NUV)$, the uncertainty dramatically increases up to $\sim$1 mag for galaxies lying between the blue and the red sequence. Thus, the transition region can only be investigated using a statistical approach and a comparison between colours and SFRs of transition galaxies could be meaningless given such large uncertainties.
Unfortunately, all dust correction recipes developed so far are calibrated on late-type/star-forming galaxies whereas in the case of early-type/quiescent systems (the dominant population in the red sequence) it is not a priori appropriate to apply such corrections. The UV emission coming from old stellar populations should in fact be less affected by dust. For example, a systematic (but likely unreal) shift of $\sim$0.5 mag is applied to the red sequence if the extinction corrections calibrated on late-types are applied to early-type galaxies [@afuv_luca]. In order to avoid any systematic overestimate of the SFR in early-type galaxies (and also to be consistent with previous works) here we corrected all galaxies for Galactic extinction according to [@schlegel98], but we applied internal extinction corrections only to late-type galaxies. In detail, the internal dust attenuation was determined using the total infrared (TIR) to UV luminosity ratio method (e.g., [@xu95]) and the age-dependent relations of [@afuv_luca]. The TIR luminosity is obtained from IRAS 60 and 100 $\mu$m fluxes or, in the few cases when IRAS observations are not available, using the empirical recipes described in [@COdust05].
Although this technique is likely to overestimate the colour of the few early-type galaxies lying in the blue cloud, we can exclude that it significantly affects the properties of transition region early-type galaxies and peculiar gas-rich ellipticals studied in the following sections. To test this hypothesis, we applied the dust attenuation corrections described above to these objects finding that, even after correcting for dust, transition early-types still lie in the transition region and peculiar red ellipticals are still in the red sequence. However, in the following, we will try to discuss as much as possible any systematic error that could be introduced by problems in the extinction corrections and all error-bars shown in our figures take into account the uncertainty in the estimate of $A(UV)$.
![\[typedistr\] The morphological type distribution of galaxies in our sample (dotted histogram). The filled histogram represents galaxies in the transition region, while red and blue sequence galaxies are shown with the empty and dashed histogram, respectively. ](histo_type_all.epsi){width="8.5cm"}
Where does the blue cloud end? {#deftrans}
==============================
Contrary to what is observed in optical [@baldry04], at fixed stellar mass, the UV-optical colour distribution of galaxies is not best fitted with the sum of two gaussians. An excess of galaxies is clearly present in between the two sequences [@wyder07]. This is likely due to galaxies with suppressed star formation perhaps migrating from the blue to the red sequence. However, it is not straightforward to clearly define the range of colours characterizing galaxies which do not belong either to the blue cloud or the red sequence.
On one side, it is difficult to determine (both observationally and theoretically) where the blue cloud ends. In this paper we decide to use the blue cloud of [H[i]{}]{}-normal galaxies to determine the range of colours typical of unperturbed galaxies. This is motivated by the fact, discussed in HC09, that the transition region is mainly populated by [H[i]{}]{}-deficient galaxies in high-density environments. [H[i]{}]{}-normal late-type galaxies form a blue cloud clearly separated from the red sequence at all masses, as shown in the right panel of Fig. \[CMall\]. We thus define galaxies with suppressed star formation as those objects with $NUV-H>$4.5 mag (corresponding to the 90th percentile of the colour of [H[i]{}]{}-normal spirals with M$_{star}>$10$^{10}$ M$_{\odot}$), consistently with [@martin07b]. This colour-cut roughly corresponds to star formation histories (SFHs) having e-folding time-scale of $\sim$3 Gyr (assuming a galaxy age of 13.7 Gyr, solar metallicities and the models of [@BC2003]) e.g., the typical e-folding time dividing local late- and early-type galaxies [@gav02]. We note that, adopting this convention, M31 would be classified as ‘normal’ blue-cloud galaxy ($NUV-H\sim$4.1 mag). Being calibrated on massive galaxies, our colour-cut must be considered as a conservative upper-limit in the case of dwarf systems.
{width="12.5cm"}
On the other side, UV-to-near-infrared colours typical of the red sequence ($NUV-H \gtrsim$6) can either indicate low residual star formation activity or old, evolved stellar populations (e.g. [@bosell05]). The phenomenon of the UV-upturn [@connell] makes colours redder than $NUV-H \sim$6 difficult to interpret so that the $NUV-H$ colour cannot be considered anymore as a good proxy for the specific star formation rate (SSFR). Following [@kaviraj07], we use observations of well known strong UV-upturn galaxies to derive a lower limit on the $NUV-H$ colour typical of evolved stellar populations. In details, given the typical colour observed in M87 ($NUV-H\sim$ 6.1 mag) and NGC4552 ($NUV-H\sim$ 6.4 mag), we assumed $NUV-H$=6 mag as a conservative lower limit to discriminate between residual star formation and UV-upturn. The validity of this colour-cut is confirmed by a visual inspection of GALEX colour images, which indicates that only 6% (i.e. 6 objects) of galaxies redder than $NUV-H$=6 mag show clear evidence of residual SF (e.g., blue star-forming knots).
The morphological type distributions for galaxies belonging to the three groups here considered (i.e., blue cloud, red sequence and transition region) are shown in Fig. \[typedistr\]. It clearly emerges that red and blue galaxies are two disjoint families not only in colour, but also in shape.
Of course, the criteria described above are arbitrary and vary according to the colour adopted and to the stellar mass range investigated. This can be clearly seen in Fig. \[CMall2\], where the position of ‘transition galaxies’ in different UV and optical colour-stellar-mass diagrams is highlighted. Although it is indisputable that not all transition galaxies are outside the red sequence in a $FUV-H$ and $NUV-B$ colour diagram (in particular at low stellar-masses), it emerges that the definition here adopted is able to select a statistically representative sample of galaxies with suppressed star formation. Moreover, the comparison between the top and bottom row in Fig.\[CMall2\] highlights the necessity of UV colours to select fair samples of transition galaxies: e.g., a simple $B-H$ colour-cut would significantly contaminate our sample with star-forming blue-sequence and quiescent red-sequence systems. Finally, it is interesting to note the wide range in $FUV-NUV$ colour spanned by the transition galaxies, suggesting different current star formation rates.
As shown in the following sections, the sample of transition galaxies selected using these criteria is not significantly contaminated either by active star-forming or quiescent galaxies erroneously classified as transition systems. Thus, in the rest of this paper we will refer to the colour interval 4.5$<NUV-H<$6 mag as the ‘transition region’.
Finally, it is worth reminding the reader that, in all the figures presented in this paper, the UV-near-infrared colour is directly related to the SSFR only outside the red sequence. For colours redder than $NUV-H\sim$6 mag, UV magnitudes cannot be blindly used to quantify current SFRs and the presence of a sequence does not imply that all red galaxies have the same SSFR, as shown in the following sections.
{width="17.5cm"}
The link between atomic hydrogen content and colour
===================================================
In HC09 we showed that late-type galaxies outside the blue cloud appear to have lost at least $\sim$70% of their atomic hydrogen content, when compared with isolated galaxies of similar size and morphology. We commented this result as a strong evidence of a physical relation between the loss of gas and quenching of the star formation. This interpretation is confirmed and reinforced in Fig. \[colgfrac\]. The left panel shows the correlation between colour and [H[i]{}]{} deficiency: galaxies outside the blue cloud have not only lost a significant fraction of their atomic hydrogen content, but we also find a correlation between [H[i]{}]{} deficiency and $NUV-H$ colour, although with large scatter. At least part of the scatter is due to the large uncertainty in the estimate of [H[i]{}]{} deficiency ($\sim$0.3 dex). The same relation can be expressed in terms of gas-fraction (here defined as the ratio of the [H[i]{}]{} to the stellar mass), as discussed by [@kannappan04] and shown in the central and right panels of Fig. \[colgfrac\]: lower gas-fractions correspond to redder colours. The best linear fit to the relation (excluding upper-limits) is $log(M(HI)/M_{star})$=$-0.35\times(NUV-H)$+0.19, with a dispersion of $\sim$0.43 dex. However, from the three panels in Fig. \[colgfrac\] it clearly emerges that not all transition region and red-sequence galaxies are [H[i]{}]{}-deficient, but a number of systems have an amount of hydrogen typically observed in objects lying in the blue cloud. This is particularly interesting if we look at the colour-gas-fraction relations (central and right panels). Outside the blue cloud, galaxies lie mainly at the two edges of the relation depending on whether they are [H[i]{}]{}-deficient (empty circles) or not and the colour-gas-fraction relation appears more scattered[^2]. This suggests that, for $NUV-H>$4.5 mag, the gas-fraction is not a good proxy of the UV-optical colour anymore and vice-versa. The dispersion in the colour-gas-fraction relation increases from $\sim$0.35 dex (consistent with [@kannappan04] and [@cheng_gfrac09]) to $\sim$0.54 dex when we move from [H[i]{}]{}-normal blue-cloud galaxies to transition and red-sequence objects. This is in reality a lower-limit on the real scatter increase since upper-limits are not included in the calculation.
The results shown in Fig. \[colgfrac\] are strongly suggestive of a) a different evolutionary path followed by [H[i]{}]{}-deficient and [H[i]{}]{}-normal galaxies outside the blue cloud and b) of a weaker link between [H[i]{}]{}-content and colour than the one typically observed in star-forming galaxies. Therefore, in order to gain additional insights on the evolution of galaxies in the transition region, in the following we divide transition galaxies into two families according to their gas content and investigate separately their properties.
Before investigating the detailed properties of transition galaxies, it is worth adding a few notes about the validity of the classification for [H[i]{}]{}-normal transition systems. The low number of objects in this category and the large uncertainties in the estimate of gas fractions and UV dust attenuation might suggest that these are just random outliers, not different from the bulk of the [H[i]{}]{}-deficient population. Although we cannot exclude the presence of a few misclassified galaxies in both the [H[i]{}]{}-deficient and [H[i]{}]{}-normal population, it is very unlikely that all (and only) the [H[i]{}]{}-normal galaxies outside the blue sequence are affected by a large ($>$0.5 dex) systematic underestimate of dust attenuation, and/or gas fraction. More importantly, the analysis presented in the next sections will clearly show that these two families have reached the transition region following different evolutionary paths.
![\[trdistr\] The stellar mass distribution of transition galaxies (filled circles). Left: Spirals (empty squares), lenticulars (triangles) and E+dE (asterisks) are indicated. Right: Galaxies are highlighted according to their [H[i]{}]{} content. Galaxies for which the estimate of [H[i]{}]{} deficiency is unsure are indicated with asterisks. ](TR_massdistr.epsi){width="8.5cm"}
{width="17.5cm"}
The properties of transition galaxies
=====================================
In total, 67 galaxies in our sample, corresponding to $\sim$17% in both number and total stellar-mass lie in the transition region as defined in § 3. In Fig. \[trdistr\], we show the stellar-mass distribution of transition galaxies divided according to their morphological type (left panel) and gas content (right panel). For ${\mbox{$M_{*}$}}\gtrsim 10^{10}$ , galaxies with 4.5$<NUV-H<$6 mag are mainly spirals, whereas at lower stellar masses they are preferentially dwarf elliptical systems. More importantly, the majority of transition galaxies have $\gtrsim$70% less atomic hydrogen content than isolated galaxies of similar optical size and morphological type. However, as already noted in HC09, not all galaxies in the transition region are [H[i]{}]{}-deficient. For ${\mbox{$M_{*}$}}\gtrsim 10^{10}$ , $\sim$30% (8 galaxies) of the transition galaxies have [H[i]{}]{} deficiency lower than 0.5. For lower stellar masses, it is difficult to quantify the number of gas-rich objects. [H[i]{}]{} observations are not available for 9 of our galaxies and for 3 additional objects the lower limits obtained for the [H[i]{}]{} deficiency are below our threshold of 0.5. These are mainly dE cluster galaxies (Fig. \[trdistr\], left panel), suggesting that their evolution is related to the cluster environment [@dEale]. However, to be conservative, in the following analysis we will focus our attention on the 55 galaxies for which the classification as [H[i]{}]{}-deficient or [H[i]{}]{}-normal galaxy is reliable.
[H[i]{}]{}-deficient systems {#hidef}
----------------------------
Overall, sure [H[i]{}]{}-deficient galaxies represent $\sim$70% (47 galaxies) in number and $\sim$63% in stellar mass of the transition region[^3]. All except four galaxies lie in the Virgo cluster region (as defined in [@goldmine]) suggesting that the cluster environment is playing an important role in quenching the star formation. Additional support to this scenario is obtained when we consider the properties of galaxies divided according to their gas content and their position in the colour-mass diagram. We compared the median projected distance from the cluster center of the different populations. Given the large asymmetry of Virgo and the presence of two main sub-clusters (Virgo A and Virgo B, at $\sim$1 virial radii projected-distance), for each galaxy we computed the projected-distance from the center of both clouds and adopted the smallest of the two values. Our results do not qualitatively change if just the distance from M87 is adopted. The median cluster-centric distance decreases from $\sim$0.83 virial radii (R$_{vir}$), in case of [H[i]{}]{}-normal blue-cloud galaxies, to 0.51, 0.52 and 0.42 R$_{vir}$ for [H[i]{}]{}-deficient blue-cloud, [H[i]{}]{}-deficient transition and red-sequence objects, respectively (see Fig. \[cluster\], left panel). Similarly, the difference between the 25th and 75th percentiles of the line-of-sight velocity distribution (i.e. a good estimate of the velocity dispersion in case of non gaussian distributions) increases in the blue cloud from $\sim$760 [km s$^{-1}$ ]{}to $\sim$1400 [km s$^{-1}$ ]{}when we consider [H[i]{}]{}-normal and [H[i]{}]{}-deficient galaxies respectively. Then, the typical velocity dispersion gradually decreases to $\sim$915 [km s$^{-1}$ ]{}and $\sim$610 [km s$^{-1}$ ]{}when we consider the transition region and the red sequence respectively (see Fig. \[cluster\], right panel). The gradual variation in projected distance and velocity distribution when moving in the colour magnitude diagram from blue, [H[i]{}]{} normal systems to red, quiescent objects supports the idea that [H[i]{}]{}-deficient galaxies represent a population of galaxies recently infalled into the cluster and not yet virialized. For example, a free-falling population is expected to have a velocity dispersion $\sqrt[]{2}$ times larger than the virialized population. Moreover, the velocity dispersion profile for [H[i]{}]{}-deficient galaxies decreases with cluster-centric distance consistent with isotropic velocities in the center and radial velocities in the external regions, as expected in the case of galaxy infall onto the cluster [@GIRARD98; @COGA04]. The opposite trend (i.e., increasing with cluster-centric distance) is observed for red-sequence galaxies, as expected in a relaxed cluster undergoing two-body relaxation in the dense central region, with circular orbits in the center and more isotropic velocities in the external regions. Finally, a visual investigation of UV images reveals that in at least 50% of star-forming Virgo galaxies in the transition region the star formation is only present well within the optical radius, completing the collection of evidence supporting environmental effects behind the quenching of the star formation in [H[i]{}]{}-deficient Virgo galaxies.
Less clear is the origin of the [H[i]{}]{} deficiency in galaxies outside Virgo: 4 objects in total, namely NGC4684, UGC8756, UGC8032 and NGC5566. NGC5566 is the brightest member of a galaxy triplet while UGC8032 lies just $\sim$1.1 virial radii from the center of Virgo. Thus for these two galaxies it is still possible that environmental effects are playing a role in the gas stripping. The fact that UGC8032 is not included in our Virgo sample despite its small distance from M87 is due to the fact that it just lies outside the Virgo boundaries defined by [@goldmine]. The origin of the [H[i]{}]{} deficiency in NGC4684 and UGC8756 remains a puzzle. UGC8756 has in fact no nearby companions or any clear sign of interaction. NGC4684 is a lenticular galaxy with very strong UV nuclear emission, probably related to the extended H$\alpha$ outflow discovered by [@bettoni93]. The outflow has been interpreted as related to bar instability and it is not clear whether such process can be responsible for the [H[i]{}]{} deficiency observed in this object. Thus, the origin of the [H[i]{}]{} deficiency in these two objects still remains unclear.
### Time-scales for the migration
Combining the observational evidence presented above with previous works on the Virgo cluster (e.g. [@review] and references therein), we favor hydrodynamical interactions like ram-pressure as the main process responsible for the suppression of the star formation. However, we note that gravitational interactions cannot be excluded in at least one case (NGC4438, e.g., [@kenney08; @n4438; @vollmer05]).
Only recently, has it become possible to accurately quantify the time-scale for the quenching of the star formation after the stripping event [@n4569; @crowl2008]. Luckily, almost all the objects for which a stripping time-scale has been computed are included in our sample. In Fig. \[timescale\], we highlight the position of these six objects in the colour-mass diagram according to the age of the stripping event: hexagons and triangles indicate galaxies in which the star formation in the outer regions has been suppressed less or more than $\sim$300 Myr, respectively. Interestingly, there is a difference in the average colour between very recent quenching ($t<$300 Myr) and older events, so that only galaxies with quenching time-scale $\sim$400-500 Myr are at the edge or have already reached the transition region. This is consistent with the fact that the Virgo transition galaxy population is not virialized, implying a recent ($\leq$ 1.7 Gyr, i.e., the Virgo crossing time; [@review]) infall into the cluster center. Thus, we can conclude that once the [H[i]{}]{} has been stripped from the disk, a galaxy moves from the blue cloud to the transition region in a time-scale roughly $\sim$0.5-1 Gyr.
It is more difficult to predict the future evolution of transition galaxies and in particular to estimate how long they will remain in the transition region and whether they will eventually join red-sequence galaxies. As already pointed out by [@crowl2008] and [@dEale], one cluster-crossing is not sufficient to completely halt the star formation in massive (M$_{star}$[[$ \stackrel{>}{\sim}$]{}]{}10$^{10}$ M$_{\odot}$) galaxies. In fact, while the outer disk is completely deprived of its gas content and star formation is quickly stopped, in the central regions the restoring force is too strong, keeping the atomic hydrogen reservoir necessary to sustain continuos star formation. Moreover, [@dEale] showed that two cluster crossings ($\sim$2-3 Gyr) are already necessary to move the brightest dE from the blue to the red sequence, suggesting that the transition galaxies described here will take at least the same amount of time to have their star formation completely quenched. Assuming that the [@GUNG72] formalism for ram-pressure is still valid after the first passage and that the galaxy’s orbit does not change significantly, we can expect that very little additional gas will be stripped during the second passage by ram-pressure. Significant stripping would occur only if the galaxy’s restoring force is lowered by gravitational interactions with other members and the cluster potential well. Other intra-cluster medium related environmental effects, like viscous stripping [@NUL82] and thermal evaporation [@COWS77], may thus play an important role in the complete suppression of the star formation.
![\[timescale\] Same as Fig. \[CMall\] (left panel). Large symbols indicates galaxies in the Crowl & Kenney (2008) sample for which a stripping time-scale estimate is available. Stripping time-scale shorter than 300 Myr and between 300-500 Myr are shown with hexagons and triangles respectively. ](CM_timescale.epsi){width="6.cm"}
An upper limit to the time spent by [H[i]{}]{}-deficient galaxies in the transition region can be obtained if we assume that all the remaining gas will be consumed by star formation. It is in fact plausible that the intra-cluster medium will prevent additional infall of cold gas. In this case, the ‘Roberts’ time’ ([@roberts63], defined as the ratio of the gas mass to the current star formation rate: i.e., M(gas)/SFR) can be used to obtain a rough estimate of the gas consumption time. Assuming that 15% of the total gas is in the molecular state [@boselligdust] and $\sim$30% is composed by helium and heavy elements [@boselli], we find that the ‘Roberts’ time’ is already $\sim$2.2 Gyr[^4]. This is in reality a lower limit to the real value since it does not take into account gas recycling. As shown by [@kennbirth], the real gas consumption time is 1.5-4 times longer then the time scale calculated above. Thus, although they have lost a significant amount of their original gas content, [H[i]{}]{}-deficient transition galaxies still have enough fuel to sustain star formation at the current rate for at least a couple of Gyr.
Detailed simulations focused on the effect of the cluster environment after the first passage will thus be extremely interesting to understand the future evolution of these systems. At this stage, the main conclusion we can draw from our analysis is that at least $\sim$3 Gyr seem to be necessary for the complete migration of a galaxy from the blue to the red sequence when gas stripping via the intra-cluster medium is involved.
[l@c@c@c@c@c@c@c@c@c@c@c@c@c@c@]{}\
NAME & TYPE & D &AGN & B-V & FUV & NUV & H & F$_{60 \mu m}$ & F$_{100 \mu m}$ & M(HI) & M(HI)/M$_{star}$ & C$_{31}$(H) & Merging/Accretion? & Ref.\
& & Mpc & & & m$_{AB}$ & m$_{AB}$ & m$_{AB}$ & Jy & Jy & 10$^{8}$ M$_{\odot}$ & & & &\
NGC3619 & S0 & 20.7 & - & 0.86& 16.86 & 16.14 & 7.40 & 0.43 & 1.61 & 7.08 & 0.04 & 6.2 & yes & 1,2\
NGC3898 & Sa & 15.7 & Lin& 0.79& - & 15.03 & 6.25 & 0.42 & 2.02 & 26.9 & 0.09 & 3.1 & may be & 3\
NGC4324 & Sa & 17.0 & Lin/Sey& 0.87& 16.62 & 15.99 & 7.29 & 0.45 & 1.96 & 6.76 & 0.05 & 3.5 & - & -\
NGC4370 & Sa & 23.0 & NoL & 0.70& - & 18.26 &8.28 & 0.94 & 3.27 & 4.0 & 0.04 & 3.8 & yes & 4,5\
NGC4378 & Sa & 17.0 & Sey& 0.81& 16.00 & 15.50 & 7.23 & 0.36 & 1.45 & 10.0 & 0.07 & 4.9 & - & -\
NGC4772 & Sa & 17.0 & Lin& 0.87& 17.07 & 16.16 & 7.12 & 0.38 & 1.32 & 6.17 & 0.04 & 5.1 & yes & 6\
NGC5701 & Sa & 20.1 & Lin& 0.84& 15.48 & 15.11 & 6.74 & 0.27 & 1.36 & 61.7 & 0.20 & 4.0 & - & -\
\
NAME & TYPE & D &AGN & B-V & FUV & NUV & H & F$_{60 \mu m}$ & F$_{100 \mu m}$ & M(HI) & M(HI)/M$_{star}$ & C$_{31}$(H) & Merging/Accretion? & Ref.\
& & Mpc & & & m$_{AB}$ & m$_{AB}$ & m$_{AB}$ & Jy & Jy & 10$^{8}$ M$_{\odot}$ & & & &\
NGC4203 & S0 & 17.0 & Lin. & 0.99 &16.94 & 15.87 & 6.26 & 0.59 & 2.16 & 33.1 & 0.09 & 6.8 & yes & 7\
NGC4262$^{a}$ & S0 & 17.0 & NoL & 0.83 &- & 16.9 & 7.20 & - & 0.50 & 5.13 & 0.04 & 6.0 & yes & 8\
NGC4698$^{a}$ & Sa & 17.0 & Sey & 0.83 &16.64 & 15.71 & 6.16 & 0.63 & 1.89 & 17.0 & 0.04 & 4.2 & yes & 9\
NGC4866 & S0 & 17.0 & Lin & 0.96 &17.26 & 16.27 & 6.76 & - & - & 13.5 & 0.06 & 3.7 & - & -\
NGC5103 & Sab & 17.0 & - & - &19.49 & 17.76 & 8.29 & - & - & 2.40 & 0.05 & 6.3 & - & -\
$^{a}$ UV fluxes increase significantly if the UV rings outside the optical radius are included.\
References: (1) [@vandriel89]; (2) [@howell06]; (3) [@noordermeer05]; (4) [@bertola88]; (5) [@patil09]; (6) [@haynes00]; (7) [@vandriel88]; (8) [@krumm85]; (9) [@bertola99]\
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[H[i]{}]{}-normal galaxies
--------------------------
Contrary to [H[i]{}]{}-deficient objects, sure [H[i]{}]{}-normal transition galaxies are equally distributed between the field and cluster environments. They are $\sim$12 % in number[^5] (8 objects in total) and $\sim$34% in stellar mass of the whole transition galaxy population in our sample. We note that NGC4565 is the only example of perfectly edge-on transition object, in this case we cannot exclude that the corrections adopted still underestimate the real UV extinction [@panuzzo03] making this galaxy possibly an erroneous transition object. Therefore, we will exclude NGC4565 from the following analysis. The properties of the remaining 7 objects are listed in Table \[hirich\] as follows: Col. 1: Name. Col.2: Morphological type. Col. 3: Distance in Mpc. Col.4 AGN classification (following the criteria described in [@decarli07]): Lin=LINER, Sey=Seyfert, NoL= No emission lines. Col.5: $B-V$ colour corrected for dust extinction. Col.6-8: FUV, NUV and H AB magnitudes. Col.9-10: IRAS fluxes at 60 and 100 $\mu m$. Col. 11: HI mass. Col. 12: [H[i]{}]{}- to stellar-mass ratio. Col. 13: concentration index in H band taken from 2MASS ($C_{31}(H)$ defined as the ratio between the radii containing 75% and 25% of the total H-band light). Col. 14-15: Note regarding any evidence (and relative reference) supporting an external origin for the [H[i]{}]{}. In Fig. \[FCtrans\], we show SDSS optical and GALEX UV colour images for each galaxy. In Appendix A, we describe the properties of each object in order to investigate the possible origins for the normal [H[i]{}]{} content and low SSFR. From this analysis, it emerges that [H[i]{}]{}-normal transition galaxies are a heterogeneous class of objects going from merger remnants (e.g., NGC3619) with star-formation activity limited to the center, to satellites of big ellipticals (NGC4370), with no evident signs of recent star formation. Contrary to the [H[i]{}]{}-deficient family, gas stripping by environmental effect seems not to be playing any role in their recent evolution. As expected in the case of massive ($M_{star}\geq10^{10}$ M$_{\odot}$) galaxies, the majority of these objects host an ‘optical’ AGN. More surprisingly, our analysis shows that, although they have likely followed different evolutionary paths, a significant fraction of galaxies (at least 4 out of 7) has recent star formation mainly in form of one or more UV rings. As shown in Fig. \[FCtrans\], the UV rings have different morphologies going from inner rings (NGC3898, NGC4324, NGC4772) to Hoag-like objects (NGC5701). Resolved [H[i]{}]{} maps, available for two of our objects (NGC3898, NGC4772), reveal that the [H[i]{}]{} is distributed in extended low surface density disks, exceeding significantly the typical column density of 1-2 M$_{\odot}$ pc$^{-2}$ only in correspondence of the star forming rings. Thus, in these cases, star formation is reduced not because the [H[i]{}]{} has been stripped but just because the gas is not able to collapse into stars efficiently. This is likely due to the fact that the gas reservoir has a typical column density well below the critical density necessary to convert [H[i]{}]{} into molecular hydrogen and onset the star formation [@krumholz09].
The presence of UV rings becomes more intriguing when [H[i]{}]{}-normal galaxies in the ‘NUV red sequence’ are also taken into account (5 galaxies in total, see Table \[hirich\], Fig. \[FCred\] and Appendix A for a description of these objects). The four galaxies with clear star formation activity (NGC4203, NGC4262, NGC4698, NGC4866) have star-forming regions mainly arranged in structures which are suggestive of one or multiple rings (see Fig. \[FCred\]). In the case of NGC4203 and NGC4698, [H[i]{}]{} maps reveal a morphology similar to the one observed in [H[i]{}]{}-normal transition objects with extended low surface density [H[i]{}]{} disks and peaks of column density in correspondence of UV star-forming regions. The only known exception is represented by NGC4262, where the [H[i]{}]{} is mainly segregated in the UV star forming ring. Interestingly, NGC4262 and NGC4698 could be immediately reclassified as transition region galaxies if the outer UV rings are included in the estimate of the UV flux. All this observational evidence strongly suggests that the evolutionary paths leading these objects to the transition region have been significantly different from the one followed by [H[i]{}]{}-deficient galaxies.
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### Past and future evolution in the colour-mass diagram
The main piece of information necessary to understand the recent evolutionary history of these unusual ‘gas-rich’ systems, is the origin of their gas reservoir. Has the gas an external origin (e.g, accretion, infall, merging, shells, etc.) or was it always part of the galactic halo but has not been efficiently converted into stars? The origin of the gas can in fact not only provide clues on the past history of these objects (i.e., whether they are really migrating from the blue to the red sequence) but also may help in predicting their future evolution in the colour-mass diagram. In the following, we combine the information available for HI-normal transition and red sequence galaxies to determine whether these systems have recently left the blue sequence after a quenching episode or are migrating back from the red sequence to the transition region thanks to a recent accretion event.
As discussed in Appendix A, many galaxies in our sample show direct or indirect evidence of past gas accretion/infall events (e.g, warps, counter-rotating or decoupled components, stellar shells). Among the best candidates for an external origin of all the [H[i]{}]{} observed, there are NGC4262 and NGC4203 in the red sequence and NGC3619 in the transition region [@vandriel91]. However, it is interesting to note that the acquisition mechanisms (and therefore the evolution) of the three systems is likely to be different. In the case of NGC3619, we are likely witnessing a minor merger with a gas-rich satellite. The [H[i]{}]{} is segregated well within the optical radius, roughly coinciding with the star forming disk observed in UV, suggesting that a satellite has sunk into the center initiating an episode of star formation. Given that the stellar populations have ages and metallicities typical of unperturbed ellipticals [@howell06], the most plausible scenario is that NGC3619 has left the red sequence after the merging event. Interestingly, at the current SFR ($\sim$0.1 M$_{\odot}$ yr$^{-1}$), the amount of atomic hydrogen present within the optical disk ($\sim$7$\times$10$^{8}$ M$_{\odot}$) is sufficient to sustain the star formation for several billion years. Thus, NGC3619 will either remain in the transition region for a long time or, in case of a significant increase of the SFR, may be able to temporarily rejoin the blue cloud in a UV-near-infrared colour magnitude diagram. A similar evolutionary path could also have been followed by the dust-lane early-type NGC4370. However, the lack of detailed [H[i]{}]{} maps prevent us from drawing any conclusion.
On the contrary, the infall of [H[i]{}]{} into NGC4203 and NGC4262 has likely followed less ‘violent’ paths. In NGC4262, the presence of a ring composed only of [H[i]{}]{} and newly formed stars is strongly suggestive of recent accretion, apparently ruling out that the ring has been formed from galactic material through bar instability. An interesting possibility is that the bar could still be responsible for the peculiar configuration of the [H[i]{}]{}, preventing the newly accreted gas to collapse into the center. What remains unclear is whether the gas in the ring has been accreted from the intergalactic medium (as proposed in the case of polar ring galaxies; e.g., [@maccio06]) or during an interaction with another galaxy [@vollmer05b]. As for NGC3619, the gas reservoir in the ring is sufficient to keep the galaxy in the transition region for several Gyr or to move it back to the blue cloud, building-up a new stellar disk/ring. To this regard, it is tempting to consider NGC4262 the ancestor of Hoag-type objects like NGC5701, thus implying that these two systems may be on their way back to the UV blue cloud. However, at this stage it is impossible to determine whether these two systems are at different stages of the same evolutionary path.
A migration back to the transition region appears instead very unlikely in the case of NGC4203. Despite its huge [H[i]{}]{} reservoir, this galaxy shows only weak traces of recent star formation activity and at this rate the integrated colour will not be significantly affected, leaving this object in the red sequence. A similar scenario could also be valid for NGC5107 and NGC4886 which already are in the red sequence. However, additional observations are required to unveil the evolutionary history of these systems.
In summary, for at least a few cases, observations seem to suggest that [H[i]{}]{}-normal red galaxies have recently acquired atomic hydrogen and have started a new cycle of star formation activity leaving, at least temporarily, the red sequence.\
For other transition galaxies, this scenario appears extremely unlikely. This is particularly the case of NGC3898, NGC4772 and NGC4698. These three systems have very similar properties: i.e., Sa/Sab type with a significant bulge component (bulge-to-total ratio $\sim$0.2-0.4; [@drory07]), [H[i]{}]{} mainly distributed in two rings, one inside and one outside the optical radius, corresponding to the sites of recent star formation activity. HST images reveal that all three galaxies harbor a classical bulge [@drory07], consistent with a ‘violent’ and quick bulge formation in the past through mergers or clump coalescence in primordial disks (e.g., [@noguchi99; @kormendy04; @elmegreen08]). In addition, the presence of a decoupled core (NGC4698) or a counter-rotating gas disk (NGC4772) is suggestive of a more recent accretion event (e.g., minor merger) supporting an external origin for at least part of the [H[i]{}]{} in these objects. The preferred explanation for the properties of NGC4698 is in fact a later formation of the disk through the acquisition of material by a completely formed spheroid. Thus, it would be natural to argue that we are witnessing the build-up of the disk and that these galaxies are gradually moving from the red to the blue cloud, following a path consistent with what expected by hierarchical models [@baugh96; @kauffmann96]. However, a rough time-scale argument rules out this hypothesis. All three galaxies harbor massive stellar disks (M$_{disk}\sim$1-3$\times$10$^{10}$ M$_{\odot}$ depending on the mass-to-light ratio difference between bulge and disk) and a SFR of $\sim$0.7-2 M$_{\odot}$ yr$^{-1}$ during the last Hubble time is thus necessary to form the observed disks. These SFRs are more typical of blue-cloud galaxies and a factor $\sim$10 larger than the SFR observed in these systems ($SFR\sim$0.07-0.2 M$_{\odot}$ yr$^{-1}$). While minor mergers have probably affected these systems, we can exclude that a great part of the stellar disk is composed of ‘accreted stars’, since the whole stellar component of the satellite is supposed to collapse into the center, contributing to the growth of the bulge [@hopkins09]. Although the uncertainties in the estimate of stellar masses and SFR are still quite significant, the large discrepancy between the observed and expected SFR suggests that the existing stellar disks are too massive to have been formed at the current SFR. Of course, we cannot exclude multiple transitions from the blue to the red sequence and vice-versa [@birnboim07], but given the low amount of observational constraints available, we prefer not to include such scenario in our analysis. We thus propose that the SFR in the disks was higher in the past or, in other words, that these galaxies have probably migrated from the blue cloud. A reduction in the SFR is likely due to the low [H[i]{}]{} column density in these systems: on average below the threshold for the onset of star formation. What caused this reduction is still unclear and only more detailed theoretical models and simulations will help us to solve this mystery. The same mechanism is probably behind the ring-like structures observed in both [H[i]{}]{} and UV. Both internal (e.g, bar instability) or external (e.g. accretion, merging) processes can be responsible for such features. However the absence of strong bars, the presence of decoupled/counter rotating components and the size-ratio of the inner and outer rings ($\sim$3.3-3.6, i.e. different from the typical value expected for Lindblad resonances $\sim$2.2, although not completely inconsistent; [@athanassoula82; @buta95]), favour an external mechanism behind the unusual properties of these systems. Since all these galaxies harbour an AGN, it is natural to think about AGN-feedback. However, these are generally low-energetic AGNs and no direct evidence (e.g., jets) supporting this scenario has been found so far. Moreover, it is not clear how star-forming rings and low surface density [H[i]{}]{} disks can be formed via AGN-feedback.
Finally, it is important to note that, whatever the past evolutionary history of these systems, the hydrogen reservoir available can sustain the current star formation activity in these galaxies for at least 2-5 Gyrs (ignoring molecular hydrogen, helium and recycling). Thus, if star formation will remain as efficient as it is now, it will take a long time for these galaxies to reach the red sequence in a UV-optical/near-infrared colour-mass diagram.
Unfortunately, no speculation can be made about the past evolutionary history of NGC4324 and NGC4378 given the lack of multiwavelength observations.
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Discussion
==========
Our analysis provides definitive evidence that galaxies in the transition region of the $NUV-H$ colour-mass diagram are a heterogeneous population. This result strongly suggests that galaxies lying between the blue and red sequence have followed different evolutionary paths, not always going towards redder colours.
We remind the reader that the difference in number density between [H[i]{}]{}-deficient and [H[i]{}]{}-normal transition galaxies must be taken with a grain of salt. Although our sample is magnitude- and volume-limited, it might be biased towards high-density environments. Nearly half of the galaxies in our sample lies in fact within the Virgo cluster, which might not be a fair representation of the local universe.
Interestingly, external (i.e., environmental) mechanisms are almost always behind the peculiar properties of transition galaxies, both [H[i]{}]{}-deficient and [H[i]{}]{}-normal objects. Although the majority of high-mass ($M_{star}$[[$ \stackrel{>}{\sim}$]{}]{}10$^{10}$ M$_{\odot}$) transition galaxies harbour an AGN, feedback from accreting super-massive black holes appears not to be necessary to explain their properties. As discussed in HC09, the presence of AGNs in transition galaxies does not automatically imply a physical link between nuclear activity and quenching. Moreover, we do not find any direct observational evidence (e.g., jets, radio lobes, etc.) supporting an interaction between the central black hole and the galaxy’s gas reservoir.
The migration to the red sequence of [H[i]{}]{}-deficient galaxies
------------------------------------------------------------------
[H[i]{}]{}-deficient transition galaxies constitute the majority of the transition population in our sample. By extending the analysis presented in HC09, we have shown that these galaxies not only are mainly found in the Virgo cluster but they also are the only population which is clearly migrating from the blue towards the red sequence. While environmental effects are certainly able to strip the gas from the disk, reducing the star formation in just a few hundreds million years and forcing the galaxy to leave the blue cloud, less clear is the last leg of the journey, i.e. the migration from the transition region to the red sequence. The complete suppression of the star formation requires at least a few billion years. This ‘two-step’ migration is more dramatic, and perhaps only visible, in a UV-near-infrared colour-mass diagram whereas in optical the first stripping event is sufficient to make the colours almost as red as an early-type galaxy.
At this stage, it is very tempting to use our data to quantify the current ‘migration’ rate and, consequently, the mass accretion rate of the red sequence. Unfortunately, the large uncertainties in both observables and on the typical time-scale of the migration (at least a factor 2) make this exercise not very useful. For example, a quenching time of $\sim$3 Gyr would suggest that, at the current rate, the red sequence in our sample could have been built by the migration of objects from the blue cloud in a Hubble time, consistent with previous works (e.g., [@arnouts07; @martin07; @schiminovich07]). However, at the same time, we are not able to reject scenarii in which either the observed rate is able to build-up the red sequence in half a Hubble time or the observed migration is not able to explain the growth of the quiescent galaxy population in the last 13 Gyr. Thus, no additional constraints on the evolution of the colour-stellar mass diagram are imposed by the estimate of the stellar mass accretion rate observed in our sample.
A more interesting exercise is to look for any morphological transformation during the migration towards the red sequence. The crucial question here is whether the red sequence is fed with bulge dominated or disk galaxies. The answer is clear from Fig. \[c31virgo\], where we compare the distribution of the concentration index in H-band for galaxies in the Virgo cluster. In the transition region, we show only Virgo [H[i]{}]{}-deficient galaxies. Overall (Fig. \[c31virgo\], left panel), [H[i]{}]{}-deficient transition galaxies have a concentration index much more similar to blue than red-sequence systems. However, such difference is only evident at stellar masses higher than $\sim$10$^{10}$ M$_{star}$ (Fig. \[c31virgo\], central panel) whereas for smaller galaxies (right panel) the distribution of $C_{31}(H)$ does not significantly vary across the whole range of colours, reflecting the fact that dwarf ellipticals have exponential light profiles like dwarf irregulars (e.g.,[@binggeli91]). This result implies that, while [H[i]{}]{}-deficient transition galaxies are likely the progenitors of cluster low-mass red objects (see also [@haines08; @dEale]), this is not completely true at high stellar masses. This is additionally supported by the fact that the vast majority of high-mass transition galaxies are early-type spirals and almost no ellipticals are present (Fig \[typedistr\] and \[trdistr\]). The vast majority of galaxies in the process of reaching the red sequence are thus disk systems, significantly different from ellipticals or bulge-dominated galaxies characterizing high-mass, quiescent objects at low redshift.
Since a significant fraction of [H[i]{}]{}-deficient transition galaxies appears to have recently infalled into the center of Virgo and will likely spend a few Gyr in the transition region (see § \[hidef\]), it may be possible that a morphological transformation still takes place before reaching the red sequence. Although we cannot completely exclude this scenario, we consider it unlikely. Given the long time required to halt the star formation, galaxies with increased bulge component should be present in our sample. Moreover, since gas stripping appears not to significantly increase the bulge component in early type galaxies (e.g., [@n4569]), an additional environmental effect (different from the one responsible for the quenching of the star formation) must be invoked.
Thus, the fact that transition galaxies are not morphologically transformed before reaching the red sequence may have two different implications: either 1) galaxies are morphologically transformed once already in the red sequence, or 2) the mechanism controlling the accretion of stellar mass into the red sequence at $z\sim$0 is not the one responsible for the creation of the red sequence in the first place. Although it is possible that the bulge component is enhanced in some red galaxies via gravitational interactions, such scenario seems unlikely to explain the growth of the red sequence. Firstly, the mechanism responsible for the morphological transformation should be efficient only on giants and not on dwarf systems, which seems inconsistent with what known about environmental effects [@review]. Secondly, the major mergers required to significantly increase the bulge component are extremely rare in today’s clusters of galaxies. Thirdly, the presence of red-sequence galaxies in isolation (e.g., HC09) implies that star formation has been suppressed also outside clusters of galaxies. Although our sample could be biased against isolated objects, the fact that we do not find a field population which is clearly migrating from the blue sequence may suggest that in low density environments the red sequence has been mainly populated in the past. Finally, if the morphological transformation takes place in the red sequence, the number of bulge-dominated/elliptical galaxies should decrease at increasing redshift, which does not seem to be the case [@postman05; @desai07]. Thus, the most favourite scenario emerging from our analysis is that the red sequence is currently accreting mass, at all masses, mainly via disk galaxies. No significant structural modification takes place during the journey from the blue cloud to the red sequence. The main process responsible for the suppression of the star formation in nearby galaxies is thus not the same responsible for the formation of the red sequence at high redshift.
![\[c31gasrich\]. The distribution of the H-band concentration index for Virgo [H[i]{}]{}-deficient (dashed) and [H[i]{}]{}-normal transition galaxies. All galaxies and galaxies with $M_{star}\geq10^{10}$ M$_{\odot}$ are shown in the left and right panel respectively.](c31_gasrich.epsi){width="8.5cm"}
The [H[i]{}]{}-normal side of the transition region
---------------------------------------------------
While in the case of [H[i]{}]{}-deficient objects it seems plausible to associate transition galaxies with objects that are migrating from the blue to red sequence, this is not always the case for [H[i]{}]{}-normal transition galaxies. The discovery of such systems is probably the most exciting result of this work. They represents $\sim$12% of our transition galaxy population, and are only found at high stellar masses ($M_{star}>$10$^{10}$ M$_{star}$).
As shown in Fig. \[c31gasrich\], these galaxies are mainly disks with a significant bulge component [**(they are in fact all Sa or S0 galaxies)**]{} and, despite the low number statistics, it seems clear that, , contrary to the [H[i]{}]{}-deficient population, in this family we find very few ‘disk-only’ galaxies. Despite their different properties, a great fraction of these galaxies show active star-forming regions and [H[i]{}]{} segregated in one or multiple ring-like structures. The picture emerging from our analysis is quite complex and exciting at the same time, showing that the transition region can be fed with galaxies from both sequences.
Red sequence galaxies can acquire new gas supply and restart their star formation activity, as predicted by cosmological simulations. However, it is unlikely that the red galaxies in our sample will re-build a significant stellar disk (see also [@hau08]). Merging, accretion of gas-rich satellites and exchange of material during close encounters are among the likely responsible for this rejuvenation. Such processes are more frequent in low density environments, where red-sequence galaxies are rarer, perhaps reducing the chances to observe such phenomenon.
When a suppression of the star formation is the most likely scenario to explain [H[i]{}]{}-normal transition galaxies, the process behind such migration is still unclear. In this case, star formation must be reduced by making the [H[i]{}]{} stable against fragmentation (e.g., by decreasing the [H[i]{}]{} column density below the threshold for star formation) and not via [H[i]{}]{} stripping as observed in [H[i]{}]{}-deficient objects. Starvation [@larson80] by removing any extended gaseous halo surrounding the galaxy, preventing further infall, could be a possibility. This would imply a longer time-scale (several Gyr) for the migration from the blue cloud [@n4569; @dEale] than the one observed in the case of [H[i]{}]{} stripping (a few hundreds million years). However, the evidence for accretion/interaction in some of these objects may suggest that starvation, if efficient, is not the only mechanism at work.
[@martig09] have recently proposed a ‘morphological quenching’ to explain the origin of gas-rich bulge-dominated objects. The idea behind this mechanism is that the presence of a bulge could inhibit the collapse of a gas disk. However, the ‘morphological quenching’ appears only to be effective when the disk stellar component is negligible, which is not the case for the majority of the systems in our sample.
Finally, the fact that the majority of these objects show some level of nuclear activity, might indicate a link between AGN activity and their position in the colour stellar mass diagram. However, although we cannot exclude that recent accretion events may have triggered the AGN, we do not find any direct observational evidence suggesting that AGN-feedback is playing a significant role in the recent star formation history of these objects (see also HC09).
Thus, the past evolutionary history of [H[i]{}]{}-normal transition galaxies has still to be unravelled. Whatever is the path followed to get to the transition region, [H[i]{}]{}-normal systems currently have enough fuel to sustain star formation at the current rate for almost another Hubble time. This is an unexpected result, implying that such systems could remain in the transition region for a very long time and that the colour range 4.5$<NUV-H<$6 mag does not automatically correspond to a snapshot in the star formation history of nearby galaxies.
Summary & Conclusion
====================
In this paper, we have combined UV, [H[i]{}]{} and near-infrared observations to investigate the properties of local transition galaxies. Our main results are as follows.\
i) We confirm that the reddening of the galaxy colour is accompanied by a decrease in the [H[i]{}]{} gas-fraction. However we show that, while in the blue cloud colour and gas-fraction are tightly correlated, in the transition region and red sequence such correlation is more scattered. Transition region galaxies can thus be divided into two main families according to their [H[i]{}]{}-content.\
ii) [H[i]{}]{}-deficient transition objects reside in high-density environments and environmental effects are the likely cause for the loss of gas. Star formation is rapidly suppressed after the gas stripping and the transition region is reached in less than a billion years. However, once in the transition region, massive galaxies are still forming stars and will not immediately reach the red sequence. A subsequent quenching is thus required to reach the red sequence, implying that the total migration time-scale is, at least, a few billion years.\
iii) Although [H[i]{}]{}-deficient galaxies represents the bulk of the migrating population in our sample, they [**are apparently not responsible**]{} for the formation of the red sequence in the first place. The gas stripping is in fact ‘polluting’ the red sequence mainly with disk galaxies and no morphological transformation is observed during the quenching of the star formation.\
iv) Contrary to [H[i]{}]{}-deficient systems, [H[i]{}]{}-normal transition galaxies ($\sim$12% of the whole transition population in our sample) represent an heterogeneous population of objects, at least two of which are probably following the inverse path, migrating back from the red sequence. The high hydrogen content in these systems is, at least in some cases, due to external accretion/interaction events. The detailed evolution of these objects is still unclear, but it is likely that they will remain in the transition region for several billion years. Thus, a connection between transition region and migration from the blue to the red sequence may not always be true.\
The discovery of a [H[i]{}]{}-normal population in the transition region has only been possible thanks to the combination of UV imaging (necessary to properly separate blue and red sequence at all masses) and [H[i]{}]{} single-dish observations (to quantify the atomic hydrogen content). The results here presented have inevitably lead to several questions we have not been able to answer in the current work. Is accretion really playing a crucial role in the evolution of these objects? Which mechanism suppressed star formation without removing the gas? How frequent are such objects? etc. To be able to answer these questions, both detailed theoretical and observational studies of individual objects and statistical investigations of larger samples are mandatory. Luckily, we are currently in an exciting time for UV and 21 cm astronomy. Particularly promising for this topic is the [*GALEX Arecibo SDSS Survey*]{} (GASS, [@catinella08]) which is currently making a census of the [H[i]{}]{} content in a complete sample of $\sim$1000 massive galaxies ($M_{star}\geq$10$^{10}$ M$_{\odot}$), ideal to investigate the role of gas and accretion on the evolution of transition galaxies. Thus, we are probably not too far from being able to unravel some of the mysteries still surrounding [H[i]{}]{}-normal galaxies in the transition region.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are greatly indebted to Alessandro Boselli for his support and encouragement in carrying out this analysis and for providing part of the data before publication. We wish to thank Alessandro Boselli, Barbara Catinella, Jonathan Davies and Rory Smith for useful discussions and helpful comments on the manuscript. We thank the anonymous referee for useful comments which helped to improve this paper. LC thanks the hospitality of the Max Planck Institute for Astrophysics where part of this paper was written. LC and TMH are supported by the UK Science and Technology Facilities Council.
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation, from the GALEX mission, developed in cooperation with the Centre National d’Etudes Spatiales of France and the Korean Ministry of Science and Technology, from the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration and from the GOLDMine data base.
This research has made extensive use of NASA’s Astrophysics Data System and of the astro-ph preprint aechive at http://arXiv.org.
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Appendix A: Notes on individual objects {#appendix-a-notes-on-individual-objects .unnumbered}
=======================================
[H[i]{}]{}-normal transition galaxies {#hi-normal-transition-galaxies .unnumbered}
-------------------------------------
[**NGC3619**]{} is a S0 galaxy, member of a group composed of at least five systems [@garcia03]. The [H[i]{}]{} distribution is asymmetric and much concentrated in the center [@vandriel89], where star formation is taking place and dust lanes are clearly visible. The extent of the [H[i]{}]{} roughly matches the UV star-forming disk, i.e., $\sim$half the optical radius. In optical, the galaxy shows prominent outer stellar shells, suggesting a recent minor-merging/accretion event. Numerical simulations predict that the stars from a satellite make a system of shells several 10$^{8}$ yr after the end of the merging event which can last for more than 1 Gyr [@kojima97]. The accretion scenario is also supported by the spectroscopic analysis performed by [@howell06]. He finds that the age, metallicity, and $\alpha$-enhancement ratios of NGC3619 are consistent with those of a representative sample of unperturbed nearby elliptical galaxies, contrary to what expected in the case of a recent major merger and supporting a minor merger with a gas-rich dwarf satellite. Thus, all the observational evidence is consistent with an external origin for the atomic hydrogen and residual star formation observed in NGC3619.
[**NGC3898**]{} is a Sa galaxy in the Ursa major group. In the Hubble Atlas it is described as the prototype of Sa galaxy: i.e., with a prominent bulge component and multiple spiral arms. HST observations reveal that the bulge is a ‘classical bulge’ [@drory07], consistent with a major merger/accretion event in the past. UV observations show that active star formation is mainly segregated in a ring well within the optical radius, although star-forming regions are also found at the edge of the optical disk. The atomic hydrogen is distributed in a low column density ($\sim$1-2 M$_{\odot}$ pc$^{-2}$) disk extending up to $\sim$4 times the radius containing 80% of the B-band light [@noordermeer05]. The highest [H[i]{}]{} column density is observed in correspondence with the UV star-forming regions, suggesting that only in these few regions the [H[i]{}]{} can condensate into molecular hydrogen and initiate the star formation cycle [@krumholz09]. The [H[i]{}]{} disk appears to be warped in the inner parts and shows some ‘wiggles’ with an amplitude of 30-50 [km s$^{-1}$ ]{}[@noordermeer07]. Although H$\alpha$ long-slit spectroscopy [@pignatelli01; @vega01] and 21 cm [H[i]{}]{} line interferometry [@noordermeer05; @noordermeer07] suggest that both ionized and atomic hydrogen have a regular velocity field and a smooth distribution, [@noordermeer05] suggest that this galaxy may have experienced a recent interaction/accretion event. However, the basis for this interpretation is not completely clear.
[**NGC4324**]{} is a Sa galaxy belonging to the Virgo S cloud [@goldmine]. Star formation is segregated in a ring, also visible in SDSS images, distinct from the central bulge component. Although [H[i]{}]{} maps are not available, [@duprie96] detect [H[i]{}]{} up to $\sim$2 optical diameters suggesting that the atomic hydrogen is not only concentrated in the star-forming ring.
[**NGC4370**]{} is an early type galaxy member of the Virgo B cloud [@goldmine] forming a non-interacting pair with NGC4365. Its detailed morphological classification is rather uncertain, varying from Sa [@vcc] to elliptical [@bertola88]. This is mainly due to the presence of a prominent dust lane along the equatorial plane of the galaxy. Although active star-forming regions are not clearly evident from UV and H$\alpha$ images, we cannot exclude that we are observing an analogous of the Sombrero galaxy (NGC4594) completely edge-on. Excluding the prominent dust lane, no other evidence for interaction/accretion is found and the gas and stars in the galaxy are co-rotating [@bertola88]. Surprisingly, [@bertola88] interpret the co-rotation in NGC4370 as a sign of possible accretion, suggesting that the gas present in this galaxy has an external origin. Although the basis for this interpretation may be debatable, the same conclusion has been recently reached by [@patil09], who investigated the dust content in NGC4370. They propose that the amount of dust observed in the galaxy can not have been formed in situ and that at least part of the material must have been accreted during an interaction event. An external origin for the atomic hydrogen content in this galaxy is thus not excluded, as observed in other [H[i]{}]{}-rich dust-lane elliptical galaxies [@oosterloo02].
[**NGC4378**]{} is a Sa galaxy belonging to the Virgo S cloud. It has a prominent nuclear bulge surrounded by a low surface brightness disk which contains a single, tightly-wound, spiral arm. Given the low surface brightness of the disk component, [@vandberg76] classified this object as an early-type anemic spiral, although we now know that its atomic hydrogen content is not significantly different from the one in high surface brightness disks. UV star formation is clearly visible across all the disk. Numerical simulations [@byrd94] suggest that the unusual one-armed spiral structure in NGC4378 is in reality an impulsive trailing arm created by the recent passage ($\sim$200 Myr) of a small companion ($\sim$1/30 the galaxy’s mass). Since the satellite is supposed to survive the interaction, it seems unlikely that a significant fraction of the [H[i]{}]{} present in NGC4378 has been recently accreted. The low SSFR of NGC4378 may therefore have an internal origin. By studying a sample of nearby galaxies, [@seigar05] recently found a correlation between the average shear rate in disks and their SSFR, suggesting the existence of a shear-threshold above which star formation is inhibited. Interestingly, NGC4378 has a shear rate (0.69$\pm$0.03) just below the threshold value (0.70$\pm$0.09) suggested by [@seigar05]. If this is the case, the internal dynamical properties of NGC4378 may be responsible for the low star formation efficiency observed in this object.
[**NGC4772**]{} is a Sa galaxy in the Virgo S cloud. Recently, [@haynes00] carried out a detailed multiwavelength analysis of this object, showing that it is probably the result of a minor merger event. The main observational evidence supporting such scenario is: 1) the presence of dynamically decoupled central gas and stellar components, in particular in correspondence of the star forming ring visible in UV (Fig. \[FCtrans\]) and H$\alpha$, 2) the unusual [H[i]{}]{} distribution, which is segregated in two concentric rings, the inner one where star formation is taking place and 3) the presence of a warp in the outer [H[i]{}]{} ring. The most likely scenario is that NGC4772 is at the end stage of a prograde merger in which the transfer of angular momentum leads to an outward spread of the disk [@quinn93; @haynes00].
[**NGC5701**]{} is the bluest object in the transition region. We classified this galaxy as an Sa (NED gives S0/a). However its peculiar morphology, characterized by a central bulge surrounded by a faint and apparently detached star-forming ring, makes NGC5701 a possible Hoag-type object [@hoag50]. In this case the central condensation is a bar, not a ‘bulge or elliptical’ [@hoag50] and [@gadotti03] dubbed this object ‘a barred galaxy without a disk’. The origin of Hoag-type objects is still a puzzle, but two different scenarii seem to be the most commonly accepted: a) accretion of small satellites [@schweizer87], or b) a strong bar instability [@brosh85], which destroys the disk and builds up a ring (e.g. [@schwarz84; @byrd94b]). The lack of detailed dynamical and spectroscopic data prevent us from driving a conclusion on the origin of the ring in NGC5701, but it is quite likely that the bar is playing an important role either destroying the disk or preventing the accreted gas from infalling into the center.
[H[i]{}]{}-normal red-sequence galaxies {#hi-normal-red-sequence-galaxies .unnumbered}
---------------------------------------
[**NGC4203**]{} is an, apparently isolated, barred S0 galaxy. UV images show low level star-formation at least in the south-east part of the galaxy, corresponding to the outer edge of the warm-dust spiral structure detected at 8$\mu$m in the galaxy center [@pahre04]. Despite its low star formation activity, NGC4203 has a large reservoir of atomic hydrogen, distributed in a peculiar, filamentary structure and extended up to $\sim$2.2 optical radii [@vandriel88; @noordermeer05]. The atomic gas has a general sense of motion around the center, but it is clearly not on regular circular orbits [@noordermeer05]. Again, the [H[i]{}]{} is mainly found in two ring structures: an inner ring (in correspondence with the UV features) and an outer ring which is likely rotating on a different plane [@vandriel88]. The peculiar dynamics of NGC4203 is thus suggestive of an accretion event [@vandriel88]. The [H[i]{}]{} disk of NGC4203 is also one of the nearest damped Lyman alpha systems. [@miller99] found that the metallicity of the gas is significantly lower than the one observed in galaxies of the same luminosity. If confirmed, this would represent a strong observational evidence supporting the external origin for the [H[i]{}]{}.
[**NGC4262**]{} is a barred S0 galaxy belonging to the Virgo main cloud A. No star formation is observed within the optical radius. However, UV images reveal the presence of a UV star-forming ring extended up to $\sim$2.5 optical radii. Although faint UV emission is marginally visible all across the ring, great part of the star formation is taking place in the south-west. This roughly coincides with the peak in [H[i]{}]{} distribution as revealed in the maps presented by [@krumm85] and [@minchin07]. Moreover, all the [H[i]{}]{} is segregated into a ring and no low surface density [H[i]{}]{} disk is detected. [@krumm85] and [@vandriel91] suggested that the ring has probably an external origin and more recently [@vollmer05b] tentatively proposed that the [H[i]{}]{} could have been accreted during an interaction between NGC4262 and NGC4254. Being a barred system plus a ring, NGC4262 shows some similarities with the Hoag-type object NGC5701. If this is the case, NGC4262 is probably at an earlier evolutionary stage, since its ring is only visible in UV whereas the ring in NGC5701 is evident also in optical, suggesting an older age for the stellar populations. We note that, if we integrate the UV light up to the extent of the ring, NGC4262 would move from the red sequence to the transition region.
[**NGC4698**]{} is a bulge-dominated Sa galaxy in the Virgo E cloud. The Carnegie Atlas [@sandage94] describes it as an elliptical-like bulge in which there is no evidence of recent star formation or spiral structure, plus a low-surface-brightness disk with spiral arms become prominent only in the outer part of the disk. HST observations reveal that the bulge is a ‘classical bulge’ [@drory07], consistent with a major merger/accretion event in the past. Once again, UV images reveal that star formation is segregated into two ‘almost perfect’ rings: one within and the other outside the optical radius. Interestingly, the two rings are not concentric: the inner ring is centered on the nucleus, whereas the outer one is offset $\sim$ 20 arcsec to the north. The peculiarity of NGC4698 becomes more evident when the dynamical properties of the bulge and disk component are compared. [@bertola99] showed that the rotation axis of the disk and of the bulge are almost perpendicular. They interpret this unusual geometry as an evidence of a later formation of the disk through the acquisition of material by a completely formed spheroid. Also in this case, the galaxy moves back to the transition region if the outer ring is included in the estimate of the UV flux.
[**NGC4866**]{} is an isolated, highly inclined, S0 galaxy (although [@sandage94] classify it as Sa). Very few data are available in the literature, and the only unusual feature is the presence, once again, of a UV star-forming ring at the edge of the optical disk.
[**NGC5103**]{} is an isolated Sab galaxy. As for NGC3414, no recent star formation activity is clearly visible in the UV image and the galaxy resembles an edge-on polar ring/disk system: [@whitmore90] classifies NGC5103 as possible candidate for polar-ring galaxy. No high-resolution [H[i]{}]{} maps are available for this object.
Finally, a visual inspection of galaxies in the red sequence reveals the presence of two other objects with UV rings: NGC3945 and NGC4643. Unfortunately, no [H[i]{}]{} observations are available and they are thus excluded from the discussion in the text.
[^1]: luca.cortese@astro.cf.ac.uk
[^2]: The fact that, for the same colour, [H[i]{}]{}-deficient objects have a lower gas fraction is expected, since both quantities trace the specific amount of atomic hydrogen in a galaxy.
[^3]: This fraction might increase up to $\sim$88% in case all galaxies without [H[i]{}]{} measurement are [H[i]{}]{}-deficient galaxies. We note, however, that these values may not be representative of the local universe, being our sample likely biased towards high-density environments (see § 6).
[^4]: This value decreases by a factor $\sim$1.5 if a Salpeter IMF is adopted
[^5]: This fraction might increase up to $\sim$30% in case all galaxies without [H[i]{}]{} measurement are [H[i]{}]{}-normal galaxies.
|
---
abstract: 'Deep Neural Network (DNN) is a widely used deep learning technique. How to ensure the safety of DNN-based system is a critical problem for the research and application of DNN. Robustness is an important safety property of DNN. However, existing work of verifying DNN’s robustness is time-consuming and hard to scale to large-scale DNNs. In this paper, we propose a boosting method for DNN robustness verification, aiming to find counter-examples earlier. Our observation is DNN’s different inputs have different possibilities of existing counter-examples around them, and the input with a small difference between the largest output value and the second largest output value tends to be the achilles’s heel of the DNN. We have implemented our method and applied it on Reluplex, a state-of-the-art DNN verification tool, and four DNN attacking methods. The results of the extensive experiments on two benchmarks indicate the effectiveness of our boosting method.'
author:
-
bibliography:
- 'nuthin.bib'
title: 'Boosting the Robustness Verification of DNN by Identifying the Achilles’s Heel '
---
DNN, Robustness, Verification, Adversarial Example, Boosting
|
---
author:
- |
P. Boily, `pboily@uottawa.ca`\
Department of Mathematics and Statistics, University of Ottawa\
Ottawa K1N 6N5, Canada
bibliography:
- 'spiralsp.bib'
title: Epicyclic drifting in anisotropic excitable media with multiple inhomogeneities
---
**Keywords**: symmetry-breaking, integral manifold, epicycle, spiral wave, excitable media, averaging, center bundle equation, center manifold reduction theorem.
Introduction
============
Spiral are found in numerous excitable media [@LOPS; @GZM; @ZM; @MPMPV; @WR; @YP; @Detal; @R1; @Wetal; @J; @MAK; @B1; @B2; @BKT] and they give rise to beautiful imagery. While this in itself might yield enough interest to study them, there is also (at least) one serious reason to do so: spiral waves have been linked to cardiac arrhythmias (to disruptions of the heart’s normal electrical cycle) [@W; @Wetal; @Detal; @KS]. Furthermore,
> most arrhythmias are harmless but if they are “re-entrant in nature and \[...\] occur because of the spatial distribution of cardiac tissue” they can seriously hamper the pumping mechanism of the heart and lead to death [@KS p. 401].
As a result, a fuller understanding of spiral wave dynamics in these media becomes imperative.
### The equivariant dynamical system approach {#the-equivariant-dynamical-system-approach .unnumbered}
In recent years, one of the most rewarding approach to the study of spiral waves is based on Barkley’s initial observation that the observed transition from rotating to modulated rotating wave can be explained *via* a Hopf bifurcation together with the underlying Euclidean symmetries of the governing reaction-diffusion equations [@B1; @B2] (*i.e.*: the semi-flow generated by the dynamical system commutes with the $$u(t,x)\longmapsto
u(t,x_1\cos\,\theta-x_2\sin\,\theta+p_1,x_1\sin\,\theta+
x_2\cos\,\theta+p_2), \label{uaction}$$ where $(\theta,p_1,p_2)\in\mathbb{S}^1\times\mathbb{R}^2\simeq
{\mathbb{SE}}(2)$ and $x\in {\mathbb{R}}^2$ [@Wulff; @DMcK]). This lead Barkley to formulate a simpler *ad hoc* 5-dimensional ODE system with Euclidean symmetry replicating the above transition [@BK].
Sandstede, Scheel and Wulff then proved a general center manifold reduction theorem (CMRT) for relative equilibria and relative periodic solutions in spatially extended infinite-dimensional Euclidean-equivariant dynamical systems, providing a mathematical justification of Barkley’s insight [@SSW1; @SSW2; @SSW3; @SSW4; @FSSW]. [However, this center manifold reduction theorem requires that the spiral wave satisfy certain spectral gap conditions, which often fail [@S]. Sandstede and Scheel have developed a comprehensive theory of spiral instabilities using techniques of spatial dynamics [@abscon2; @abscon1] to deal with such a situation.]{}
Other methods are also used to reduce the dynamics to finite-dimensional systems (such as the kinematic model using the curvature of the wave as a driving mechanism [@MDZ]), but the equivariant dynamical system approach has the advantage that it can often provide universal, model-independent explanations and predictions regarding the dynamics and bifurcations of spiral waves. For example, the fore-mentioned ‘Hopf bifurcation’ from rigid rotation to quasi-periodic meandering has been observed in numerically [@BKT] and experimentally [@LOPS]. Another example is provided by the anchoring/repelling of spiral waves on/from a site of inhomogeneity, which has been observed in numerical integrations of an Oregonator system [@MPMPV], in photo-sensitive chemical reactions [@ZM] and in cardiac tissue [@Detal]: using a model-independent approach based on forced symmetry-breaking, LeBlanc and Wulff showed that anchoring/repelling of rotating waves is a generic property of systems in which the translation symmetry is broken by a small perturbation [@LW].[^1] In the same vein, certain dynamics of spiral waves in anisotropic media, such as phase-locking and linear drifting of meandering spiral, have been shown to be generic consequences of rotational symmetry-breaking [@R1; @R2; @LeB].
### The basic viewpoint {#the-basic-viewpoint .unnumbered}
Consider a piece of cardiac tissue on which numerous (roughly) circular ablation have been performed, perhaps in order to treat a patient who is suffering from atrial fibrillation [@Grubb; @MDC]. These surgical procedures affects both the geometry and the excitability of the tissue. Under certain modeling assumptions, any system used to model the electrical activity of the tissue needs to incorporate translational symmetry-breaking (TSB) components to model the effects of the circular ablations, and a rotational symmetry-breaking (RSB) component to model the effects of anisotropy. Let us model the electrical properties of such a perturbed piece of anisotropic cardiac tissue using a modified version of the bidomain equations of cardiology, under the modeling assumption that the circular ablation (inhomogeneous) zones consist of a finite number of independent “sources” which are localized near distinct sites $\zeta_1,\ldots,
\zeta_n$ in the plane (see [@BLM] for a similar hypothesis). The model then has the form $$\begin{aligned}
\label{thebidomain2}
\begin{split} u_t&=\frac{1}{\varsigma}(u-\frac{u^3}{3}-v)+\nabla^2
u+\frac{\alpha \varepsilon}{1+\alpha (1-\varepsilon)}\Psi_{x_1x_1}+
\sum_{j=1}^n\mu_j g_j^{u}(\|x-\zeta_j\|^2,\mu)\\
v_t&=\varsigma(u+\delta-\gamma v)+\sum_{j=1}^n\mu_j g_j^{v}(\|x-\zeta_j\|^2,\mu), \\
\nabla^2& \Psi+\varepsilon g(\alpha,\varepsilon)\Psi_{x_2x_2}=
\varepsilon h(\alpha,\varepsilon) u_{x_2x_2},
\end{split}\end{aligned}$$ where $u$ is a transmembrane potential, $v$ controls the recovery of the action potential, $\Psi$ is an auxiliary potential (without obvious physical interpretation), $x_1$ is the preferred direction in physical space in which tissue fibers align, $\varepsilon$ is a measure of that preference, $g$ and $h$ are appropriate model functions, $\alpha,\varsigma,\delta$ and $\gamma$ are model parameters, $\mu=(\mu_1,\ldots,\mu_n)\in {\mathbb{R}}^n$ is a small parameter and $g_j^{u,v}$ are smooth functions, uniformly bounded in their variables [@R2; @LR; @BEL; @Byeah].
If the tissue has equal anisotropy ratios (*i.e.* $\varepsilon=0$) and the inhomogeneities have no effect on spiral wave dynamics (*i.e.* $\mu=0$), (\[thebidomain2\]) decouples into the FitzHugh-Nagumo equations for $u$ and $v$, and Poisson’s equation for $\Psi$ [@R2].Let $\mathbb{S}\mathbb{E}(2)$ denote the group of all planar translations and rotations, and fix an integer $1\leq \jmath^*$ and $\zeta\in {\mathbb{R}}^2$. The subgroups $\mathbb{Z}_{\jmath^*}{\dot{+}}{\mathbb{R}}^2$ (the notation will be explained later) and ${\mathbb{SO}}(2)_{\zeta}$ of ${\mathbb{SE}}(2)$ consist of all cartesian pairs of translations and rotations about the origin by an integer multiple of $2\pi/\jmath^*$ radians, and of all rotations about the point $\zeta$, respectively. Let $\Gamma={\mathbb{C}}{\dot{+}}\mathbb{Z}_{\jmath^*}$ or $\Gamma={\mathbb{SO}}(2)_{\zeta}$. Then, $\Gamma<{\mathbb{SE}}(2)$ and we will say that the semi-flow $\Psi_{t,\varepsilon,\mu}$ is $\Gamma-$equivariant if it commutes with the restriction of (\[uaction\]) to $\Gamma$.
In the equivariant dynamical system approach, the particular form of the functions $g_j^{u,v}$ is unimportant; the analysis is driven by the fact that (\[thebidomain2\]) can sustain spiral wave propagation [@R2; @LR; @BEL; @Byeah] and by the equivariance properties of the semi-flow $\Phi_{t,\varepsilon,\mu}$ generated by (\[thebidomain2\]), namely: if we neglect boundary effects, the semi-flow
(E1)
: is ${\mathbb{SE}}(2)-$equivariant when $(\varepsilon,\mu)=0$;
(E2)
: is ${\mathbb{Z}}_{2}{\dot{+}}{\mathbb{R}}^2-$equivariant when $\varepsilon\neq 0$ is small and $\mu=0$;
(E3)
: preserves rotations around $\zeta_{i}$ (but generically not translations) when $\varepsilon=0$ and $\mu_{j}=0$ for all $j\neq i$, and
(E4)
: is (generically) trivially equivariant when $(\varepsilon,\mu)$ is a generic small parameter vector.
This is but a special case of a more general family of semi-flows for which **(E2)** is replaced by the following property: the semi-flow
(E2’)
: is ${\mathbb{Z}}_{\jmath^*}{\dot{+}}{\mathbb{R}}^2-$equivariant when $\varepsilon\neq 0$ is small and $\mu=0$ for some integer $\jmath^*\geq 1$.
In [@BLM; @Byeah; @Bo1], we used the dynamical system approach to study spiral anchoring in media satisfying **(E1)**, **(E2’)**, **(E3)** and **(E4)**: the predictive power of the method was used to show that in the case $n>1$, spiral anchoring typically takes place *away* from the inhomogeneities. At the time, such a statement defied experimental wisdom.
### Epicyclic drifting {#epicyclic-drifting .unnumbered}
At this stage, nothing has been said about the nominal topic of this paper: epicyclic drifting. The various spiral motions observed in experiments and simulations have been classified according to their tip path, an arbitrary point on the wave front that is followed in time [@Detal; @LOPS]: for instance, the tip path of a (rigidly) rotating wave is a perfect circle. Barkley [@BKT] and Wulff [@Wulff] have shown that the appearance of an epicyclic tip path can be linked to a ’symmetric Hopf bifurcation:’ when that happens, every spiral wave in the excitable medium is epicyclic.
However, other epicyclic behaviour cannot be explained by this mechanism. When the sizes of the physical domain and of the spiral core are comparable, the latter is sometimes attracted to the boundary of the domain and rotates around it in a meandering fashion. This has been observed in experiments and numerical simulations in a light-sensitive BZ reaction [@YP; @ZM].
Yet another instance of epicyclic motion is shown
![Epicyclic motion on the stable epicyclic manifold $\mathcal{E}$. The arrows indicate the direction of the flow, while $\mathcal{S}$ corresponds to a repelling (perturbed) rotating wave solution pinned at the inhomogeneity indicated by the black dot (see [@BLM] for details on spiral anchoring). \[epmannn\]](figure1){width="180pt"}
------------------------------------------------------------------------
figure : in a bounded region, all solutions are attracted/repelled to/by an epicyclic solution manifold. This type of spiral wave motion is what we refer to as *epicyclic drifting*. À la Poincaré-Bendixson, if a system has a stable epicyclic manifolds (stable in the sense of Lyapunov) it will also have a repelling rotating wave (see figure ), and *vice-versa*. As such, these manifolds cannot be observed in fully Euclidean media. What then, can forced Euclidean symmetry-breaking (FESB) tell us about epicyclic drifting in systems with the equivariance properties of **(E1)**, **(E2’)**, **(E3)** and **(E4)**? The only work in this vein has been performed by LeBlanc and Wulff in [@LW], in the case $n=1$ and without rotational symmetry-breaking: unfortunately, the main tool in their analysis cannot be used in the general case.
### Article Overview {#article-overview .unnumbered}
The main object of analysis in the present paper is a finite-dimensional system of ODE that share the equivariance properties of **(E1)**, **(E2’)**, **(E3)** and **(E4)** when $n>1$: it is derived in section \[cbe\]. Then, in section \[agat\], we present a preliminary result about averaging which will subsequently be used to prove our main results: to wit, when $\varepsilon=0$ or $\jmath^*=1$ and certain conditions are satisfied, there is a (minimal) parameter wedge region in which an epicyclic manifold persists. In the case $\jmath^*>1$, the epicyclic manifold persists in a deleted neighbourhood of the origin. The parameter wedges are illustrated in
![On the left, the epicyclic parameter wedge regions for the case $n=2$ without anisotropy. On the right, the epicyclic deleted neighbourhood in parameter space for the case $n=1$ with anisotropy characterized by $\jmath^*>1$. If the semi-flow has an epicyclic solution manifold for a particular set of parameter values, then the semi-flow has an epicyclic solution manifold (of the same stability type) for all parameter values in the adjacent region. Note that these manifolds continuously deform along any path contained entirely in the parameter region. The local analysis does not provide a clear picture of the behaviour as a path leaves a parameter region. In [@Byeah], for instance, we give an example where the epicyclic solution manifold disappears as a result of a saddle-node bifurcation of rotating waves. The notation will be explained later in the paper. \[epmannnn\]](figure4){width="360pt"}
------------------------------------------------------------------------
figure , for the case $n=2$. The conditions needed depend on the kind of forced symmetry-breaking under consideration: in section 3, we study the general semi-flow under $n$ TSB terms (*i.e.* $n>1$, $\varepsilon=0$); in section 4, we study the combination of a single RSB and a single TSB term (*i.e.* $n=1$, $\varepsilon\neq 0$). We then combine these results in section 5 to obtain the epicyclic drifting theorems under general FESB. Next, we perform a simple numerical experiment on \[thebidomain2\]) with $n=1$ showing the predicted epicyclic motion: to the best of our knowledge, the figure in section 6 is the first observed instance of epicyclic drifting in a numerical simulation of excitable media. Finally, we give the proofs of two technical results in appendix A.
Preliminaries
=============
We start with a derivation of the appropriate center bundle equations describing the essential dynamics of spiral waves near a rotating wave under full Euclidean symmetry-breaking (FESB). More details on these manipulations can be found in [@Wulff; @BLM; @SSW1; @SSW2; @Byeah]. Then, in order to lighten the text, we introduce some necessary definitions. Finally, we state an averaging theorem which will be used in later sections of this work.
The Center Bundle Equations {#cbe}
---------------------------
In order to facilitate the subsequent analysis, we make the same simplifying assumptions and adopt the same notation as in [@BLM; @Bo1].
In particular, let $X$ be a Banach space, ${\mathcal
U}\subset{\mathbb{R}}\times\mathbb{R}^n$ a neighborhood of the origin, and $\Phi_{t,\varepsilon,\mu}$ be a smoothly parameterized family (parameterized by $(\varepsilon,\mu)\in {\mathcal U}$) of smooth local semi-flows on $X$, and let $$a:\mathbb{S}\mathbb{E}(2)\longrightarrow \operatorname{GL}(X) \label{a_action}$$ be a faithful and isometric representation of $\mathbb{S}\mathbb{E}(2)$ in the space of bounded, invertible linear operators on $X$. For example, if $X$ is a space of functions with planar domain, a typical $\mathbb{S}\mathbb{E}(2)$ action (such as (\[uaction\]) in the preceding section) is given by $$(a(\gamma)u)(x)=u(\gamma^{-1}(x)),\,\,\,\,\gamma\in\mathbb{S}\mathbb{E}(2).$$ In this paper, we concern ourselves with the study of epicycle drifting in the case where the two following hypotheses are satisfied. The first one is simply a re-telling of the equivariance properties **(E1)**, **(E2’)**, **(E3)** and **(E4)**, while the second postulates the existence of a rotating wave in the unperturbed ${\mathbb{SE}}(2)-$equivariant semi-flow.
\[hyp1\] There exists $1\leq \jmath^*\in {\mathbb{N}}$, distinct points $\zeta_1,\ldots,\zeta_n$ in ${\mathbb{R}}^2$ such that if $e_j$ denotes the $j^{\mbox{\footnotesize th}}$ vector of the canonical basis in $\mathbb{R}^n$, then $\forall\,u\in\,X,\ \varepsilon\neq
0,\ \alpha\neq 0,\ t>0,$ $$\begin{aligned}
\Phi_{t,\varepsilon,0}(a(\gamma)u)&=a(\gamma)\Phi_{t,\varepsilon,0}(u)
\iff
\gamma\in\,\mathbb{Z}_{\jmath^*}{\dot{+}}{\mathbb{R}}^2,\\
\Phi_{t,0,\alpha e_j}(a(\gamma)u)&=a(\gamma)\Phi_{t,0,\alpha
e_j}(u) \iff \gamma\in\,\mathbb{S}\mathbb{O}(2)_{\zeta_j}, \quad\mbox{and}\\
\Phi_{t,0,0}(a(\gamma)u)&=a(\gamma)\Phi_{t,0,0}(u), \quad \forall\,
\gamma\in\,\mathbb{S}\mathbb{E}(2).\end{aligned}$$
\[hyp2\] There exists $u^*\in X$ (with trivial isotropy subgroup) and $\Omega^*$ in the Lie algebra of $\mathbb{S}\mathbb{E}(2)$ such that $e^{\Omega^*t}$ is a rotation and $\Phi_{t,0}(u^*)=a(e^{\Omega^*t})u^*$ for all $t$. Moreover, the set $\{\,\mu\in\mathbb{C}\,\,|\,\,|\mu|\geq 1\,\}$ is a spectral set for the linearization $a(e^{-\Omega^*})D\Phi_{1,0}(u^*)$ with projection $P_*$ such that the generalized eigenspace $\mbox{\rm range}(P_*)$ is three dimensional.
As discussed previously, such semi-flows can arise from the family of perturbed reaction-diffusion systems (1.4) from [@BLM] if $\varepsilon=0$, as well as from the modified bidomain model (\[thebidomain2\]) given in the introduction if $\jmath^*=2$, for instance.
It has been shown in [@BLM; @Bo1; @Byeah] that, for small parameter vectors $(\varepsilon,\mu)\in {\mathbb{R}}\times \mathbb{R}^n$, the essential dynamics of the semi-flow $\Phi_{t,\varepsilon,\mu}$ near a (hyperbolic) rotating wave is (locally) equivalent to the semi-flow of the following ordinary differential equations on the bundle $\mathbb{C}\times\mathbb{S}^1$: $$\begin{aligned}
\label{basic12}
\dot{p}&= e^{it}\left[v+\beta G(t,\beta)+\sum_{j=1}^n\lambda_j
H_j((p-\xi_j) e^{-it},{\overline}{(p-\xi_j)} e ^{it},\lambda)\right]\end{aligned}$$ where $v\in {\mathbb{C}}$, $(\beta,\lambda)\in{\mathbb{R}}\times {\mathbb{R}}^n$, $\xi_1,\ldots,\xi_n\in {\mathbb{C}}$ are all distinct and the functions $G,H_j$ are smooth, periodic in $t$ and uniformly bounded in $p$, and $G$ is $2\pi/\jmath^*-$periodic in $t$. The specific form of the perturbations is a consequence of forced Euclidean symmetry-breaking from the ${\mathbb{SE}}(2)-$equivariance of (\[basic12\]) under the following ${\mathbb{SE}}(2)-$action on the bundle $\mathbb{C}\times\mathbb{S}^1$: $$\begin{aligned}
(x,\theta)\cdot
(p,\varphi)
=(e^{i\theta}p+x,\varphi+\theta),\label{therealaction}\end{aligned}$$ for all $(p,\varphi)\in {\mathbb{C}}\times {\mathbb{S}}^1$ and $(x,\theta)\in
{\mathbb{SE}}(2)={\mathbb{C}}\times {\mathbb{S}}^1$, where ${\mathbb{SE}}(2)={\mathbb{C}}{\dot{+}}{\mathbb{S}}^1$ with multiplication $(p_1,\varphi_1) \cdot (p_2,\varphi_2) =
(e^{i\varphi_1}p_2 + p_1, \varphi_1 + \varphi_2).$ The non-standard multiplication is made explicit by using the semi-direct product notation ${\dot{+}}$.
Let $\jmath^*\geq 1$ be an integer and $\xi\in{\mathbb{C}}$. In ${\mathbb{SE}}(2)={\mathbb{C}}{\dot{+}}{\mathbb{S}}^1$, the subgroup of rotations around $\xi$ is given by $${\mathbb{S}}^1_{\xi}=\{(\xi,0)\cdot
(0,\theta)\cdot (-\xi,0):\theta\in {\mathbb{S}}^1\},$$ while the subgroup containing all translations and rotations by angle $\frac{2\pi k}{\jmath^*}$, $k\in {\mathbb{Z}}$, is $${\mathbb{C}}{\dot{+}}{\mathbb{Z}}_{\jmath^*}=\left\{\left(x,\frac{2\pi
k}{\jmath^*}\right):k\in {\mathbb{Z}}:x\in {\mathbb{C}}\right\}.$$
Then ${\mathbb{C}}{\dot{+}}{\mathbb{Z}}_{\jmath^*}\simeq {\mathbb{Z}}_{\jmath^*}{\dot{+}}{\mathbb{R}}^2$ and ${\mathbb{S}}^1_{\xi}\simeq {\mathbb{SO}}(2)_{\zeta}$: under the action described by (\[therealaction\]), the center bundle equation (\[basic12\]) is
(C1)
: ${\mathbb{C}}{\dot{+}}{\mathbb{S}}^1-$equivariant when $(\beta,\lambda)=0$;
(C2’)
: ${\mathbb{C}}{\dot{+}}{\mathbb{Z}}_{\jmath^*}-$equivariant when $\beta\neq 0$ is small and $\lambda=0$;
(C3)
: ${\mathbb{S}}^{1}_{\xi_{\ell}}-$equivariant when $\beta=0$ and $\lambda_{j}=0$ for all $j\neq \ell$, and
(C4)
: (generically) trivially equivariant when $(\beta,\lambda)$ is a generic small parameter vector.
Clearly, (\[basic12\]) shares the equivariance properties of Hypothesis 1. As such, $G$ ‘models’ the RSB perturbation while the various $H_j$ ‘model’ the various TSB terms.
It might seem strange that the parameters $(\varepsilon,\mu)$ are replaced by $(\beta,\lambda)$ in (\[basic12\]), just as the $\zeta_j\in {\mathbb{R}}^2$ are replaced by $\xi_j\in {\mathbb{C}}$, but since the center manifold reduction theorems of [@SSW1; @SSW2; @SSW3; @SSW4] do not provide an explicit relation between the coefficients of the original system of partial differential equations and the reduced ordinary differential system of center bundle equations, one cannot conclude that the parameters are the same in both systems.
Definitions
-----------
An integral manifold is *stable* if it has a neighbourhood in which all originating positive-time solutions approach the manifold exponentially; it is *hyperbolic* if the linearization of the flow on this manifold admits no critical eigenvalues. Let $\alpha_0>0$, $\Delta>0$, $V\subseteq {\mathbb{R}}^p$, $\Sigma={\mathbb{R}}\times V\times
[0,\alpha_0]$, $f:\Sigma\to {\mathbb{R}}^q$, $g:V\to {\mathbb{R}}^q$ and $h:{\mathbb{R}}\times
V\to {\mathbb{R}}^q$. We say that $f$ is *Lipschitz in Hale’s sense*\[LipH\], which we denote by $f\in
\operatorname{Lip}(x;\Sigma,\eta(\alpha,V))$, if $f$ is continuous in all of its arguments and is Lipschitz in $x$ for $(t,x,\alpha)\in \Sigma$ with continuous Lipschitz constant.
Next we say that $g$ is *bounded by $\Delta$ over $V$*, which we denote by $g\in
\mathcal{B}(\Delta;V)$, if $\|g(x)\|\leq \Delta$ for all $x\in V$. Finally, we denote the fact that $h$ is $T-$periodic in $\phi\in
{\mathbb{R}}$ by $h\in\mathfrak{P}^T_{\phi}$. When the sets $\Sigma$ and $V$ are understood from the context, they are omitted. Finally, by abuse of notation, we shall often denote $O\left(|x_1|+\cdots+|x_m|\right)$ by $O(x_1,\ldots,x_m)$.
A Generalized Averaging Theorem {#agat}
-------------------------------
Averaging methods are used to determine whether a particular system has an non-trivial invariant integral manifold by studying an averaged system. The main theorem is a modified version of one of Hale’s averaging theorems (see [@Byeah] for details); it can easily be extended to the case where $\nu$ is a parameter vector in ${\mathbb{R}}^n$.
\[Halesavg\] <span style="font-variant:small-caps;">(</span>$\mathrm{modified\ from\ }$<span style="font-variant:small-caps;">[@H1]</span>,$\mathrm{\
theorem \ 6.1,\ pp.\ 526-527}$<span style="font-variant:small-caps;">)</span> Let $\sigma_0>0$. Consider the system of equations $$\begin{aligned}
\label{Halesavgsys}
\begin{split}
\dot{x}&=\epsilon \gamma_{\epsilon,\nu} x+\epsilon \Lambda(t,\psi,x,\epsilon,\nu)\\
\dot{\psi}&=d(\epsilon,\nu)+\Theta(t,\psi,x,\epsilon,\nu),
\end{split}\end{aligned}$$ where $\psi\in {\mathbb{R}}$, $x\in [-\sigma_0,\infty)$, $\gamma_{\epsilon,\nu}\neq 0$ depends continuously on $(\epsilon,\nu)$, and $d$ is defined over $S_0=[-\epsilon_0,\epsilon_0]\times [-\nu_0, \nu_0]$, with $d(0,0)=1$. For $\sigma>0$, let $$\begin{aligned}
\Sigma_{\sigma}&={\mathbb{R}}\times {\mathbb{R}}\times [-\sigma,\sigma]\times S_0
\quad\mbox{and}\quad \Sigma_0={\mathbb{R}}\times{\mathbb{R}}\times \{0\}\times S_0.\end{aligned}$$ Suppose $\Theta,\Lambda\in
\mathfrak{P}^{\chi}_t\cap\mathfrak{P}^{\omega}_{\psi}$ and that
1. $\Theta$ and $\Lambda$ are real-valued over $\Sigma_{\sigma_0}$;
2. $\Theta,\Lambda\in \mathcal{B}(\Xi(\epsilon,\nu);\Sigma_0)$ where $\Xi(\epsilon,\nu)=O(\epsilon,\nu)$;
3. for all $0\leq \sigma\leq\sigma_0$, $\Theta\in
\operatorname{Lip}(\psi,x;\Sigma_{\sigma},\theta(\epsilon,\nu,\sigma))$ and $\Lambda\in\operatorname{Lip}(\psi,x;\Sigma_{\sigma},\eta(\epsilon,\nu,\sigma))$, with $\theta(\epsilon,\nu,\sigma)=O(\epsilon,\nu,\sigma)$ and $\eta(\epsilon,\nu,\sigma)=O(\epsilon,\nu,\sigma)$.
Then, there exists $(\epsilon_1,\nu_1)\in (0,\epsilon_0]\times
(0,\nu_0]$ such that for all $$(\epsilon,\nu)\in
S_1=[-\epsilon_1,\epsilon_1]\times [-\nu_1,\nu_1]$$ with $\epsilon\neq 0$, $(\ref{Halesavgsys})$ has a hyperbolic integral manifold $\mathcal{T}_{\epsilon,\nu}$ which can be represented as an invariant torus $x=\Upsilon_{\epsilon,\nu}(t,\psi)$, where $$\Upsilon_{\epsilon,\nu}\in \mathcal{B}(D(\epsilon,\nu))\cap
\operatorname{Lip}(\psi,\Omega(\epsilon,\nu))\cap \mathfrak{P}^{\chi}_t\cap
\mathfrak{P}^{\omega}_{\psi},$$ with $D(\epsilon,\nu),\Omega(\epsilon,\nu)\to 0$ uniformly as $(\epsilon,\nu)\to 0$. Furthermore, the stability of $\mathcal{T}_{\epsilon,\nu}$ is exactly determined by the sign of $\epsilon \gamma_{0,0}$.
Epicyclic Drifting For $n$ Simultaneous TSB Terms
=================================================
When $\beta=0$, (\[basic12\]) gives the dynamics near a hyperbolic rotating wave for a parameterized family of semi-flows $\Phi_{t,0,\lambda}$ satisfying the forced-symmetry breaking conditions in hypothesis \[hyp1\]. We start with a brief review of epicyclic drifting in the case $n=1$, which was studied in detail in [@LW], and then present our new results in the general case $n>1$.
The Case $n=1$
--------------
Without loss of generality, we may assume $\xi_1=0$. In this case, the center bundle equation (\[basic12\]) reduce to $$\begin{aligned}
\label{basiceqs4}
\dot{p}&={\displaystyle e^{it}\left[v+ \lambda_1
H_1(pe^{-it},\overline{p}e^{it}, \beta)\right]},\end{aligned}$$where $v\in {\mathbb{C}}^{\times}$ and $\lambda_1\in\mathbb{R}$ is small. Set $\widetilde{H}(w,\overline{w},\lambda_1)=
H_1(w-iv,\overline{w}+i\overline{v},\lambda_1)$.
\[nongenericthm\] <span style="font-variant:small-caps;">([@LW],</span>$\mathrm{\ re-written\ to\ fit\ the\ current\ symbolism}$<span style="font-variant:small-caps;">)</span> Let $$I(\rho)=\operatorname{Re}\left[\int_0^{2\pi}\!\!e^{-it}\widetilde{H}\left(\rho
e^{-it},\rho e^{it},0\right)
dt\right].$$ If $\rho_0>0$ is a hyperbolic solution of $I(\rho)=0$, then for all $\lambda_1\neq 0$ small enough, the center bundle equation $(\ref{basiceqs4})$ has an integral (solution) manifold $\mathcal{E}^{1}_{\lambda_1}$ around the origin, whose stability is exactly determined by the sign of $\lambda_1 I'(\rho_0)$.
These solutions represent quasi-periodic motion around the origin in the $p-$plane and are observable as epicycle-like motion along a circular boundary in the physical space, with angular frequency $1+O(\lambda_1)$. Note that the hypotheses of theorem \[nongenericthm\] are not generic: in a random system, $I(\rho)$ may very well not have a positive hyperbolic root.
The presence of a repelling integral manifold could explain the fact that spirals are sometimes observed to be repulsed by an inhomogeneity if the spiral tip is located beyond a certain distance from the perturbation center [@MPMPV].
The Case $n>1$ {#bd}
--------------
However, the main averaging tool used in [@LW] to obtained theorem (\[nongenericthm\]) cannot be used to analyze the situation in the case $n>1$; furthermore, this difficulty yields a interesting twist, as we shall see in this section.By re-labeling the indices in (\[basic12\]) if necessary, we can temporarily shift our point of view so that $\xi_1$ plays the central role in the following analysis. Set $\Xi_j=\xi_j-\xi_1$ for $j=1,\ldots, n$. Then, under the co-rotating frame of reference $z=p-\xi_1+i e^{i t}v$, (\[basic12\]) becomes $$\begin{aligned}
\label{zdotforced2rescaled}
\dot{z}= e^{i t}\sum_{j=1}^n \lambda_j H_j \big((z-\Xi_j) e^{-i
t}\!\!-i v,{\overline}{(z-\Xi_j)} e^{i t}\!\!+i{\overline}{v},\lambda\big).\end{aligned}$$ When $\lambda_1\neq 0$ and $\lambda_2=\cdots=\lambda_n=0$, we find ourselves in the situation described in the previous subsection. Now, set $\epsilon=\lambda_1$, $\nu_1=1$ and $\lambda_j=\nu_j\epsilon$ for $j=2,\ldots, n$, and $\nu=(\nu_2,\ldots,\nu_n)\in {\mathbb{R}}^{n-1}$. Then (\[zdotforced2rescaled\]) can be viewed as a perturbation of the corresponding equation in the case $n=1$. Note that $\Xi_1=0$ and $\lambda=(1,\nu)\epsilon$. Equation (\[zdotforced2rescaled\]) rewrites as $$\label{system2}
\dot{z}=\epsilon e^{i t}\sum_{j=1}^n \nu_j H_j \big((z-\Xi_j) e^{-i
t}\!\!-i v,{\overline}{(z-\Xi_j)} e^{i t}\!\!+i{\overline}{v},(1,\nu)\epsilon\big).$$ Let $\label{funcH} \hat{H}_j(w,{\overline}{w},\epsilon,\nu)=H_j\big(w-i
v,{\overline}{w}+i{\overline}{v},(1,\nu)\epsilon\big)$ for $j=1,\ldots n$. Then (\[system2\]) becomes $$\begin{aligned}
\label{system3}
\dot{z}=\epsilon e^{i t}K(z e^{-i t},{\overline}{z} e^{i t},t,\epsilon,\nu)\end{aligned}$$ where $\displaystyle{K(w,{\overline}{w},t,\epsilon,\nu)=\sum_{j=1}^n \nu_j
\hat{H}_{j}(w-\Xi_j e^{-i t},{\overline}{w}-{\overline}{\Xi}_j e^{i
t},\epsilon,\nu)}$ is $2\pi-$periodic in $t$. Consider the near-identity change of variables $$\begin{aligned}
\label{fcov} z&=w+\epsilon \kappa
(w,{\overline}{w},t,\epsilon,\nu)\end{aligned}$$ where $\kappa\in \mathfrak{P}^{2\pi}_t$ is differentiable in all of its variables. Then $$\begin{aligned}
\dot{z}&=\dot{w}+\epsilon \left(\frac{\partial \kappa}{\partial t}+\frac{\partial \kappa}{\partial w }\dot{w}+\frac{\partial \kappa}{\partial {\overline}{w}}\dot{{\overline}{w}}\right). $$ Introducing the equivalent complex conjugate equation, this last system becomes $$\label{laprem}
\left[I_2\index{I2@$I_2$}+\epsilon\begin{pmatrix} \kappa_{w} &
\kappa_{{\overline}{w}} \\ {\overline}{\kappa}_w & {\overline}{\kappa}_{{\overline}{w}}
\end{pmatrix} \right]\begin{pmatrix} \dot{w} \\
\dot{{\overline}{w}}\end{pmatrix} =\begin{pmatrix} \dot{z} \\ \dot{{\overline}{z}}
\end{pmatrix}-\epsilon\begin{pmatrix} \kappa_t \\ {\overline}{\kappa}_t
\end{pmatrix},$$ where $\kappa_{w}$, $\kappa_{{\overline}{w}}$, $\kappa_t$, ${\overline}{\kappa}_{w}$, ${\overline}{\kappa}_{{\overline}{w}}$, ${\overline}{\kappa}_t$ are used to denote the partial derivatives of $\kappa$ and ${\overline}{\kappa}$. Set $$\mathcal{I}=I_2+\epsilon\begin{pmatrix} \kappa_{w} &
\kappa_{{\overline}{w}} \\ {\overline}{\kappa}_w & {\overline}{\kappa}_{{\overline}{w}}
\end{pmatrix}.$$ Combining (\[laprem\]) with (\[system3\]) yields $$\label{theeq}
\begin{pmatrix} \dot{w} \\ \dot{{\overline}{w}}\end{pmatrix}=
\epsilon\mathcal{I}^{-1}
\begin{pmatrix} e^{it}K\left((w+\epsilon\kappa)e^{-it},({\overline}{w}+\epsilon{\overline}{\kappa})e^{it},t,\epsilon,\nu\right)- \kappa_t \\ e^{-it}{\overline}{K}\left((w+\epsilon\kappa)e^{-it},({\overline}{w}+\epsilon{\overline}{\kappa})e^{it},t,\epsilon,\nu\right)- {\overline}{\kappa}_t \end{pmatrix}$$ By Taylor’s theorem, there are appropriate continuous bounded functions $A_1,$ $A_2$ and $A_3\in \mathfrak{P}^{2\pi}_t$ satisfying $$\begin{aligned}
e^{it}K\left((w+\epsilon\kappa)e^{-it},({\overline}{w}+\epsilon{\overline}{\kappa})e^{it},t,\epsilon,\nu\right)&=e^{it}K\left(we^{-it},{\overline}{w}e^{it},t,0,\nu\right)
+\epsilon
A_1(w,{\overline}{w},t,\epsilon,\nu) \\
\kappa_t(w,{\overline}{w},t,\epsilon,\nu)&=\kappa_t(w,{\overline}{w},t,0,\nu)+\epsilon
A_2(w,{\overline}{w},t,\epsilon,\nu)\end{aligned}$$ and $$\mathcal{I}^{-1}=
\begin{pmatrix}
1-\epsilon \kappa_{w}^0 & -\epsilon \kappa_{{\overline}{w}}^0 \\
-\epsilon {\overline}{\kappa}_{w}^0 & 1-\epsilon {\overline}{\kappa}_{{\overline}{w}}^0 \\
\end{pmatrix} +\epsilon^2A_3(w,{\overline}{w},t,\epsilon,\nu),$$ where $$\kappa_{w}^0=\kappa_w(w,{\overline}{w},t,0,\nu)\quad \mbox{and} \quad
\kappa_{{\overline}{w}}^0=\kappa_{{\overline}{w}}(w,{\overline}{w},t,0,\nu).$$With these, (\[theeq\]) re-writes (upon dropping the equivalent complex conjugate equation) as $$\begin{aligned}
\label{listen} \dot{w}&=\epsilon
\left(e^{it}K(we^{-it},{\overline}{w}e^{it},t,0,\nu)-\kappa_t(w,{\overline}{w},t,0,\nu)\right)+\epsilon^2
\mathcal{H}(w,{\overline}{w},t,\epsilon,\nu),\end{aligned}$$ where $\mathcal{H}\in \mathfrak{P}^{2\pi}_t$ is bounded and continuous in all its variables. Denote the average value of \[pappequiv\]$e^{it}K(we^{-it},{\overline}{w}e^{it},t,0,\nu)$ by $$\label{avgM}M^1(w,{\overline}{w},\nu)=\frac{1}{2\pi}\int_{0}^{2\pi} \!\!\!\!e^{it}K(we^{-it},{\overline}{w}e^{it},t,0,\nu)\, dt.\index{average value!Ma1@$M^1$}\index{Maaaa1@$M^1$}$$ Then $$e^{it}K(we^{-it},{\overline}{w}e^{it},t,0,\nu)= M^1(w,{\overline}{w},\nu)+F(w,{\overline}{w},t,\nu),$$ where $F\in \mathfrak{P}^{2\pi}_t$ is uniformly continuous and $$\label{noconstant}\int_{0}^{2\pi}\!\!\!\!F(w,{\overline}{w},t,\nu)\, dt=0.$$ Let $\kappa$ be an antiderivative of $F$ with respect to $t$. Then $\kappa\in \mathfrak{P}^{2\pi}_t$ by (\[noconstant\]) and $$F(w,{\overline}{w},t,\nu)-\kappa_t(w,{\overline}{w},t,0,\nu)=0.$$ With such a $\kappa$, (\[listen\]) simplifies to $$\begin{aligned}
\label{avg} \dot{w}&=\epsilon M^1(w,\overline{w},\nu)+\epsilon^2
\mathcal{H}(w,{\overline}{w},t,\epsilon,\nu).\end{aligned}$$ It is easy to see that $M^1(w,\overline{w},0)$ is ${\mathbb{S}}^1-$equivariant (see appendix for details); as such, there is a continuous function $L_1:{\mathbb{R}}\to {\mathbb{C}}$ such that $M^1(w,{\overline}{w},0)=wL_1(w{\overline}{w})$ [@GSS p. 360].
By Taylor’s theorem, there are appropriate continuous bounded functions $M_j$, for $j=2,\ldots, n$, such that $$M^1(w,{\overline}{w},\nu)=M^1(w,{\overline}{w},0)+\sum_{j=2}^n\nu_j
M_{j}(w,{\overline}{w},\nu)$$ and so (\[avg\]) becomes $$\label{avg2}
\dot{w}=\epsilon wL_1(w{\overline}{w})+\epsilon W(w,{\overline}{w},t,\epsilon,\nu),$$ where $$\label{funcW}W(w,{\overline}{w},t,\epsilon,\nu)=\sum_{j=2}^n\nu_j
M_j(w,{\overline}{w},\nu)+\epsilon
\mathcal{H}(w,{\overline}{w},t,\epsilon,\nu).$$ Differentiating the polar coordinates $w=\rho e^{-i(\psi-t)}$ yields $$\begin{aligned}
\dot{\rho}&= \operatorname{Re}\left[\dot{w}e^{i(\psi-t)} \right]\\
\dot{\psi}&= 1-\frac{1}{\rho}\operatorname{Im}\left[\dot{w}e^{i(\psi-t)}\right].\end{aligned}$$ But $$\begin{aligned}
\dot{w}e^{i(\psi-t)}&=\left(\epsilon wL_1(w{\overline}{w})+\epsilon W(w,{\overline}{w},t,\epsilon,\nu)\right)e^{i(\psi-t)}\\
&=\left(\epsilon \rho e^{-i(\psi-t)}L_1(\rho^2)+\epsilon W(\rho e^{-i(\psi-t)},\rho e^{i(\psi-t)},t,\epsilon,\nu)\right)e^{i(\psi-t)}\\
&=\epsilon\rho L_1(\rho^2)+\epsilon e^{i(\psi-t)} W(\rho
e^{-i(\psi-t)},\rho e^{i(\psi-t)},t,\epsilon,\nu)\end{aligned}$$ and so $$\begin{aligned}
\label{polavg}
\begin{split}
\dot{\rho}&= \epsilon R^1_0(\rho)+ \epsilon R(t,\psi,\rho,\epsilon,\nu)\\
\dot{\psi}&= 1+\epsilon\Psi_0(\rho)+\epsilon
\Psi(t,\psi,\rho,\epsilon,\nu),
\end{split}\end{aligned}$$ where $R^1_0(\rho)= \rho\operatorname{Re}\left[L_1(\rho^2)\right]$, $\Psi_0(\rho)=-\operatorname{Im}\left[L_1(\rho^2)\right]$ and $$\begin{aligned}
\label{thewhat2}
\begin{split}
R(t,\psi,\rho,\epsilon,\nu)&=\operatorname{Re}\left[e^{i(\psi-t)} W(\rho e^{-i(\psi-t)},\rho e^{i(\psi-t)},t,\epsilon,\nu)\right]\\
\Psi(t,\psi,\rho,\epsilon,\nu)&=-\frac{1}{\rho}\operatorname{Im}\left[e^{i(\psi-t)}
W(\rho e^{-i(\psi-t)},\rho e^{i(\psi-t)},t,\epsilon,\nu)\right].
\end{split}\end{aligned}$$ Note that $R,\Psi\in
\mathfrak{P}^{2\pi}_t\cap\mathfrak{P}^{2\pi}_{\psi}$ and that $\Psi$ is not defined at $\rho=0$. We now give sufficient conditions for the existence of an integral manifold in (\[polavg\]).
\[thmtorus\] Assume that $R$ and $\Psi$, as defined in $(\ref{thewhat2})$, are $C^1$ on intervals away from $\rho=0$ and that the averaged equation $$\label{averaged}
\dot{\rho}=\epsilon R^1_0(\rho)$$ has an equilibrium $\rho_1>0$ with $D_{\rho}R_0^1(\rho_1)=\gamma_1\neq 0$. If the parameters are small enough to satisfy the conditions outlined in the proof below, then $(\ref{polavg})$ has an invariant torus $\hat{\cal
E}_{\epsilon,\nu}$, whose stability is exactly determined by the sign of $\epsilon\gamma_1$.
[**Proof:**]{}By the implicit function theorem, there is a neighbourhood $$U=(-\epsilon_*,\epsilon_*)\times \prod_{j=2}^n (-\nu_{j,*},\nu_{j,*})$$ in parameter space and a continuous function $\rho:U\to {\mathbb{R}}^+$ such that $\rho(0,0)=\rho_1$, $$\epsilon R_0^1(\rho(\epsilon,\nu))\equiv
0\quad\mbox{and}\quad
D_{\rho}R_0^1(\rho(\epsilon,\nu))=\gamma_{\epsilon,\nu}\neq 0,$$ where $\gamma_{\epsilon,\nu}\gamma_1>0$ for all $(\epsilon,\nu)\in
U$, *i.e.* the stability of the equilibria $\rho(\epsilon,\nu)$ is the same as that of $\rho_0$ for all $(\epsilon,\nu)\in U$.
When $\epsilon=0$, the phase space of (\[polavg\]) is foliated by invariant tori and so, from now on, we will assume that $\epsilon\neq 0$. Consider the change of variables $\rho=\rho(\epsilon,\nu)+x$ in (\[polavg\]). Differentiating the new coordinates, we get $\dot{x}=\dot{\rho}$ and the equivalent system $$\begin{aligned}
\begin{split}
\dot{x}&=\epsilon R_0^1(\rho(\epsilon,\nu)+x)+\epsilon R(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)\\
\dot{\psi}&=1+\epsilon\Psi_0(\rho(\epsilon,\nu)+x)+\epsilon\Psi(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu).
\end{split}\end{aligned}$$ By Taylor’s theorem, there are continuously differentiable functions $B_1$ and $B_2$ such that $$\begin{aligned}
R_0^1(\rho(\epsilon,\nu)+x)&=R_0^1(\rho(\epsilon,\nu))+D_{x} R_0^1(\rho(\epsilon,\nu))x+B_1(x,\epsilon,\nu)x^2\\
\Psi_0(\rho(\epsilon,\nu)+x)&=\Psi_0(\rho(\epsilon,\nu))+
B_2(x,\epsilon,\nu)x.\end{aligned}$$ Since $R_0^1(\rho(\epsilon,\nu))\equiv 0$ and $D_{x}
R_0^1(\rho(\epsilon,\nu))=\gamma_{\epsilon,\nu}$, we obtain the new system $$\begin{aligned}
\begin{split}\label{thexeqs}
\dot{x}&=\epsilon\gamma_{\epsilon,\nu}x+\epsilon\Lambda(t,\psi,x,\epsilon,\nu) \\
\dot{\psi}&=d(\epsilon,\nu)+\Theta(t,\psi,x,\epsilon,\nu),
\end{split}\end{aligned}$$ where $$\begin{aligned}
\Lambda(t,\psi,x,\epsilon,\nu)&=B_1(x,\epsilon,\nu)x^2+R(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)\\
\Theta(t,\psi,x,\epsilon,\nu)&=\epsilon B_2(x,\epsilon,\nu)x+\epsilon\Psi(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)\\
d(\epsilon,\nu)&=1+\epsilon\Psi_0(\rho(\epsilon,\nu))\end{aligned}$$ are at least $C^1$ by hypothesis.
Let $U^+=\{\varsigma\in U:
\varsigma_i>0 \mbox{ for all }i=1,\ldots,n\}$ and $(\epsilon_0,\nu_0)\in U^+$. Define $$S_0=[-\epsilon_0,\epsilon_0]\times
\prod_{j=2}^n[-\nu_{0,j},\nu_{0,j}].$$ As $\rho_1>0$ and $\rho(\epsilon,\nu)$ is continuous on $U$, it is possible to chose $(\epsilon_0,\nu_0)$ in such a way that $$\sigma_0=\min_{(\epsilon,\nu)\in S_0}\left\{\rho(\epsilon,\nu)\right\}-\textstyle{\frac{1}{2}}\rho_1>0.$$ If $x\geq -\sigma_0$, then $\rho=\rho(\epsilon,\nu)+x\geq \rho(\epsilon,\nu)-\sigma_0\geq
\textstyle{\frac{1}{2}}\rho_1$ for all $(\epsilon,\nu)\in S_0$. In that case, $\Theta$ and $\Lambda$ are continuously differentiable, as $R$ and $\Psi$ are continuously differentiable in $\rho$ on $[\frac{1}{2}\rho_1,\infty)$. Note further that $\Theta,\Lambda\in
\mathfrak{P}^{2\pi}_t\cap \mathfrak{P}^{2\pi}_{\psi}$.
Set $\Sigma_0={\mathbb{R}}\times {\mathbb{R}}\times \{0\}\times S_0$, and $\Sigma_{\sigma}={\mathbb{R}}\times{\mathbb{R}}\times [-\sigma,\sigma]\times S_0.$ Then \[pappH1H2\]
1. $\Theta$ and $\Lambda$ are bounded by a function $\Xi(\epsilon,\nu)=O(\epsilon,\nu_2,\ldots,\nu_n)$ over $\Sigma_0$ (see appendix for details), and
2. for all $0\leq \sigma\leq \sigma_0$, $\Theta$ and $\Lambda$ are Lipschitz in Hale’s sense (with Lipschitz constants $\theta(\epsilon,\nu,\delta)=O(\epsilon,\nu_2,\ldots,\nu_n,\delta)$ and $\eta(\epsilon,\nu,\delta)=O(\epsilon,\nu_2,\ldots,\nu_n,\delta)$, respectively) over $\Sigma_{\sigma}$ (see appendix for details).
Accordingly, theorem \[Halesavg\] can be applied to show there is a neighbourhood $S_1\subseteq S_0$ of the origin in parameter space for which (\[polavg\]) (since it is equivalent to (\[thexeqs\])) has an invariant torus $\hat{\mathcal{T}}_{\epsilon,\nu}$ when $(\epsilon,\nu)\in S_1$. Furthermore, the stability of $\hat{\mathcal{T}}_{\epsilon,\nu}$ is the same as that of the hyperbolic equilibrium $\rho(\epsilon,\nu)$, which is given by $\epsilon\gamma_1$. The invariant torus $\hat{\mathcal{T}}_{\epsilon,\nu}$ appearing in the proof of Theorem \[thmtorus\] can be parameterized by a relation of the form $x=\Upsilon_{\epsilon,\nu}(\theta_1,\theta_2)$, where $\theta_1,\theta_2\in {\mathbb{S}}^1$. Let $$\begin{aligned}
\big\langle\hat{\mathcal{T}}_{\epsilon,\nu}\big\rangle=\frac{1}{4\pi^2}\int_{0}^{2\pi}\!\!\!\!\int_{0}^{2\pi}\!\!\!\! \Upsilon_{\epsilon,\nu}(\theta_1,\theta_2)\,d\theta_1d\theta_2\end{aligned}$$ denote the *center* of $\hat{\cal
T}_{\epsilon,\nu}$, and let $\hat{\cal E}_{\epsilon,\nu}$ be the corresponding *epicyclic manifold* of (\[system3\]), in which all solutions are epicycles when projected upon the $z-$plane.
Define the average value $$\begin{gathered}
\index{[]D@$[-]_{\operatorname{D}}$}[\hat{\cal
E}_{\epsilon,\nu}]_{\operatorname{D}}=\frac{1}{4\pi^2}\int_{0}^{2\pi}\!\!\!\!\int_{0}^{2\pi}\!\!\!\!\big(\big(\rho(\epsilon,\nu)+\big\langle\hat{\mathcal{T}}_{\epsilon,\nu}\big\rangle\big)e^{-i(\psi-t)}
\\ +\epsilon\kappa\big(\big(\rho(\epsilon,\nu)+\big\langle\hat{\mathcal{T}}_{\epsilon,\nu}\big\rangle\big)e^{-i(\psi-t)},\mbox{c.c.},t,\epsilon,\nu\big) \big)\, d\psi dt.\label{cd}\end{gathered}$$ If $\mathcal{\hat{T}}_{\epsilon,\nu}$ is stable (in the sense of theorem \[Halesavg\]), we shall say that $[\hat{\mathcal{E}}_{\epsilon,\nu}]_{\operatorname{D}}$ is the *center of drifting*\[cdriftoo\] of $\hat{\mathcal{E}}_{\epsilon,\nu}$.
\[thmnld1\] Suppose the hypotheses of theorem $\ref{thmtorus}$ are satisfied. Then there exists a wedge-shaped region near $\lambda=0$ of the form $${\cal V}_{1}=\{(\lambda_1,\ldots,\lambda_n)\in
{\mathbb{R}}^n\,:\,|\lambda_j|<V_{1,j}|\lambda_1|,\,\,\,V_{1,j}>0,\,\,\mbox{\rm
for $j\neq 1$ and $\lambda_1$ near}\,\,0\,\}$$ such that for all $0\neq \lambda\in {\cal V}_{1}$, $(\ref{basic12})$ has an epicycle manifold ${\cal E}_{\lambda}^1$, with $[{\cal
E}^1_{\lambda}]_{\operatorname{D}}$ near, but generically not at, $\xi_1$. Furthermore, $[{\cal E}^1_{\lambda}]_{\operatorname{D}}$ is a center of drifting when $\lambda_1\gamma_1<0$.
[**Proof:**]{}According to theorem \[thmtorus\], there are constants $\epsilon_1,\nu_{1,2},\ldots,\nu_{1,n}>0$ and a neighbourhood $$S_1=[-\epsilon_1,\epsilon_1]\times
\prod_{j=2}^n[-\nu_{1,j},\nu_{1,j}]$$ such that (\[polavg\]) has an integral manifold $\hat{\mathcal{E}}_{\epsilon,\nu}$ whenever $(\epsilon,\nu)\in S_1$.
For $j\neq 1$, set $\lambda_1=\epsilon\neq 0$, $\lambda_j=\nu_j\epsilon$ and $V_{1,j}=\nu_{1,j}$. Then $\lambda\in {\cal V}_1$ as $$|\lambda_j|\leq |\nu_j|\cdot |\lambda_1|\leq
V_{1,j}|\lambda_1|\quad \mbox{for }j\neq 1,$$ and (\[basic12\]) has an integral manifold ${\cal
E}_{\lambda}^1=\xi_1-ie^{it}v+\hat{\mathcal{E}}_{\epsilon,\nu}$. Furthermore, $[{\cal E}_{\lambda}^1]_{\operatorname{D}}=\xi_1+O(\lambda_1)$ and so $[\mathcal{E}_{\lambda}^1]_{\operatorname{D}}\neq \xi_1$ for a generic $0\neq\lambda\in \mathcal{V}_1$. The conclusion on the stability of ${\cal E}^1_{\lambda}$ then follows directly from theorem \[thmtorus\].
1. These isolated epicycle manifolds need not in general be unique for a given $\lambda\in {\cal V}_1$ as $R_0^1(\rho)=0$ may have any number of hyperbolic solutions.
2. In generic semi-flows, all that can be said with certainty from the analysis when the parameter values stray outside of ${\cal V}_{1}$ is that the epicycle manifolds in (\[basic12\]) drift away from $\xi_1$, which cannot then be a center of drifting. This is not unlike the situation with regards to spiral anchoring [@BLM]. Richer dynamics and interactions with rotating waves can also take place; for instance in [@Byeah], we gave an example in which the epicyclic manifold collapses at a saddle-node bifurcation of rotating waves.
3. Note that the actual parameter region in which epicyclic drifting is observed may be much larger than $\mathcal{V}_1$: however, our local analysis cannot be used to obtain global results.
The preceding results have been achieved by considering (\[basic12\]) under a co-rotating frame of reference around $\xi_1$. Of course, since the choice for $\xi_1$ was arbitrary, corresponding results must also be achieved, in exactly the same manner, when the viewpoint shifts to another $\xi_k$. Indeed, for $j=1,\ldots,n$, define the average functions $$M^{j}(w,{\overline}{w})=\frac{1}{2\pi}\int_{0}^{2\pi}\!\!\!\!e^{it}\hat{H}_j(we^{-it},{\overline}{w}e^{it},0,0)\,
dt;$$ as before, each $M^j$ is ${\mathbb{S}}^1-$equivariant and so there are continuous functions $L_j:{\mathbb{R}}\to {\mathbb{C}}$ such that $M^{j}(w,{\overline}{w})=wL_j(w{\overline}{w})$. We will call $$R^{j}_0(\rho)=\rho\operatorname{Re}\left[L_j(\rho^2)\right]$$ the *epicycle functions of* (\[basic12\]).
\[thmnld2\] Let $k\in \{1,\ldots, n\}$. If $\rho_*>0$ is such that $$R_0^k(\rho_*)=0 \quad\mbox{and}\quad D_{\rho}R_0^k(\rho_*)=\gamma_*\neq 0,$$ then there exists a wedge-shaped region near $\lambda=0$ of the form $${\cal V}_{k}=\{(\lambda_1,\ldots,\lambda_n)\in
{\mathbb{R}}^n\,:\,|\lambda_j|<V_{k,j}|\lambda_k|,\,\,\,V_{k,j}>0,\,\,\mbox{\rm
for $j\neq k$ and $\lambda_k$ near}\,\,0\,\}$$ such that for all $0\neq \lambda\in {\cal V}_{k}$, $(\ref{basic12})$ has an epicycle manifold ${\cal E}_{\lambda}^k$, with $[{\cal
E}^k_{\lambda}]_{\operatorname{D}}$ near, but generically not at, $\xi_k$. Furthermore, $[{\cal E}^k_{\lambda}]_{\operatorname{D}}$ is a center of drifting when $\lambda_k\gamma_*<0$.
[**Proof:**]{}The epicycle function $R_0^k$ is exactly the function that would appear in (\[averaged\]) had the preceding analysis been done around $\xi_k$. Theorems \[thmtorus\] and \[thmnld1\] can then be applied directly to obtain the desired result.Clearly, the remarks appearing after the proof of theorem \[thmnld1\] still hold. There is one last statement to be made concerning epicycle manifolds: theorem \[thmnld2\] only gives sufficient conditions for their existence in (\[basic12\]). In section [@Byeah], we have provided an example that shows that they are not, in fact, necessary conditions.
Epicyclic Drifting For Combined RSB-TSB Terms
=============================================
In this section, we investigate another way in which the Euclidean symmetry can be broken: by combining rotational and translational symmetry breaking. In effect, we are lifting the restriction $\beta=0$, with $n=1$ in (\[basic12\]).
It turns out that the value of $\jmath^*$ plays a crucial role in the analysis: the cases $\jmath^*=1$ and $\jmath^*>1$ are essentially different. The general lines are very similar to those of the preceding section, as such, the proofs are omitted in order to avoid tedious repetitions. In either case, however, me assume without loss of generality that $\xi_1=0$.
The Case $\jmath^*=1$
---------------------
Let $F_G:{\mathbb{R}}\times{\mathbb{R}}^2\to {\mathbb{C}}$ be defined by $$F_G(t,\beta)=e^{it}\Big[-iv+\beta \sum_{m\neq
-1}\,\frac{g_m(\beta)e^{im t}}{i(m+1)}\Big],$$ where the $g_{m}(\beta)$ are the Fourier coefficients of $G\in
\mathfrak{P}^{2\pi}_t$. Set $z=p-F_G$. Then, (\[basic12\]) rewrites as $$\begin{aligned}
\label{zdotforcedyeah1}
\begin{split}
\dot{z}=\beta g_{-1}(\beta)+\beta e^{it}
H((z+F_G(t,\beta)e^{-it},\mbox{c.c.},\lambda_1),
\end{split}\end{aligned}$$ where represents throughout the complex conjugate of the preceding term. Generically, $g_{-1}(0)\neq 0$. Set $\epsilon=\beta$, $\nu=\lambda_1$ and $\epsilon=\hat{\epsilon}\lambda_1$. Then (\[zdotforcedyeah1\]) transforms to $$\dot{z}=\nu H_*(ze^{-it},{\overline}{z}e^{it},t,\hat{\epsilon},\nu),
\label{newzdotforced1}$$ where $$\label{thefirstH}H_*(w,{\overline}{w},t,\hat{\epsilon},\nu)=\hat{\epsilon}g_{-1}(\hat{\epsilon}\nu)+e^{it}
H(w+F_G(t,\hat{\epsilon}\nu)e^{-it},\operatorname{c.c.},\nu)$$ is $2\pi-$periodic in $t$, smooth and uniformly bounded in $w$. Consider the near-identity change of variables $$\begin{aligned}
\label{ahahfcov}
z&=w+\nu \varrho (w,{\overline}{w},t,\epsilon,\nu)\end{aligned}$$ where $\varrho\in \mathfrak{P}^{2\pi}_t$ is continuous in all of its variables and to be determined later. This change of variables transform (\[newzdotforced1\]) into the equivalent system $$\begin{aligned}
\label{ahahlisten} \dot{w}&=\nu
\left(H_*(we^{-it},{\overline}{w}e^{it},t,\hat{\epsilon},0)-\varrho_t(w,{\overline}{w},t,\hat{\epsilon},0)\right)+\nu^2
\mathcal{H}_*(w,{\overline}{w},t,\hat{\epsilon},\nu),\end{aligned}$$ where $\mathcal{H}_*\in \mathfrak{P}^{2\pi}_t$ is bounded and continuous in all its variables. Denote the average value of \[ahahpappequiv\]$H_*(we^{-it},{\overline}{w}e^{it},t,\hat{\epsilon},0)$ by $$\label{ahahavgM}M_*(w,{\overline}{w},\hat{\epsilon})=\frac{1}{2\pi}\int_{0}^{2\pi} \!\!\!\!H_*(we^{-it},{\overline}{w}e^{it},t,\hat{\epsilon},0)\, dt.$$ Since $$\begin{aligned}
H_*(we^{-it},{\overline}{w}e^{it},t,\hat{\epsilon},0)&=\hat{\epsilon}g_{-1}(0)+e^{it}H\left((w+F_G(t,0))e^{-it}, \operatorname{c.c.},0\right) \\
&=\hat{\epsilon}g_{-1}(0)+e^{it}H(we^{-it}\!\!-iv,{\overline}{w}e^{it}\!\!+i{\overline}{v},0),\end{aligned}$$ we have $$M_*(w,{\overline}{w},\hat{\epsilon})=\hat{\epsilon}g_{-1}(0)+\frac{1}{2\pi}\int_{0}^{2\pi}\!\!\!\!e^{it} H(we^{-it}\!\!-iv,{\overline}{w}e^{it}\!\!+i{\overline}{v},0)\, dt.$$ Then $$H_*(we^{-it},{\overline}{w}e^{it},t,\hat{\epsilon},0)= M_*(w,{\overline}{w},\hat{\epsilon})+F_*(w,{\overline}{w},t,\hat{\epsilon}),$$ where $F_*\in \mathfrak{P}^{2\pi}_t$ is uniformly continuous and $$\label{ahahnoconstant}\int_{0}^{2\pi}\!\!\!\!F(w,{\overline}{w},t,\hat{\epsilon})\, dt=0.$$ Let $\varrho$ be an antiderivative of $F_*$ with respect to $t$. Then $\varrho\in \mathfrak{P}^{2\pi}_t$ by (\[ahahnoconstant\]) and $$F_*(w,{\overline}{w},t,\hat{\epsilon})-\varrho_t(w,{\overline}{w},t,\hat{\epsilon},0)=0.$$ With such a $\varrho$, (\[ahahlisten\]) simplifies to $$\begin{aligned}
\label{ahahavg} \dot{w}&=\nu
M_*(w,\overline{w},\hat{\epsilon})+\nu^2
\mathcal{H}_*(w,{\overline}{w},t,\hat{\epsilon},\nu).\end{aligned}$$ As $M_*(w,\overline{w},0)$ is also ${\mathbb{S}}^1-$equivariant, there is a continuous function $L_*:{\mathbb{R}}\to {\mathbb{C}}$ such that $M_*(w,{\overline}{w},0)=wL_*(w{\overline}{w})$, and so (\[ahahavg\]) becomes $$\label{ahahavg2}
\dot{w}=\nu wL_*(w{\overline}{w})+\nu W_*(w,{\overline}{w},t,\hat{\epsilon},\nu),$$ where $$\label{ahahfuncW}W_*(w,{\overline}{w},t,\hat{\epsilon},\nu)=\hat{\epsilon}g_{-1}(0)+\nu
\mathcal{H}_*(w,{\overline}{w},t,\hat{\epsilon},\nu).$$ Differentiating the polar coordinates $w=\rho e^{-i(\psi-t)}$ and substituting in \[ahahavg2\] yields $$\begin{aligned}
\label{ahahpolavg}
\begin{split}
\dot{\rho}&= \nu R_*(\rho)+ \nu R(t,\psi,\rho,\hat{\epsilon},\nu)\\
\dot{\psi}&= 1+\nu\Psi_*(\rho)+\nu
\Psi(t,\psi,\rho,\hat{\epsilon},\nu),
\end{split}\end{aligned}$$ where $R_*(\rho)= \rho\operatorname{Re}\left[L_*(\rho^2)\right]$, $\Psi_*(\rho)=-\operatorname{Im}\left[L_*(\rho^2)\right]$, $R,\Psi\in \mathfrak{P}^{2\pi}_t\cap
\mathfrak{P}^{2\pi}_{\psi}$ and $\Psi$ is not continuous at $\rho=0$. We now provide sufficient conditions for the existence of an epicycle manifold in (\[basic12\]).
\[ahahthmtorus\] Assume that $R$ and $\Psi$, as defined in $(\ref{ahahpolavg})$, are $C^1$ on intervals away from $\rho=0$ and that the averaged equation $$\label{ahahaveraged}
\dot{\rho}=\epsilon R_*(\rho)$$ has an equilibrium $\rho_0>0$ with $D_{\rho}R_*(\rho_0)=\gamma_0\neq
0$. If the parameters are small enough, there exists a wedge-shaped region near $(\beta,\lambda_1)=(0,0)$ of the form $${\cal V}=\{(\beta,\lambda_1)\in {\mathbb{R}}^2\,:\,|\beta|<K|\lambda_1|,\,\,\,K>0,\,\,\mbox{$\lambda_1$ near}\,\,0\,\}$$ such that for all $(\beta,\lambda_1)\in {\cal V}$, $\beta\neq 0$, $(\ref{basic12})$ has an epicycle manifold ${\cal
G}_{\beta,\lambda_1}^1$, with $[{\cal G}^1_{\beta,\lambda_1}]_{\operatorname{D}}$ near, but generically not at, the origin. Furthermore, $[{\cal
G}^1_{\beta,\lambda_1}]_{\operatorname{D}}$ is a center of drifting when $\lambda_1\gamma_0<0$.
The remarks after theorem \[thmnld1\] still hold after having been suitably modified.
The Case $\jmath^*>1$
---------------------
The case $\jmath^*>1$ is handled slightly differently. Let $C^{0}_{{\mathbb{R}}}({\mathbb{C}})$ and $C^{1}_{{\mathbb{R}}}({\mathbb{C}})$ be the spaces of continuous and continuously differentiable functions from ${\mathbb{R}}$ to ${\mathbb{C}}$, respectively. Then $$C^0_{2\pi/{\jmath^*}}=\{f: f\in \mathfrak{P}^{2\pi/{\jmath^*}}_t\cap
C^0_{{\mathbb{R}}}({\mathbb{C}})\}\quad\mbox{and}\quad C^1_{2\pi/{\jmath^*}}=\{f:f\in
\mathfrak{P}^{2\pi/{\jmath^*}}_t\cap C^1_{{\mathbb{R}}}({\mathbb{C}})\}$$ are Banach spaces when endowed with the respective norms $$||u||_0=\sup\{|u(x)|:x\in [0,\textstyle{2\pi/{\jmath^*}}]\}\quad
\mbox{and}\quad ||u||_1=||u||_0+||u'||_0,$$ and the linear operator ${\cal Y}:C^1_{2\pi/{\jmath^*}}\to C^0_{2\pi/{\jmath^*}}$ defined by ${\cal Y}(u)=iu+u'$ is bounded, invertible and has bounded inverse. Define the nonlinear operator ${\cal
H}_G:C^1_{2\pi/{\jmath^*}}\times {\mathbb R}^2\to
C^0_{2\pi/{\jmath^*}}$ by $$\label{HG}
{\cal H}_G(u,\beta,\lambda_1)={\cal Y}(u)-\lambda_1 H\left(u-iv+
\beta\sum_{m\in {\mathbb Z}}\,\frac{g_m(\beta)e^{im{\jmath^*}
t}}{i(m{\jmath^*}+1)}, \mbox{c.c.},\lambda_1\right),$$ where the $g_{m}(\beta)$ are as they were in the previous section. But ${\cal H}_G(0,0,0)=0$ and $D_1{\cal H}_G(0,0,0)=i\neq 0$ and so, by the implicit function theorem, there is a neighbourhood ${\cal N}$ of the origin in ${\mathbb R}^2$ and a unique smooth function $U:{\mathbb R}^2\to
C^1_{2\pi/{\jmath^*}}$ satisfying $U(0,0)=0$ and ${\cal
H}_G(U(\beta,\lambda_1),\beta,\lambda_1)\equiv 0$ for all $(\beta,\lambda_1)\in {\cal N}$.Let $F_G:{\mathbb{R}}\times{\mathbb{R}}^2\to
{\mathbb{C}}$ be defined by $$\label{FG}
F_G(t,\beta,\lambda_1)=e^{it}\Big[-iv+\beta \sum_{m\in {\mathbb
Z}}\,\frac{g_m(\beta)e^{im{\jmath^*} t}}{i(m{\jmath^*}+1)}+
U(\beta,\lambda_1)(t)\Big].$$ Then ${\cal Y}(U(\beta,\lambda_1))(t)=\lambda_1
H\left(F_G(t,\beta,\lambda_1)e^{-it},\mbox{c.c.},\lambda_1\right),$ and, upon setting $z=p-F_G$, (\[basic12\]) rewrites as $$\begin{aligned}
\begin{split}
\dot{z}=\lambda_1 e^{it}\big[
H((z+F_G(t,\beta,\lambda_1))e^{-it},\mbox{c.c.},\lambda_1)-H(F_G(t,\beta,\lambda_1)e^{-it},\mbox{\mbox{c.c.}},\lambda_1))\big],
\end{split}\end{aligned}$$ which reduces to $$\dot{z}=\lambda_1
e^{it}\widehat{H}(ze^{-it},{\overline}{z}e^{it},t,\beta,\lambda_1),
\label{newzdotforced}$$ where $$\label{thesecondH}\widehat{H}(w,{\overline}{w},t,\beta,\lambda_1)=H(w+F_G(t,\beta,\lambda_1)e^{-it},\mbox{c.c.},\lambda_1)-H(F_G(t,\beta,\lambda_1)e^{-it},\mbox{c.c.},\lambda_1)$$ is $2\pi/{\jmath^*}-$periodic in $t$. Then, (\[newzdotforced\]) becomes $$\begin{aligned}
\label{ahahahpolavg}
\begin{split}
\dot{\rho}&= \lambda_1 R_*(\rho)+ \lambda_1 R(t,\psi,\rho,\beta,\lambda_1)\\
\dot{\psi}&= 1+\lambda_1\Psi_*(\rho)+\lambda_1
\Psi(t,\psi,\rho,\beta,\lambda_1),
\end{split}\end{aligned}$$ where $R_*(\rho)= \rho\operatorname{Re}\left[L_*(\rho^2)\right]$, $\Psi_*(\rho)=-\operatorname{Im}\left[L_*(\rho^2)\right]$ for some continuous function $L_*:{\mathbb{R}}\to {\mathbb{C}}$, $R,\Psi\in
\mathfrak{P}^{2\pi}_t\cap\mathfrak{ P}^{2\pi}_{\psi}$ and $\Psi$ is not continuous at $\rho=0$.
\[ahahahthmtorus\] Assume that $R$ and $\Psi$, as defined in $(\ref{ahahahpolavg})$, are $C^1$ on intervals away from $\rho=0$ and that the averaged equation $$\label{ahahahaveraged}
\dot{\rho}=\epsilon R_*(\rho)$$ has an equilibrium $\rho_0>0$ with $D_{\rho}R_*(\rho_0)=\gamma_0\neq
0$. If the parameters are in a (small enough) deleted neighbourhood $\mathcal{V}^{\jmath^*}$ of the origin, $(\ref{basic12})$ has an epicyclic manifold ${\cal G}^{{\jmath^*}}_{\beta,\lambda_1}$, with $[{\cal G}^{{\jmath^*}}_{\beta,\lambda_1}]_{\operatorname{D}}=0$. Furthermore, the origin is a center of drifting when $\lambda_1\gamma_0<0$.
1. The small term $\hat{\epsilon}g_{-1}(0)$ in (\[thefirstH\]) and the absence of a corresponding term in (\[thesecondH\]) are responsible for the different form of the regions $\mathcal{V}$ (in theorem \[ahahaveraged\]) and $\mathcal{V}^{\jmath^*}$ (in theorem \[ahahahaveraged\]), as well as for the location of the center of drifting.
2. The remarks made after theorem \[thmnld1\] still hold, when suitably modified.
3. There are a lot of similarities between our analysis and the results obtained during the analysis of spiral anchoring in [@BLM; @Bo1], such as the presence of wedge-shaped regions or the deleted neighbourhoods, depending on the nature of $\jmath^*$. In particular, one might hope that the epicyclic manifolds would possess ${\mathbb{Z}}_{{\jmath^*}}-$spatio-temporal symmetry; however, this is not the case as the averaged system defined by (\[ahahahpolavg\]) is generally only $2\pi/\jmath^*-$periodic in $t$ when $R\equiv 0$ and $\Psi\equiv 0$. That being said, the epicycles themselves possess this symmetry in an appropriate co-rotating frame of reference.
Epicyclic Drifting For General ESB Terms
========================================
Lifting all restrictions on $\beta$ and $\lambda$ in (\[basic12\]), and combining the methods of the preceding section, we obtain the following general epicyclic drifting theorems for (\[basic12\]).
\[noclueagainconj\] Let $n>1$ and $k\in
\{1,\ldots,n\}$. Given a hyperbolic equilibrium $\rho_*$ of an appropriate averaged equation $\dot{\rho}=\beta\tilde{R}_0^k(\rho)$ with eigenvalue $\gamma_k^*$ (derived as in section 3), there is a region in parameter space near $(\beta=\lambda_0,\lambda_1,\ldots,\lambda_n)=0$ of the form $${\mathcal V}^1_k=\{(\lambda_0,\lambda_1,\ldots,\lambda_n)\in
{\mathbb{R}}^{n+1}\,:\,|\lambda_j|<V_{j,k}|\lambda_k|,\,\,\,V_{j,k}>0,\,\,\mbox{\rm
for $j\neq k$ and $\lambda_k$ near}\,\,0\,\}$$ when ${\jmath^*}=1$, or of the form $${\mathcal
V}^{\jmath^*}_k=\{(\beta,\lambda_1,\ldots,\lambda_n)\in
{\mathbb{R}}^{n+1}\,:\,|\beta|<\beta_0,\,\,\,|\lambda_j|<V_{j,k}|\lambda_k|,\,\,\,V_{j,k}>0,\,\,\mbox{\rm
for $j\neq k$ and $\lambda_k$ near}\,\,0\,\}$$ for some constant $\beta_0>0$, when $\jmath^*>1$, such that for all $0\neq (\beta,\lambda)\in {\mathcal V}_{k}^{\jmath^*}$, with the additional condition that $\beta\neq 0$ when $\jmath^*=1$, $(\ref{basic12})$ has an epicyclic manifold ${\cal
E}^{\jmath^*;k}_{\beta,\lambda}$, with $[{\cal
E}^{\jmath^*;k}_{\beta,\lambda}]_{\operatorname{D}}$ near, but generically not at, $\xi_k$. Furthermore, $[{\cal
E}^{\jmath^*;k}_{\beta,\lambda}]_{\operatorname{D}}$ is a center of drifting when $\lambda_k\gamma_k^*<0$.
Since the hypotheses of this theorems are not generic, it is not clear that such integral epicyclic manifolds are common, and their existence must sometimes be inferred in the physical space, especially if they are repelling, such as appears to be the case in [@MPMPV].
An Epicyclic Manifold in Physical Space
=======================================
In this section, we provide what we believe to be the first observed example of epicyclic drifting in a modified bidomain system. Our system is a TSB perturbation of the bidomain equations and parameter values found in [@R1]: $$\begin{aligned}
\label{thebid}\begin{split} u_t &= \frac{1}{\varsigma} \left(u-\frac{u^3}{3}-v\right)+ \Delta
u+\frac{\alpha \varepsilon
}{1+\alpha(1-\varepsilon)}\psi_{xx} \\
u_{yy}&= \left[\left(2+\alpha+\frac{1}{\alpha}\right)\psi_{xx}+\left(2+\alpha(1-\varepsilon)+\frac{1}{\alpha(1-\varepsilon)}\right)\psi_{yy}\right]\left(1+\frac{1}{\alpha(1-\varepsilon)}\right)^{-1}\frac{1}{\varepsilon} \\
v_t &= \varsigma(u+\delta-\gamma v)+\phi(x-35,y-35)
\end{split}\end{aligned}$$ where $\varsigma = 0.3$, $\alpha = 1.0$, $\varepsilon = 0.75$, $\delta = 0.8$, $\gamma = 0.5$ and $$\phi(z_1,z_2)= -0.03\exp\left(-0.085(z_1^2+z_2^2)\right).$$ The TSB term $\phi(x-35,y-35)$ is uniformly bounded and goes to $0$ as $\|(x,y)\|\to\infty$. Futhermore, it preserves rotations around the point $(35,35)$.
As our emphasis lies with qualitative observations rather than with precise numerical analysis, the numerical perspective is somewhat naive. The computations are carried out on a two-dimensional square domain $[10,60]^2$ with 120 grid points to a side and Neumann boundary condition, using a 5-point Laplacian, continuous linear finite elements on square meshes and the fully implicit second order Gear finite difference integrator. The tip path of the $u$ component of two solutions are shown in
![Tip path of the $u$ component of two solutions of (\[thebid\]). The solution in red is attracted by an epicyclic manifold. The solution in blue shows the effect of the boundary: the two solutions are clearly not of the same nature. Compare the epicyclic manifold with the image in figure \[epmannn\].](figure6){width="222pt"}
------------------------------------------------------------------------
figure .
Summary and Concluding Remarks
==============================
[Recently, equivariant dynamical systems theory has been used to provide an approach to the study of spiral wave dynamics and bifurcations, in particular, it has provided mechanisms for such behaviour as spiral tip meandering and resonant growth [@B2; @SSW1; @SSW2; @SSW3; @SSW4; @Wulff], spiral anchoring/repelling and boundary drifting [@LW], which have been explained as consequences of forced Euclidean symmetry breaking.]{}
[In this paper, we have used this model-independent approach to analyze epicyclic drifting of the spiral tip in media with several localized inhomogeneities, with or without anisotropy. The result of a simple numerical experiment is in agreement with our theoretical conclusions. The RSB terms are characterized by an integer $\jmath^*>1$. It is important to note that, as of now, only the integers $\jmath^*=2$ (anisotropic cardiac tissue, say) and $\jmath^*=1$ (an excitable medium that in which there is a directed current, such as a reaction-diffusion-advection system) have easy interpretations in the context of RSB.]{}
[It should be recalled that our analysis rests on certain simplifying modeling assumptions which may not be valid in some realistic physical systems: namely concerning the discrete nature of the inhomogeneities (i.e. finite number of inhomogeneity sites) and the hypothesis of local rotational symmetry of the individual inhomogeneities. In a realistic model of excitable media such as the bidomain model describing electrical conduction of cardiac tissue, with or without advection, actual inhomogeneities may lack this local rotational symmetry, or may even be distributed smoothly and non-symmetrically over the medium (*i.e.* an inhomogeneity field).]{}
Should the discrete localized inhomogeneities not possess circular symmetry, we believe that our results would still describe the essential qualitative features of epicyclic drifting, even though our analysis does not technically apply, as epicyclic drifting is linked to hyperbolic fixed points. However, for a smoothly distributed non-symmetric inhomogeneity field, our techniques are unlikely to yield meaningful results.Finally, we would like to point out that our existence results do not provide a description of the flow on the epicyclic manifolds: we gave two examples of center bundle systems with essentially different flows in [@Byeah]. In the first example, the flow is “ergodic” (see the first figure of this article): if the spiral tip lies in the epicyclic manifold, the tip path eventually fills the entire manifold. In the second example, the flow on the epicyclic manifold is dictated by the stability of rotating waves located *on* the manifold (see
![Projection (in the $z-$plane) of a stable epicycle manifold $\hat{\cal E}$ containing three rotating waves $\mathfrak{s}$, $\mathfrak{n}$ and $\mathfrak{f}$. The arrows indicate the flow on the manifold. \[epmannnnn\]](figure2){width="240pt"}
------------------------------------------------------------------------
figure ).
Acknowledgements
================
[The author is grateful to the referees for suggestions that have made this paper much more readable. The author would also like to thank Prof. Victor G. LeBlanc for the helpful advice and feedback he has provided over the last 10 years, and Marc Ethier and Eric Matsui for lending a hand with some of the numerics.]{}
Technical Results
=================
**Proposition A.1** *The function $M^1(w,{\overline}{w},0)$ defined in $(\ref{avgM})$ is ${\mathbb{S}}^1-$equivariant.*[**Proof:**]{}Recall that $\hat{H}_1$ is ${\mathbb{S}}^1-$equivariant by construction. Then $$\begin{aligned}
M^1(we^{-i\theta},{\overline}{w}e^{i\theta},0)&=\frac{1}{2\pi}\int_{0}^{2\pi}e^{it}K(we^{-i\theta}e^{-it},{\overline}{w}e^{i\theta}e^{it},t,0,0)\, dt \\
&=\frac{1}{2\pi}\int_{0}^{2\pi}e^{it}\hat{H}_1(we^{-it}e^{-i\theta},{\overline}{w}e^{it}e^{i\theta},0,0)\, dt \\
&=\frac{1}{2\pi}\int_{0}^{2\pi}e^{it}e^{i\theta}\hat{H}_1(we^{-it},{\overline}{w}e^{it},0,0)\, dt \\
&=e^{i\theta}M^1(w,{\overline}{w},0),\end{aligned}$$ that is, $M^1$ is ${\mathbb{S}}^1-$equivariant. **Proposition A.2** *Let all terms, variables and functions be as in Theorem \[thmtorus\]. \[appH1H2a\] In particular, the functions $R$ and $\Psi$ are $C^1$ on intervals away from $\rho=0$. Denote $$\begin{aligned}
\Sigma^r_0&=[0,2\pi]\times[0,2\pi]\times\{0\}\times S_0 \\
\Sigma^r_{\sigma}&=[0,2\pi]\times[0,2\pi]\times [-\sigma,\sigma]\times S_0.\end{aligned}$$ Note that these spaces, as well as the spaces $\Sigma_0$ and $\Sigma_{\sigma}$ from Theorem \[Halesavg\], are convex. The functions $\Theta$ and $\Lambda$ satisfy the following conditions:*
1. *$\Theta$ and $\Lambda$ are bounded by a function $\Xi(\epsilon,\nu)=O(\epsilon,\nu_2,\ldots,\nu_n)$ over $\Sigma_0,$ and*
2. *for all $0\leq \sigma\leq \sigma_0$, $\Theta$ and $\Lambda$ are Lipschitz in Hale’s sense (with Lipschitz constants $\theta(\epsilon,\nu,\sigma)=O(\epsilon,\nu_2,\ldots,\nu_n,\sigma)$ and $\nu(\epsilon,\nu,\sigma)=O(\epsilon,\nu_2,\ldots,\nu_n,\sigma)$, respectively) over $\Sigma_{\sigma}$.* (p. )
[**Proof:**]{}Since $\Theta,\Lambda\in \mathfrak{P}^{2\pi}_t\cap
\mathfrak{P}^{2\pi}_{\psi}$, we need only show that the first statement holds over $\Sigma^r_0$ and the second over $\Sigma^r_{\sigma}$, for all $0\leq
\sigma\leq \sigma_0$. For $j=1, \ldots, n$, there are appropriate functions $R_j,\Psi_j\in \mathfrak{P}^{2\pi}_t\cap \mathfrak{P}^{2\pi}_{\psi}$, $C^1$ on intervals away from $\rho=0$, such that $$\begin{aligned}
\label{thedefin}
\begin{split}
\Psi(t,\psi,\rho,\epsilon,\nu)&=\epsilon \Psi_1(t,\psi,\rho,\epsilon,\nu)+\sum_{j=2}^n\nu_j\Psi_j(t,\psi,\rho,\nu) \\
R(t,\psi,\rho,\epsilon,\nu)&=\epsilon
R_1(t,\psi,\rho,\epsilon,\nu)+\sum_{j=2}^n\nu_jR_j(t,\psi,\rho,\nu),
\end{split}\end{aligned}$$ according to (\[funcW\]).
1. Over $\Sigma^r_0$, we have $x=0$ and so $$\begin{aligned}
\label{upthere1}\begin{split}
\Theta(t,\psi,0,\epsilon,\nu)&=\epsilon\Psi(t,\psi,\rho(\epsilon,\nu),\epsilon,\nu)\\&
=\epsilon^2\Psi_1(t,\psi,\rho(\epsilon,\nu),\epsilon,\nu)+\epsilon
\sum_{j=2}^n\nu_j\Psi_j(t,\psi,\rho(\epsilon,\nu),\epsilon,\nu)
\end{split}\end{aligned}$$ and $$\begin{aligned}
\label{upthere2}\begin{split}
\Lambda(t,\psi,0,\epsilon,\nu)&=R(t,\psi,\rho(\epsilon,\nu),\epsilon,\nu)\\&
=\epsilon R_1(t,\psi,\rho(\epsilon,\nu),\epsilon,\nu)+
\sum_{j=2}^n\nu_j
R_j(t,\psi,\rho(\epsilon,\nu),\epsilon,\nu)\end{split}\end{aligned}$$ according to (\[thedefin\]).
For $j=1,\ldots, n$, the continuous functions $|R_j|$ and $|\epsilon\Psi_j|$ reach their maximum $C_j$ and $E_j$, respectively, on the compact set $[0,2\pi]\times \{0\}$ for $j=1,\ldots, n$. Then $$\left|\epsilon\Psi_j(t,\psi,\rho(\epsilon,\nu),\epsilon,\nu)\right|\leq E_j\quad\mbox{and}\quad \left|R_j(t,\psi,\rho(\epsilon,\nu),\epsilon,\nu)\right|\leq C_j$$ over $\Sigma_0^r$ for $j=1,\ldots,n$. According to (\[upthere1\]) and (\[upthere2\]), $$\begin{aligned}
|\Theta(t,\psi,0,\epsilon,\nu)|&\leq |\epsilon| E_1+\sum_{j=2}^n |\nu_j|E_j=Q_1(\epsilon,\nu) \\
|\Lambda(t,\psi,0,\epsilon,\nu)|&\leq |\epsilon| C_1
+\sum_{j=2}^n|\nu_j| C_j=Q_2(\epsilon,\nu)\end{aligned}$$ over $\Sigma^r_0$. Set $$\Xi(\epsilon,\nu)=\max\{Q_1(\epsilon,\nu),Q_2(\epsilon,\nu)\}.$$ Then $\Lambda$ and $\Theta$ are bounded by $\Xi(\epsilon,\nu)=O(\epsilon,\nu_2,\ldots,\nu_n)$ over $\Sigma^r_0$.
2. \[appH1H2\] Let $(\psi_1,x_1),(\psi_2,x_2)\in {\mathbb{R}}\times [-\sigma,\sigma]$ for $0\leq\sigma\leq\sigma_0$. By one of the mean value theorems, there exist points $(\psi^*,x^*),(\psi_*,x_*)\in [0,2\pi]\times [-\sigma,\sigma]$ on the line joining $(\psi_1,x_1)$ and $(\psi_2,x_2)$ such that $$\begin{aligned}
|\Theta(t,\psi_1,x_1,\epsilon,\nu)-\Theta(t,\psi_2,x_2,\epsilon,\nu) |&=|\widehat{\Theta}(t,\psi^*,x^*,\epsilon,\nu)| \big[|\psi_1-\psi_2|+|x_1-x_2|\big] \\
|\Lambda(t,\psi_1,x_1,\epsilon,\nu)-\Lambda(t,\psi_2,x_2,\epsilon,\nu)
|&=|\widehat{\Lambda}(t,\psi_*,x_*,\epsilon,\nu)|\big[|\psi_1-\psi_2|+|x_1-x_2|\big]
,\end{aligned}$$where $$\begin{aligned}
\widehat{\Theta}(t,\psi,x,\epsilon,\nu)&=D_x\Theta(t,\psi,x,\epsilon,\nu)+D_{\psi}\Theta(t,\psi,x,\epsilon,\nu) \\ &=x K_0^{\Psi}(x,\epsilon,\nu)+\epsilon K_1^{\Psi}(t,\psi,x,\epsilon,\nu)+\sum_{j=2}^n\nu_j K_j^{\Psi}(t,\psi,x,\epsilon,\nu) \\
\widehat{\Lambda}(t,\psi,x,\epsilon,\nu)&=D_x\Lambda(t,\psi,x,\epsilon,\nu)+D_{\psi}\Lambda(t,\psi,x,\epsilon,\nu)
\\ &=x K_0^{R}(x,\epsilon,\nu)+\epsilon
K_1^{R}(t,\psi,x,\epsilon,\nu)+\sum_{j=2}^n\nu_j
K_j^{R}(t,\psi,x,\epsilon,\nu), \end{aligned}$$ where $$\begin{aligned}
K_0^{\Psi}(x,\epsilon,\nu)&=\epsilon D_x B_2(x,\epsilon,\nu) \\ K_0^{R}(x,\epsilon,\nu)&=\epsilon \left(D_x B_1(x,\epsilon,\nu)x + 2B_1(x,\epsilon,\nu)\right)\\
K_1^{\Psi}(t,\psi,x,\epsilon,\nu)&=B_2(x,\epsilon,\nu)+\epsilon\left(D_x \Psi_1(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)+D_{\psi} \Psi_1(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)\right) \\ K_1^{R}(t,\psi,x,\epsilon,\nu)&=\epsilon\left(D_x R_1(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)+D_{\psi} R_1(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)\right) \\ \intertext{and}
K_j^{\Psi}(t,\psi,x,\epsilon,\nu)&=\epsilon\left(D_x \Psi_j(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)+D_{\psi} \Psi_j(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)\right) \\ K_j^{R}(t,\psi,x,\epsilon,\nu)&=\epsilon\left(D_x R_j(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)+D_{\psi} R_j(t,\psi,\rho(\epsilon,\nu)+x,\epsilon,\nu)\right),\end{aligned}$$ where $\Psi_j$ and $R_j$ are as in (\[thedefin\]). Since $\Theta$ and $\Lambda$ are continuously differentiable, $\widehat{\Theta}$ and $\widehat{\Lambda}$ are continuous on $\Sigma^r_{\sigma}$, as are $K_j^{\Psi}$ and $K_j^R$ for $j=0,\ldots, n$.
In particular, the functions $|K_j^{\Psi}|$ and $|K_j^{\Psi}|$ each reach their respective maximum $k_j^{\Psi}$ and $k_j^{R}$ on $\Sigma^r_{\sigma}$ for $j=0,\ldots, n$. Then, note that $|x^*|,|x_*|\leq \sigma$, $$\begin{aligned}
|\widehat{\Theta}(t,\psi^*,x^*,\epsilon,\nu)|&\leq
|x^*| |K_0^{\Psi}(x^*,\epsilon,\nu)|+|\epsilon|
|K_1^{\Psi}(t,\psi^*,x^*,\epsilon,\nu)|+\sum_{j=2}^n|\nu_j|
|K_j^{\Psi}(t,\psi^*,x^*,\epsilon,\nu)| \\ &\leq |x^*|
k_0^{\Psi}+|\epsilon| k_1^{\Psi}+\sum_{j=2}^n |\nu_j| k_j^{\Psi}
\\ &\leq \sigma k_0^{\Psi}+|\epsilon| k_1^{\Psi}+\sum_{j=2}^n
|\nu_j| k_j^{\Psi}=\theta(\epsilon,\nu,\sigma)\end{aligned}$$ and $$\begin{aligned}
|\widehat{\Lambda}(t,\psi_*,x_*,\epsilon,\nu)|&\leq
|x_*| |K_0^{R}(x_*,\epsilon,\nu)|+|\epsilon|
|K_1^{R}(t,\psi_*,x_*,\epsilon,\nu)|+\sum_{j=2}^n|\nu_j|
|K_j^{R}(t,\psi_*,x_*,\epsilon,\nu)| \\ &\leq |x_*|
k_0^{R}+|\epsilon| k_1^{R}+\sum_{j=2}^n |\nu_j| k_j^{R} \\ &\leq
\sigma k_0^{R}+|\epsilon| k_1^{R}+\sum_{j=2}^n |\nu_j|
k_j^{R}=\nu(\epsilon,\nu,\sigma).\end{aligned}$$ In particular $$\begin{aligned}
|\Theta(t,\psi_1,x_1,\epsilon,\nu)-\Theta(t,\psi_2,x_2,\epsilon,\nu) |&\leq \theta(\epsilon,\nu_2,\ldots,\nu_n,\sigma)\big[|\psi_1-\psi_2|+|x_1-x_2|\big] \\ |\Lambda(t,\psi_1,x_1,\epsilon,\nu)-\Lambda(t,\psi_2,x_2,\epsilon,\nu) |&\leq \nu(\epsilon,\nu_2,\ldots,\nu_n,\sigma)\big[|\psi_1-\psi_2|+|x_1-x_2|\big], \end{aligned}$$ where $\theta(\epsilon,\nu,\sigma),\nu(\epsilon,\nu,\sigma)=O(\epsilon,\nu_2,\ldots,\nu_n,\sigma)$. Hence $\Theta$ and $\Lambda$ are Lipschitz in Hale’s sense.
This completes the proof.
[^1]: In this paper, we use the terms ‘generic’ and ‘typical’ interchangeably: the set of coefficient values for which the anchoring/repelling property fails to hold has measure zero in the complete coefficient space.
|
---
abstract: 'Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.'
author:
- |
Janusz Mierczyński [^1]\
Institute of Mathematics\
Wroc[ł]{}aw University of Technology\
Wybrzeże Wyspiańskiego 27\
PL-50-370 Wroc[ł]{}aw\
Poland\
\
and\
\
Wenxian Shen [^2]\
Department of Mathematics\
Auburn University\
Auburn University, AL 36849\
USA
title: 'Time averaging for nonautonomous/random linear parabolic equations [^3]'
---
[**AMS Subject Classification.**]{} Primary: 35K15, 35P15; Secondary: 35R60, 37H15
[**Key Words.**]{} Nonautonomous linear partial differential equation of parabolic type, random linear partial differential equation of parabolic type, principal spectrum, principal Lyapunov exponent, averaging.
Introduction
============
It is well known that parabolic equations can be used to model many evolution processes in science and engineering. Parabolic equations with general time dependence are gaining more and more attention since they can take various time variations of the underlying processes into account in modeling the processes. A great amount of research work has been carried out toward the existence, uniqueness, and regularity of solutions of general linear, semilinear, quasilinear parabolic equations (see [@Ama], [@Ama1], [@Dan2], [@DL], [@Fri1], [@LaSoUr], [@Lie], [@Yag], etc.). As a basic tool for nonlinear problems, it is of great significance to study the spectral theory for linear parabolic equations.
Spectral theory, in particular, principal spectrum theory (i.e.,principal eigenvalues and principal eigenfunctions theory) for time independent and time periodic parabolic equations is well understood (see, for example, [@Hes]). For such an equation, its principal eigenvalue provides the growth rate of the evolution operator and hence a least upper bound of the growth rates of all the solutions. Recently much effort has been devoted to the extension of principal eigenvalue and principal eigenfunction theory of time independent and periodic parabolic equations to general time dependent and random parabolic equations. See, for example, [@Hu1], [@Hu2], [@HuPo], [@HuPoSa1], [@HuPoSa2], [@HuShVi], [@Mi1], [@Mi2], [@MiSh1], [@Po], [@PoTer], [@ShVi], etc.
In the current paper, we focus on time dependent parabolic equations of the form $$\label{nonauton-eq1}
\begin{cases}
\disp\frac{\p u}{\p t} = \sum_{i,j=1}^{N} a_{ij}(x)\frac{\p^{2}u}{\p
x_{i} \p x_{j}} + \sum_{i=1}^{N}
a_i(x)\frac{\p u}{\p x_i} + c(t,x)u, \quad & t > 0,\ x \in D, \\[1.5ex]
\mathcal{B}(t)u = 0, \quad & t > 0,\ x \in \p D,
\end{cases}$$ where $D\subset {\ensuremath{\mathbb{R}}}^N$, $$\mathcal{B}(t)u =
\begin{cases}
u & \text{(Dirichlet)}
\\[1.5ex]
\disp \sum_{i=1}^{N} b_i(x)\frac{\p u}{\p x_i} & \text{(Neumann)} \\[1.5ex]
\disp \sum_{i=1}^{N} b_i(x)\frac{\p u}{\p x_i} + d(t,x)u &
\text{(Robin)},
\end{cases}$$ and random parabolic equations of the form $$\label{random-eq1}
\begin{cases}
\disp\frac{\p u}{\p t} = \sum_{i,j=1}^{N} a_{ij}(x)\frac{\p^{2}u}{\p
x_{i} \p x_{j}} + \sum_{i=1}^{N} a_i(x) \frac{\p u}{\p x_i} +
c(\theta_{t}\omega,x)u, \quad & t > 0,\ x \in D,
\\[1.5ex]
\mathcal{B}(\theta_t\omega)u = 0, \quad & t > 0,\ x \in \p D,
\end{cases}$$ where $$\mathcal{B}(\theta_t\omega)u =
\begin{cases}
u & \text{(Dirichlet)}
\\[1.5ex]
\disp
\sum_{i=1}^{N} b_i(x) \frac{\p u}{\p x_i} & \text{(Neumann)} \\[1.5ex]
\disp \sum_{i=1}^{N} b_i(x) \frac{\p u}{\p x_i} +
d(\theta_{t}\omega,x)u & \text{(Robin)},
\end{cases}$$ and $({(\Omega,\mathcal{F},{\ensuremath{\mathbb{P}}})}, \{\theta_t\}_{t\in{\ensuremath{\mathbb{R}}}})$ is an ergodic metric dynamical system (see Section 2 for definition).
Our objective is to study the influence of time variations of the zeroth order terms on the so-called principal spectrum and principal Lyapunov exponent of and (which are analogs of principal eigenvalue of time independent and periodic parabolic equations), respectively. To do so, we first study the existence and uniqueness of globally positive solutions via the skew-product semiflows on (a subspace of) $C^1(\bar{D})$ generated by and . Next we define the principal spectrum and principal Lyapunov exponent of and in terms of the globally positive solutions. We then compare the principal spectrum and principal Lyapunov exponent of and with those of their averaged equations.
To be more precise, we first introduce some notations and state some basic assumptions.
In the Dirichlet or Neumann type boundary conditions we assume $d(\cdot,\cdot) \equiv 0$. In the case of we write $c^{\omega}(t,x)$ for $c(\theta_{t}\omega,x)$, $\omega \in
\Omega$, $t \in {\ensuremath{\mathbb{R}}}$, $x \in \bar{D}$, and $d^{\omega}(t,x)$ for $d(\theta_{t}\omega,x)$, $\omega \in \Omega$, $t \in {\ensuremath{\mathbb{R}}}$, $x \in \p
D$.
For $m_1, m_2 \in {\ensuremath{\mathbb{N}}}\cup \{0\}$ and $\beta \in [0,1)$ the symbol $C^{m_1+\beta, m_2+\beta}({\ensuremath{\mathbb{R}}}\times \bar{D})$ denotes the Banach space consisting of functions $h \colon {\ensuremath{\mathbb{R}}}\times \bar{D} \to {\ensuremath{\mathbb{R}}}$ whose mixed derivatives of order up to $m_1$ in $t$ and up to $m_2$ in $x$ are bounded, and whose mixed derivatives of order $m_1$ in $t$ and $m_2$ in $x$ are globally Hölder continuous with exponent $\beta$, uniformly in $(t,x) \in {\ensuremath{\mathbb{R}}}\times \bar{D}$ (provided that the boundary $\p D$ of $D$ is of class $C^{m_2+\beta}$, at least).
Similarly, for $m_1, m_2 \in {\ensuremath{\mathbb{N}}}\cup \{0\}$ and $\beta \in [0,1)$ the symbol $C^{m_1+\beta, m_2+\beta}({\ensuremath{\mathbb{R}}}\times \p D)$ denotes the Banach space consisting of functions $h \colon {\ensuremath{\mathbb{R}}}\times \p D \to
{\ensuremath{\mathbb{R}}}$ whose mixed derivatives of order up to $m_1$ in $t$ and up to $m_2$ in $x$ are bounded, and whose mixed derivatives of order $m_1$ in $t$ and $m_2$ in $x$ are globally Hölder continuous with exponent $\beta$, uniformly in $(t,x) \in {\ensuremath{\mathbb{R}}}\times \p D$ (provided that $\p D$ is of class $C^{m_2+\beta}$, at least).
Throughout the paper, we assume the following smoothness conditions on the domain and the coefficients in and (the nonsmooth case will be considered in the monograph [@MiSh2]).
- $D \subset {\ensuremath{\mathbb{R}}}^{N}$ is a bounded domain, with boundary $\p D$ of class $C^{3+\alpha}$, for some $\alpha > 0$.
- The functions $a_{ij}$, $a_{i}$ belong to $C^{2}(\bar{D})$ and the functions $b_i$ belong to $C^2(\p D)$.
- - $c \in C^{2+\alpha,1+\alpha}({\ensuremath{\mathbb{R}}}\times \bar{D})$ (in the case of ),
- $c^{\omega} \in C^{2+\alpha,1+\alpha}({\ensuremath{\mathbb{R}}}\times \bar{D})$ for all $\omega \in \Omega$, with the $C^{2+\alpha,1+\alpha}({\ensuremath{\mathbb{R}}}\times \bar{D})$-norm bounded uniformly in $\omega \in \Omega$ (in the case of ).
- - $d \in C^{2+\alpha,3+\alpha}({\ensuremath{\mathbb{R}}}\times \p D)$ (in the case of ),
- $d^{\omega} \in C^{2+\alpha,3+\alpha}({\ensuremath{\mathbb{R}}}\times \p D)$ for all $\omega \in \Omega$, with the $C^{2+\alpha,3+\alpha}({\ensuremath{\mathbb{R}}}\times \p D)$-norm bounded uniformly in $\omega \in \Omega$ (in the case of ).
We also assume the following uniform ellipticity condition and the complementing boundary condition:
- $a_{ij}(x) = a_{ji}(x)$ for $i,j = 1,2,\dots,N$ and $x \in \bar
D$, and there is $\alpha_0 > 0$ such that $$\sum_{i,j=1}^{N} a_{ij}(x)\,\xi_i\,\xi_j \ge \alpha_{0}
\sum_{i=1}^{N}\xi_i^2, \quad x \in \bar{D}, \ \xi \in {\ensuremath{\mathbb{R}}}^{N}.$$
- There is $\alpha_1 > 0$ such that $$\sum_{i=1}^{N} b_{i}(x)\nu_{i}(x) \ge \alpha_{1}, \quad x \in \p D,$$ where $\nu(x) = (\nu_1(x),\nu_2(x),\cdots, \nu_N(x))$ is the unit outer normal vector of $\p D$ at $x \in \p D$.
In the case of let $$\label{Y(a)-def}
Y(c,d) := \operatorname{cl}\{\,(c,d) \cdot t:t \in {\ensuremath{\mathbb{R}}}\,\},$$ be equipped with the open-compact topology, where $((c, d) \cdot
t)(s,x):= (c(s+t,x),d(s+t,x))$, $s \in {\ensuremath{\mathbb{R}}}$, $x \in \p D$, and the closure is taken in the open-compact topology of ${\ensuremath{\mathbb{R}}}\times \bar D$.
In the case of let $$\label{Y(omega)-def}
Y(\Omega) := \operatorname{cl}\{\,(c^{\omega},d^{\omega}):\omega \in \Omega\,\}$$ be equipped with the open-compact topology, where the closure is also taken in the open-compact topology. We will write $Y$ instead of $Y(c,d)$ (for the case of ) or instead of $Y(\Omega)$ (for the case of ).
For given $(\tilde c,\tilde d) \in Y$ and $u_0 \in L_p(D)$, consider $$\label{nonauton-eq2}
\begin{cases}
\disp\frac{\p u}{\p t} = \sum_{i,j=1}^{N} a_{ij}(x)\frac{\p^{2}u}{\p
x_{i} \p x_{j}} + \sum_{i=1}^{N}
a_i(x)\frac{\p u}{\p x_i} + \tilde c(t,x)u, \quad & t > 0,\ x \in D, \\[1.5ex]
\tilde{\mathcal{B}}(t)u = 0, \quad & t > 0,\ x \in \p D,
\end{cases}$$ where $$\tilde{\mathcal{B}}(t)u =
\begin{cases}
u & \text{(Dirichlet)}
\\[1.5ex]
\disp \sum_{i=1}^{N} b_i(x)\frac{\p u}{\p x_i} & \text{(Neumann)} \\[1.5ex]
\disp \sum_{i=1}^{N} b_i(x)\frac{\p u}{\p x_i} + \tilde d(t,x)u &
\text{(Robin)},
\end{cases}$$ with the initial condition $$\label{initial-condition}
u(0,x) = u_0(x), \quad x \in D.$$ Applying the theory presented by H. Amann in [@Ama], we have that + has a unique [*$L_{p}(D)$-solution*]{} $U_{(\tilde{c},\tilde{d}),p}(\cdot,0)u_0
\colon [0,\infty) \to L_{p}(D)$ ($p > 1$) (see Proposition \[existence-prop\]).
Note that $U_{(\tilde{c},\tilde{d}),p}(\cdot,0)u_0$ is also a classical solution of + (see Section \[skew-product\] for more detail). We may therefore write $U_{(\tilde{c},\tilde{d}),p}(t,0)u_0$ as $U_{(\tilde{c},\tilde{d})}(t,0)u_0$ for $u_0 \in L_p(D)$. In the present paper we further assume the following continuous dependence.
- For any $T > 0$ the mapping $$[\, Y \ni (\tilde{c},\tilde{d}) \mapsto [\,[0,T] \ni t \mapsto
U_{(\tilde{c},\tilde{d})}(t,0)\,] \in
B([0,T],\mathcal{L}(L_2(D),L_2(D))) \,]$$ is continuous, where $\mathcal{L}(L_2(D),L_2(D))$ represents the space of all bounded linear operators from $L_2(D)$ into itself, endowed with the norm topology, and $B(\cdot,\cdot)$ stands for the Banach space of bounded functions, endowed with the supremum norm.
It should be pointed out that in [@Ama2] and [@Pen] conditions, for some special cases (for example, the Dirichlet boundary condition case and the case with infinitely differentiable coefficients), are given that guarantee the continuous dependence of $[\,[0,T] \ni t \mapsto U_{(\tilde{c},\tilde{d})}(t,0)\,] \in
B([0,T],\mathcal{L}(L_2(D),L_2(D)))$ on the coefficients. For the general case, the continuous dependence of $[\,[0,T] \ni t \mapsto
U_{(\tilde{c},\tilde{d})}(t,0)\,] \in
B([0,T],\mathcal{L}(L_2(D),L_2(D)))$ on the coefficients is not covered in [@Ama2] and [@Pen]. We will not investigate the conditions under which (A7) is satisfied in this paper.
Then () generates the following skew-product semiflow (see Section \[skew-product\] for detail) $$\Pi_t \colon X \times Y \to X \times Y,$$ $$\Pi_t(u_0,(\tilde{c},\tilde{d})) =
(U_{(\tilde{c},\tilde{d})}(t,0)u_0,(\tilde{c},\tilde{d}) \cdot t),$$ where $$\label{X-eq}
X :=
\begin{cases}
\overset{\circ}{C^{1}}(\bar{D}) & \quad \text{(Dirichlet)} \\[2ex]
C^{1}(\bar{D}) & \quad \text{(Neumann or Robin),}
\end{cases}$$ $$\overset{\circ}{C^{1}}(\bar{D}) := \{\,u \in C^{1}(\bar{D}): u(x) =
0 \text{ for each } x \in \p D\,\}.$$
Throughout the paper, we denote ${\ensuremath{\lVert\cdot\rVert}}$ as the norm in $L_2(D)$ (see Section 2 for other notations).
Among others, we prove
**
- $\Pi_t$ is strongly monotone see Theorem \[strong-positivity-class\].
- has a unique up to multiplication by positive scalars globally positive solution $v(t,x;\tilde{c},\tilde{d})$ which is an analog of a principal eigenfunction see Theorem \[globally-positive-existence\] we denote $v((\tilde{c},\tilde{d}))(\cdot)$ as $v(0,\cdot;\tilde{c},\tilde{d})/
{\ensuremath{\lVertv(0,\cdot;\tilde{c},\tilde{d})\rVert}}$.
- Consider . Then the set $\Sigma(c,d)$ consisting of all limits $$\label{nonauton-principal-spectrum-eq0}
\lim\limits_{n\to\infty} \frac{\ln{\ensuremath{\lVertU_{(c,d) \cdot
S_n}(T_n-S_n,0)v((c,d) \cdot S_n)\rVert}}} {T_n-S_n}$$ where $T_n - S_n \to \infty$ as $n \to \infty$, is a compact interval see Theorem \[principal-spectrum-thm2\].
- Consider . Then for a.e. $\omega \in \Omega$ $$\label{random-lyapunov-exponent-eq0}
\lim\limits_{T\to\infty}\frac{\ln{\ensuremath{\lVertU_\omega(T,0)v(\omega)\rVert}}}{T} =
{\ensuremath{\mathrm{const}}}$$ where $U_\omega (t,0) = U_{(c^\omega,d^\omega)}(t,0)$ and $v(\omega) = v((c^\omega,d^\omega))$ see Theorem \[principal-exponent-thm\].
Denote the compact interval in 3) by $[{\ensuremath{\lambda_{\mathrm{inf}}}}(c,d),
{\ensuremath{\lambda_{\mathrm{sup}}}}(c,d)]$ and the constant in 4) by $\lambda(c,d)$. We call $[{\ensuremath{\lambda_{\mathrm{inf}}}}(c,d),{\ensuremath{\lambda_{\mathrm{sup}}}}(c,d)]$ the [*principal spectrum*]{} of (see Definition \[principal-spectrum-def\]) and call $\lambda(c,d)$ the [*principal Lyapunov exponent*]{} of (see Definition \[principal-exponent-def\]).
Observe that if $c(t,x)$ and $d(t,x)$ in are independent of $t$ or are periodic in $t$, then ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d)( =
{\ensuremath{\lambda_{\mathrm{sup}}}}(c,d))$ is the principal eigenvalue of and $v(t,\cdot;c,d)$ is an eigenfunction associated with ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d)$ (called a principal eigenfunction). As in the time independent and periodic cases, the principal spectrum of and principal Lyapunov exponent of provide upper bounds of growth rates of the solutions of and , respectively. This can indeed be easily seen from the fact that $$\begin{aligned}
\lim_{n\to\infty} \frac{\ln{\ensuremath{\lVertU_{(c,d) \cdot
S_n}(T_n-S_n,0)v((c,d) \cdot S_n)\rVert}}} {T_n-S_n} & =
\lim_{n\to\infty}\frac{\ln{\ensuremath{\lVertU_{(c,d)\cdot
S_n}(T_n-S_n,0)\rVert}}}{T_n-S_n} \\
& = \lim_{n\to \infty}\frac{\ln{\ensuremath{\lVertU_{(c,d)\cdot
S_n}(T_n-S_n,0)u_0\rVert}}}{T_n-S_n}\end{aligned}$$ for any nontrivial $u_0\in X$ with $u_0(x) \ge 0$ for $x \in D$ as long as the limits exist (the existence of one of the limits implies the existence of the others), and $$\begin{aligned}
\lim_{T\to\infty} \frac{\ln{\ensuremath{\lVertU_\omega(T,0)v(\omega)\rVert}}}{T} & =
\lim_{T\to\infty} \frac{\ln{\ensuremath{\lVertU_{\omega}(T,0)\rVert}}}{T} \\
& = \lim_{T\to\infty} \frac{\ln{\ensuremath{\lVertU_{\omega}(T,0)u_0\rVert}}}{T}\end{aligned}$$ for any nontrivial $u_0 \in X$ with $u_0(x) \ge 0$ for $x \in D$ as long as the limits exist (again the existence of one of the limits implies the existence of the others) (this fact follows from Theorem \[globally-positive-existence\]).
We remark that the existence and uniqueness of globally positive solutions to nonautonomous parabolic equations with time independent boundary conditions were studied in [@Mi1], [@Mi2], [@Po]. In [@Hu1] the author studied the uniqueness of globally positive solutions to nonautonomous parabolic equations with time dependent boundary conditions. When the boundary conditions are time independent, the results 3) and 4) are proved in [@MiSh1]. The results 3), 4), and the existence part of 2) for time dependent boundary conditions are new. The strong monotonicity result 1) basically follows from [@Ama3 Theorem 11.6] and strongly maximum principal and the Hopf boundary point principle for classical solutions of parabolic equations.
We now consider the averaged equations of and in the following sense:
In the case of we call $(\hat{c}(\cdot),
\hat{d}(\cdot))$ an [*averaged function*]{} of $(c,d)$ if $$\hat{c}(x) = \lim_{n\to\infty}\frac{1}{T_n-S_n}
\int_{S_n}^{T_n}c(t,x)\,dt \quad\text{for } x \in D$$ and $$\hat{d}(x) =
\lim_{n\to\infty}\frac{1}{T_n-S_n}\int_{S_n}^{T_n}d(t,x)\,dt
\quad\text{for } x \in \p D$$ for some $T_n - S_n \to \infty$, where the limit is uniform in $x \in
\bar{D}$ (resp. in $x \in \p D$).
In the case of we call $(\hat{c}(\cdot),
\hat{d}(\cdot))$ the [*averaged function*]{} of $(c,d)$ if $$\hat{c}(x) = \int_\Omega c(\omega,x)\,d{\ensuremath{\mathbb{P}}}(\omega) \quad \text{for }
x \in D$$ and $$\hat{d}(x) = \int_\Omega d(\omega,x)\,d{\ensuremath{\mathbb{P}}}(\omega) \quad \text{for }
x \in \p D.$$
The equation $$\label{nonauton-or-random-avg}
\begin{cases}
\disp\frac{\p u}{\p t} = \sum_{i,j=1}^{N} a_{ij}(x)\frac{\p^{2}u}{\p
x_{i} \p x_{j}} + \sum_{i=1}^{N}
a_i(x)\frac{\p u}{\p x_i} + \hat{c}(x)u, \quad & x \in D, \\[1.5ex]
\hat{\mathcal{B}}u = 0, \quad & x \in \p D,
\end{cases}$$ where $$\hat{\mathcal{B}}u =
\begin{cases}
u & \text{(Dirichlet)}
\\[1.5ex]
\disp \sum_{i=1}^{N} b_i(x)\frac{\p u}{\p x_i} & \text{(Neumann)}\\[1.5ex]
\disp \sum_{i=1}^{N} b_i(x)\frac{\p u}{\p x_i}+ \hat{d}(x)u &
\text{(Robin)},
\end{cases}$$ is called [*an averaged equation*]{} of ([*the averaged equation*]{} of ) if $(\hat{c},\hat{d})$ is an averaged function of $(c,d)$ (the averaged function of $(c,d)$).
Denote $\lambda(\hat{c},\hat{d})$ to be the principal eigenvalue of . We then have the following main results of the paper.
Hence time variations cannot reduce the principal spectrum and principal Lyapunov exponent (or the principal eigenvalues of the time averaged equations give lower bounds of principal spectrum and principal Lyapunov exponent of non-averaged equations). Indeed, the time variations increase the principal spectrum and principal Lyapunov exponents except in the degenerate cases. In the biological context these results mean that invasion by a new species (see [@CC], p. 220) is always easier in the time-dependent case or that time variations favor persistence (viewing both and as linear population growth models, then by 5), positive solutions of all averaged equations of bounded away from zero implies positive solutions of also bounded away from zero, but not vice versa in general).
It should be pointed out that the results 5), 6) have been proved in [@HuShVi] and [@MiSh1] when the boundary conditions are time independent. They are new when the boundary conditions are time dependent and the proof presented in this paper is not the same as those in [@HuShVi] and [@MiSh1].
It should be also pointed out that the results 1)–4) apply to fully time dependent/random parabolic equations (i.e., equations in which all the coefficients can depend on $t$/$\theta_t\omega$). But 5) and 6) are mainly for equations of form and , respectively.
The rest of the paper is organized as follows. In Section \[preliminaries\] we collect several elementary lemmas and introduce some standing notations for future reference. We review some existence and regularity theorems and construct the skew-product semiflow generated by and in Section \[skew-product\]. Section \[strong-monotonicity\] is devoted to the study of the monotonicity of the skew-product semiflow constructed in Section \[skew-product\] and the existence of global positive solutions of . Definition and basic properties of principal spectrum and principal Lyapunov exponents are discussed in Section \[principal-spectrum\]. We prove the time averaging results in Section \[main-results\].
The authors are grateful to the referees for their remarks.
Elementary Lemmas and notations {#preliminaries}
===============================
We collect first, for further reference, some elementary results.
First of all, let $Z$ be a compact metric space and ${\mathcal B}(Z)$ be the Borel $\sigma$-algebra of $Z$. $(Z,{\ensuremath{\mathbb{R}}}) :=
(Z,\{\sigma_t\}_{t\in{\ensuremath{\mathbb{R}}}})$ is called a [*compact flow*]{} if $\sigma_t \colon Z \to Z$ ($t \in {\ensuremath{\mathbb{R}}}$) satisfies: $[\, (t,z)
\mapsto \sigma_{t}z\,]$ is jointly continuous in $(t,z) \in {\ensuremath{\mathbb{R}}}\times Z$, $\sigma_0 = \mathrm{id}$, and $\sigma_s \circ \sigma_t =
\sigma_{s+t}$ for any $s, t \in {\ensuremath{\mathbb{R}}}$. We may write $z \cdot t$ or $(z,t)$ for $\sigma_{t}z$. A probability measure $\mu$ on $(Z,
\mathcal{B}(Z))$ is called an [*invariant measure*]{} for $(Z,\{\sigma_t\}_{t\in{\ensuremath{\mathbb{R}}}})$ if for any $E \in \mathcal{B}(Z)$ and any $t \in {\ensuremath{\mathbb{R}}}$, $\mu(\sigma_t(E)) = \mu(E)$. An invariant measure $\mu$ for $(Z,\{\sigma_t\}_{t\in{\ensuremath{\mathbb{R}}}})$ is said to be [*ergodic*]{} if for any $E \in \mathcal{B}(Z)$ satisfying $\mu(\sigma_t^{-1}(E)
\bigtriangleup E) = 0$ for all $t \in {\ensuremath{\mathbb{R}}}$, $\mu(E) = 1$ or $\mu(E) =
0$. The compact flow $(Z,\{\sigma_t\}_{t\in{\ensuremath{\mathbb{R}}}})$ is said to be [*uniquely ergodic*]{} if it has a unique invariant measure (in such case, the unique invariant measure is necessarily ergodic). We say that $(Z,\{\sigma_t\}_{t\in{\ensuremath{\mathbb{R}}}})$ is [*minimal*]{} or [*recurrent*]{} if for any $z \in Z$, the orbit $\{\,\sigma_{t}z:
t\in{\ensuremath{\mathbb{R}}}\,\}$ is dense in $Z$.
Let ${(\Omega,\mathcal{F},{\ensuremath{\mathbb{P}}})}$ be a probability space, $\{\theta_t\}_{t\in{\ensuremath{\mathbb{R}}}}$ be a family of ${\ensuremath{\mathbb{P}}}$-preserving transformations (i.e., ${\ensuremath{\mathbb{P}}}(\theta_t^{-1}(F)) = {\ensuremath{\mathbb{P}}}(F)$ for any $F \in \mathcal{F}$ and $t
\in {\ensuremath{\mathbb{R}}}$) such that $(t,\omega) \mapsto \theta_{t}\omega$ is measurable, $\theta_0 = \mathrm{id}$, and $\theta_{t+s} = \theta_{t}
\circ \theta_{s}$ for all $t, s \in {\ensuremath{\mathbb{R}}}$. Thus $\{\theta_t\}_{t\in{\ensuremath{\mathbb{R}}}}$ is a flow on $\Omega$ and $({(\Omega,\mathcal{F},{\ensuremath{\mathbb{P}}})},\{\theta_t\}_{t\in{\ensuremath{\mathbb{R}}}})$ is called a [*metric dynamical system*]{}. $({(\Omega,\mathcal{F},{\ensuremath{\mathbb{P}}})},\{\theta_t\}_{t\in{\ensuremath{\mathbb{R}}}})$ is said to be [*ergodic*]{} if for any $F \in \mathcal{F}$ satisfying ${\ensuremath{\mathbb{P}}}(\theta_t^{-1}(F) \bigtriangleup F) = 0$ for any $t \in {\ensuremath{\mathbb{R}}}$, ${\ensuremath{\mathbb{P}}}(F) = 1$ or ${\ensuremath{\mathbb{P}}}(F) = 0$.
In the following, we assume that $({(\Omega,\mathcal{F},{\ensuremath{\mathbb{P}}})},\{\theta_t\}_{t\in{\ensuremath{\mathbb{R}}}})$ is an ergodic metric dynamical system.
\[ch5-pre-holder-lm\]
- Let $h_i \colon [0,T] \times D \to {\ensuremath{\mathbb{R}}}$ $i = 1,2,\dots,
N$ be square-integrable in $t \in [0,T]$ and $a_{ij} =
a_{ji} \colon D \to {\ensuremath{\mathbb{R}}}$ $i, j = 1,2,\dots,N$ satisfy $$\sum_{i,j=1}^N a_{ij}(x)\xi_i\xi_j \ge \alpha_0 \sum_{i=1}^N \xi_i^2$$ for some $\alpha_0 > 0$ and any $x \in \bar{D}$, $\xi =
(\xi_1,\xi_2,\dots,\xi_N)^{\top} \in {\ensuremath{\mathbb{R}}}^N$. Then for any $x \in
D$, $$\begin{aligned}
& \sum_{i,j=1}^N a_{ij}(x)\; \frac {1}{T}\int_0^T h_i(t,x)\,dt \;
\frac{1}{T}\int_0^T h_j(t,x)\,dt \\
\le & \sum_{i,j=1}^N a_{ij}(x) \; \frac{1}{T}\int_0^T
h_i(t,x)h_j(t,x)\,dt.\end{aligned}$$ Moreover, the equality holds at some $x_0 \in D$ if and only if $h_i(t,x_0) = \tilde{h}_i(x_0)$ for some $\tilde{h}_i(x_0)$ $i = 1,2,\dots,N$ and a.e. $t \in [0,T]$.
- Let $h_i \colon \Omega \times D \to {\ensuremath{\mathbb{R}}}$ $i =
1,2,\dots,N$ be square-integrable in $\omega \in
\Omega$ and $a_{ij} = a_{ji} \colon D \to {\ensuremath{\mathbb{R}}}$ $(i,j =
1,2,\dots,N)$ satisfy $$\sum_{i,j=1}^Na_{ij}(x)\xi_i\xi_j \ge \alpha_0 \sum_{i=1}^N\xi_i^2$$ for some $\alpha_0 > 0$ and any $x \in \bar{D}$, $\xi =
(\xi_1,\xi_2,\dots,\xi_N)^\top \in {\ensuremath{\mathbb{R}}}^N$. Then for any $x \in
D$, $$\begin{aligned}
& \sum_{i,j=1}^N a_{ij}(x)\int_{\Omega}
h_i(\omega,x)\,d{\ensuremath{\mathbb{P}}}(\omega)\int_\Omega h_j(\omega,x)\,d{\ensuremath{\mathbb{P}}}(\omega)
\\
\le & \sum_{i,j=1}^{N} a_{ij}(x)\int_\Omega h_i(\omega,x)
h_j(\omega,x)\,d{\ensuremath{\mathbb{P}}}(\omega).\end{aligned}$$ Moreover, the equality holds at some $x_0 \in D$ if and only if $h_i(\omega,x_0) = \tilde{h}_i(x_0)$ for some $\tilde{h}_i(x_0)$ $i = 1,2,\dots, N$ and a.e. $\omega \in
\Omega$.
See [@HuShVi Lemma 2.2] for (1) and [@MiSh1 Lemma 3.5] for (2).
\[ch5-pre-ergodic-lm\] Let $h \in L^1{(\Omega,\mathcal{F},{\ensuremath{\mathbb{P}}})}$. Then there is an invariant measurable set $\Omega_0 \subset \Omega$ such that ${\ensuremath{\mathbb{P}}}(\Omega_0) = 1$ and $$\lim\limits_{T\to\infty} \frac{1}{T}\int\limits_0^{T}
h(\theta_t\omega)\,dt = \int\limits_\Omega h(\cdot)\,d{\ensuremath{\mathbb{P}}}(\cdot)$$ for any $\omega \in \Omega_0$.
See [@Arn] or references therein.
\[averaging-uniform\] Assume that $h \colon \Omega \times D \to {\ensuremath{\mathbb{R}}}$ resp. $h
\colon \Omega \times \bar{D} \to {\ensuremath{\mathbb{R}}}$ has the following properties:
1. $h(\cdot,x)$ belongs to $L^{1}(\Omega)$, for each $x \in D$,
2. for each $x \in D$ resp. $x \in \bar{D}$ and each $\epsilon > 0$ there is $\delta > 0$ such that if $y \in D$ resp. $y \in \bar{D}$, $\omega \in \Omega$ and ${\ensuremath{\lvertx - y\rvert}} < \delta$ then ${\ensuremath{\lverth(\omega,x) - h(\omega,y)\rvert}}
< \epsilon$, where ${\ensuremath{\lvert\cdot\rvert}}$ stands for the norm in ${\ensuremath{\mathbb{R}}}^{N}$ or the absolute value, depending on the context.
Denote, for each $x \in D$ resp. $x \in \bar{D}$, $$\hat{h}(x) := \int\limits_{\Omega} h(\omega,x)\, d{\ensuremath{\mathbb{P}}}(\omega).$$ Then
1. for any $x \in D$ resp. $x \in \bar{D}$ and any $\epsilon > 0$ there is $\delta
> 0$ the same as in such that if $y \in
D$ resp. $y \in \bar{D}$, $\omega \in \Omega$ and ${\ensuremath{\lvertx - y\rvert}} < \delta$ then ${\ensuremath{\lvert\hat{h}(x) - \hat{h}(y)\rvert}} <
\epsilon$,
2. there is a measurable $\Omega' \subset \Omega$ with ${\ensuremath{\mathbb{P}}}(\Omega')
= 1$ such that $$\lim\limits_{T\to\infty} \frac{1}{T} \int\limits_{0}^{T}
h(\theta_{t}\omega,x)\, dt = \hat{h}(x)$$ for all $\omega \in \Omega'$ and all $x \in D$ resp. $x
\in \bar{D}$. Moreover the convergence is uniform in $x
\in D_0$, for any compact $D_0 \Subset D$ resp. uniform in $x \in \bar{D}$.
Part (a) follows easily by the fact that the continuity is uniform in $\omega \in \Omega$. To prove (b), take a countable dense set $\{x_l\}_{l=1}^{\infty}$ in $D$. By Birkhoff’s Ergodic Theorem (Lemma \[ch5-pre-ergodic-lm\]) for each $l \in {\ensuremath{\mathbb{N}}}$ there is a measurable $\Omega_l \subset \Omega$ with ${\ensuremath{\mathbb{P}}}(\Omega_l) = 1$ such that $$\lim\limits_{T\to\infty} \frac{1}{T} \int\limits_{0}^{T}
h(\theta_{t}\omega,x_l)\, dt = \hat{h}(x_l)$$ for each $\omega \in \Omega_l$. Take $\Omega' :=
\bigcap_{l=1}^{\infty} \Omega_l$.
Fix $x \in D$ (resp. $x \in \bar{D}$). For $\epsilon > 0$ take $\delta > 0$ such that if ${\ensuremath{\lvertx - y\rvert}} < \delta$ then ${\ensuremath{\lverth(\omega,x) - h(\omega,y)\rvert}} < \epsilon/3$ and ${\ensuremath{\lvert\hat{h}(x) -
\hat{h}(y)\rvert}} < \epsilon/3$. Let $x_l$ be such that ${\ensuremath{\lvertx - x_l\rvert}} <
\delta$, and let $T_0 > 0$ be such that $$\left\lvert \frac{1}{T} \int\limits_{0}^{T}
h(\theta_{t}\omega,x_l)\, dt - \hat{h}(x_l) \right\rvert <
\frac{\epsilon}{3}$$ for all $T > T_0$. Then $$\left\lvert \frac{1}{T} \int\limits_{0}^{T} h(\theta_{t}\omega,x)\,
dt - \hat{h}(x) \right\rvert < \epsilon$$ for all $T > T_0$. (b) then follows.
\[averaging-derivative\] Assume that $h \colon \Omega \times D \to {\ensuremath{\mathbb{R}}}$ resp. $h
\colon \Omega \times \bar{D} \to {\ensuremath{\mathbb{R}}}$ has the following properties:
1. $h(\cdot,x)$ belongs to $L^{1}(\Omega)$, for each $x \in D$,
2. $(\p h/\p x_i)(\omega,x)$ exists for each $\omega \in \Omega$ and each $x \in D$ resp. each $x \in \bar{D}$; further, $(\p h/\p x_i)(\cdot,x)$ belongs to $L^{1}(\Omega)$, for each $x \in D$,
3. there exists $\alpha \in (0,1]$ such that for each $x \in D$ resp. each $x \in \bar{D}$ there are $L > 0$ and $\delta_0 > 0$ with the property that $$\left\lvert \frac{\p h}{\p x_i}(\omega,x) - \frac{\p h}{\p
x_i}(\omega,y) \right\rvert \le L {\ensuremath{\lvertx - y\rvert}}^{\alpha}$$ for any $\omega \in \Omega$ and any $y \in D$ resp. any $y \in \bar{D}$ with $|x - y| < \delta_0$.
Denote, for each $x \in D$ resp. $x \in \bar{D}$, $$\hat{h}(x) := \int\limits_{\Omega} h(\omega,x)\, d{\ensuremath{\mathbb{P}}}(\omega).$$ Then
1. for each $x \in D$ resp. each $x \in \bar{D}$ the derivative $(\p \hat{h}/\p x_i)(x)$ exists, and the equality $$\frac{\p \hat{h}}{\p x_i}(x) = \int\limits_{\Omega} \frac{\p h}{\p
x_i}(\omega,x)\, d{\ensuremath{\mathbb{P}}}(\omega)$$ holds,
2. for each $x \in D$ resp. each $x \in \bar{D}$ there are $L > 0$ and $\delta_0 > 0$ the same as in ii with the property that $$\left\lvert \frac{\p \hat{h}}{\p x_i}(x) - \frac{\p \hat{h}}{\p
x_i}(y) \right\rvert \le L {\ensuremath{\lvertx - y\rvert}}^{\alpha}$$ for any $y \in D$ resp. any $y \in \bar{D}$ with ${\ensuremath{\lvertx - y\rvert}} < \delta_0$,
3. there is a measurable $\Omega' \subset \Omega$ with ${\ensuremath{\mathbb{P}}}(\Omega')
= 1$ such that $$\frac{\p \hat{h}}{\p x_i}(x) = \lim\limits_{T\to\infty} \frac{1}{T}
\int\limits_{0}^{T} \frac{\p h}{\p x_i}(\theta_{t}\omega,x)\, dt$$ for all $\omega \in \Omega'$ and all $x \in D$ resp. $x
\in \bar{D}$. Moreover, the convergence is uniform in $x \in D_{0}$, for any compact $D_0 \Subset D$ resp. uniform in $x \in \bar{D}$.
Parts (a) and (b) follow in a standard way. Part (c) follows by an application of Lemma \[averaging-uniform\](b) to the function $(\p
h/\p x_i)(\omega,x)$.
From now on we assume that (A1)–(A6) are satisfied.
Consider the space $H$ consisting of $(\tilde{c}(\cdot,\cdot),
\tilde{d}(\cdot,\cdot))$, where $\tilde{c} \colon {\ensuremath{\mathbb{R}}}\times \bar{D}
\to {\ensuremath{\mathbb{R}}}$ and $\tilde{d} \colon {\ensuremath{\mathbb{R}}}\times \p D \to {\ensuremath{\mathbb{R}}}$ are bounded continuous. The set $H$ endowed with the topology of uniform convergence on compact sets (the open-compact topology) becomes a Fréchet space.
For $(\tilde{c}, \tilde{d}) \in H$ and $t \in {\ensuremath{\mathbb{R}}}$ we define the [*time-translate*]{} as $(\tilde{c}, \tilde{d}) \cdot t := ((\tilde{c}
\cdot t)(\cdot,\cdot), (\tilde{d} \cdot t)(\cdot,\cdot))$, where $(\tilde{c} \cdot t)(s,x) := \tilde{c}(s+t,x)$, $s \in {\ensuremath{\mathbb{R}}}$, $x \in
\bar{D}$, and $(\tilde{d} \cdot t)(s,x) := \tilde{d}(s+t,x)$, $s \in
{\ensuremath{\mathbb{R}}}$, $x \in \p D$. It is well known that $(\tilde{c},\tilde{d})
\cdot t \in H$ whenever $(\tilde{c}, \tilde{d}) \in H$ and $t \in
{\ensuremath{\mathbb{R}}}$, and that the mapping $[\, {\ensuremath{\mathbb{R}}}\times H \ni
(t,(\tilde{c},\tilde{d})) \mapsto (\tilde{c},\tilde{d}) \cdot t \in H
\,]$ is continuous.
In the case of let $Y = Y(c,d) := \operatorname{cl}\{\,(c,d)
\cdot t:t \in {\ensuremath{\mathbb{R}}}\,\}$. In the case of let $Y =
Y(\Omega) := \operatorname{cl}\{(c^{\omega},d^{\omega}):\omega \in \Omega\}$ (see Section \[introduction\] for detail).
The following result is a consequence of the Ascoli–Arzelà theorem.
\[lemma:properties-of-Y\]
- $Y$ is a compact subset of $H$.
- For any $(\tilde{c},\tilde{d}) \in Y$ and any $t \in {\ensuremath{\mathbb{R}}}$ there holds $(\tilde{c},\tilde{d}) \cdot t \in Y$.
- For any $(\tilde{c},\tilde{d}) \in Y$, $\tilde{c} \in
C^{2+\alpha,1+\alpha}({\ensuremath{\mathbb{R}}}\times \bar{D})$. Moreover, the $C^{2+\alpha,1+\alpha}({\ensuremath{\mathbb{R}}}\times \bar{D})$-norms are bounded uniformly in $Y$ by the same bound as in .
- For any $(\tilde{c},\tilde{d}) \in Y$, $\tilde{d} \in
C^{2+\alpha,3+\alpha}({\ensuremath{\mathbb{R}}}\times \p D)$. Moreover, the $C^{2+\alpha,3+\alpha}({\ensuremath{\mathbb{R}}}\times \p D)$-norms are bounded uniformly in $Y$ by the same bound as in .
- For a sequence $(\tilde{c}^{(n)},\tilde{d}^{(n)}) \to
(\tilde{c},\tilde{d})$ in $Y$, the mixed derivatives of $\tilde{c}^{(n)}$ of order up to $2$ in $t$ and up to $1$ in $x$ converge to the respective derivatives of $\tilde{c}$, uniformly on compact subsets of ${\ensuremath{\mathbb{R}}}\times \bar{D}$.
- For a sequence $(\tilde{c}^{(n)},\tilde{d}^{(n)}) \to
(\tilde{c},\tilde{d})$ in $Y$, the mixed derivatives of $\tilde{d}^{(n)}$ of order up to $2$ in $t$ and up to $3$ in $x$ converge to the respective derivatives of $\tilde{d}$, uniformly on compact subsets of ${\ensuremath{\mathbb{R}}}\times \p D$.
We write $\sigma_{t}(\tilde{c},\tilde{d})$ for $(\tilde{c},\tilde{d})
\cdot t \in Y$. We will denote by $(Y,{\ensuremath{\mathbb{R}}})$ the compact flow $(Y,
\{\sigma_{t}\}_{t\in{\ensuremath{\mathbb{R}}}})$.
Consider . For $x \in \bar{D}$ and $S < T$ we denote $$\bar{c}(x;S,T) := \frac{1}{T-S}\int\limits_{S}^{T} c(t,x) \, dt.$$ Similarly, for $x \in \p D$ and $S < T$ we denote $$\bar{d}(x;S,T) := \frac{1}{T-S}\int\limits_{S}^{T} d(t,x) \, dt.$$ Let $$\begin{aligned}
\label{hat-Y-eq}
\hat{Y}(c,d) := \{\,(\hat{c},\hat{d}):\ & \exists\, S_n < T_n \
\text{with} \ T_n-S_n \to \infty
\ \text{such that} \nonumber \\
& \ (\hat{c},\hat{d}) = \lim_{n\to\infty}
(\bar{c}(\cdot;S_n,T_n),\bar{d}(\cdot;S_n,T_n))\,\},\end{aligned}$$ where the convergence is in $C(\bar{D}) \times C(\p D)$.
The following result is a consequence of the Ascoli–Arzelà theorem (compare Lemma \[lemma:properties-of-Y\]).
\[lemma:properties-of-Y(hat-a)\]
- $\hat{Y}(c,d)$ is a nonempty compact subset of $C(\bar{D}) \times
C(\p D)$.
- For any $(\hat{c},\hat{d}) \in \hat{Y}(c,d)$, $\hat{c} \in
C^{1}(\bar{D})$. Moreover, the $C^{1}(\bar{D})$-norms are bounded uniformly in $\hat{Y}(c,d)$.
- For any $(\hat{c},\hat{d}) \in \hat{Y}(c,d)$, $\hat{d} \in
C^{3}(\p D)$. Moreover, the $C^{3}(\p D)$-norms are bounded uniformly in $\hat{Y}(c,d)$.
2ex
\[ch5-pre-def\]
- Let $a$ be as in . We say $(c,d)$ is [*uniquely ergodic*]{} if the compact flow $(Y(c,d),{\ensuremath{\mathbb{R}}})$ is uniquely ergodic.
- Let $a$ be as in . We say $(c,d)$ is [*minimal*]{} or [*recurrent*]{} if $(Y(c,d),{\ensuremath{\mathbb{R}}})$ is minimal.
<!-- -->
- If $c(t,x)$ and $d(t,x)$ are almost periodic in $t$ uniformly with respect to $x \in \bar{D}$ and $x \in \p D$, respectively, then $(c,d)$ is both uniquely ergodic and minimal.
- If $c(t,x)$ and $d(t,x)$ are almost automorphic in $t$ uniformly with respect to $x \in \bar{D}$ and $x \in \p D$, respectively, then $(c,d)$ is minimal, but it may not be uniquely ergodic see [@Jon] for examples.
- There is $(c,d)$ which is neither uniquely ergodic nor minimal. For example, let $c(t,x) = \tan^{-1}(t)$ and $d(t,x) \equiv 1$, then $\{(\pi/2,1)\}$ and $\{(-\pi/2,1)\}$ are two minimal invariant subsets of $Y(c,d)$, and hence $Y(c,d)$ is neither uniquely ergodic nor minimal.
\[ch5-pre-existence-avg-lm\] Consider with $(c,d)$ uniquely ergodic, $\mu$ being the unique ergodic measure. For $(\tilde{c},\tilde{d}) \in
Y(c,d)$ put $\tilde{c}_0(x) := \tilde{c}(0,x)$ and $\tilde{d}_0(x) :=
\tilde{d}(0,x)$. Then $$\label{time-space-avg-eq1}
\lim\limits_{T\to\infty} \frac{1}{T}\int\limits_0^T c(t,x)\,dt =
\int\limits_{Y(c,d)} \tilde{c}_0(x) \, d\mu((\tilde{c},\tilde{d}))$$ uniformly for $x \in D$, and $$\label{time-space-avg-eq2} \lim\limits_{T\to\infty}
\frac{1}{T}\int_0^T d(t,x)\,dt = \int\limits_{Y(c,d)} \tilde{d}_0(x)
\, d\mu((\tilde{c},\tilde{d}))$$ uniformly for $x \in \p D$.
We prove only , the other proof being similar. It follows via the Ascoli–Arzelà theorem that the set $\{\, (1/T)\int_{0}^{T} c(t,\cdot)\,dt : T > 0\,\} =
\{\,\bar{c}(\cdot;0,T) : T > 0 \,\}$ has compact closure in $C(\bar{D})$, consequently from any sequence $(T_n)$ with $\lim_{n\to\infty}T_n = \infty$ one can extract a subsequence $(T_{n_k})$ such that $\bar{c}(\cdot;0,T_{n_k})$ converges uniformly in $x \in \bar{D}$ to some $\check{c}$ (depending perhaps on the subsequence).
On the other hand, as $(Y(c,d),{\ensuremath{\mathbb{R}}})$ is uniquely ergodic, for each continuous $g \colon Y(c,d) \to {\ensuremath{\mathbb{R}}}$ there holds $$\lim\limits_{T\to\infty} \frac{1}{T} \int\limits_{0}^{T}g((c,d)
\cdot t)\,dt = \int\limits_{Y(c,d)} g(\cdot) \, d\mu(\cdot)$$ (compare, e.g., Oxtoby [@Oxt]). Fix $x \in \bar{D}$ and take $g((\tilde{c},\tilde{d})) := \tilde{c}_0(x)$. We have thus obtained that if $\bar{c}(x;0,T_n)$ converges, for some $T_n \to \infty$, uniformly in $x \in \bar{D}$, then the limit is always equal to $\check{c}(x) = \int_{Y(c,d)} \tilde{c}_0(x) \, d\mu$.
We introduce the following standing notations ($X_1$, $X_2$ are Banach spaces):
$\mathcal{L}(X_1,X_2)$ represents the space of all bounded linear operators from $X_1$ to $X_2$, endowed with the norm topology;
$\lVert \cdot \rVert_{X_1}$ denotes the norm in $X_1$;
$X_1^{*}$ denotes the Banach space dual to $X_1$;
$(\cdot,\cdot)_{X_1,X_1^{*}}$ stands for the duality pairing between $X_1$ and $X_1^{*}$;
${\ensuremath{\lVert\cdot\rVert}}$ denotes the norm in $L_2(D)$ or the norm in $\mathcal{L}(L_2(D),L_2(D))$;
$\langle \cdot, \cdot \rangle$ stands for the standard inner product in $L_2(D)$;
$\lVert \cdot \rVert_{X_1,X_2}$ indicates the norm in $\mathcal{L}(X_1,X_2)$;
$[\cdot,\cdot]_\theta$ is a complex interpolation functor;
$(\cdot,\cdot)_{\theta,p}$ is a real interpolation functor (see [@BeLo], [@Tri] for more detail);
${\ensuremath{\mathbb{Z}}}$ denotes the set of integers;
${\ensuremath{\mathbb{N}}}$ denotes the set of nonnegative integers.
Skew-product semiflows {#skew-product}
======================
We construct in this section a linear skew-product semiflow on $X$ generated by or by , where $X$ is as in .
To do so, we first use the theory presented by H. Amann in [@Ama] to consider the existence of solution of + for any $(\tilde{c},
\tilde{d}) \in Y$ and any $u_0 \in L_p(D)$. Recall that we assume (A1)–(A6) throughout.
Let $\mathcal{A}(\tilde{c})$ denote the operator given by $$\mathcal{A}(\tilde{c})u = \sum_{i,j=1}^{N} a_{ij}(x)
\frac{\p^{2}u}{\p x_{i} \p x_{j}} + \sum_{i=1}^{N} a_i(x) \frac{\p
u}{\p x_i} + \tilde{c}(0,x)u, \quad x \in D,$$ and let $\mathcal{B}(\tilde{d})$ denote the boundary operator given by $$\mathcal{B}(\tilde{d})u =
\begin{cases}
u & x \in \p D \quad \text{(Dirichlet)}
\\[1.5ex]
\disp \sum_{i=1}^{N} b_i(x) \frac{\p u}{\p x_i} & x \in \p D \quad
\text{(Neumann)}\\[1.5ex]
\disp \sum_{i=1}^{N} b_i(x) \frac{\p u}{\p x_i} + \tilde{d}(0,x)u &
x \in \p D \quad \text{(Robin)}.
\end{cases}$$ Let $$V_{p}^{1}(\tilde{d}) := \{\,u \in W_{p}^{2}(D):
\mathcal{B}(\tilde{d})u = 0\,\}.$$ For given $0 < \theta < 1$ and $1 < p < \infty$, let $${\ensuremath{V_{p}^{\theta}}}:=
\begin{cases}
(L_p(D),W_{p}^{2}(D))_{\theta,p} \quad
\text{if }2\theta \not \in {\ensuremath{\mathbb{N}}}\\[2ex]
[L_p(D),W_{p}^{2}(D)]_{\theta} \quad \text{if } 2\theta \in {\ensuremath{\mathbb{N}}}\end{cases}$$ and $${\ensuremath{V_{p}^{\theta}}}(\tilde{d}) :=
\begin{cases}
(L_p(D),V_{p}^{1}(\tilde{d}))_{\theta,p}
\quad \text{if } 2\theta \not \in {\ensuremath{\mathbb{N}}}\\[2ex]
[L_p(D), V_{p}^{1}(\tilde{d})]_{\theta} \quad \text{if } 2\theta \in
{\ensuremath{\mathbb{N}}}.
\end{cases}$$
\[X-theta-spaces-prop\]
- ${\ensuremath{V_{p}^{\theta}}}= W_{p}^{2\theta}$.
- If $2\theta - \frac{1}{p} \ne 0, 1$ then ${\ensuremath{V_{p}^{\theta}}}(\tilde{d})$ is a closed subspace of ${\ensuremath{V_{p}^{\theta}}}$.
\(1) follows from [@Ama Theorem 11.6].
\(2) follows from [@Ama Lemma 14.4].
Recall the following compact embedding: $$\label{sobolev-embed-eq1}
W_{p}^{j+m}(D) \hookrightarrow C^{j,\lambda}(\bar{D})$$ if $mp > N > (m-1)p$ and $ 0 < \lambda < m - (N/p)$, and $$\label{sobolev-embed-eq2}
W_{p}^{2}(D) \hookrightarrow {\ensuremath{V_{p}^{\theta}}}$$ $$\label{sobolev-embed-eq3}
V_{p}^{1}(\tilde{d}) \hookrightarrow {\ensuremath{V_{p}^{\theta}}}(\tilde{d})$$ for any $0 \leq \theta < 1$ and $\tilde{a} \in Y$, where $V_p^0,
V_p^0(\tilde d)=L_p(D)$.
Let $$A_{(\tilde{c},\tilde{d}),p}(t) := \mathcal{A}(\tilde{c} \cdot
t)|_{V_p^1(\tilde{d} \cdot t)}.$$ Then + can be written as $$\label{evolution-eq1}
\begin{cases} u_t = A_{(\tilde{c},\tilde{d}),p}(t) u \\
u(0) = u_0.
\end{cases}$$
\[lp-solution-def\] $u = u(t,x)$ is called an [*$L_p$-solution*]{} of if it is a solution of the evolution equation in $L_p(D)$.
\[classical-solution-def\] $u = u(t,x)$ defined on $(t_0,t_1)
\times \bar{D}$, $t_0 < t_1$, is a [*classical solution*]{} of on $(t_0,t_1)$ if it is continuous on $(t_0,t_2)\times \bar D$, it satisfies the differential equation in for all $t \in (t_0,t_1)$ and all $x \in D$, and it satisfies the boundary conditions for all $t \in (t_0,t_1)$ and all $x \in \p D$.
The following existence result follows from [@Ama Theorem 15.1].
\[existence-prop\] For each $(\tilde{c},\tilde{d}) \in Y$ and each $u_0 \in L_{p}(D)$ there exists a unique $L_{p}(D)$-solution $U_{(\tilde{c},\tilde{d}),p}(\cdot,0)u_0 \colon [0,\infty) \to
L_{p}(D)$ of .
It follows from the uniqueness of $L_p$-solutions that the following [*cocycle property*]{} for the solution operator holds: $$\label{cocycle0}
U_{(\tilde{c},\tilde{d}),p}(t+s,0) = U_{(\tilde{c},\tilde{d}) \cdot
s, p}(t,0) U_{(\tilde{c},\tilde{d}),p}(s,0) \qquad \text{for any }
(\tilde{c},\tilde{d}) \in Y, \ s, t \ge 0.$$
We collect now the regularity properties of the $L_{p}(D)$-solutions which will be useful in the sequel.
\[evolution-op-in-smooth-case-prop1\] For any $1 < p < \infty$, $(\tilde{c},\tilde{d}) \in Y$ and $u_0 \in
L_2(D)$ there holds $U_{(\tilde{c},\tilde{d}),2}(t,0)u_0 \in
V_{p}^{1}(\tilde{d} \cdot t)$ for $t > 0$. Moreover, for any fixed $0 < t_1 \le t_2$ there is $C_{p} = C_{p}(t_1,t_2) > 0$ such that $$\lVert U_{(\tilde{c},\tilde{d}),2}(t,0)
\rVert_{L_{2}(D),W_{p}^{2}(D)} \le C_{p}$$ for all $(\tilde{c},\tilde{d}) \in Y$ and $t_1 \le t \le t_2$.
First of all, by [@Ama Lemma 6.1 and Theorem 14.5], for any $1
< p < \infty$, any $(\tilde{c},\tilde{d}) \in Y$, and any $u_0 \in
L_p(D)$, $$\label{smoothness-of-solution1}
U_{(\tilde{c},\tilde{d}),p}(t,0)u_0 \in V_p^1(\tilde{d} \cdot
t)\quad \text{for } t > 0.$$ Moreover, for any $t_2 > 0$, there is $C_p = C_p(t_2) > 0$ such that $$\label{smoothness-of-solution2}
\lVert U_{(\tilde{c},\tilde{d}),p}(t,0)
\rVert_{L_{p}(D),W_{p}^{2}(D)} \le \frac{C_p}{t}$$ for all $(\tilde{c},\tilde{d}) \in Y$ and $0 < t \le t_2 $.
Next, note that if $1 < p \le 2$, then we have $L_2(D) \subset
L_p(D)$, $V_2^1(\tilde{d} \cdot t)\subset V_p^1(\tilde{d} \cdot t)$, and $W_2^2(D) \subset W_p^2(D)$. The proposition then follows from and .
Now, assume $2 < p < \infty$. If $4 \ge N$, then by Sobolev embeddings (see [@Ada Theorem 6.2], we have $$\label{smoothness-of-solution3}
W_2^2(D)\hookrightarrow C(\bar{D}).$$ Then it follows with the help of that $U_{(\tilde{c},\tilde{d}),2}(t/2,0)u_0 \in L_p(D)$ for all $t > 0$. gives that $U_{(\tilde{c},\tilde{d}),2}(t,0)u_0 \in V_p^1(\tilde{d} \cdot t)$ for all $t > 0$. We estimate $$\begin{aligned}
& \lVert U_{(\tilde{c},\tilde{d}),2}(t,0)
\rVert_{L_{2}(D),W_{p}^{2}(D)} \\
\le& \tilde{C} \lVert U_{(\tilde{c},\tilde{d}) \cdot
(t_1/2),p}(t-t_1/2,0) \rVert_{L_p(D),W_p^2(D)} \cdot \lVert
U_{(\tilde{c},\tilde{d}),2}(t_1/{2},0)u_0 \rVert_{L_2(D),W_2^2(D)} \\
\le& \tilde{C} \cdot \frac{C_p(t_2-t_1/2)}{t-t_1/2} \cdot
\frac{C_2(t_1/2)}{t_1/2}\end{aligned}$$ for all $(\tilde{c},\tilde{d}) \in Y$ and $t_1 \le t \le t_2$, where $\tilde{C}$ denotes the norm of the embedding $W_2^2(D)
\hookrightarrow L_p(D)$. Hence the proposition also holds.
Finally, assume $p > 2$ and $N > 4$. There are $l \in {\ensuremath{\mathbb{N}}}$ and $p_0 =
2 < p_1 < p_2 < \dots < p_l$ such that $p_{i-1} < p_i <
\frac{Np_{i-1}}{N-2p_{i-1}}$ for $i = 1, 2, \dots, l$, and $2p_l
> N$. For any $\delta > 0$, let $0 = \tau_0 < \tau_1 < \tau_2 < \dots
< \tau_l = \frac{\delta}{2}$. By , $$\lVert U_{(\tilde{c},\tilde{d}) \cdot \tau_i,p_i}
(\tau_{i+1}-\tau_i,0) \rVert_{L_{p_i}(D),W_{p_i}^2(D)} \le
\frac{C_{p_i}(\tau_{i+1}-\tau_i)}{\tau_{i+1}-\tau_i}$$ for $i = 0, 1, 2, \dots, l-1$. By Sobolev embeddings (see [@Ada Theorem 6.2]), $$W_{p_i}^2(D) \hookrightarrow L_{p_{i+1}}(D)$$ for $i = 0, 1, 2, \dots, l-1$. We then have $$\label{smoothness-aux}
\lVert U_{(\tilde{c},\tilde{d}) \cdot
\tau_{i-1},p_i}(\tau_{i+1}-\tau_i,0)
\rVert_{L_{p_i}(D),L_{p_{i+1}}(D)} \le \tilde{C}_{p_i}$$ for some $\tilde{C}_{p_i} > 0$. Further, since $p_l > \frac{N}{2}$, by Sobolev embeddings (see [@Ada Theorem 6.2]), $$W_{p_l}^2(D) \hookrightarrow C(\bar{D}).$$ Consequently, we have an embedding $W_{p_l}^2(D) \hookrightarrow
L_p(D)$ (denote its norm by $\bar{C}$). It then follows that for any $u_0 \in L_2(D)$, $$\begin{aligned}
U_{(\tilde{c},\tilde{d}),2}(\tau_l,0)u_0 & =U_{(\tilde{c},\tilde{d})
\cdot \tau_1, p_1}(\tau_l-\tau_1,0)U_{(\tilde{c},\tilde{d}),2}(\tau_1,0)u_0 \\
& = U_{(\tilde{c},\tilde{d}) \cdot
\tau_{l-1},p_{l-1}}(\tau_l-\tau_{l-1},0) \,
U_{(\tilde{c},\tilde{d}) \cdot
\tau_{l-2},p_{l-2}}(\tau_{l-1}-\tau_{l-2},0) \\
& \quad \dots U_{(\tilde{c},\tilde{d}) \cdot
\tau_1,p_1}(\tau_2-\tau_1,0)
U_{(\tilde{c},\tilde{d}),2}(\tau_1,0)u_0 \\
& \in L_p(D).\end{aligned}$$ This implies, via , that $$U_{(\tilde c,\tilde d),2}(t,0)u_0 \in V_p^1(\tilde{d} \cdot t)$$ for any $t \ge \delta$ (and hence for any $t > 0$, since $\delta > 0$ is arbitrary). Now we take $\delta = t_1$. It follows from and that $$\lVert U_{(\tilde{c},\tilde{d}),2}(t,0)
\rVert_{L_{2}(D),W_{p}^{2}(D)} \le \bar{C} \tilde{C}_{p_0} \dots
\tilde{C}_{p_{l-1}} \frac{C_{p}(t_2-t_1/2)}{t_1/2}$$ for all $(\tilde{c},\tilde{d}) \in Y$ and $t_1 \le t \le t_2$.
\[evolution-op-in-smooth-case-prop2\] Suppose that $2\theta -
1/p \notin {\ensuremath{\mathbb{N}}}$. Then for any $t \ge 0$ and $u_0 \in
{\ensuremath{V_{p}^{\theta}}}(\tilde{d})$ there holds $U_{(\tilde{c},\tilde{d}),p}(t,0)u_0
\in {\ensuremath{V_{p}^{\theta}}}(\tilde{d} \cdot t)$. Moreover, for any $T > 0$ there is $C_{p,\theta} = C_{p,\theta}(T) > 0$ such that $${\ensuremath{\lVertU_{(\tilde{c},\tilde{d}),p}(t,0)u_0\rVert_{{\ensuremath{V_{p}^{\theta}}}}}} \le C_{p,\theta}
{\ensuremath{\lVertu_0\rVert_{{\ensuremath{V_{p}^{\theta}}}}}}$$ for any $(\tilde{c},\tilde{d}) \in Y$, $0 \le t \le T$, and $u_0 \in
{\ensuremath{V_{p}^{\theta}}}(\tilde{d})$.
See [@Ama Theorems 7.1 and 14.5].
\[classical-solution-prop\] For any $u_0\in L_p(D)$, $U_{(\tilde{c},\tilde{d}),p}(t,0)u_0$ is a classical solution of on $(0,\infty)$.
It follows from Proposition \[evolution-op-in-smooth-case-prop1\] and [@Ama Corollary 15.3]).
Proposition \[classical-solution-prop\] allows us to write $U_{(\tilde{c},\tilde{d})}(t,0)u_0$ ($t > 0$) instead of $U_{(\tilde{c},\tilde{d}),p}(t,0)u_0$. In case of we write $U_{\omega}(t,0)$ instead of $U_{(c^{\omega},d^{\omega)}}(t,0)$.
\[global-solution-def\] A [*global solution*]{} of is a classical solution of on $(-\infty,\infty)$.
Observe that $v = v(t,x)$ is a global solution of if and only if $$U_{(\tilde{c},\tilde{d}) \cdot t}(s,0)v(t,\cdot) =
v(t+s,\cdot)\quad\text{for any }t \in {\ensuremath{\mathbb{R}}}\text{ and any }s \ge 0.$$ 2ex From now on, we assume (A7). For any sequence $(\tilde{c}^{(n)},
\tilde{d}^{(n)})_{n=1}^{\infty} \subset Y$, we write $\lim_{n\to\infty}(\tilde{c}^{(n)}$, $\tilde{d}^{(n)}) =
(\tilde{c},\tilde{d})$ if $(\tilde{c}^{(n)},\tilde{d}^{(n)})$ converges to $(\tilde{c},\tilde{d})$ in $Y$ as $n \to \infty$ (here the convergence is uniform in the space variable and uniform on compact sets in the time variable). We then present various continuous dependence propositions.
\[joint-continuity-in-X-theta\] For any sequence $((\tilde{c}^{(n)},\tilde{d}^{(n)}))_{n=1}^{\infty} \subset Y$, any sequence $(t_n)_{n=1}^{\infty} \subset (0,\infty)$ and any sequence $(u_n)_{n=1}^{\infty} \subset L_2(D)$, if $\lim_{n\to\infty}(\tilde{c}^{(n)},\tilde{d}^{(n)})$ $=
(\tilde{c},\tilde{d})$, $\lim_{n\to\infty}t_n = t$, where $t > 0$, and $\lim_{n\to\infty}u_n = u_0$ in $L_2(D)$, then the following holds.
- $U_{(\tilde{c}^{(n)},\tilde{d}^{(n)})}(t_n$, $0)u_n$ converges in ${\ensuremath{V_{p}^{\theta}}}$ to $U_{(\tilde{c},\tilde{d})}(t,0)u_0$, where $0 \le
\theta < 1$ and $1 < p < \infty$ with $2\theta - 1/p \notin {\ensuremath{\mathbb{N}}}$.
- $U_{(\tilde{c}^{(n)},\tilde{d}^{(n)})}(t_n,0)u_n$ converges in $C^1(\bar{D})$ to $U_{(\tilde{c},\tilde{d})}(t,0)u_0$.
\(1) Proposition \[evolution-op-in-smooth-case-prop1\] and Eq. imply that there is a subsequence $(n_k)_{k=1}^{\infty}$ such that $U_{(\tilde{c}^{(n_k)},\tilde{d}^{(n_k)})}(t_{n_k},0)u_{n_k}$ converges, as $k \to \infty$, in $V_{p}^{\theta}$ to some $u^{*}$. Note that $${\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t_n,0)u_0 -
U_{(\tilde{c},\tilde{d})}(t,0)u_0\rVert}} \to 0$$ and $${\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t_n,0)u_n -
U_{(\tilde{c},\tilde{d})}(t_n,0)u_0\rVert}} \to 0.$$ By (A7) we have that $${\ensuremath{\lVertU_{(\tilde{c}_n,\tilde{d}_n)}(t_n,0)u_n -
U_{(\tilde{c},\tilde{d})}(t_n,0)u_n\rVert}} \to 0$$ and hence $${\ensuremath{\lVertU_{(\tilde{c}_n,\tilde{d}_n)}(t_n,0)u_n -
U_{(\tilde{c},\tilde{d})}(t,0)u_0\rVert}} \to 0$$ as $n \to \infty$. As $V_{p}^{\theta}$ embeds continuously in $L_2(D)$, we must have $u^{*} = U_{(\tilde{c},\tilde{d})}(t,0)u_0$ and the sequence $U_{(\tilde{c}^{(n)},\tilde{d}^{(n)})}(t_{n},0)u_{n}$ converges, as $n \to \infty$, in ${\ensuremath{V_{p}^{\theta}}}$, to $U_{(\tilde{c},\tilde{d})}(t,0)u_0$.
\(2) It follows by (1) and Eq. .
\[norm-continuity-prop-in-X-theta\]
- Let $1 < p < \infty$ and $2\theta - 1/p \not\in {\ensuremath{\mathbb{N}}}$. The mapping $$[\,Y \times (0,\infty) \ni ((\tilde{c},\tilde{d}),t) \mapsto
U_{(\tilde{c},\tilde{d})}(t,0) \in \mathcal{L}(L_2(D),{\ensuremath{V_{p}^{\theta}}})\,]$$ is continuous.
- The mapping $$[\,Y \times (0,\infty) \ni ((\tilde{c},\tilde{d}),t) \mapsto
U_{(\tilde{c},\tilde{d})}(t,0) \in
\mathcal{L}(L_2(D),C^1(\bar{D}))\,]$$ is continuous. Moreover, for any $t > 0$ and any $(\tilde{c},\tilde{d}) \in Y$ the linear operator $U_{(\tilde{c},\tilde{d})}(t,0)$ is compact completely continuous.
\(1) Assume that $(\tilde{c}^{(n)},\tilde{d}^{(n)})$ converges to $(\tilde{c},\tilde{d})$ in $Y$ and that $t_n$ converges to $t > 0$. Suppose to the contrary that $$\lVert U_{(\tilde{c}^{(n)},\tilde{d}^{(n)})}(t_n,0) -
U_{(\tilde{c},\tilde{d})}(t,0) \rVert_{L_2(D),{\ensuremath{V_{p}^{\theta}}}} \not\to 0$$ as $n \to \infty$. Then there are $\epsilon_0 > 0$ and a sequence $(u_n)_{n=1}^{\infty} \subset L_2(D)$ with ${\ensuremath{\lVertu_n\rVert}} = 1$ such that $${\ensuremath{\lVertU_{(\tilde{c}^{(n)},\tilde{d}^{(n)})}(t_n,0)u_n -
U_{(\tilde{c},\tilde{d})}(t,0)u_n\rVert_{{\ensuremath{V_{p}^{\theta}}}}}} \ge \epsilon_0$$ for all $n$. By Proposition \[evolution-op-in-smooth-case-prop1\], there are $u^*$, $u^{**} \in {\ensuremath{V_{p}^{\theta}}}$ such that (after possibly extracting a subsequence) $$U_{(\tilde{c}^{(n)},\tilde{d}^{(n)})}(t_n,0)u_{n} \to u^{*}$$ and $$U_{(\tilde{c},\tilde{d})}(t,0)u_{n} \to u^{**}$$ in ${\ensuremath{V_{p}^{\theta}}}$, as $n \to \infty$. Without loss of generality, we may assume that there is $\tilde u^*\in V_p^\theta$ such that $$U_{(\tilde{c},\tilde{d})}(t/2,0)u_n \to \tilde{u}^*$$ in $V_p^\theta$ as $n \to \infty$. Then by Proposition \[joint-continuity-in-X-theta\], we have $$\begin{aligned}
&{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t_n,0)u_n -
U_{(\tilde{c},\tilde{d})}(t,0)u_n\rVert}} \\
=& {\ensuremath{\lVertU_{(\tilde{c},\tilde{d}) \cdot
{t/2}}(t_n-t/2,0)U_{(\tilde{c},\tilde{d})}(t/2,0)u_n -
U_{(\tilde{c},\tilde{d}) \cdot
{t/2}}(t/2,0)U_{(\tilde{c},\tilde{d})}(t/2,0)u_n\rVert}} \\
\to& {\ensuremath{\lVertU_{(\tilde{c},\tilde{d}) \cdot t/2}(t/2,0) \tilde{u}^* -
U_{(\tilde{c},\tilde{d}) \cdot t/2}(t/2,0) \tilde{u}^*\rVert}} = 0\end{aligned}$$ as $n \to \infty$. By the property (A7) we have $${\ensuremath{\lVertU_{(\tilde{c}^{(n)},\tilde{d}^{(n)})}(t_n,0) -
U_{(\tilde{c},\tilde{d})}(t_n,0)\rVert}} \to 0$$ as $n \to \infty$. Then we must have $u^{*} = u^{**}$, hence $${\ensuremath{\lVertU_{(\tilde{c}^{(n)},\tilde{d}^{(n)})}(t_{n},0)u_{n} -
U_{(\tilde{c},\tilde{d})}(t,0)u_{n}\rVert_{{\ensuremath{V_{p}^{\theta}}}}}} \to 0$$ as $n \to \infty$, a contradiction.
\(2) It follows by (1) and Eq. .
We are now ready to construct the skew-product semiflow on $X$ ($X$ is as in ) generated by or . For $t \ge 0$, $(\tilde{c},\tilde{d}) \in Y$, $u_0 \in X$, put $$\label{skew-product-semiflow-in-X}
\Pi_t(u_0,(\tilde{c},\tilde{d})) = \Pi(t;u_0,(\tilde{c},\tilde{d}))
:= (U_{(\tilde{c},\tilde{d})}(t,0)u_0,(\tilde{c},\tilde{d}) \cdot
t).$$ $\Pi = \{\,\Pi_{t}\,\}_{t \ge 0}$ satisfies the usual algebraic properties of a semiflow on $X$ : $\Pi_0$ equals the identity on $X$, and $\Pi_{t} \circ \Pi_{s} = \Pi_{s+t}$ for any $s, t \ge 0$. Moreover, the continuity of $\Pi$ restricted to $(0,\infty) \times X
\times Y$ follows by Proposition \[joint-continuity-in-X-theta\] and the embedding $X \hookrightarrow L_2(D)$. (However, we need not have continuity at $t = 0$.)
Sometimes we write $U_{(\tilde{c},\tilde{d})}(t,s)$ instead of $U_{(\tilde{c},\tilde{d}) \cdot s}(t-s,0)$, $s \le t$. The semigroup property $\Pi_{t} \circ \Pi_{s} = \Pi_{s+t}$ takes in that notation the following form (see the cocycle property ): $$\label{cocycle}
U_{(\tilde{c},\tilde{d})}(t,r) = U_{(\tilde{c},\tilde{d})}(t,s)
U_{(\tilde{c},\tilde{d})}(s,r), \qquad r \le s \le t.$$
\[continuity-at-time-0\] Let $\theta \in (1/2,1)$ and $p > 1$ be such that $2\theta - p
\not\in {\ensuremath{\mathbb{N}}}$ and $V_p^{\theta} \hookrightarrow C(\bar{D})$. Then for any $(\tilde{c},\tilde{d}) \in Y$ and $u_0 \in
V_p^{\theta}(\tilde{d})$, $$\lVert U_{(\tilde{c},\tilde{d})}(t,0)u_0 - u_0 \rVert_{C(\bar{D})}
\to 0$$ as $t \to 0$.
It follows from [@Ama Theorem 15.1] and Eq. .
Throughout the rest of this paper, we assume (A1)–(A7).
Strong monotonicity and globally positive solutions {#strong-monotonicity}
===================================================
In this section, we first show that the skew-product semiflow $\Pi_t$ constructed in the previous section is strongly monotone and then show that has a unique globally positive solution, which will be used in next section to define the principal spectrum and principal Lyapunov exponent of and .
Let $X$ be as in . The Banach space $X$ is ordered by the standard cone $$X^{+} := \{\,u \in X: u(x) \ge 0 \text{ for each } x \in D\,\}.$$ The interior $X^{++}$ of $X^+$ is nonempty, where $$X^{++} = \{\,u \in X: u(x) > 0 \text{ for } x \in D \text{ and } (\p
u/\p \nu)(x) < 0 \text{ for } x \in \p D \,\}$$ for the Dirichlet boundary conditions, and $$X^{++} = \{\,u \in X: u(x) > 0 \text{ for } x \in \bar{D}\,\}$$ for the Neumann or Robin boundary conditions.
For $u_1, u_2 \in X$, we write $u_1 \le u_2$ if $u_2 - u_1 \in
X^{+}$, $u_1 < u_2$ if $u_1 \le u_2$ and $u_1 \ne u_2$, and $u_1 \ll
u_2$ if $u_2 - u_1 \in X^{++}$. The symbols $\ge$, $>$ and $\gg$ are used in the standard way.
We proceed now to investigate the strong monotonicity property of the solution operator $U_{(\tilde{c},\tilde{d})}(t,0)$. When the equations and are in divergence form, the monotonicity of $U_{(\tilde{c},\tilde{d})}(t,0)$ follows from [@Ama3 Theorem 11.6]. But the strong monotonicity is not included in [@Ama3 Theorem 11.6]. Though the monotonicity for equations in non-divergence form can also be proved by [@Ama3 Theorem 11.6] after verifying certain conditions, however for convenience we will give a proof for the monotonicity directly. We will prove the strong monotonicity by using the strong maximum principle and the Hopf boundary point principle for classical solutions. But before we do that we have to analyze whether the existing theory (as presented, e.g., in [@Fri1]) can be applied: notice that in the Robin case $\tilde{d}$ may change sign. We show that coefficient can be made nonnegative by an appropriate change of variables.
Indeed, consider $$\label{aux-eq}
\begin{cases}
\displaystyle \frac{\p u^{*}}{\p t} = \sum_{i,j=1}^{N}
a_{ij}(x)\frac{\p^2 u^{*}} {\p x_i\p x_j}, & \quad t > -1,\ x \in D, \\[2ex]
\displaystyle \sum_{i=1}^{N} b_{i}(x) \frac{\p u^{*}}{\p x_{i}} +
u^{*} = 0, & \quad t > -1,\ x \in \p D.
\end{cases}$$ Let $p > 1$ and $\theta \in (1/2,1)$ be as in Proposition \[continuity-at-time-0\]. By the $C^{\infty}$ Urysohn Lemma (see [@Fol Lemma 8.18]), there is a nonzero $C^{\infty}$ function $u_0 \colon {\ensuremath{\mathbb{R}}}^{N} \to {\ensuremath{\mathbb{R}}}$ such that $0 \le u_0 \le 1$ on $D$ and $\operatorname{supp}{u_0} \Subset D$. Then $u_0 \in V_p^{\theta}(1)$. Let $u^*(t,x)$ be the solution of with $u^*(-1,x) =
u_0(x)$. By Proposition \[continuity-at-time-0\] $$\lVert u^{*}(t,\cdot) - u_0 \rVert_{C(\bar{D})} \to 0 \quad
\text{as} \quad t \to -1^{+}.$$ Hence, the function $u^{*}$ is continuous on $[-1,\infty) \times
\bar{D}$ and satisfies, by Proposition \[classical-solution-prop\], the equation in pointwise on $(-1,\infty) \times D$ and the boundary condition in pointwise on $(-1,\infty) \times \p D$. Consequently, it follows from the strong maximum principle and the Hopf boundary point principle for parabolic equations that $u^{*}(t,x) > 0$ for all $t > -1$ and all $x \in
\bar{D}$.
Now, let $v(t,x) := e^{M u^{*}(t,x)}u(t,x)$, where $M$ is a positive constant (to be determined later). Then becomes $$\label{general-eq-smooth1}
\begin{cases}
\disp \frac{\p v}{\p t} = \sum_{i,j=1}^{N} a_{ij}(x)\frac{\p^2 v}{\p
x_i \p x_j} + \sum_{i=1}^{N} \check{a}_{i}(x) \frac{\p v}{\p
x_i} + \check{c}(t,x)v, & \quad t > 0,\ x \in D, \\[2ex]
\disp \sum_{i=1}^{N} b_{i}(x) \frac{\p v}{\p x_{i}} + \check{d}(t,x)
v = 0 & \quad t > 0,\ x \in \p D,
\end{cases}$$ where $$\begin{aligned}
\check{a}_i(x) & := a_{i}(x) - M \biggl(\sum_{j=1}^{N}
\Bigl(a_{ij}(x) \frac{\p u^*}{\p x_j} + a_{ji}(x) \frac{\p u^*}{\p
x_j}\Bigr) \biggr),
\\
\check{c}(t,x) &:= \tilde{c}(t,x) - M \sum_{i=1}^{N} a_i(t,x)
\frac{\p u^*}{\p x_i} + M^{2} \sum_{i,j=1}^{N} a_{ij}(x) \frac{\p
u^*}{\p x_i} \frac{\p u^*}{\p x_j}, \\
\check{d}(t,x) &:= \tilde{d}(t,x) + M u^{*}(t,x).\end{aligned}$$ We see that for any $(\tilde{c},\tilde{d}) \in Y$ and any $T > 0$, there is $M = M(T) > 0$ such that $\check{d}(t,x) > 0$ for $t \in
[0,T]$ and $x \in \bar{D}$. Observe that, since the mapping $[\,[0,\infty) \ni t \mapsto u^{*}(t,\cdot) \in C^{1}(\bar{D})\,]$ is continuous by Proposition \[joint-continuity-in-X-theta\], the coefficients $\check{a}_i$ and $\check{c}$ are bounded on $[0,T]
\times \bar{D}$ and the coefficient $\check{d}$ is bounded on $[0,T]
\times \p D$.
Consequently, we have the following result.
\[strong-positivity-class\] Let $u_1, u_2 \in L_2(D)$. If $u_1 \ne u_2$ and $u_1(x) \le u_2(x)$ for a.e. $x \in D$, then
- $$(U_{(\tilde{c},\tilde{d})}(t,0)u_1)(x) <
(U_{(\tilde{c},\tilde{d})}(t,0)u_2)(x) \quad \text{for }
(\tilde{c},\tilde{d}) \in Y,\ t
> 0 \text{ and }x \in D$$
and $$\frac{\p}{\p \nu}(U_{(\tilde{c},\tilde{d})}(t,0)u_1)(x) >
\frac{\p}{\p \nu} (U_{(\tilde{c},\tilde{d})}(t,0)u_2)(x) \quad
\text{for } (\tilde{c},\tilde{d}) \in Y,\ t > 0 \text{ and }x \in \p
D$$ in the Dirichlet case,
- $$(U_{(\tilde{c},\tilde{d})}(t,0)u_1)(x) <
(U_{(\tilde{c},\tilde{d})}(t,0)u_2)(x) \quad \text{for }
(\tilde{c},\tilde{d}) \in Y,\ t
> 0 \text{ and }x \in \bar{D}$$
in the Neumann or Robin case.
Fix $(\tilde{c},\tilde{d}) \in Y$. Assume that $u_1, u_2 \in X$ and $u_1 < u_2$. For any given $T > 0$, in the case of the Robin boundary conditions let $M > 0$ be such that $\check{d}(t,x) > 0$ for all $t
\in [0,T]$ and all $x \in \bar{D}$, where $\check{d}$ is as in the reasoning above the statement of the present proposition (in the case of the Dirichlet or Neumann boundary conditions put $M = 0$). Define $v_0(x) := e^{M u^{*}(0,x)}(u_2(x) - u_1(x))$, $x \in \bar{D}$.
Let $\theta \in (1/2,1)$ and $p > 1$ be as in Proposition \[continuity-at-time-0\]. We claim that there is a sequence $(v^{(n)})_{n=1}^{\infty} \subset V_p^{\theta}(\check{d})$ such that $v^{(n)}(x) \ge 0$ ($n = 1,2,\dots$, $x \in D$), $v^{(n)}
\not\equiv 0$ ($n = 1,2,\dots$), and $\lim_{n\to\infty}{\ensuremath{\lVertv^{(n)}
- v_0\rVert}} = 0$.
First note that there is a sequence $(v_0^{(n)})_{n=1}^\infty$ of simple functions such that $$0 \le v_0^{(1)}(x) \le v_0^{(2)}(x) \le \dots \le v_0(x) \quad
\text{for~a.e. } x \in D,$$ and $$v_0^{(n)}(x) \to v_0(x) \quad \text{as} \quad n \to \infty, \quad
\text{for~a.e. } x \in D,$$ and $v_0^{(n)} \to v_0$ uniformly on any set on which $v_0$ is bounded. It is therefore sufficient to prove the claim for the case that $v_0 = \chi_E$, where $E \subset D$ is a Lebesgue measurable set.
Now assume $v_0 = \chi_E$, where $E \subset D$ is a Lebesgue measurable set. For $\epsilon_n := \frac{1}{4n^2}$, choose a compact set $K \subset E$ and an open set $U \supset K$ such that $U \Subset
D$, ${\ensuremath{\lvertE \setminus K\rvert}} < \epsilon_n$ and ${\ensuremath{\lvertU \setminus K\rvert}} <
\epsilon_n$, where here ${\ensuremath{\lvert\cdot\rvert}}$ denotes the Lebesgue measure of a set. Then, by the $C^{\infty}$ Urysohn Lemma (see [@Fol Lemma 8.18]), there is a $C^{\infty}$ function $v^{(n)} \colon {\ensuremath{\mathbb{R}}}^{N}
\to {\ensuremath{\mathbb{R}}}$ such that $0 \le v^{(n)} \le 1$ on $D$, $v^{(n)} \equiv 1$ on $K$ and $\operatorname{supp}{v^{(n)}} \subset U$. It then follows that $${\ensuremath{\lVertv^{(n)} - v_0\rVert}} \le {\ensuremath{\lvertU \setminus K\rvert}}^{1/2} + {\ensuremath{\lvertE \setminus
K\rvert}}^{1/2} < \frac{1}{n} \to 0$$ as $n \to \infty$. Moreover, since $\operatorname{supp}{v^{(n)}} \subset U \Subset
D$, we also have $v^{(n)} \in V_p^\theta(\check{d})$. The claim is thus proved. Denote by $v(t,\cdot;v_0)$ and $v(t,\cdot;v^{(n)})$ the solutions of with $v(0,\cdot;v_0) = v_0(\cdot)$ and $v(0,\cdot;v^{(n)}) = v^{(n)}(\cdot)$ ($n = 1,2,\dots$), respectively.
By Proposition \[continuity-at-time-0\], $$\lVert v(t,\cdot;v^{(n)}) - v^{(n)} \rVert_{C(\bar{D})} \to 0$$ as $t \to 0^{+}$. We can thus apply the strong comparison principle for parabolic equations to conclude that $$v(t,x;v^{(n)}) > 0 \quad \text{for} \quad t \in (0,T],\ x \in D, \ n
= 1,2,\dots.$$ This together with Proposition \[norm-continuity-prop-in-X-theta\] implies that $$v(t,x;v_0) \ge 0 \quad \text{for} \quad t \in (0,T],\ x \in D.$$
By Proposition \[classical-solution-prop\], for any $n = 2,3,
\dots$ the function $v(\cdot,\cdot;v_0)$ is continuous on $[T/n,T]$, satisfies the equation in pointwise on $(T/n,T] \times D$ and satisfies the boundary condition in pointwise on $(T/n,T] \times \p D$. Further, from Proposition \[existence-prop\] and the nonnegativity of $v$ it follows that for $n$ sufficiently large there is $x_n \in
D$ such that $v(T/n,x_n;v_0) > 0$. An application of the strong maximum principle for parabolic equations gives $v(t,x;v_0) > 0$ for each $t \in (0,T]$ and each $x \in D$.
In the Dirichlet boundary condition case, suppose to the contrary that there are $t^{*} \in (0,T]$ and $x^{*} \in \p D$ such that $\frac{\p}{\p \nu}(v(t^{*},x^{*};v_0)) = 0$. But this contradicts the Hopf boundary point principle applied to $v$ restricted to $[t^{*}/2,t^{*}] \times \bar{D}$. Hence $\frac{\p}{\p
\nu}(v(t,x;v_0)) < 0$ for any $t \in (0,T]$ and any $x \in \p D$. This completes the proof in that case, since $v(t,x;v_0) =
(U_{\tilde{a}}(t,0)u_2)(x) - (U_{\tilde{a}}(t,0)u_1)(x)$ for any $t
\in (0,T]$ and $x \in \bar{D}$.
Suppose to the contrary that, in the Neumann or Robin boundary condition case, there are $t^{*} \in (0,T]$ and $x^{*} \in \p D$ such that $v(t^{*},x^{*};v_0) = 0$. It follows from the Hopf boundary point principle (applied to $v$ restricted to $[t^{*}/2,t^{*}] \times
\bar{D}$) that $\sum_{i=1}^{N} b_{i}(x^{*}) \frac{\p}{\p
x_{i}}v(t^{*},x^{*};v_0) < 0$, which is incompatible with the boundary condition. Hence $v(t,x;v_0) > 0$ for all $t \in (0,T]$ and all $x \in \bar{D}$. Since $v(t,x;v_0) = e^{M u^{*}(t,x)} (
(U_{\tilde{a}}(t,0)u_2)(x) - (U_{\tilde{a}}(t,0)u_1)(x))$ for any $t
\in (0,T]$ and $x \in \bar{D}$, this completes the proof.
(It is to be remarked that in Eqs. and their coefficients may not belong to $Y$, so [*formally*]{} we cannot apply propositions from Section \[skew-product\] in those cases. This should not cause any misunderstanding.)
By Theorem \[strong-positivity-class\] we have the following [*strong monotonicity*]{}:
1.5ex [*For $(\tilde{c},\tilde{d}) \in Y$, $u_1, u_2 \in X$ and $t > 0$, if $u_1 < u_2$ then $U_{(\tilde{c},\tilde{d})}(t,0)u_1
\ll U_{(\tilde{c},\tilde{d})}(t,0)u_2$.*]{}
3ex The theory of existence and uniqueness of globally positive solutions can then be extended to our case. Below, we collect its basic concepts and facts.
\[globally-positive-def\] For $(\tilde{c},\tilde{d}) \in Y$, we say that a global solution $v =
v(t,x)$ of is a [*globally positive solution*]{} of if $v(t,x) > 0$ for all $t \in
{\ensuremath{\mathbb{R}}}$ and all $x \in D$.
We shall consider now the problem of existence of globally positive solutions.
\[globally-positive-existence\] There exist
- a continuous function $w \colon Y \to X^{++}$, ${\ensuremath{\lVertw((\tilde{c},\tilde{d}))\rVert}} = 1$ for each $(\tilde{c},\tilde{d}) \in Y$, and
- a continuous function $w^{*} \colon Y \to L_2(D)$, ${\ensuremath{\lVertw^{*}((\tilde{c},\tilde{d}))\rVert}} = 1$ for each $(\tilde{c},\tilde{d}) \in Y$, and such that for each $(\tilde{c},\tilde{d}) \in Y$, $w^{*}((\tilde{c},\tilde{d}))(x) >
0$ for a.e. $x \in D$,
having the following properties:
- For each $(\tilde{c},\tilde{d}) \in Y$ the function $v_{(\tilde{c},\tilde{d})} = v(t,x;\tilde{c},\tilde{d})$ given by $$\label{globally-positive-formula}
v(t,\cdot;\tilde{c},\tilde{d}) :=
\begin{cases}
U_{(\tilde{c},\tilde{d})}(t,0) w((\tilde{c},\tilde{d}))
& \quad \text{for } t \ge 0, \\[1ex]
\disp \frac{w((\tilde{c},\tilde{d}) \cdot
t)}{{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})\cdot
t}(-t,0)w((\tilde{c},\tilde{d}) \cdot t)\rVert}}} & \quad \text{for } t <
0,
\end{cases}$$ is a globally positive solution of .
- Let, for some $(\tilde{c},\tilde{d}) \in Y$, $v = v(t,x)$ be a globally positive solution of . Then there exists a constant $\beta > 0$ such that $v(t,x) = {\beta}
v(t,x;\tilde{c},\tilde{d})$ for each $t \in {\ensuremath{\mathbb{R}}}$ and each $x \in
D$.
- There are constants $C > 0$ and $\mu > 0$ such that $$\label{exp-sep-formula-1} {\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0) u_0\rVert}}
\le C e^{{-\mu}t} {\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0)
w((\tilde{c},\tilde{d}))\rVert}}$$ for any $(\tilde{c},\tilde{d}) \in Y$, $t > 0$ and $u_0 \in
L_2(D)$ with ${\ensuremath{\lVertu_0\rVert}} = 1$ and $\langle u_0,
w^{*}((\tilde{c},\tilde{d})) \rangle = 0$.
- There are constants $C' > 0$ and $\mu > 0$ such that $$\label{exp-sep-formula-2} \lVert U_{(\tilde{c},\tilde{d})}(t,0) u_0
\rVert_{X} \le C' e^{{-\mu}t} \lVert U_{(\tilde{c},\tilde{d})}(t,0)
w((\tilde{c},\tilde{d})) \rVert_{X}$$ for any $(\tilde{c},\tilde{d}) \in Y$, $t \ge 1$ and $u_0 \in
L_2(D)$ with ${\ensuremath{\lVertu_0\rVert}} = 1$ and $\langle u_0,
w^{*}((\tilde{c},\tilde{d})) \rangle = 0$.
We start by considering a discrete-time dynamical system on the product bundle $X \times Y$ ($X$ is a fiber, $Y$ is the base space): $$\Pi_n(u_0,(\tilde{c},\tilde{d})) :=
(U_{(\tilde{c},\tilde{d})}(n,0)u_0,(\tilde{c},\tilde{d}) \cdot n),
\qquad u_0 \in X,\ (\tilde{c},\tilde{d}) \in Y,\ n = 1,2,3,\dots.$$ Proposition \[norm-continuity-prop-in-X-theta\] and Theorem \[strong-positivity-class\] allow us to use the results contained in [@PoTer] to conclude that there are continuous functions $\tilde{w} \colon Y \to X$, $\tilde{w}^{*} \colon Y \to
X^{*}$, $\lVert \tilde{w}((\tilde{c},\tilde{d}))\rVert_{X} = \lVert
\tilde{w}^{*}((\tilde{c},\tilde{d}))\rVert_{X^{*}} = 1$ for each $\tilde{a} \in Y$, such that (we write $X_1((\tilde{c},\tilde{d})) :=
\operatorname{span}{\tilde{w}((\tilde{c},\tilde{d}))}$, $X_2((\tilde{c},\tilde{d})) :=
\operatorname{\mathcal{N}}{(\tilde{w}^{*}((\tilde{c},\tilde{d})))}$, where $\operatorname{\mathcal{N}}$ stands for the nullspace of an element of $X^{*}$)
- $\tilde{w}((\tilde{c},\tilde{d})) \in X^{++}$, for each $(\tilde{c},\tilde{d}) \in Y$.
- $(v, \tilde{w}^{*}((\tilde{c},\tilde{d})))_{X,X^{*}} > 0$ for each $(\tilde{c},\tilde{d}) \in X$ and each nonzero $v \in
X^{+}$. It follows that $X_2((\tilde{c},\tilde{d})) \cap X^{+} =
\{0\}$, for each $(\tilde{c},\tilde{d}) \in Y$.
- For each $(\tilde{c},\tilde{d}) \in Y$ there is $d_1 =
d_1((\tilde{c},\tilde{d})) > 0$ such that $U_{(\tilde{c},\tilde{d})}(1,0)\tilde w((\tilde{c},\tilde{d})) =
d_{1}\tilde w((\tilde{c},\tilde{d}) \cdot 1)$. It follows that $U_{(\tilde{c},\tilde{d})}(1,0) X_1((\tilde{c},\tilde{d})) =
X_1((\tilde{c},\tilde{d}) \cdot 1)$.
- For each $(\tilde{c},\tilde{d}) \in Y$ there is $d_1^{*} =
d_1^{*}((\tilde{c},\tilde{d})) > 0$ such that$(U_{(\tilde{c},\tilde{d})}(1,0))^{*} \tilde
w^{*}((\tilde{c},\tilde{d}) \cdot 1) = d_1^{*} \tilde
w^*((\tilde{c},\tilde{d}))$, where $(U_{(\tilde{c},\tilde{d})}(1,0))^{*} \colon X^{*} \to X^{*}$ stands for the linear operator dual to $U_{(\tilde{c},\tilde{d})}(1,0)$. It follows that $U_{(\tilde{c},\tilde{d})}(1,0) X_2((\tilde{c},\tilde{d}))
\subset X_2((\tilde{c},\tilde{d}) \cdot 1)$, for any $(\tilde{c},\tilde{d}) \in Y$.
- There are constants $\tilde{C} > 0$ and $0 < \gamma < 1$ such that $$\label{exp-sep-1}
\lVert U_{(\tilde{c},\tilde{d})}(n,0) u_0 \rVert_{X} \le \tilde{C}
{\gamma}^{n} \lVert U_{(\tilde{c},\tilde{d})}(n,0)
\tilde{w}((\tilde{c},\tilde{d})) \rVert_{X}$$ for any $(\tilde{c},\tilde{d}) \in Y$, any $u_0 \in
X_2((\tilde{c},\tilde{d}))$ with $\lVert u_0 \rVert_{X} = 1$ and any $n \in {\ensuremath{\mathbb{N}}}$.
Put $w((\tilde{c},\tilde{d})) := \tilde{w}((\tilde{c},\tilde{d}))/
{\ensuremath{\lVert\tilde{w}((\tilde{c},\tilde{d}))\rVert}}$, $(\tilde{c},\tilde{d}) \in
Y$. As $X$ embeds continuously in $L_2(D)$, the function $w \colon Y
\to X$ is continuous. Further, put $w^{*}((\tilde{c},\tilde{d})) :=
\tilde{w}^{*}((\tilde{c},\tilde{d}))/
{\ensuremath{\lVert\tilde{w}^{*}((\tilde{c},\tilde{d}))\rVert}}$, $(\tilde{c},\tilde{d})
\in Y$. From Proposition \[norm-continuity-prop-in-X-theta\] it follows that the mapping $[\, Y \ni (\tilde{c},\tilde{d}) \mapsto
(U_{(\tilde{c},\tilde{d})}(1,0))^{*} \in \mathcal{L}(X^{*},L_2(D))
\,]$ is continuous, too, so we obtain with the help of (d) that $w^{*} \colon Y \to L_2(D)$ is well defined and continuous.
By the definition of the dual operator, $$\begin{aligned}
d_1^{*}((\tilde{c},\tilde{d})) \cdot (v,
\tilde{w}^{*}((\tilde{c},\tilde{d})))_{X,X^{*}} & = (v,
(U_{(\tilde{c},\tilde{d})}(1,0))^{*}
\tilde{w}^{*}((\tilde{c},\tilde{d}) \cdot 1))_{X,X^{*}} \\
& = (U_{(\tilde{c},\tilde{d})}(1,0)v,
\tilde{w}^{*}((\tilde{c},\tilde{d}) \cdot 1))_{X,X^{*}}\end{aligned}$$ for each $(\tilde{c},\tilde{d}) \in Y$ and each $v \in X$. As $\tilde{w}^{*}((\tilde{c},\tilde{d}))$ is a bounded linear functional on $L_2(D)$ and $X$ is dense in $L_2(D)$, we conclude that $$\begin{aligned}
d_1^{*}((\tilde{c},\tilde{d})) \cdot \langle v,
\tilde{w}^{*}((\tilde{c},\tilde{d})) \rangle & = \langle v,
(U_{(\tilde{c},\tilde{d})}(1,0))^{*}\tilde{w}^{*}((\tilde{c},\tilde{d})
\cdot 1) \rangle \\
& = \langle U_{(\tilde{c},\tilde{d})}(1,0)v,
\tilde{w}^{*}((\tilde{c},\tilde{d}) \cdot 1) \rangle\end{aligned}$$ for each $(\tilde{c},\tilde{d}) \in Y$ and each $v \in L_2(D)$.
We prove now that $w^{*}((\tilde{c},\tilde{d}))(x) > 0$ for a.e. $x
\in D$, or, which is equivalent, that $\tilde{w}^{*}((\tilde{c},\tilde{d}))(x) > 0$ for a.e. $x \in D$. Suppose first that for some $(\tilde{c},\tilde{d}) \in X$ there are $D_{+}, D_{-} \subset D$ of positive Lebesgue measure such that $\tilde{w}^{*}((\tilde{c},\tilde{d}))(x) > 0$ for $x \in D_{+}$, $\tilde{w}^{*}((\tilde{c},\tilde{d}))(x) < 0$ for $x \in D_{-}$, and $\tilde{w}^{*}((\tilde{c},\tilde{d}))(x) = 0$ for $x \in D \setminus
(D_{+} \cup D_{-})$. Define $v \in L_2(D)$ to be the simple function equal to $1/\int_{D_{+}}\tilde{w}^{*}((\tilde{c},\tilde{d}))(x)\,dx$ on $D_{+}$, equal to $-1/\int_{D_{-}}\tilde{w}^{*}((\tilde{c},\tilde{d}))(x)\,dx$ on $D_{-}$, and equal to zero elsewhere. We have $$\begin{gathered}
0 = d_1^{*}((\tilde{c},\tilde{d})) \cdot \langle v,
\tilde{w}^{*}((\tilde{c},\tilde{d})) \rangle = \langle v,
(U_{(\tilde{c},\tilde{d})}(1,0))^{*}
\tilde{w}^{*}((\tilde{c},\tilde{d}) \cdot 1)
\rangle \\
= \langle U_{(\tilde{c},\tilde{d})}(1,0)v,
\tilde{w}^{*}((\tilde{c},\tilde{d}) \cdot 1) \rangle =
(U_{(\tilde{c},\tilde{d})}(1,0)v, \tilde{w}^{*}((\tilde{c},\tilde{d})
\cdot 1))_{X,X^{*}}.\end{gathered}$$ By Theorem \[strong-positivity-class\], $U_{(\tilde{c},\tilde{d})}(1,0)v \in X^{++}$. This contradicts (b). Suppose now that for some $(\tilde{c},\tilde{d}) \in X$ there are $D_{+}, D_{0} \subset D$ of positive Lebesgue measure such that $\tilde{w}^{*}((\tilde{c},\tilde{d}))(x) > 0$ for $x \in D_{+}$ and $\tilde{w}^{*}((\tilde{c},\tilde{d}))(x) = 0$ for $x \in D_{0}$, and the complement of the union $D_{+} \cup D_{0}$ in $D$ has Lebesgue measure zero. We repeat the above construction, this time with $v$ equal to zero on $D_{+}$ and equal to one on $D_{0}$.
Fix $(\tilde{c},\tilde{d}) \in Y$. The fact that if there exists a globally positive solution of then it is unique up to multiplication by a positive constant is proved for the Dirichlet case in [@HuPoSa2], and for the Neumann and Robin case in [@Hu1]. We proceed now to the construction of a globally positive solution.
We define first the trace of a positive solution $v(t,x;\tilde{c},\tilde{d})$ on ${\ensuremath{\mathbb{Z}}}$: $$v(k,\cdot;\tilde{c},\tilde{d}) :=
\begin{cases}
U_{(\tilde{c},\tilde{d})}(k,0) w((\tilde{c},\tilde{d})) & \quad
\text{for } k = 0,1,2,3, \dots,
\\[1ex]
\disp \frac{w((\tilde{c},\tilde{d}) \cdot
k)}{{\ensuremath{\lVertU_{(\tilde{c},\tilde{d}) \cdot
k}(-k,0)w((\tilde{c},\tilde{d}) \cdot k)\rVert}}} & \quad \text{for } k =
\dots, -3, -2, -1.
\end{cases}$$ It follows from (a) and (c) that $$\label{glob-exist-1} U_{(\tilde{c},\tilde{d})}(l+k,k) v(k,\cdot;
\tilde{c},\tilde{d}) = v(k+l,\cdot; \tilde{c},\tilde{d})$$ for any $k \in {\ensuremath{\mathbb{Z}}}$ and any nonnegative integer $l$. Also, ${\ensuremath{\lVertv(0,\cdot; \tilde{c},\tilde{d})\rVert}} = 1$. We extend $v$ to a function defined on $(-\infty,\infty)$ by putting $$\label{glob-exist-2} v(t,\cdot;\tilde{c},\tilde{d}) :=
U_{(\tilde{c},\tilde{d})}(t,{\ensuremath{\lfloort\rfloor}})
v({\ensuremath{\lfloort\rfloor}},\cdot;\tilde{c},\tilde{d}), \qquad t \in {\ensuremath{\mathbb{R}}}\setminus
{\ensuremath{\mathbb{Z}}},$$ where ${\ensuremath{\lfloort\rfloor}}$ denotes the greatest integer less than or equal to $t$. To check that the function so defined is indeed a global solution we need to show that $$\label{glob-exist-3} v(s+t,\cdot;\tilde{c},\tilde{d}) =
U_{(\tilde{c},\tilde{d})}(s+t,t) v(t,\cdot;\tilde{c},\tilde{d})
\qquad \text{for any } t \in {\ensuremath{\mathbb{R}}}\text{ and any }s \ge 0$$ (see Definition \[global-solution-def\] and Eq. ). We write $$\begin{aligned}
v(s+t,\cdot;\tilde{c},\tilde{d}) & =
U_{(\tilde{c},\tilde{d})}(s+t,{\ensuremath{\lfloors+t\rfloor}})
v({\ensuremath{\lfloors+t\rfloor}},\cdot;\tilde{c},\tilde{d}) &\text{by \eqref{glob-exist-2}}\\
& = U_{(\tilde{c},\tilde{d})}(s+t,{\ensuremath{\lfloors+t\rfloor}})
U_{(\tilde{c},\tilde{d})}({\ensuremath{\lfloors+t\rfloor}},{\ensuremath{\lfloort\rfloor}})
v({\ensuremath{\lfloort\rfloor}},\cdot;\tilde{c},\tilde{d}) &\text{by \eqref{glob-exist-1}}\\
& = U_{(\tilde{c},\tilde{d})}(s+t,t)
U_{(\tilde{c},\tilde{d})}(t,{\ensuremath{\lfloort\rfloor}})
v({\ensuremath{\lfloort\rfloor}},\cdot;\tilde{c},\tilde{d}) &\text{by \eqref{cocycle}} \\
& = U_{(\tilde{c},\tilde{d})}(s+t,t) v(t,\cdot;\tilde{c},\tilde{d})
&\text{by \eqref{glob-exist-2}}.\end{aligned}$$ The fact that $v(t,\cdot;\tilde{c},\tilde{d}) \in X^{++}$ for each $t
\in {\ensuremath{\mathbb{R}}}$ is a consequence of the construction of $v$ and of Theorem \[strong-positivity-class\].
Formula for $t \ge 0$ is straightforward. It follows from the uniqueness of globally positive solutions that $$v(t,\cdot;\tilde{c},\tilde{d}) =
{\ensuremath{\lVertv(t,\cdot;\tilde{c},\tilde{d})\rVert}} \, w((\tilde{c},\tilde{d})
\cdot t), \qquad t \in (-\infty,\infty).$$ From we obtain, for any $t < 0$, that $$\begin{aligned}
1 = {\ensuremath{\lVertv(0,\cdot;\tilde{c},\tilde{d})\rVert}} & =
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})\cdot t}(-t,0)
v(t,\cdot;\tilde{c},\tilde{d})\rVert}} \\
& = {\ensuremath{\lVertv(t,\cdot;\tilde{c},\tilde{d})\rVert}} \,
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})\cdot t}(-t,0) w((\tilde{c},\tilde{d})
\cdot t)\rVert}},\end{aligned}$$ which concludes the proof of formula .
We proceed now to the proof of part (iii). Denote by $M_1$ the norm of the embedding $X \hookrightarrow L_2(D)$. Moreover, by the compactness of $Y$ and the continuity of $\tilde{w}$ there is $M_2 >
0$ such that $\lVert \tilde{w}((\tilde{c},\tilde{d})) \rVert_{X} \le
M_2 {\ensuremath{\lVertw((\tilde{c},\tilde{d}))\rVert}}$ for all $(\tilde{c},\tilde{d})
\in Y$.
Take $u_0 \in L_2(D)$ such that ${\ensuremath{\lVertu_0\rVert}} = 1$ and $\langle u_0,
w^{*}((\tilde{c},\tilde{d})) \rangle = 0$. It follows from (d) that $\langle U_{(\tilde{c},\tilde{d})}(1,0)u_0,
w^{*}((\tilde{c},\tilde{d}) \cdot 1) \rangle = 0$. As $U_{(\tilde{c},\tilde{d})}(1,0)u_0 \in X$, one has $U_{(\tilde{c},\tilde{d})}(1,0)u_0 \in X_2(\tilde{a} \cdot 1)$. This allows us to estimate, for $n = 2,3,4, \dots$, $$\begin{aligned}
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(n,0) u_0\rVert}} & \le M_1 \lVert
U_{(\tilde{c},\tilde{d})}(n,1) (U_{(\tilde{c},\tilde{d})}(1,0)u_0)
\rVert_{X} & \text{ by \eqref{cocycle}}
\\
& \le M_1 \tilde{C} {\gamma}^{n-1} \lVert
U_{(\tilde{c},\tilde{d})}(n,1) \tilde{w}((\tilde{c},\tilde{d}) \cdot
1) \rVert_{X} \lVert U_{(\tilde{c},\tilde{d})}(1,0)u_0
\rVert_{X} & \text{ by \eqref{exp-sep-1}} \\
& = \frac{M_1 \tilde{C}}{\gamma} {\gamma}^n \frac{\lVert
U_{(\tilde{c},\tilde{d})}(n,0) \tilde{w}((\tilde{c},\tilde{d}))
\rVert_{X}}{\lVert U_{(\tilde{c},\tilde{d})}(1,0)
\tilde{w}((\tilde{c},\tilde{d})) \rVert_{X}} \lVert
U_{(\tilde{c},\tilde{d})}(1,0)u_0 \rVert_{X} & \text{ by
\eqref{cocycle}} \\
& \le \frac{M_{1} M_{2} D_{1} \tilde{C}}{D_2 \gamma} {\gamma}^{n}
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(n,0) w((\tilde{c},\tilde{d}))\rVert}},\end{aligned}$$ where $D_1 := \sup\{\,\lVert U_{(\tilde{c},\tilde{d})}(1,0)
\rVert_{L_2(D),X}: (\tilde{c},\tilde{d}) \in Y \,\} < \infty$, $D_2 := \inf\{\,\lVert
U_{(\tilde{c},\tilde{d})}(1,0)\tilde{w}((\tilde{c},\tilde{d}))
\rVert_{X}: (\tilde{c},\tilde{d}) \in Y \,\} > 0$.
Clearly, ${\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(1,0) u_0\rVert}} \le
\frac{M_1M_2D_1}{D_2} {\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(1,0)
w((\tilde{c},\tilde{d}))\rVert}}$ for all $(\tilde{c},\tilde{d}) \in Y$ and all $u_0 \in L_2(D)$ with ${\ensuremath{\lVertu_0\rVert}} = 1$ and $\langle u_0,
w^{*}((\tilde{c},\tilde{d})) \rangle = 0$.
As a consequence we obtain the existence of $\bar{C} = \frac{M_{1}
M_{2} D_{1} }{D_{2} \gamma} \max\{\tilde{C}, 1\}$ such that $$\label{exp-sep-discrete} {\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(n,0) u_0\rVert}}
\le \bar{C} \gamma^n {\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(n,0)
w((\tilde{c},\tilde{d}))\rVert}}$$ for any $(\tilde{c},\tilde{d}) \in Y$, any $n \in {\ensuremath{\mathbb{N}}}$ and any $u_0
\in L_2(D)$ satisfying ${\ensuremath{\lVertu_0\rVert}} = 1$ and $\langle u_0,
w^*((\tilde{c},\tilde{d})) \rangle = 0$.
To show we notice that $$\begin{aligned}
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0) u_0\rVert}} & =
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,{\ensuremath{\lfloort\rfloor}})
(U_{(\tilde{c},\tilde{d})}({\ensuremath{\lfloort\rfloor}},0) u_0)\rVert}} \\
& \le D_3 {\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}({\ensuremath{\lfloort\rfloor}},0) u_0\rVert}} \\
& \le D_3 \bar{C} \gamma^{{\ensuremath{\lfloort\rfloor}}}
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}({\ensuremath{\lfloort\rfloor}},0)
w((\tilde{c},\tilde{d}))\rVert}} \qquad \qquad \qquad \qquad \qquad
\text{by \eqref{exp-sep-discrete}} \\
& \le D_{3} \bar{C} {\gamma}^{{\ensuremath{\lfloort\rfloor}}}
\frac{1}{{\ensuremath{\lVertU_{(\tilde{c},\tilde{d}) \cdot
{\ensuremath{\lfloort\rfloor}}}(t-{\ensuremath{\lfloort\rfloor}},0) w((\tilde{c},\tilde{d}) \cdot {\ensuremath{\lfloort\rfloor}})\rVert}}}
\, {\ensuremath{\lVertU_{\tilde{a}}(t,0) w((\tilde{c},\tilde{d}))\rVert}} \\
& \le \frac{D_{3} \bar{C}}{{\gamma} D_4} \gamma^{t}
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0) w((\tilde{c},\tilde{d}))\rVert}}\end{aligned}$$ for any $(\tilde{c},\tilde{d}) \in Y$, $t \ge 1$ and any $u_0 \in
L_2(D)$ with ${\ensuremath{\lVertu_0\rVert}} = 1$ and $\langle u_0,
w^*((\tilde{c},\tilde{d})) \rangle = 0$, where $D_3 :=
\sup\{\,{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0)\rVert}}: t \in [0,1],\
(\tilde{c},\tilde{d}) \in Y\,\} < \infty$ and $D_4 := \inf\{\,
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0)\tilde{w}((\tilde{c},\tilde{d}))\rVert}}:
t \in [0,1],\ (\tilde{c},\tilde{d}) \in Y \,\} > 0$.
Clearly, ${\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0) u_0\rVert}} \le
\frac{D_3}{D_4} {\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0)
w((\tilde{c},\tilde{d}))\rVert}}$ for all $(\tilde{c},\tilde{d}) \in Y$, all $t \in [0,1]$ and all $u_0 \in L_2(D)$ with ${\ensuremath{\lVertu_0\rVert}} = 1$ and $\langle u_0, w^*((\tilde{c},\tilde{d})) \rangle = 0$.
This proves , with $C = \frac{D_3}{D_4
\gamma} \max\{\bar{C}, 1\}$ and $\mu = -\ln{\lambda}$.
To prove we estimate, for $u_0 \in L_2(D)$ with ${\ensuremath{\lVertu_0\rVert}} = 1$ and $\langle u_0, w^{*}((\tilde{c},\tilde{d}))
\rangle = 0$, and $t \ge 1$, $$\begin{aligned}
\lVert U_{(\tilde{c},\tilde{d})}(t,0) u_0 \rVert_{X} & = \lVert
U_{(\tilde{c},\tilde{d})}(t,t-1)
(U_{(\tilde{c},\tilde{d})}(t-1,0)u_0) \rVert_{X} \\
& \le D_{1} {\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t-1,0)u_0\rVert}} \\
& \le D_{1} C e^{-\mu(t-1)}
{\ensuremath{\lVertU_{\tilde{a}}(t-1,0)w((\tilde{c},\tilde{d}))\rVert}}
& \qquad \text{ by \eqref{exp-sep-formula-1}} \\
& \le \frac{D_1 C e^{\mu}}{D_5} e^{-{\mu}t}
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0)w((\tilde{c},\tilde{d}))\rVert}} \\
& \le \frac{D_1 M_1 C e^{\mu}}{D_5} e^{-{\mu}t} \lVert
U_{(\tilde{c},\tilde{d})}(t,0)w((\tilde{c},\tilde{d}))\rVert_{X},\end{aligned}$$ where $M_1 := \sup\{\,{\ensuremath{\lVertu\rVert}}: u \in X, \lVert u \rVert_{X} \le
1\,\}$, $D_1 := \sup\{\,\lVert U_{(\tilde{c},\tilde{d})}(1,0)
\rVert_{L_2(D),X}: $ $(\tilde{c},\tilde{d}) \in Y \,\}$ and $D_5 :=
\inf\{\,{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(1,0)w((\tilde{c},\tilde{d}))\rVert}}:
(\tilde{c},\tilde{d}) \in Y \,\}$.
For other approaches to the question of existence and/or uniqueness of globally positive solutions the reader can consult also [@Hu2], [@HuPo], [@HuPoSa1], [@Mi1], [@Mi2], [@Po].
\[pre-bounds-of-w-thm\]
- In the Dirichlet boundary condition case, there is $M > 0$ such that $$w((\tilde{c},\tilde{d}))(x) \le M \quad \text{for any } x \in D
\text{ and any } (\tilde{c},\tilde{d}) \in Y.$$ Further, for each compact $D_0 \Subset D$ there is $m = m(D_0) >
0$ such that $$w((\tilde{c},\tilde{d}))(x) \ge m(D_0) \quad \text{for any } x \in
D_0 \text{ and any } (\tilde{c},\tilde{d}) \in Y.$$
- In the Neumann or Robin boundary condition case, there are $M, m
> 0$ such that $$m \le w((\tilde{c},\tilde{d}))(x) \le M \quad \text{for any } x \in
D \text{ and any } (\tilde{c},\tilde{d}) \in Y.$$
It follows in a standard way from the compactness of $Y$ and from the fact that $w((\tilde{c},\tilde{d})) \in X^{++}$ for each $(\tilde{c},\tilde{d}) \in Y$.
\[Holder-continuity-thm\] The first order derivatives of $w$ are Hölder in $x$ uniformly in $(\tilde{c},\tilde{d}) \in Y$ and in $x
\in \bar{D}$, and the second order derivatives of $w$ are Hölder in $x$ uniformly in $(\tilde{c},\tilde{d}) \in Y$ and locally uniformly in $x \in D$.
By Theorem \[globally-positive-existence\], for each $(\tilde{c},\tilde{d}) \in Y$ there holds $$w((\tilde{c},\tilde{d})) = \frac{U_{(\tilde{c},\tilde{d}) \cdot
(-1)}(1,0) w((\tilde{c},\tilde{d}) \cdot (-1))}
{{\ensuremath{\lVertU_{(\tilde{c},\tilde{d}) \cdot (-1)}(1,0)
w((\tilde{c},\tilde{d}) \cdot (-1))\rVert}}}.$$ It is a consequence of the continuity of $w$, the compactness of $Y$ and Proposition \[joint-continuity-in-X-theta\] that the denominators on the right-hand side are positive and bounded away from zero, uniformly in $Y$. Now we apply the parabolic regularity estimates [@Fri1 Theorem 5, Chapter 3].
Principal spectrum and principal Lyapunov exponent {#principal-spectrum}
==================================================
In this section, we collect the basic concepts and facts about the principal spectrum and principal Lyapunov exponent of and .
\[principal-spectrum-def\] In case of we define its [*principal spectrum*]{} to be the set of all limits $$\lim\limits_{n\to\infty} \frac{\ln{\ensuremath{\lVertU_{(c,d) \cdot
S_n}(T_n-S_n,0)w((c,d) \cdot S_n)\rVert}}} {T_n-S_n},$$ where $T_n - S_n \to \infty$ as $n \to \infty$.
The following proposition follows from the results contained in [@JPSe] (cp., e.g., [@Mi3 Thm. 2.10]).
\[principal-spectrum-thm2\] The principal spectrum of is a compact interval $[{\ensuremath{\lambda_{\mathrm{inf}}}}(c,d)$, ${\ensuremath{\lambda_{\mathrm{sup}}}}(c,d)]$. Moreover, if $(c,d)$ is uniquely ergodic and minimal then ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) = {\ensuremath{\lambda_{\mathrm{sup}}}}(c,d)$.
2ex In the case of , for $\omega \in \Omega$ we write $U_\omega(t,0)$ for $U_{(c^\omega,d^\omega)}(t,0)$ and $w(\omega)$ for $w((c^{\omega},d^{\omega}))$.
\[principal-exponent-thm\] For , there exists $\lambda (c,d)\in {\ensuremath{\mathbb{R}}}$ such that $$\lambda (c,d)= \lim\limits_{T\to\infty}
\frac{\ln\|U_\omega(T,0)w(\omega)\|}{T}$$ for a.e. $\omega \in \Omega$.
It follows from subadditive ergodic theorems (see [@Krengel]).
\[principal-exponent-def\] The $\lambda(c,d)$ as in Theorem \[principal-exponent-thm\] is called the [*principal Lyapunov exponent*]{} of .
In the existing literature, the principal spectrum is either defined precisely as in Definition \[principal-spectrum-def\] see [@Mi3] or with the $L_2(D)$-norm replaced by the norm in some fractional power space that embeds continuously into $C^1(\bar{D})$ see, e.g., [@MiSh1]. In our setting, as $X_1$ is a one-dimensional invariant subbundle spanned by a continuous function from $Y$ into $X$, we can replace the $L_2(D)$-norm in Definition \[principal-spectrum-def\] with the $X$-norm.
Similarly, in the Definition \[principal-exponent-def\] the $L_2(D)$-norm can be replaced with the $X$-norm. Further, in [@MiSh1] the principal Lyapunov exponent was introduced as the a.e. constant limit $$\lim\limits_{T\to\infty} \frac{\ln{\lVert U_{\omega}(T,0)}
\rVert_{X, X}}{T},$$ where $X$ is some fractional power space that embeds continuously into $C^1(\bar{D})$. With the help of one can prove that for those $\omega \in \Omega$ for which $\lambda (c,d)
= \lim_{T\to\infty} \frac{\ln{\ensuremath{\lVertU_\omega(T,0)w(\omega)\rVert}}}{T}$ there holds also $\lambda (c,d) = \lim_{T\to\infty}
\frac{\ln{{\ensuremath{\lVertU_{\omega}(T,0)\rVert}}}}{T}$ see the proof of [@Mi1 Thm. 3.2(2)].
For the $L_2(D)$-theory of the principal spectrum and principal Lyapunov exponents see the upcoming monograph [@MiSh2].
We introduce now a useful concept. For $(\tilde{c},\tilde{d}) \in Y$ put $$\begin{aligned}
\label{introduction-kappa-eq} \kappa((\tilde{c},\tilde{d})) :=&
\int\limits_{D} \Bigl( \sum\limits_{i,j=1}^{N} a_{ij}(x) \frac{\p^2
w((\tilde{c},\tilde{d}))(x)}{\p x_{i} \p x_{j}} \Bigr)
w((\tilde{c},\tilde{d}))(x) \, dx \nonumber\\
&+ \int\limits_{D} \Bigl( \sum\limits_{i=1}^{N} a_{i}(x) \frac{\p
w((\tilde{c},\tilde{d}))(x)}{\p x_{i}} + \tilde{c}(0,x)
w((\tilde{c},\tilde{d}))(x) \Bigr) w((\tilde{c},\tilde{d}))(x) \,
dx.\end{aligned}$$ By Proposition \[evolution-op-in-smooth-case-prop1\], $w((\tilde{c},\tilde{d})) \in W^{2}_{2}(D)$, so $\kappa((\tilde{c},\tilde{d}))$ is well defined.
The function $\kappa \colon Y \to {\ensuremath{\mathbb{R}}}$ is continuous. Indeed, notice that applying integration by parts we can write $$\begin{aligned}
\label{property-kappa-eq} \kappa((\tilde{c},\tilde{d})) = &-
\int\limits_{D} \sum\limits_{i,j=1}^{N} a_{ij}(x) \frac{\p
w((\tilde{c},\tilde{d}))(x)}{\p x_{i}} \frac{\p
w((\tilde{c},\tilde{d}))(x)}{\p x_{j}} \, dx \nonumber\\
&- \int\limits_{D} \sum\limits_{i=1}^{N} \Bigl(
\sum\limits_{j=1}^{N} \frac{\p a_{ij}(x)}{\p x_{i}} \frac{\p
w((\tilde{c},\tilde{d}))(x)}{\p x_j}\Bigr)
w((\tilde{c},\tilde{d}))(x) \, dx \nonumber\\
&+ \int\limits_{D} \Bigl( \sum\limits_{i=1}^{N} a_{i}(x) \frac{\p
w((\tilde{c},\tilde{d}))(x)}{\p x_{i}} + \tilde{c}(0,x)
w((\tilde{c},\tilde{d}))(x) \Bigr)
w((\tilde{c},\tilde{d}))(x) \, dx \nonumber\\
&+ \int\limits_{\p D} \sum\limits_{i=1}^{N} \Bigl(
\sum\limits_{j=1}^{N} a_{ij}(x) \frac{\p
w((\tilde{c},\tilde{d}))(x)}{\p x_{j}} \Bigr)
w((\tilde{c},\tilde{d}))(x) \nu_{i}(x) \, dS.\end{aligned}$$ As $w \colon Y \to X$ is continuous, the above expression depends continuously on $(\tilde{c},\tilde{d})$, too.
We point out that the function $\kappa((\tilde{c},\tilde{d}))$ introduced in is a very useful quantity in the investigation of various properties of principal spectrum and principal Lyapunov exponents. This quantity will be heavily used in next section. In the rest of this section, we discuss how to use the function $\kappa$ to characterize the principal spectrum and principal Lyapunov exponents.
Let $\eta_{(\tilde{c},\tilde{d})}(t) :=
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0)w((\tilde{c},\tilde{d}))\rVert}}$ $( >
0)$. Then $\eta_{(\tilde{c},\tilde{d})}(t)$ is differentiable and $U_{(\tilde{c},\tilde{d})}(t,0)w((\tilde{c},\tilde{d})) =
\eta_{(\tilde{c},\tilde{d})}(t)w((\tilde{c},\tilde{d}) \cdot t)$. Hence $w((\tilde{c},\tilde{d}) \cdot t)$ is also differentiable in $t$. By , we have $$\begin{aligned}
&\dot{\eta}_{(\tilde{c},\tilde{d})}(t) w((\tilde{c},\tilde{d}) \cdot
t) + \eta_{(\tilde{c},\tilde{d})}(t)
\frac{\p}{\p t}w((\tilde{c},\tilde{d}) \cdot t) \\
=& \sum\limits_{i,j=1}^{N}
a_{ij}(x)\eta_{(\tilde{c},\tilde{d})}(t)\frac{\p ^2
w((\tilde{c},\tilde{d}) \cdot t)}{\p x_i\p
x_j}+\sum\limits_{i=1}^{N} a_{i}(x)\eta_{(\tilde{c},\tilde{d})}(t)
\frac{\p w((\tilde{c},\tilde{d}) \cdot t)}{\p x_i}\\
&+ \tilde{c}(t,x)\eta_{(\tilde{c},\tilde{d})}(t)
w((\tilde{c},\tilde{d}) \cdot t).\end{aligned}$$ Taking the inner product of the above equation with $w((\tilde{c},\tilde{d}) \cdot t)$ and observing that $\langle
w((\tilde{c},\tilde{d}) \cdot t), w((\tilde{c},\tilde{d}) \cdot t)
\rangle \equiv 1$ and $\langle \frac{\p}{\p t}w((\tilde{c},\tilde{d})
\cdot t), w((\tilde{c},\tilde{d}) \cdot t) \rangle \equiv 0$ we get $\dot{\eta}_{(\tilde{c},\tilde{d})}(t) = \kappa((\tilde{c},\tilde{d})
\cdot t) \eta_{(\tilde{c},\tilde{d})}(t)$, that is, $$\label{kappa-exponent} \frac{d}{dt}
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0) w((\tilde{c},\tilde{d}))\rVert}} =
\kappa((\tilde{c},\tilde{d}) \cdot t)
{\ensuremath{\lVertU_{(\tilde{c},\tilde{d})}(t,0) w((\tilde{c},\tilde{d}))\rVert}}$$ for any $(\tilde{c},\tilde{d}) \in Y$ and any $t \ge 0$.
By , we have $$\label{evolution-kappa}
\ln{\ensuremath{\lVertU_{(\tilde c,\tilde d) \cdot S}(T-S,0)
w((\tilde{c},\tilde{d})\cdot S)\rVert}} = \int_S^ T
\kappa((\tilde{c},\tilde{d}) \cdot t) \,dt$$ for any $(\tilde{c},\tilde{d}) \in Y$ and $S < T$. Then following from Definition \[principal-spectrum-def\] we have
\[principal-spectrum-thm3\] Let $[{\ensuremath{\lambda_{\mathrm{inf}}}}(c,d),{\ensuremath{\lambda_{\mathrm{sup}}}}(c,d)]$ be the principal spectrum interval of . Then $$\label{minimum-spectrum}
{\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) = \liminf_{T-S\to\infty} \frac{1}{T-S}\int_S^T
\kappa((c,d) \cdot t)\,dt$$ and $$\label{maximum-spectrum}
{\ensuremath{\lambda_{\mathrm{sup}}}}(c,d) = \limsup_{T-S\to\infty} \frac{1}{T-S}\int_S^T
\kappa((c,d) \cdot t)\,dt.$$
2ex In the case of we write $\kappa(\omega)$ instead of $\kappa((c^{\omega},d^{\omega}))$. We have
\[kappa-exponent-hm\] Consider . Then $$\lambda = \lim\limits_{T\to \infty}\frac{1}{T}\int_0^T
\kappa(\theta_t\omega) \,dt = \int_{\Omega} \kappa(\cdot)
\,d{\ensuremath{\mathbb{P}}}(\cdot)$$ for a.e. $\omega \in \Omega$.
By the arguments of Lemma 3.4 in [@MiSh1], the map $[\,\Omega \ni
\omega \mapsto (c^{\omega},d^{\omega}) \in Y\,]$ is measurable. The theorem is then a consequence of Theorem \[principal-exponent-thm\], Eq. and Birkhoff’s Ergodic Theorem (Lemma \[ch5-pre-ergodic-lm\]).
We remark that if $c(t,x)$ and $d(t,x)$ are independent of $t$, then $(c,d) = (\hat{c},\hat{d})$ and ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) = {\ensuremath{\lambda_{\mathrm{sup}}}}(c,d) =
\lambda(c,d)$. Moreover, we have the following easy theorem about the continuous dependence of $\lambda(c,d)$ on $(c,d)$.
\[continuous-dependence-of-principal-eigenvalue-thm\] If $c^{(n)}$ converges in $C(\bar{D})$ to $c$ and $d^{(n)}$ converges in $C(\p D)$ to $d$ then $\lambda(c^{(n)},d^{(n)}) \to \lambda(c,d)$.
Time averaging {#main-results}
==============
In this section we state and prove our results on the influence of time variations on principal spectrum and principal Lyapunov exponent of and .
Consider . Let $\Sigma(c,d) :=
[{\ensuremath{\lambda_{\mathrm{inf}}}}(c,d),{\ensuremath{\lambda_{\mathrm{sup}}}}(c,d)]$ be the principal spectrum interval of . For $(\hat{c},\hat{d}) \in \hat{Y}(c,d)$ let $\lambda(\hat{c},\hat{d})$ denote the principal eigenvalue of an averaged equation . Recall that $$\begin{aligned}
\hat{Y}(c,d) = \{\,(\hat{c},\hat{d}):\, & \exists\, S_n < T_n \
\text{with} \ T_n-S_n \to\infty
\ \text{such that} \nonumber \\
& \ (\hat{c},\hat{d}) = \lim_{n\to\infty}
(\bar{c}(\cdot;S_n,T_n),\bar{d}(\cdot;S_n,T_n))\,\},\end{aligned}$$ where $\bar{c}(x;S_n,T_n) :=
\frac{1}{T_n-S_n}\int_{S_n}^{T_n}c(t,x)\,dt$, $\bar{d}(x;S_n,T_n) :=
\frac{1}{T_n-S_n}\int_{S_n}^{T_n}d(t,x)\,dt$, and the convergence is in $C(\bar{D}) \times C(\p D)$.
Consider . Let $\lambda(c,d)$ be the principal Lyapunov exponent. Let $$\hat{c}(x) := \int_{\Omega} c(\omega,x)\,d{\ensuremath{\mathbb{P}}}(\omega),\quad
\hat{d}(x) = \int_{\Omega} d(\omega,x)\,d{\ensuremath{\mathbb{P}}}(\omega).$$ Let $\lambda(\hat{c},\hat{d})$ be the principal eigenvalue of the averaged equation .
Then we have
\[ch5-smoothlb-thm1\]
- Consider . There is $(\hat{c},\hat{d}) \in
\hat{Y}(c,d)$ such that ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) \ge
\lambda(\hat{c},\hat{d})$ and ${\ensuremath{\lambda_{\mathrm{sup}}}}(c,d) \ge
\lambda(\hat{c},\hat{d})$ for any $(\hat{c},\hat{d}) \in
\hat{Y}(c,d)$.
- Consider . $\lambda(c,d) \ge
\hat{\lambda}(\hat c,\hat d)$.
\[ch5-smoothlb-thm2\]
- Consider . If $(c,d)$ is uniquely ergodic and minimal, then ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) = {\ensuremath{\lambda_{\mathrm{sup}}}}(c,d)$ and ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) = \lambda(\hat{c},\hat{d})$ for $(\hat{c},\hat{d}) \in \hat{Y}(\hat{c},\hat{d})$ $(\hat{Y}(c,d)$ is necessarily a singleton$)$ if and only if $c(t,x) = c_{1}(x) +
c_{2}(t)$ and $d(t,x) = d(x)$.
- Consider . $\lambda(c,d) = \hat{\lambda}(c,d)$ if and only if there is $\Omega^{*} \subset \Omega$ with ${\ensuremath{\mathbb{P}}}(\Omega^{*}) = 1$ such that $c(\theta_{t}\omega,x) = c_{1}(x)
+ c_{2}(\theta_{t}\omega)$ for any $\omega \in \Omega^{*}$, $t
\in {\ensuremath{\mathbb{R}}}$ and $x \in \bar{D}$, and $d(\theta_{t}\omega,x) = d(x)$ for any $\omega \in \Omega^{*}$, $t \in {\ensuremath{\mathbb{R}}}$ and $x \in \p D$.
In the case that the boundary condition is of the Dirichlet or Neumann type or of the Robin type with $d$ independent of $t$, the above theorems have been proved in [@MiSh1]. For completeness, we will provide proofs of the theorems including the case that the boundary condition is of the Robin type with $d$ depending on $t$. We note that the proof in the following for Theorem \[ch5-smoothlb-thm1\] is not the same as that in [@MiSh1] even in the case $d$ is independent of $t$.
First of all, let $(\tilde{c},\tilde{d}) = (c,d)$ in the case of and $(\tilde{c},\tilde{d})=
(c^\omega,d^{\omega})$ in the case of for some given $\omega \in \Omega$. For given $S$ and $T > 0$, let $$\eta(t;\tilde{c},\tilde{d}, S) := {\ensuremath{\lVertU_{(\tilde{c},\tilde{d})
\cdot S}(t,0)w((\tilde{c},\tilde{d}) \cdot S)\rVert}}, \qquad t \ge 0,$$ and $$\hat{w}(x;\tilde{c},\tilde{d}, S, T) :=
\exp\Bigl(\frac{1}{T}\int_0^T \ln{w((\tilde{c},\tilde{d}) \cdot
(t+S))(x)}\,dt\Bigr)$$ for $x\in D$ and $$\hat w(x;\tilde{c},\tilde{d},S,T)=0$$ for $x\in\p D$ in the Dirichlet boundary condition case, and $$\hat{w}(x;\tilde{c},\tilde{d}, S, T) :=
\exp\Bigl(\frac{1}{T}\int_0^T \ln{w((\tilde{c},\tilde{d}) \cdot
(t+S))(x)}\,dt\Bigr)$$ for $x\in \bar D$ in the Neumann and Robin boundary conditions cases. Note that $\hat w(x;\tilde c,\tilde d, S,T)\in C(\bar D)$.
Let $\bar{v}(t,x;\tilde{c},\tilde{d},S) :=
w((\tilde{c},\tilde{d}) \cdot (t+S))(x)$. We have that $\eta(t;\tilde{c},\tilde{d}, S)$ satisfies $$\label{eta-nonauton}
\eta_t(t;\tilde{c},\tilde{d}, S) = \kappa((\tilde{c},\tilde{d})
\cdot(t+S)) \eta(t;\tilde{c},\tilde{d}, S),$$ and $\bar{v}(t,x;\tilde{c},\tilde{d}, S)$ satisfies $$\label{nonauton-hat-v-eq1}
\begin{cases}
\disp\frac{\p \bar{v}}{\p t} = \sum_{i,j=1}^{N} a_{ij}(x)\frac{\p^2
\bar{v}}{\p x_{i} \p x_{j}} + \sum_{i=1}^{N} a_i(x) \frac{\p
\bar{v}}{\p x_i} \\[2ex]
\qquad\quad {} + \tilde{c}(t+S,x)\bar{v} -
\kappa((\tilde{c},\tilde{d}) \cdot (t+S)) \bar{v}, &
\quad x \in D \\[2ex]
\mathcal{\tilde B}(t+S) \bar{v} = 0, & \quad x \in \p D,
\end{cases}$$ where $\mathcal{\tilde B}(\cdot)$ is as in . Theorem \[Holder-continuity-thm\] allows us to differentiate sufficiently many times to obtain that for any $x \in {D}$ ($x$ can also be in $\bar D$ in the Neumann and Robin boundary conditions cases) we have $$\begin{aligned}
&\label{nonauton-hat-w-eq1}
\frac{\p\hat{w}}{\p x_i}(x;\tilde{c},\tilde{d},S,T) \nonumber\\
= \null & \hat{w}(x;\tilde{c},\tilde{d}, S,T) \frac{1}{T}\int_0^T
\Bigl(\frac{1}{w((\tilde{c},\tilde{d}) \cdot (t+S))(x)} \frac{\p
w((\tilde{c},\tilde{d}) \cdot (t+S))(x)}{\p x_i}\Bigr)\,dt,\end{aligned}$$ and that for any $x \in D$ we have $$\begin{aligned}
& \label{nonauton-hat-w-eq2} \frac{\p^2\hat{w}}{\p x_i\p x_j} \nonumber\\
= \null &
\hat{w}(x;\tilde{c},\tilde{d}, S,T) \Bigg( \frac{1}{T^2}\int_{0}^{T}
\Bigl(\frac{1}{w((\tilde{c},\tilde{d}) \cdot (t+S))(x)} \frac{\p
w((\tilde{c},\tilde{d}) \cdot (t+S))(x)}{\p
x_i}\Bigr)\,dt \cdot \nonumber\\
&\qquad\qquad\qquad\qquad\qquad\int_{0}^{T} \Bigl(
\frac{1}{w((\tilde{c},\tilde{d}) \cdot (t+S))(x)}\frac{\p
w((\tilde{c},\tilde{d}) \cdot (t+S))(x)}{\p x_j}\Bigr)\,dt\Bigg ) \nonumber\\
\null &+ \hat{w}(x;\tilde{c},\tilde{d}, S,T) \frac{1}{T}
\int_0^T\Bigl(\frac{1}{w((\tilde{c},\tilde{d}) \cdot (t+S))(x)}
\frac{\p ^2 w((\tilde{c},\tilde{d}) \cdot (t+S))(x)}{\p x_i\p
x_j} \nonumber\\
\null &- \frac{1}{w^2((\tilde{c},\tilde{d}) \cdot (t+S))(x)} \frac{\p
w((\tilde{c},\tilde{d}) \cdot (t+S))(x)}{\p x_i} \frac{\p
w((\tilde{c},\tilde{d}) \cdot (t+S))(x)}{\p x_j}\Bigr)\,dt.\end{aligned}$$ Then by , $\hat{w} =
\hat{w}(x;\tilde{c},\tilde{d},S,T)$ satisfies $$\begin{aligned}
\label{nonauton-hat-w-eq3} &\sum_{i,j=1}^{N} a_{ij}(x) \frac{\p^2
\hat{w}}{\p x_{i} \p x_{j}} + \sum_{i=1}^{N} a_i(x) \frac{\p
\hat{w}}{\p x_i}
\nonumber \\
= \null & \Bigl(\frac{1}{T}\int\limits_{0}^{T} \frac{1}{\bar{v}} \frac{\p
\bar{v}}{\p t}(t,x;\tilde{c},\tilde{d},S)\,dt\Bigr ) \hat{w} \nonumber \\
& \quad + \Bigl(\frac{1}{T} \int_0^T \kappa((\tilde{c},\tilde{d})
\cdot (t+S))\,dt -
\frac{1}{T} \int_0^T \tilde{c}(t+S,x)\,dt\Bigr) \hat{w}\nonumber \\
& \quad + \hat{w}\sum_{i,j=1}^N a_{ij}(x) \Biggl(
\frac{1}{T}\int_0^T \Bigl(\frac{1}{ w((\tilde{c},\tilde{d}) \cdot
(t+S))} \frac{\p w((\tilde{c},\tilde{d})
\cdot (t+S))}{\p x_i} \Bigr)\,dt \cdot \nonumber \\
&\quad\quad \frac{1}{T} \int_0^T \Bigl(\frac
{1}{w((\tilde{c},\tilde{d}) \cdot (t+S))}\frac{\p w(\tilde{a} \cdot
(t+S))}{\p x_j} \Bigr)
\,dt \Biggr ) \\
& \quad - \hat{w} \sum_{i,j=1}^N a_{ij}(x) \cdot
\nonumber \\
& \quad\quad\quad\frac{1}{T}\int_0^T
\Bigl(\frac{1}{w^2((\tilde{c},\tilde{d}) \cdot (t+S))} \frac{\p
w((\tilde{c},\tilde{d}) \cdot (t+S))}{\p x_i} \frac{\p
w((\tilde{c},\tilde{d}) \cdot (t+S))}{\p x_j} \Bigr)\,dt\nonumber\end{aligned}$$ for $x \in D$, and $\mathcal{\hat{B}}_{S,T} \hat{w} = 0$ for $x \in
\p D$, where $$\label{nonauton-hat-w-eq4}
\mathcal{\hat{B}}_{S,T}\hat{w} :=
\begin{cases}
\hat{w} & \text{(Dirichlet)} \\[2ex]
\disp \sum_{i=1}^N b_i(x)\frac{\p \hat{w}}{\p x_i} &
\text{(Neumann)} \\[2ex]
\disp \sum_{i=1}^N b_i(x)\frac{\p \hat{w}}{\p x_i} +
\left(\frac{1}{T}\int_0^T \tilde{d}(t+S,x)\,dt \right) \hat{w} &
\text{(Robin)}.
\end{cases}$$ By Lemma \[ch5-pre-holder-lm\](1), $$\begin{aligned}
\label{nonauton-hat-w-eq5} &\sum_{i,j=1}^{N} a_{ij}(x) \frac{\p^2
\hat{w}}{\p x_{i} \p x_{j}} + \sum_{i=1}^{N} a_i(x) \frac{\p
\hat{w}}{\p x_i} \nonumber \\
\le \null & \Bigl(\frac{1}{T}\int\limits_{0}^{T} \frac{1}{\bar{v}}
\frac{\p \bar{v}}{\p t}(t,x;\tilde{c},\tilde{d},S)\,dt \Bigr )
\hat{w}
\nonumber \\
& + \Bigl(\frac{1}{T}\int_0^T \kappa((\tilde{c},\tilde{d}) \cdot
(t+S))\,dt - \frac{1}{T}\int_{0}^{T} \tilde{c}(t+S,x)\,dt \Bigr)
\hat{w}.\end{aligned}$$
Note that $\bar{v}(t,x;\tilde{c},\tilde{d},S) =
w((\tilde{c},\tilde{d}) \cdot (t+S))(x)$ and by Theorem \[pre-bounds-of-w-thm\], for a fixed compact $D_0 \Subset
D$ there are $0 < m(D_0) < M$ such that $m(D_0) \le
\bar{v}(t,x;\tilde{c},\tilde{d}, S) \le M$ for any $(\tilde{c},\tilde{d}) \in Y$, $t,S \in {\ensuremath{\mathbb{R}}}$, and $x \in D_0$. Hence $$\begin{aligned}
&\label{nonauton-bar-v-eq3} \lim\limits_{T\to\infty} \frac{1}{T}
\int\limits_{0}^{T}
\frac{1}{\bar{v}} \frac{\p \bar{v}}{\p t}(t,x;\tilde{c},\tilde{d}, S)\,dt \nonumber\\
=& \lim\limits_{T\to\infty} \frac{1}{T}
(\ln\bar{v}(T,x;\tilde{c},\tilde{d},S) -
\ln\bar{v}(0,x;\tilde{c},\tilde{d},S)) = 0\end{aligned}$$ for any $(\tilde{c},\tilde{d}) \in Y$, $S \in {\ensuremath{\mathbb{R}}}$, and $x \in D$. Moreover, the limits are uniform in $(x,S) \in D_0 \times {\ensuremath{\mathbb{R}}}$ for any compact $D_0 \Subset D$.
\(1) We first prove that ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) \ge
\lambda(\hat{c},\hat{d})$ for some $(\hat{c},\hat{d}) \in
\hat{Y}(c,d)$.
Note that for given $S$, $T > 0$, $$\eta(t;c,d, S) = {\ensuremath{\lVertU_{(c,d) \cdot S}(t,0)w((c,d) \cdot S)\rVert}},$$ $$\hat{w}(x;c,d, S,T) = \exp\Bigl(\frac{1}{T}\int_S^{T+S}\ln{w((c,d)
\cdot t)(x)}\,dt\Bigr),$$ $$\frac{1}{T} \int_0^T \kappa((c,d) \cdot (t+S)) \,dt =
\frac{1}{T}\int_S^{T+S}\kappa((c,d) \cdot t)dt,$$ and $$\frac{1}{T} \int_{0}^{T} c(t+S,x)\,dt = \frac{1}{T}
\int_S^{T+S}c(t,x)dt.$$
By Theorem \[principal-spectrum-thm3\] there are $(S_n)$, $(T_n)$ with $T_n \to \infty$ such that $$\frac{1}{T_n} \int_{S_n}^{T_n+S_n}\kappa((c,d) \cdot t)\,dt =
\frac{\ln{\eta(T_n;c,d,S_n)}}{T_n} \to {\ensuremath{\lambda_{\mathrm{inf}}}}(c,d).$$ Without loss of generality we may assume that the limits$\lim_{n\to\infty} \frac{1}{T_n} \int_{S_n}^{T_n+S_n}c(t,x) \,dt$ and $\lim_{n\to\infty}\frac{1}{T_n}\int_{S_n}^{T_n+S_n}d(t,x)\,dt$ exist, uniformly in $x \in \bar{D}$ (resp. in $x \in \p D$). Denote these limits by $(\hat{c},\hat{d})$.
In the Dirichlet case, it is a consequence of Theorems \[pre-bounds-of-w-thm\] and \[Holder-continuity-thm\] that for each compact $D_0 \Subset D$ the sets $\{\,\hat{w}(\cdot;c,d,S_n,T_n)|_{D_0}: n = 1, 2, \dots\,\}$, $\{\,(\p \hat{w}/\p x_i)(\cdot;c,d$, $S_n$, $T_n)|_{D_0}: n = 1, 2,
\dots\,\}$ ($i = 1, \dots, N$) and $\{\,(\p^2 \hat{w}/\p x_{i}\p
x_{j})(\cdot;c,d,S_n,T_n)|_{D_0}: n = 1, 2, \dots\,\}$ ($i, j = 1,
\dots, N$) have compact closures in $C(D_0)$.
In the Neumann and Robin cases it is a consequence of Theorems \[pre-bounds-of-w-thm\] and \[Holder-continuity-thm\] that the sets $\{\,\hat{w}(\cdot;c,d,S_n,T_n): n = 1, 2, \dots\,\}$ and $\{\,(\p \hat{w}/\p x_i)(\cdot;c,d,S_n,T_n): n = 1, 2, \dots\,\}$ ($i = 1, \dots, N$) have compact closures in $C(\bar{D})$, and that for each compact $D_0 \Subset D$ the sets $\{\,(\p^2 \hat{w}/\p
x_{i}\p x_{j})(\cdot;c,d,S_n,T_n)|_{D_0}: n = 1, 2, \dots\,\}$ ($i, j
= 1, \dots, N$) have compact closures in $C(D_0)$.
We may thus assume that there is $w^{*} = w^*(x)$ such that $$\label{limit1}
\lim_{n\to\infty} \hat{w}(x;c,d,S_n,T_n) = w^*(x)$$ $$\label{limit2}
\lim_{n\to\infty} \frac{\p \hat{w}(x;c,d,S_n,T_n)}{\p x_i} =
\frac{\p w^*(x)}{\p x_i}$$ $$\label{limit3}
\lim_{n\to\infty}\frac{\p^2 \hat{w}(x;c,d,S_n,T_n)}{\p x_i\p x_j} =
\frac{\p^2 w^*(x)}{\p x_i\p x_j}$$ for $i, j = 1,2, \dots, N$ and $x \in D$. In the Dirichlet boundary conditions case, it follows from Theorems \[pre-bounds-of-w-thm\] and \[Holder-continuity-thm\] that $w^{*}$ can be extended to a function continuous on $\bar{D}$ by putting $w^{*}(x) = 0$ for $x \in
\p D$. Moreover, by Theorem \[pre-bounds-of-w-thm\], $w^*(x)
> 0$ for $x \in D$.
Regarding the uniformity of convergence, in the Dirichlet case, the limit in is uniform for $x$ in $\bar D$ and the limits in and are uniform for $x$ in any compact subset $D_0 \Subset D$, and in the Neumann and Robin cases, the limits in and are uniform for $x\in
\bar D$ and the limit is uniform for $x$ in any compact subset $D_0 \Subset D$.
We claim that ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) \ge \lambda(\hat{c},\hat{d})$. In fact, by –, $$\begin{cases}
\disp \sum_{i,j=1}^{N} a_{ij}(x) \frac{\p^2 {w}^*}{\p x_{i} \p
x_{j}} + \sum_{i=1}^{N} a_i(x) \frac{\p {w}^*}{\p x_i} + (\hat{c}(x)
- \lambda_{\rm min}(c,d)){w}^* \leq 0, & x \in D,
\\[2.5ex]
\hat{\mathcal{B}}{w}^* = 0, & x \in \p D,
\end{cases}$$ where $$\mathcal{\hat{B}}w^* :=
\begin{cases}
w^{*} \qquad & \text{(Dirichlet)} \\[1.5ex]
\disp \sum_{i=1}^{N} b_{i}(x) \frac{\p w^*}{\p x_{i}}\qquad &
\text{(Neumann)}
\\[1.5ex]
\disp \sum_{i=1}^{N} b_{i}(x) \frac{\p w^*}{\p x_{i}} + \hat{d}(x)
w^* \qquad & \text{(Robin)}.
\end{cases}$$ This implies that $w(t,x) = {w}^*(x)$ is a supersolution of $$\label{nonauton-eq3}
\begin{cases}
w_t = \disp \sum_{i,j=1}^{N} a_{ij}(x) \frac{\p^2 {w}}{\p x_{i} \p
x_{j}} + \sum_{i=1}^{N} a_i(x) \frac{\p {w}}{\p
x_i}\\\qquad\qquad {} + (\hat{c}(x) - {\ensuremath{\lambda_{\mathrm{inf}}}}(c,d)){w},
\quad & x \in D,
\\[2.5ex]
\hat{\mathcal{B}}{w} = 0, \quad & x \in \p D.
\end{cases}$$ Let $w(t,x;\hat{w})$ be the solution of with initial condition $w(0,x;\hat{w}) = {w}^*(x)$. Then we have $$\label{useful-eq1} w(t,x;\hat{w}) \le {w}^*(x)$$ for $x \in D$ and $t \ge 0$. Note that $\lambda(\hat{c},\hat{d}) -
{\ensuremath{\lambda_{\mathrm{inf}}}}(x,d)$ is the principal eigenvalue of with $(\hat{c},\hat{d})$ being replaced by $(\hat{c} - {\ensuremath{\lambda_{\mathrm{inf}}}}(c,d),\hat{d})$. It then follows from together with the positivity of ${w}^*(x)$ that $$\label{useful-eq2}
\lambda(\hat{c},\hat{d}) - {\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) \le 0.$$ This implies that $$\lambda(\hat{c},\hat{d}) \le {\ensuremath{\lambda_{\mathrm{inf}}}}(c,d).$$
Next, we prove ${\ensuremath{\lambda_{\mathrm{sup}}}}(c,d) \ge \lambda(\hat{c},\hat{d})$ for any $(\hat{c},\hat{d}) \in \hat{Y}(c,d)$. For any $(\hat{c},\hat{d}) \in
\hat{Y}(c,d)$ there are $(S_n), (T_n)$ with $T_n \to \infty$ such that $$\frac{1}{T_n} \int_{S_n}^{T_n+S_n}c(t,x)\,dt \to \hat{c}(x)$$ and $$\frac{1}{T_n} \int_{S_n}^{T_n+S_n} d(t,x)\,dt \to \hat{d}(x),$$ uniformly in $x \in \bar{D}$ (resp. uniformly in $x \in \p D$). Without loss of generality, assume that $$\frac{1}{T_n} \int_{S_n}^{T_n+S_n} \kappa((c,d) \cdot t)\,dt \to
\lambda_0.$$ By arguments similar to the above, $\lambda_0 \ge
\lambda(\hat{c},\hat{d})$. Note that ${\ensuremath{\lambda_{\mathrm{sup}}}}(c,d) \ge \lambda_0$. Then we have ${\ensuremath{\lambda_{\mathrm{sup}}}}(c,d) \ge \lambda(\hat{c},\hat{d})$.
\(2) By Lemma \[averaging-uniform\], there is $\Omega_1 \subset
\Omega$ with ${\ensuremath{\mathbb{P}}}(\Omega_1) = 1$ such that $$\hat{c}(x) = \lim_{T\to\infty}\frac{1}{T}\int_0^{T}
c(\theta_t\omega,x)\,dt$$ for any $\omega \in \Omega_1$ and any $x \in \bar{D}$, uniformly in $\bar{D}$, and $$\hat{d}(x) = \lim_{T\to\infty}\frac{1}{T}\int_0^{T}
d(\theta_{t}\omega,x)\,dt$$ for any $\omega \in \Omega_1$ and any $x \in \p D$, uniformly in $\p
D$.
By Theorem \[principal-exponent-thm\], there is $\Omega_2 \subset
\Omega$ with ${\ensuremath{\mathbb{P}}}(\Omega_2)=1$ such that $$\lambda = \lim\limits_{T\to\infty}
\frac{\ln{\ensuremath{\lVertU_\omega(T,0)w(\omega)\rVert}}}{T}$$ for any $\omega \in \Omega_2$.
Take an $\omega \in \Omega_1 \cap \Omega_2$. Then for any $T_n \to
\infty$, $$\frac{1}{T_n}\int_0^{T_n} c(\theta_t\omega,x)\,dt =
\frac{1}{T_n}\int_0^{T_n}c^\omega(t,x)dt \to \hat{c}(x) \quad \text{
uniformly for } x \in \bar{D},$$ $$\frac{1}{T_n}\int_0^{T_n} d(\theta_{t}\omega,x)\,dt =
\frac{1}{T_n}\int_0^{T_n}d^\omega(t,x)dt\to \hat{d}(x)\quad
\text{uniformly for } x \in \p D,$$ and $$\frac{\ln\eta(T_n;c^{\omega},d^{\omega},0)}{T_n} \to \lambda.$$ By arguments as in the proof of Part (1), we must have $\lambda \ge
\hat{\lambda}$.
We first prove (2) for the reason that (2) will be used in the proof of (1).
First, suppose that $c(\omega,x) = c_{1}(x) +
c_{2}(\theta_{t}\omega)$ for any $x \in \bar{D}$, any $t \in {\ensuremath{\mathbb{R}}}$ and any $\omega \in \Omega^*$. Without loss of generality, we may assume $\int_{\Omega} c_{2}(\omega) \,d{\ensuremath{\mathbb{P}}}(\omega) = 0$ and ${\ensuremath{\mathbb{P}}}(\Omega^*) =
1$ (for otherwise, we change $c_1(x)$ to $c_1(x)+\int_{\Omega}c_2(\omega)\,d{\ensuremath{\mathbb{P}}}(\omega)$ and change $c_2(\omega)$ to $c_2(\omega) - \int_\Omega
c_2(\omega)\,d{\ensuremath{\mathbb{P}}}(\omega)$). Suppose also that $d(\theta_{t}\omega,x) = d(x)$. One has $\hat{c}(x) = c_{1}(x)$ for $x \in \bar{D}$, and $\hat{d}(x) = d(x)$ for $x \in \p D$. Let $u(x)$ be the positive principal eigenfunction of normalized so that its $L_2(D)$-norm equals $1$, and let $$v(t,x;\omega) := u(x)\exp{\Bigl(\hat{\lambda} t + \int_0^t
c_{2}(\theta_s\omega)\,ds\Bigr)}$$ for $t \in {\ensuremath{\mathbb{R}}}$, $x \in \bar{D}$ and $\omega \in \Omega^*$. It is then not difficult to see that for any $\omega \in \Omega^{*}$ the function $[\,{\ensuremath{\mathbb{R}}}\ni t \mapsto v(t,\cdot;\omega) \in L_2(D)\,]$ is the (necessarily unique) normalized globally positive solution of . For a.e. $\omega \in \Omega$, $\lambda =
\lim_{t\to\infty}(1/t)\ln{{\ensuremath{\lVertv(t,\cdot;\omega)\rVert}}}$. It follows with the help of Birkhoff’s Ergodic Theorem (Lemma \[ch5-pre-ergodic-lm\]) that the last term equals $\hat{\lambda}$ for a.e. $\omega \in \Omega^{*}$. Consequently, $\lambda = \hat{\lambda}$.
Conversely, let $\Omega_1$ and $\Omega_2$ be as in the proof of Theorem \[ch5-smoothlb-thm1\](2). We write $\eta(t;\omega)$ for $\eta(t;c^{\omega},d^{\omega},0)$ and $\hat{w}(x;\omega,T)$ for $\hat{w}(x;c^{\omega},d^{\omega},0,T)$, respectively. Then $$\eta(t;\omega) = {\ensuremath{\lVertU_{\omega}(t,0)w(\omega)\rVert}}$$ and $$\hat{w}(x;\omega,T) = \exp\Bigl(\frac{1}{T}\int_0^T \ln
w(\theta_t\omega)(x) \,dt \Bigr).$$
Let $$\phi(x) := \exp{\int\limits_{\Omega}\ln{w(\omega)(x)}\,d{\ensuremath{\mathbb{P}}}(\omega)}
\quad\text{for}\quad x \in \bar{D}$$ in the case of Neumann or Robin boundary condition, and $$\phi(x) :=
\begin{cases}
\disp \exp{\int\limits_{\Omega}\ln{w(\omega)(x)}
\,d{\ensuremath{\mathbb{P}}}(\omega)} \quad & \text{for}\quad x \in D
\\
0 \quad & \text{for}\quad x \in \p D
\end{cases}$$ in the case of Dirichlet boundary condition. By Lemma \[averaging-uniform\], there is $\Omega_3 \subset \Omega$ with ${\ensuremath{\mathbb{P}}}(\Omega_3) = 1$ such that $$\label{random-phi-eq1} \phi(x) = \lim_{T\to \infty}
\exp\Bigl(\frac{1}{T} \int_0^{T} \ln{w(\theta_{t}\omega)(x)}\,dt
\Bigr) = \lim_{T\to\infty} \hat{w}(x;\omega,T)$$ for any $\omega \in \Omega_3$ and $x \in D$. Clearly, $\phi(x)
> 0$ for $x \in D$.
Observe that by Theorems \[pre-bounds-of-w-thm\] and \[Holder-continuity-thm\], $\disp \frac{\p w(\omega)(x)}{\p x_i}$ ($i = 1,2, \dots, N$) ($\disp\frac{\p ^2 w(\omega)(x)}{\p x_i\p
x_j}$, $i,j = 1,2,\dots,N$) are locally Hölder continuous in $x \in
\bar{D}$ ($x \in D$) uniformly in $\omega \in \Omega$, and for a fixed $x \in \bar{D}$ ($x \in D$) they are bounded in $\omega \in
\Omega$. Hence, Lemma \[averaging-derivative\] together with Eqs. and gives us the existence of $\Omega_4 \subset \Omega$ with ${\ensuremath{\mathbb{P}}}(\Omega_4) =
1$ such that $$\begin{aligned}
\label{random-phi-eq2}
\frac{\p\phi}{\p x_i}(x)&
= \lim_{T\to\infty}\frac{\p \hat{w}(x;\omega,T)}{\p x_i} \nonumber \\
& = \phi(x)\int_\Omega \Bigl( \frac{1}{w(\cdot)(x)}\frac{\p
w(\cdot)(x)}{\p x_i}\Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot),\end{aligned}$$ $$\begin{aligned}
\label{random-phi-eq3}
\frac{\p^2\phi}{\p x_i\p x_j}&
= \lim_{T\to\infty}\frac{\p^2 \hat{w}(x;\omega,T)}{\p x_i\p x_j} \nonumber \\
& = \phi(x)\int_\Omega \Bigl(\frac{1}{w(\cdot)(x)} \frac{\p
w(\cdot)(x)}{\p x_i}\Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot)
\int_\Omega\Bigl(\frac{1}{w(\cdot)(x)} \frac{\p w(\cdot)(x)}{\p x_j}
\Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot)
\nonumber\\
&\quad + \phi(x) \int_\Omega\Bigl(\frac{1}{w(\cdot)(x)} \frac{\p ^2
w(\cdot)(x)}{\p x_i\p x_j} - \frac{1}{w^2(\cdot)(x)} \frac{\p
w(\cdot)(x)} {\p x_i} \frac{\p w(\cdot)(x)}{\p
x_j}\Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot)\end{aligned}$$ and $$\lim_{T\to\infty}\frac{1}{T}\int_0^T
\frac{1}{w(\theta_t\omega)(x)}\frac{\p w(\theta_t\omega)(x)}{\p
x_i}\,dt = \int_\Omega\frac{1}{w(\omega)(x)}\frac{\p
w(\omega)(x)}{\p x_i} \,d{\ensuremath{\mathbb{P}}}(\cdot),$$ $$\begin{aligned}
&\lim_{T\to\infty}\frac{1}{T}\int_0^T\frac{1}{w^2(\theta_t\omega)(x)}
\frac{\p w(\theta_t(\omega)(x)}{\p x_i} \frac{\p
w(\theta_t\omega)(x)}{\p x_j}\,dt \\
=& \int_{\Omega}\frac{1}{w(\omega)(x)}\frac{\p w(\omega)(x)}{\p
x_i}\frac{\p w(\omega)(x)}{\p x_j}\, d{\ensuremath{\mathbb{P}}}(\cdot)\end{aligned}$$ for $\omega \in \Omega_4$, $x \in D$, and $$\mathcal{\hat{B}} \phi = 0$$ for $\omega \in \Omega_4$, $x \in \p D$, where $$\label{random-phi-eq4}
\mathcal{\hat{B}}\phi :=
\begin{cases}
\phi & \text{(Dirichlet)} \\[1.5ex]
\disp \sum_{i=1}^{N} b_{i}(x) \frac{\p \phi}{\p x_{i}} &
\text{(Neumann)}
\\[1.5ex]
\disp \sum_{i=1}^{N} b_{i}(x) \frac{\p \phi}{\p x_{i}} +
\hat{d}(x)\phi & \text{(Robin)}.
\end{cases}$$
Let $\Omega_0 := \Omega_1 \cap \Omega_2 \cap \Omega_3 \cap \Omega_4$. Then – hold for any $\omega \in \Omega_0$.
Put $\bar{v}(t,x;\omega) := w(\theta_t\omega)(x)$. Since, by Theorem \[pre-bounds-of-w-thm\], for a fixed $x \in D$ there are $0
< m < M$ such that $m \le w(\omega)(x) \le M$ for any $\omega \in
\Omega$, we have $$\label{random-bar-v-eq3}
\lim\limits_{T\to\infty} \frac{1}{T} \int\limits_{0}^{T}
\frac{1}{\bar{v}} \frac{\p \bar{v}}{\p t}(t,x;\omega)\,ds =
\lim\limits_{T\to\infty} \frac{1}{T} (\ln{w(\theta_{T}\omega)(x)} -
\ln{w(\omega)(x)}) = 0$$ for any $\omega \in \Omega_0$ and $x \in D$.
Consequently, by and we have $$\begin{aligned}
\label{ch5-smooth-phi-eq7} &\sum_{i,j=1}^N a_{ij}(x)
\frac{\p^{2}\phi}{\p x_{i} \p x_{j}} + \sum_{i=1}^{N} a_{i}(x)
\frac{\p \phi}{\p x_i}
\nonumber \\
=& (\lambda - \hat{c}(x))\phi\nonumber\\
& + \phi\sum_{i,j=1}^N a_{ij}(x) \int_\Omega
\Bigl(\frac{1}{w(\cdot)} \frac{\p w(\cdot)}{\p
x_i}\Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot) \int_\Omega \Bigl(\frac
{1}{w(\cdot)}\frac{\p w(\cdot)}{\p x_j}\Bigr) \,d{\ensuremath{\mathbb{P}}}(\cdot)
\nonumber\\
& - \phi \sum_{i,j=1}^N a_{ij}(x)
\int_\Omega\Bigl(\frac{1}{w^2(\cdot)} \frac{\p w(\cdot)}{\p x_i}
\frac{\p w(\cdot)}{\p x_j}\Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot)\end{aligned}$$ for $x \in D$, and $$\hat{\mathcal{B}}\phi = 0 \quad \text{for} \quad x \in \p D.$$
Suppose that $\lambda = \hat{\lambda}$. Consider $$\label{additional-eq}
\begin{cases}
\disp u_t = \sum_{i,j=1}^N a_{ij}(x)\frac{\p^2 u}{\p x_i\p x_j} +
\sum_{i=1}^Na_i(x) \frac{\p
u}{\p x_i} + (\hat{c}(x) - \lambda)u,\quad x\in D, \\
\\
\hat{\mathcal{B}}u = 0.
\end{cases}$$
We have that $0$ is the principal eigenvalue of . Let $\hat{\phi}$ be a positive principal eigenfunction of . Let $u(t,x;\phi)$ be the solution of with initial condition $u(0,x;\phi) = \phi(x)$. By Lemma \[ch5-pre-holder-lm\](2), $$\begin{aligned}
&\quad\sum_{i,j=1}^N a_{ij}(x) \int_\Omega \Bigl(\frac{1}{w(\cdot)}
\frac{\p w(\cdot)}{\p x_i}\Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot) \int_\Omega
\Bigl(\frac {1}{w(\cdot)}\frac{\p w(\cdot)}{\p x_j}\Bigr)
\,d{\ensuremath{\mathbb{P}}}(\cdot)\\
& - \sum_{i,j=1}^N a_{ij}(x) \int_\Omega\Bigl(\frac{1}{w^2(\cdot)}
\frac{\p w(\cdot)}{\p x_i} \frac{\p w(\cdot)}{\p
x_j}\Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot) \le 0\end{aligned}$$ for all $x \in D$. This together with implies that $\phi(x)$ is a supersolution of and hence $$\label{additional-eq1}
u(t,x;\phi) \le \phi(x) \quad \text{ for } \quad x \in D, \quad t
\ge 0.$$ We apply now Theorem \[globally-positive-existence\] to the autonomous problem . In this case, $Y$ is a singleton, $w = \hat{\phi}$, and $w^{*}(x) > 0$ for a.e. $x \in D$. It follows then that $\langle \phi, w^{*} \rangle > 0$ and $\langle
\hat{\phi}, w^{*} \rangle > 0$. By taking $\alpha := \langle \phi,
w^{*} \rangle / \langle \hat{\phi}, w^{*} \rangle$ ($ > 0$) we see that $$\phi = \alpha\hat{\phi} + \hat{\psi},$$ where $\hat \psi \in X$ is such that $\langle \hat \psi, w^{*}
\rangle = 0$. Note that $u(t,x;\phi) = \alpha \hat{\phi}(x) +
u(t,x;\hat{\psi})$, where $u(t,x;\hat \psi)$ is the solution of with $u(0,x;\hat{\psi}) = \hat{\psi}(x)$. Theorem \[globally-positive-existence\](iii) gives that ${\ensuremath{\lVertu(t,\cdot;\hat{\psi})\rVert}} \to 0$ as $t \to \infty$. It then follows from that $$\alpha \hat{\phi}(x) \le \phi(x)\quad \text{ for }\quad x \in D,$$ and then $$\hat{\psi}(x) \ge 0 \quad \text{ for } \quad x \in D.$$ This implies that $$\hat{\psi}(x) = 0 \quad \text{ for } \quad x \in D,$$ hence $$\alpha \hat{\phi}(x) = \phi(x) \text{ for } \quad x \in D.$$ Therefore we must have $$\begin{aligned}
&\sum_{i,j=1}^N a_{ij}(x) \int_{\Omega} \Bigl(
\frac{1}{w(\cdot)}\frac{\p w(\cdot)}{\p x_i} \Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot)
\int_{\Omega} \Bigl( \frac{1}{w(\cdot)}\frac{\p w(\cdot)}{\p x_j}
\Bigr) \,d{\ensuremath{\mathbb{P}}}(\cdot) \\
= &\sum_{i,j=1}^N a_{ij}(x) \int_{\Omega} \Bigl(
\frac{1}{w^2(\cdot)}\frac{\p w(\cdot)}{\p x_i} \frac{\p w(\cdot)}{\p
x_j}\Bigr)\,d{\ensuremath{\mathbb{P}}}(\cdot)\end{aligned}$$ for all $x \in D$.
Let $\{\,x^{(n)}: n \in {\ensuremath{\mathbb{N}}}\,\}$ be a countable dense subset of $D$. By Lemma \[ch5-pre-holder-lm\](2), for each $n \in {\ensuremath{\mathbb{N}}}$ there is $\Omega^{(n)}$ with ${\ensuremath{\mathbb{P}}}(\Omega^{(n)}) = 1$ such that $\disp
\frac{1}{w(\omega)(x^{(n)})}\frac{\p w(\omega)(x^{(n)}) }{\p x_i}$ is independent of $\omega \in \Omega^{(n)}$. Consequently, from the continuity of $\disp \frac{1}{w(\omega)(x)}\frac{\p w(\omega)(x) }{\p
x_i}$, for a fixed $\omega \in \Omega$, in $x \in D$, there are $\Omega_5 \subset \Omega_0$, $\Omega_5 := \Omega_0 \cap
\bigcap_{n=1}^{\infty}\Omega^{(n)}$, with ${\ensuremath{\mathbb{P}}}(\Omega_5) = 1$, and functions $f_i(x)$ such that $$\frac{1}{w(\omega)(x)} \frac{\p w(\omega)(x) }{\p x_i} = f_i(x)$$ for $i = 1,2,\dots,N$, $\omega \in \Omega_5$ and $x \in D$. Hence $$\nabla{\ln{w}}(\omega)(x) = (f_1(x),f_2(x),\dots,f_N(x))^\top$$ for $\omega \in \Omega_5$ and $x \in D$. This implies that $w(\omega)(x) = F(x) G(\omega)$ for some continuous $F(x) > 0$, measurable $G(\omega) > 0$ and any $\omega \in \Omega_5$, $x \in D$. Let $\Omega^{*} := \bigcap_{r\in{\ensuremath{\mathbb{Q}}}}\theta_r\Omega_5$, where ${\ensuremath{\mathbb{Q}}}$ is the set of all rational numbers. Clearly, ${\ensuremath{\mathbb{P}}}(\Omega^{*}) = 1$ and $w(\theta_t\omega)(x) = F(x)G(\theta_t\omega)$ for $t \in {\ensuremath{\mathbb{Q}}}$, $\omega \in \Omega^{*}$ and $x \in D$. The continuity of $w(\theta_t\omega)(x)$ in $t \in {\ensuremath{\mathbb{R}}}$ then implies that the function $[\, {\ensuremath{\mathbb{R}}}\ni t \mapsto w(\theta_t\omega)(x)/F(x) \in {\ensuremath{\mathbb{R}}}]$ is continuous. Hence, for each $\omega \in \Omega^{*}$ and each $t \in
{\ensuremath{\mathbb{R}}}$ we can safely write $G(\theta_t\omega)$ for $w(\theta_t\omega)(x)/F(x)$. Therefore, by , $$\begin{aligned}
&\label{auxiliary}
F(x)\frac{dG(\theta_t\omega)}{dt} \nonumber\\
=\null & \Bigl( \sum_{i,j=1}^N a_{ij}(x) \frac{\p^2 F}{\p x_i \p x_j} +
\sum_{i=1}^N a_i(x) \frac{\p F}{\p x_i} + c(\theta_t\omega,x)F -
\kappa(\theta_t\omega)F\Bigr) G(\theta_t\omega)\end{aligned}$$ for $t \in {\ensuremath{\mathbb{R}}}$, $\omega \in \Omega^{*}$ and $x \in D$, and $$\mathcal{B}(\theta_t\omega) F = 0$$ for $t \in {\ensuremath{\mathbb{R}}}$, $\omega \in \Omega^{*}$ and $x \in \p D$. By dividing both sides of by $F(x) G(\theta_t\omega)$ we obtain $$c(\theta_t\omega,x) = \frac{\frac{dG(\theta_t\omega)}{dt}}
{G(\theta_{t}\omega)} + \kappa(\theta_{t}\omega) -
\frac{\sum_{i,j=1}^N a_{ij}(x) \frac{\p^2 F}{\p x_i \p x_j}(x) +
\sum_{i=1}^N a_i(x) \frac{\p F}{\p x_i}(x)}{F(x)}$$ for $t \in {\ensuremath{\mathbb{R}}}$, $\omega \in \Omega^{*}$ and $x \in \p D$. We can write $c(\theta_{t}\omega,x) = c_{1}(x) + c_{2}(\theta_{t}\omega)$ for some integrable $c_{2}(\omega)$ with $\int_{\Omega}
c_{2}(\cdot)\,d{\ensuremath{\mathbb{P}}}(\cdot) = 0$, any $x \in D$, $t \in {\ensuremath{\mathbb{R}}}$ and $\omega \in \Omega^{*}$. Similarly, by taking the boundary condition $\mathcal{B}(\theta_t\omega) F = 0$ we obtain that $$d(\theta_{t}\omega,x) = - \frac{\sum_{i=1}^{N}b_i(x)\frac{\p F}{\p
x_i}(x)}{F(x)}$$ for $t \in {\ensuremath{\mathbb{R}}}$, $\omega \in \Omega^{*}$ and $x \in \p D$, that is, $d(\theta_{t}\omega,x) = d(x)$ for each $x \in \p D$, each $t \in
{\ensuremath{\mathbb{R}}}$ and each $\omega \in \Omega^{*}$.
\(1) Let ${\ensuremath{\mathbb{P}}}$ be the unique ergodic measure on $Y(c,d)$. By Lemma \[ch5-pre-existence-avg-lm\], $$\hat{c}(x) := \lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}c(s,x)\,ds$$ exists for $x \in D$, and $$\hat{d}(x) := \lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}d(s,x)\,ds$$ exists for $x \in \p D$.
Assume that the equality ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) = \lambda(\hat{c},\hat{d})$ holds. By Theorem \[ch5-smoothlb-thm2\](2), there is $Y_0(c,d)
\subset Y(c,d)$ with ${\ensuremath{\mathbb{P}}}(Y_0(c,d)) = 1$ such that for any $(\tilde{c},\tilde{d}) \in Y_0(c,d)$, $\tilde{c}(t,x) = c_{1}(x) +
\check{c}_2((\tilde{c},\tilde{d}) \cdot t)$ for some ${\ensuremath{\mathbb{P}}}$-integrable $\check{c}_2$ with $\int_{Y(c,d)}\check{c}_2(\tilde{c},\tilde{d})
\,d{\ensuremath{\mathbb{P}}}(\tilde{c},\tilde{d}) = 0$, any $t \in {\ensuremath{\mathbb{R}}}$, $x \in D$, and $\tilde{d}(t,x) = d(x)$ for $x \in \p D$. Take a $(\tilde{c},\tilde{d}) \in Y_0(c,d)$. Since $Y(c,d)$ is minimal, there is a sequence $(s_n)$ such that $(\tilde{c},\tilde{d}) \cdot
s_n$ converges in $Y(c,d)$ to $(c,d)$ as $n \to \infty$. This implies that $c(t,x) = c_{1}(x) + c_{2}(t)$ and $d(t,x) = d(x)$. By unique ergodicity, $\lim_{t\to\infty} \frac{1}{t}\int_{0}^{t}
c_{2}(s)\,ds = \int_{Y(c,d)}\check{c}_2(\tilde{c},\tilde{d})
\,d{\ensuremath{\mathbb{P}}}(\tilde{c},\tilde{d}) = 0$.
If $c(t,x) = c_{1}(x)+ c_{2}(t)$ and $d(t,x) = d(x)$, then the equality ${\ensuremath{\lambda_{\mathrm{inf}}}}(c,d) = \bar{\lambda}$ follows clearly.
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[^1]: Supported by the research funds for 2005–2008 (grant MENII 1 PO3A 021 29, Poland) and NSF grant INT-0341754.
[^2]: Partially supported by NSF grants DMS-0504166 and INT-0341754.
[^3]: This is a pre-copy-editing, author-produced PDF of an article accepted for publication in *Discrete and Continuous Dynamical Systems Series B* following peer review. The definitive publisher-authenticated version *Discrete and Continuous Dynamical Systems Series B* **9**(3/4) (2008), pp. 661–699, is available online at: http://dx.doi.org/10.3934/dcdsb.2008.9.661
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abstract: 'Recent years have seen major advances in understanding the state of the intergalactic medium (IGM) at high redshift. Some aspects of this understanding are reviewed here. In particular, we discuss: (1) Different probes of IGM like Gunn-Peterson test, CMBR anisotropies, and neutral hydrogen emission from reionization, and (2) some models of reionization of the universe.'
author:
- 'Shiv K. Sethi'
title: 'High Redshift Intergalactic Medium: Probes and Physical Models'
---
Indroduction
============
One of the outstanding issues in cosmology is to understand the reionization of the universe. Following recombination of primordial plasma at redshift $z \simeq 1000$, the universe is mostly neutral with an ionization fraction $\simeq 10^{-4}$ (see e.g. [@peebles1], [@padmanabhan1; @padmanabhan2]). The Jean’s mass at recombination is $\simeq 10^6 \, \rm M_\odot$. At $z \la 100$, the plasma thermally decouples from CMBR and its temperature decreases adiabatically, $T_m \simeq 1/a^2$, which leads to a further decrease in Jean’s mass. During this ’dark and cold age’ the density perturbations at scales above the Jean’s scale can grow. Figure 1 shows the ionization and thermal history of the universe along with the evolution of Jeans’ mass. This age comes to an end when the first structures can collapse and the light from these objects can reionize and reheat the universe (for a recent review see [@barkana] and references therein) . Therefore the epoch of re-ionization holds important clues about the way first structures formed and can potentially distinguish between different models of structure formation. In the standard $\Lambda\rm CDM$ models the first structures to collapse would be just above the Jean’s length (see e.g. [@padmanabhan1]). Another crucial question in this regard is whether these structures could cool fast enough to form stars. Many of these issues will be discussed in this review.
Different ongoing and potential probes of intergalactic medium can reveal the nature of the re-ionization of the universe. One of the most important and the oldest is the Gunn-Peterson test (see e.g. [@peebles1] and reference therein), which is very sensitive to the neutral fraction in the intergalactic medium. CMBR anisotropy measurements are another powerful and complementary probe, as they are sensitive to the ionized component of the intergalactic medium (see [@hu1], [@bond] and references therein). In future, it might be possible to directly observe the first sources that re-ionize the universe. In addition, the transition from neutral to ionized universe might also be detected in neutral hydrogen emission (see e.g. [@madau], [@shaver], [@tozzi]).
To sum up the observational status: Recent detection of temperature-polarization cross-correlation in CMBR suggests that the redshift of reionization $z_{\rm reion} \simeq 17\pm 4$ [@kogut]. Gunn-Peterson probes suggest that the universe is highly ionized upto $z \simeq 5$, but might be making a transition from highly ionized to neutral for $5 \la z \la 6$ ([@fan], [@djorgovski], [@becker]) . These two observations together throw open the interesting possibility that the universe went through two phases of re-ionization.
This article is divided into two parts. In the first part, probes that give a clue about the reionization epoch will be discussed. In the second part, we will discuss the nature of ionizing sources. Throughout this review we use the currently-favoured FRW model: spatially flat with $\Omega_m = 0.3$ and $\Omega_\Lambda = 0.7$ ([@spergel], [@perlmutter], [@riess]) with $\Omega_b h^2 = 0.02$ [@spergel], [@tytler]) and $h = 0.7$ [@freedman].
Probes of Ionization at high redshifts
======================================
Gunn-Peterson effect
--------------------
High redshift sources should show absorption at frequencies close to Lyman-$\alpha$ (1216 $\rm \AA$) line owing to scattering from IGM neutral hydrogen (HI). This test applied to high redshift quasars which generally have a strong Lyman-$\alpha$ emission line means that the blueward side of the Lyman-$\alpha$ line should show strong absorption as compared to the redward side (Gunn-Peterson (GP) test). The optical depth to the Lyman-$\alpha$ scattering from HI can be calculated (see e.g. [@peebles1]): $$\tau_{\rm \scriptscriptstyle GP}(\nu_0) \simeq 4 \times 10^{5} x_{\rm \scriptscriptstyle HI} \left (h \over 0.7 \right ) \left (\Omega_b \over 0.045 \right ) \left (0.3 \over \Omega_m \right )^{1/2} \left (1+z \over 6 \right )^{1.5}
\label{eq:gpt}$$ Here $x_{\rm \scriptscriptstyle HI} \equiv n_{\rm \scriptscriptstyle HI}/n_{\rm \scriptscriptstyle H}$ ($n_{\rm \scriptscriptstyle H} \simeq 0.92 n_b$) is the neutral fraction of hydrogen and $\nu_0 = \nu_\alpha/(1+z)$. Two points worth noting in Eq. (\[eq:gpt\]) are: (1) Owing to the resonant nature of Lyman-$\alpha$ scattering, optical depth at observed frequency $\nu_0$ gives direct information about the neutral fraction at redshift $(1+z) = \nu_\alpha/\nu_0$, and (2) more importantly, this test is extremely sensitive to the neutral fraction of hydrogen, even a neutral fraction as small as a part in hundred thousand can fully absorb light shortward of Lyman-$\alpha$ in the quasar spectrum.
Since 1960s when the first high redshift quasars whose Lyman-$\alpha$ emission could be detected from ground-based telescopes were discovered, GP test has been applied to study the ionization of the universe at high redshifts. Till 2000, none of the observed quasars upto a redshift $\simeq 5$ showed any GP absorption, which means the universe is ionized to better than a part in a million upto $z \simeq 5$. The discovery of several quasars at redshifts above six by SDSS survey made it possible to apply GP tests at even higher redshifts. GP absorption has been detected in several quasars at redshifts $z \ga 5.7$ ([@becker], [@djorgovski], [@fan]) . Observations suggest that the neutral fraction increases rapidly between $z \simeq 5.5$ to $z \simeq 6.2$ (see e.g. [@fan]). These observations might mean that the universe is becoming ionized owing to the formation of the first structures at $z \simeq 6$. However a straightforward interpretation of these results is not easy. Even for the best quasar spectrum and using the Lyman-$\beta$ line, which has a smaller oscillator strength, the GP optical depth $\tau_{\rm \scriptscriptstyle GP} \ga 25$ ([@white]). Using Eq. (\[eq:gpt\]), this implies that $x_{\rm \scriptscriptstyle HI} \ga 10^{-4}$ i.e. the universe can be almost fully ionized. Use of semi-analytic models which take into account the clumpiness of the IGM give more stringent constraints on the neutral fraction and give $x_{\rm \scriptscriptstyle HI} \ga 10^{-3}$ at $z \simeq 6$ [@fan]. Even though the observations are unable to conclude that the universe made a transition from fully ionized to almost fully neutral between redshift of five and six, the ionized fraction certainly evolves very rapidly in this redshift range, and this can have important implications for the models of structure formation.
CMBR anisotropies
-----------------
CMBR anisotropies provide a complementary approach to the ionization history of the universe as compared to the GP test, as they are sensitive to the ionized component of the universe. The physics of CMBR temperature anisotropies at the last scattering surface and various data analysis issue are covered elsewhere in this volume (Subramanian, this volume). Here we shall discuss the implications of reionization on the CMBR temperature anisotropies. To highlight the effect of reionization on the polarization anisotropies we discuss more fully the CMBR polarization anisotropies.
The universe recombines at $z \simeq 1000$. Following recombination the ionized fraction ($x_e \equiv n_e/n_{\rm \scriptscriptstyle H}$) in the universe is $\simeq 10^{-4}$ for $z \simeq 100$ (see e.g. [@peebles1]). The mean free path of the CMBR photons to Thompson scattering exceeds the local Hubble radius in the post-recombination era and therefore the universe is ’transparent’ to the CMBR photons. Following reionization, the ionized fraction might reach nearly unity and a small fraction of CMBR photons might re-scatter again. An important quantity in studying CMBR quantity is visibility function which is the normalized probability that the photon scattered in a range $z$ and $z +dz$. It is defined as: $V(\eta_0,\eta) = \dot \tau \exp(-\tau)$, here $d\eta = dt/a$ is the conformal time and $\tau = \int_{\eta_0}^{0} x_e n_{\rm \scriptscriptstyle H} a \sigma_t cd\eta$. In Figure 2 we show the visibility function for two models. For the model with early reionization, the visibility function get important contribution from redshift of reionization. One can define the optical depth to the reionization surface: $\tau_{\rm reion} = \int_0^{z_{\rm reion}} x_e n_{\rm \scriptscriptstyle H} \sigma_t cdt$; this is the fraction of photons that re-scattered in the reionized universe for $\tau_{\rm reion} < 1$. For $z_{\rm reion} \simeq 6$, $\tau_{\rm reion} \simeq 0.05$ and $\tau_{\rm reion} \propto (1+z_{\rm reion})^{3/2}$.
The temperature and polarization anisotropies can be computed by solving the Boltzmann equations for the photon distribution function. Equations appropriate for studying the effect of reionization for scalar perturbations, for a given wavenumber ${\bf k}$ and line of sight ${\bf n}$, are (see e.g. [@hu1], Zalddariaga 1997, [@bond],[@dodelson]): $$\begin{aligned}
\dot \Delta_T + ik\mu \Delta_T & \simeq & \dot \tau (\mu v - \Delta_T) \nonumber \\
\dot \Delta_P + ik\mu \Delta_P &= & \dot \tau (\Pi(\mu)[\Delta_{T2} +\Delta_{P2}- \Delta_{P0}] -\Delta_P)
\label{eq:tpeq}\end{aligned}$$ Here $\mu = {\bf k.n}$, $\dot \Delta_T {} \equiv \partial \Delta_T/\partial \eta$, $ \Delta_T \equiv \Delta T/T(k,\mu,\eta)$. The polarization anisotropies $\Delta_P \equiv \Delta P/T(k,\mu,\eta)$ are in the plane perpendicular to the ${\bf k}$ vector, which is taken to be parallel to the z-axis, to exploit the axial symmetry of the problem. In this case, only one Stokes parameter $Q$ is non-zero and $\Delta P = Q T$. Also $\dot \tau = n_b x_e \sigma_T a$, $\Pi(\mu) = 0.5(1-P_2(\mu))$. $v$ is the electron velocity; we also assume curl-free velocity fields which allows us to express: ${\bf v.n} = {\bf v.k}$. The angular moments of temperature and polarization anisotropies are: $\Delta_{\ell} = {1\over 2} \int_{-1}^1 d\mu P_\ell(\mu) \Delta(k,\mu)$. Even though the temperature equation is only appropriate for studying the effect of reionization on anisotropies generated at last scattering surface, the polarization equation is exact and can also be used to study the generation of perturbations at the last scattering surface.
To understand the essential physics of CMBR anisotropies in reionized models, we only solve for anisotropy evolution for one wavenumber ${\bf k}$. The quantity measured by experiments is the two point function of this quantity summed over all the wave-numbers (for details see articles by Subramanian, this volume).
We begin by studying the effect of reionization on the temperature anisotropies. Eq (\[eq:tpeq\]) can be solved to give: $$\Delta_T(\eta) = \Delta_T(\eta_{\rm rec})\exp(ik\mu(\eta_{\rm rec}- \eta))\exp(-\tau(\eta_{\rm rec},\eta)) + \mu \int d\eta'v(\eta')V(\eta,\eta') \exp(ik\mu(\eta'- \eta))
\label{eq:tesol}$$ The first term in the solution means the anisotropies generated at the last scattering surface are exponentially damped by reionization (the solution is only correct for scales smaller than the size of local horizon, i.e. $k \ga \eta^{-1}(z_{\rm reion})$. It is not reflected in the solution owing to dropping a term $\propto
\Delta_{T0}$ in the temperature equation (for details see [@hu3])). For example, if the universe reionized at $z \simeq 50$ which gives $\tau_{\rm reion} \simeq 1$ .i.e. all the photons from the last scattering surface are re-scattered following reionizatio, this means that all anisotropies at scales smaller than the angular scale correspond to $\ell \simeq \eta_0/\eta(z_{\rm reion}) \simeq 10$ are wiped out. As we shall see the second term in the solution doesn’t contribute much to the anisotropies at small scales either. This means that for a reionization redshift $\simeq 50$ no anisotropies should be observed for $\ell \ga 10$, which is in direct contradiction with observations (e.g. WMAP observations detect anisotropies for $\ell \simeq 600$). Therefore the redshift of reionization should be small enough such that only a small fraction of CMBR photons are re-scattered. To compute the second term in Eq. (\[eq:tesol\]), we can assume the visibility to be a normalized Gaussian with a width $\Delta\eta_{\rm reion}$, this gives $$\Delta_T \propto \mu \tau_{\rm reion} v \exp\left[-(k\mu \Delta\eta_{\rm reion}/2)^2\right ]$$ This shows that CMBR anisotropies generated during the epoch of reionization owing to Doppler scattering off electrons are suppressed for $k \ga 1/\Delta\eta_{\rm reion}$. The generated signal as expected is $\propto \tau_{\rm reion}$ For realistic ionization histories $\Delta\eta_{\rm reion} \simeq \eta_{\rm reion}$. We show in Figure 3, the effect of reionization on the temperature anisotropies. The net effect of reionization on the CMBR temperature anisotropies can be summarized as: (a) anisotropies at small scales are suppressed exponentially as $\exp(-2 \tau_{\rm reion})$, (b) anisotropies at scales corresponding to $\ell \la 10$ escape this exponential damping and also new anisotropies are generated at these scales. As seen in Figure 3, reionization causes a relative decrease in the small scale anisotropies. The minimum error in detecting the angular power spectrum of CMBR anisotropies at any $\ell$ is $\Delta C_\ell \simeq \sqrt{2/(2\ell+1)} C_\ell$ (Cosmic variance) and for all CMBR anisotropy experiments the error on $C_\ell$ is dominated by the cosmic variance for $\ell \la 300$. The reionization signal is very difficult to detect owing to uncertainty in the overall normalization of the CMBR anisotropies which is compounded by cosmic variance. From WMAP data all the information about the reionization comes from the polarization signal.
In second order in perturbation theory, reionization causes potentially detectable anisotropies for $\ell \ga 1000$; for $\ell \ga 2000\hbox{--}3000$ this signal can dominate the signal from primary anisotropies. The most important second order contribution to temperature anisotropies comes from Vishniac effect (for details see [@vishniac], also see [@hu1], [@dodelson], [@hu3] for host of other second order effects).
From Eq. (\[eq:tpeq\]), it can be seen that the source of polarization anisotropy is the quadrupole of the temperature anisotropy. The generation of temperature and polarization anisotropies at the last scattering surface is generally an involved process and Eqs.(\[eq:tpeq\]) have to be solved numerically (see e.g. [@seljak]). However, several relevant assumptions allows one to get an analytic insight into the problem ([@hu2], [@zaldarriaga2]). First simplification occurs because the photon-baryon plasma can be treated as tightly-coupled for most of the period during which the recombination lasts. The tight coupling approximation is valid for scales corresponding to $k \la l_f^{-1}$, where $l_f$ is the (comoving) mean free path of the photons for Thompson scattering; $l_f \simeq 1 \, \rm Mpc$ at the last scattering surface for a fully ionized universe, which means that tight coupling approximation is valid for most scales of interest. (The scale of interest for tight-coupling in not $l_f$ but the scale of photon diffusion at recombination which for a fully ionized plasma is roughly 10 times larger than $l_f$; free-streaming of photons at recombination further increase this scale; for detailed discussions and implications of this for CMBR anisotropies see [@hu2]). A second simplification, closely related to the first but physically distinct, occurs because the width of visibility function at the last scattering surface corresponds to scales $\la 10 \, \rm Mpc$, and hence for studying physical processes at much larger scales the recombination can be treated as instantaneous (for caveats see [@hu2]). Out interest here is in scales that are super-horizon at the last scattering surface ($k^{-1} \ga 100 \, \rm Mpc$, comoving), corresponding to angular scales $\ell \la 200$. Therefore we will use tightly coupled approximation and not discuss the effects of photon diffusion. Using these approximations, adiabatic initial conditions and also the fact that the ratio of baryon to photon energy density $\rho_b/\rho_\gamma \simeq 25 \Omega_b h^2 \ll 1$ at the epoch of decoupling for acceptable models of primordial nucleosynthesis, the temperature and polarization anisotropies generated at the last scattering surface are: (for details see [@hu2], [@zaldarriaga2]): $$\begin{aligned}
\Delta_T(k,{\bf n},\eta) & = & {1 \over 3} \Phi({\bf k},\eta_{\rm rec})\cos(kc_s\eta_{\rm rec})\exp[ik\mu(\eta_{\rm rec} - \eta)] \nonumber \\
\Delta_P(k,{\bf n},\eta) & = & 0.17(1- \mu^2)kc_s \Delta\eta_{\rm rec}\Phi({\bf k},\eta_{\rm rec}) \sin(kc_s\eta_{\rm rec})\exp[ik\mu(\eta_{\rm rec} - \eta)]
\label{tpeqsol}\end{aligned}$$ Here $\Phi({\bf k},\eta_{\rm rec})$ is the Fourier component of the Newtonian potential at the last scattering surface; $\Delta\eta_{\rm rec} \simeq 10 \, \rm Mpc$ is the comoving width of LSS; $c_s \simeq 1/\sqrt{3}$ is the sound velocity in the coupled photon-baryon fluid. These large scale solutions show that: (a) temperature and polarization anisotropies are correlated with each other, (b) the amplitude of polarization anisotropies is suppressed by a factor $kc_s \Delta\eta_{\rm rec}$; e.g. for $k \simeq 0.01 \, \rm Mpc^{-1}$ this factor is roughly $1/10$. This suppression is owing to the fact that polarization anisotropies $\propto \Delta_{T2}$. In the strict tight coupling approximation, only the temperature monopole and dipole are non-zero in the comoving frame. A small quadrupole is generated owing to free streaming of photons before they scatter for the last time; and this quadrupole is suppressed with respect to the monopole and dipole, which constitute the primary sources of temperature anisotropies, by the factor $\simeq kc_s \Delta\eta_{\rm rec}$. As we shall see below, in the reionized models this suppression is absent. In Figure 4 we show the CMBR temperature and polarization anisotropies, generated at the last scattering surface.
In the reionized models, a fraction CMBR photons scatter again after the epoch of reionization. As argued above, this fraction is small as many of the features of the primary anisotropies have already been detected. We have already discussed the effect of reionization on temperature anisotropies and argued that the reionization signal is very difficult to discern from temperature anisotropies alone. Like temperature anisotropies, one of the effects of reionization would be to wipe out polarization anisotropies generated at the last scattering surface for scales $\la \eta_{\rm reion}$. From Eq. (\[eq:tpeq\]), the generation of new polarization anisotropies will be proportional to the value of $\Delta_{T2}$ at the epoch of reionization. Eq. (\[eq:tpeq\]) can simplified further by dropping terms of polarization monopole and quadrupole of polarization anisotropies in the RHS of the equation, as these terms are negligible compared to the temperature quadrupole (see below). Eq. (\[eq:tpeq\]) then give the following solution for the generation of polarization anisotropies following reionization: $$\Delta_P = \int d\eta' \Pi(\mu) \Delta_{T2}(\eta')V(\eta,\eta')\exp(ik\mu(\eta'- \eta))$$ Most of the contribution to the integral will come from close to the reionization epoch (typically $\simeq \Delta\eta_{\rm reion} \simeq \eta_{\rm reion}$, the width of visibility function if it is approximated as a Gaussian as above). The amplitude of the contribution is proportional to the temperature quadrupole at the epoch of reionization. As argued above the temperature quadrupole is suppressed at the last scattering surface owing to tight coupling approximation that holds at that epoch. Following recombination the photons free stream which allows to temperature quadrupole to increase. From Eq. (\[tpeqsol\]), we can get the temperature quadrupole at the reionization epoch from taking the angular moment of the equation (for details see e.g. [@zaldarriaga1]): $$\Delta_T^{(2)}(\eta_{\rm reion}) = {1 \over 3} \Phi(k,\eta_{\rm rec})\cos(kc_s\eta_{\rm rec}) j_2[k(\eta_{\rm reion}-\eta_{\rm rec})]$$ Using the fact that the maximum value of $j_2(x) \simeq 0.3$, and comparing with the polarization anisotropies generated at the last scattering surface,Eq. (\[tpeqsol\]), we can see that $\Delta_T^{(2)}(\eta_{\rm reion})$ doesn’t suffer the $k$ dependent suppression, and is of the order of temperature anisotropies at the last scattering surface. At $k \ll 0.1$, it can be several orders of magnitude more than the polarization anisotropies generated at the last scattering surface. The polarization anisotropies at the reionization epoch peak at the characteristic scale $k \simeq 2/\eta_{\rm reion}$, which corresponds to an angular scale $l \simeq k \eta_0 \simeq 5\hbox{--}10$ for $z_{\rm reion} \simeq 15\hbox{--}50$.
In Figure 5 we show the effect of reionization on the temperature-polarization cross- correlation power spectrum; also shown are the observations of WMAP. Figure 5 shows that the enhancement of power at $\ell \le 10$ cannot be explained within the framework of no reionization models. Also shown are the predictions of a model in which the universe reionized at $z \simeq 5.5$, which the GP observations discussed above might be suggesting and the best fit model to the WMAP observations. The best fit model requires $\tau_{\rm reion}\simeq 0.15$, which implies the epoch of reionization corresponds to $z_{\rm reion} \simeq 15$.
High Redshift HI
----------------
Another possible probe of the reionization epoch is to observe the neutral component of hydrogen thorough the epoch of reionization. The neutral fraction of hydrogen changes from near unity to zero during the epoch of reionization. This change can potentially be observed using the hyperfine transition of the hydrogen atom at $\nu_\star = 1420 \, \rm MHz$. The quantity of interest here is the spin temperature of hydrogen defined as: $${n_2 \over n_1} = 3 \exp(-T_\star/T_s)$$ Here $n_2$ and $n_1$ are the populations of the hyperfine states. $T_\star = h \nu_\star/k = 0.06 \, \rm K$. As the only radio source at the high redshift is CMBR, HI in hyperfine transition can seen against the CMBR in emission or absorption. The observed quantity then is the deviation of CMBR from a black body at radio frequencies. The observed difference is: $$\Delta T_{\rm CMBR} = -\tau_{\rm \scriptscriptstyle HI}(T_{\rm CMBR} -T_s)
\label{eqh1}$$ Here $\tau_{\rm \scriptscriptstyle HI} = \sigma_\nu N_{\rm HI} T_\star/T_s$, with the HI column density $N_{\rm HI} = \int n_{\rm \scriptscriptstyle HI} d\ell$; $\sigma_\nu = c^2 A_{21}\phi_\nu/(4\pi \nu_\star)$; $A_{21} \simeq 1.8 \times 10^{-15} \, \rm sec^{-1}$ and $\phi_\nu$ is the line response function. The spin temperature is determined from detailed balancing between various processes that can alter the relative populations of the two levels [@field1; @field2]: $$T_s = {T_{\rm cmbr} + y_c T_K + y_\alpha T_\alpha \over 1 + y_c + y_\alpha}
\label{eqts}$$ Here $y_c \propto n_{\rm \scriptscriptstyle H}$ and $y_\alpha \propto n_{\rm \scriptscriptstyle \alpha}$, with $n_\alpha$ being the number density of Lyman-$\alpha$ photons correspond, respectively, to relative probabilities with which collisions between atoms and the presence of Lyman-$\alpha$ photons determine the level populations; $T_K$ is the matter temperature. In the pre-reionization era, there are no Lyman-$\alpha$ photons, and therefore $y_\alpha = 0$.
From Eq. (\[eqh1\]) it is clear that HI can be observed in either absorption or emission against CMBR depending on whether $T_s$ is less than or exceeds $T_{\rm CMBR}$. At $z \simeq 1000$, $T_{\rm CMBR} = T_K$ and it follows from Eq. (\[eqts\]) that $\Delta T_{\rm CMBR} = 0$. As seen in Figure 1, for $z \la 100$, $T_K < T_{\rm CMBR}$ and therefore HI can be observed in absorption against CMBR if $y_c \ga 1$. During reionization, $y_\alpha$ term can become important and owing to thermalization at frequencies close to Lyman-$\alpha$, $T_\alpha \simeq T_K$ [@field2]. The temperature of the medium can also exceed $T_{\rm CMBR}$ from X-ray and Lyman-$\alpha$ heating (for details see [@madau], [@sethi1]) which implies that $\Delta T_{\rm CMBR} > 0$. $\Delta T_{\rm CMBR}$ approaches zero as the reionization is completed. In Figure 6, we show $\Delta T_{\rm CMBR}$ as a function of the observed frequency for an ionization history in which the universe becomes fully ionized at $z = 15$ (more details in next section and [@sethi1]). If the universe re-ionized at $z \simeq 15$, then $\Delta T_{\rm CMBR} \simeq 0.05 \, \rm K$ at frequencies $\nu \simeq 50\hbox{--}80 \, \rm MHz$. In addition there is a signal from pre-reionization epoch with $\Delta T_{\rm CMBR} \simeq -0.05 \, \rm K$ at $\nu \simeq 30 \, \rm MHz$. In addition to the average signal there will be fluctuations in the temperature difference owing to fluctuations in HI density from primordial density perturbations and also from the patchiness of reionization (for details see e.g. [@tozzi]). Currently the prospects of detecting this signal are being studied by using both single dish and interferometric experiments at low radio frequencies ([@shaver], see also [@lofar]).
Re-ionization of the Universe at High Redshift
==============================================
In the previous section we discussed various probes of the high redshift universe and the epoch of reionization. In this section we take up the issue whether it is possible to explain the observed ionization structure of the universe within the framework of currently-favoured $\rm \Lambda CDM$ models of formation of structures. In these models, the observed structures in the universe grew from gravitational instability of density fluctuations which originated during inflationary epoch in the very early universe (see .e.g [@peebles1], [@padmanabhan1; @padmanabhan2]). The gravitational collapse of these structures might either set off star-formation (alternatively some material might end up in black holes which by accreting more matter will radiate with harder spectrum than first star-forming galaxies, see e.g. [@ricotti]) which will emit UV light and ionize the IGM. The process of re-ionization of the universe is generally quite complicated and not well understood. However it is possible to study it within the framework of simple models which might give important clues about the details of this process. Important ingredients of this problem are: (a) Halo population at high redshift, (b) molecular and atomic cooling in Haloes, (c) Initial Mass Function of stars and star formation rate, (d) Escape fraction of UV photons from Haloes, (e) clumpiness of the IGM. Of these the most uncertain are (c) and (d) and have to be modelled using simple parameterized model.
[*Halo Population*]{}: The number density of dark matter haloes per unit mass $M$ at any redshift can be obtained from the Press-Schechter method (see .e.g [@peebles1], [@padmanabhan1; @padmanabhan2]). It is given by: $${dn \over dM} = \sqrt{{2\over\pi}}{\rho_m \over M}\delta_c(z) \left | {d\sigma \over dM} \right | {1 \over \sigma^2(M)} \exp\left [-\delta_c(z)^2/(2\sigma^2(M)) \right ]
\label{eq:psme}$$ Here $\sigma(M)$ is the mass dispersion filtered at the scale corresponding to mass $M$ in the linear theory; $\sigma(M)$ for length scale corresponding to $8 h^{-1} \, \rm Mpc$ is $\simeq 0.9$ [@spergel]; $\delta_c(z) \simeq 1.7 D(0)/D(z)$, with $D(z)$ being the solution of growing mode in linear theory; in sCDM model $D(0)/D(z) = (1+z)$ (for details see [@padmanabhan1; @padmanabhan2], [@peebles2]). The first baryonic structures to collapse would have masses exceeding the Jeans mass of the IGM, whose evolution is shown in Figure 1. From Figure 1 it is seen that the first structures might have masses $\simeq 10^4 \, \rm M_\odot$ (for non-linear extension of the concept of Jeans mass see e.g. [@barkana]; the modified mass has values similar to the one obtained using linear theory). The collapsed fraction of all structures can be calculated from Eq. (\[eq:psme\]) and is $\simeq 2 \times 10^{-3}$ at $z \ simeq 20$. However to form stars, baryons need to cool sufficiently rapidly in the dark matter haloes to collapse to density higher than initial virialized density.
[*Atomic and molecular cooling*]{}: Assuming a spherical, top-hat collapse (see e.g. [@padmanabhan1; @padmanabhan2]) the density of the collapsed structure is $\simeq 170$ times the background density at that redshift. If a virial equilibrium is reached inside the halo, the baryon temperature is raised to the virial temperature given by (e.g. [@padmanabhan1; @padmanabhan2]): $$T_{\rm vir} \simeq 800 \, {\rm K} \left ({M \over 10^6 M_\odot} \right )^{2/3} \left ({1+z \over 20}\right) \left ( { \Omega_m \over 0.3} \right )^{1/3}\left ( {h \over 0.7} \right )^{2/3} \left ( { \mu \over 1.22} \right )$$ Here molecular weight $\mu = 1.22$ for a fully neutral halo (haloes with masses $\la 10^8\, \rm M_\odot$ at $z \simeq 20$); $\mu = 0.57$ for a fully ionized halo. An important criterion is whether baryons can cool rapidly enough so that they collapse to higher densities, fragment and form stars. In the primordial gas, the cooling in haloes with virial temperature $\ga 10^4$ is dominated by atomic hydrogen and singly-ionized helium. But the smaller haloes can only cool further by molecular hydrogen. A small fraction $\simeq 10^{-6}$ of molecular hydrogen is formed at $z \simeq 1000$ in the IGM following recombination of the universe (see e.g. [@peebles1]). Such a small fraction doesn’t suffice to cause rapid enough cooling. However the collapse of halo can cause the formation of molecular hydrogen, resulting in a molecular fraction $\simeq 5 \times 10^{-4}$ by the time the halo virializes (see e.g. [@tegmark] and reference therein). In Figure 7, we show the cooling time (defined as $t_{\rm cool} = k T_K/\dot E$, where $\dot E$ is the rate at which the halo loses energy from various processes) for haloes of different masses at $z = 20$. The criterion for runaway cooling resulting in fragmentation and star formation is that cooling time be less than the dynamical time ($t_d = 1/\sqrt{G\rho}$) of the halo. From Figure 7 it is seen than haloes of masses $\simeq 10^6 \, \rm M_\odot$ can collapse to form stars at $z \simeq 20$. This result is only approximate as the haloes will not have constant density. For instance, the more dense central parts of the halo can form both more molecular hydrogen and cool more rapidly. Bromm et al. [@bromm] simulated the collapse of haloes of masses $\simeq 10^{6} \rm M_\odot $ with substructure. They concluded that the halo fragments into many clumps with typical masses $10^2 \hbox{--}10^3 \, \rm M_\odot$; their analysis suggests that this mass scale is a result of molecular hydrogen chemistry and therefore should correspond to the masses of the first stars. Abel et al. [@abel] reached similar conclusions from their simulations.
From information about the halo population and cooling arguments it is possible to speculate on the ionization history of the universe from photo-ionization. The main uncertainty is the hydrogen-ionizing luminosity (and its evolution) of each halo, which has to be parameterized. Assuming that halo of mass $M$ emits isotropically the hydrogen-ionizing luminosity $\dot N_\gamma$ (in photons $\rm sec^{-1}$), the radius of ionizing sphere around the source will satisfy the equation (Stromgren Sphere, see e.g. [@shu], [@shapiro]): $${dR \over dt} -H R={(\dot N_\gamma - 4\pi/3 R^3 \alpha_B C n_b^2 x_{\rm \scriptscriptstyle HI}) \over (\dot N_\gamma + 4 \pi R^2 x_{\rm \scriptscriptstyle HI} n_b)}$$ Here $C\equiv \langle n_b^2 \rangle/\langle n_b \rangle^2$ is the clumping factor of the IGM. Using Eq. (\[eq:psme\]), the fraction of the universe that is ionized as a given redshift is (see e.g. [@haiman1], [@sethi1]): $$f_{\rm ion}(z) = {4 \pi \over 3}\int_{0}^z dz' \int dM {dn \over dM}(M,z') R^3(M,z,z')$$ Further assuming that the luminosity of the source is $\propto M$, the ionized fraction can be calculated in terms of the evolution of the photon luminosity of a single halo of some fiducial mass and the evolution of the clumping factor. For simplicity we assume the clumping factor to have a constant value between one and five, and take the luminosity evolution of a halo to have the form of a typical star-burst galaxy: $\dot N_\gamma(t) = \dot N_\gamma(0)\exp(-t/{10^7 \, \rm years})$. In Figure 8 we show several ionization histories for different values of $N_\gamma(0)$ for $M = 5 \times 10^7 \, \rm M_\odot$ and clumping factor $C$. In Figure 8 we only plot ionization fraction upto $z \simeq 9$ at which $f_{\rm ion} \simeq 1$; further evolution keeps the universe fully ionized upto the present. These ionization histories are consistent with WMAP observations. (If the universe becomes fully ionized at $z \simeq 8$ if will remain ionized upto the present even in the absence of ionizing sources as the typical recombination time $1/(\alpha_B C n_b)$ already exceeds the age of the universe at this epoch for $C \la 2$.) It appears that $ N_\gamma(0) \simeq 10^{50}$ is required to ionize the universe early enough to satisfy WMAP observations. This is just three orders of magnitude below the photon luminosity of a typical star-burst galaxy (see .e.g.[@leitherer]). Alternatively one can infer that the efficiency of the first star formation was very high (see e.g. [@haiman1], [@chui]). There are other uncertainties like feedback from supernova, and photo-dissociation of molecular hydrogen which indicate this estimate is a lower limit (see e.g. [@barkana], [@haiman1], [@haiman2], [@dekel]). Alternatively it is possible that the collapsed fraction of the universe far exceeded the value given by $\rm \Lambda CDM$ models and was caused by some other physical process like tangled magnetic fields [@sethi2].
Is it possible to simultaneously satisfy the WMAP and the GP observations? It appears difficult to achieve it unless the efficiency of star formation decreased rapidly from $z \simeq 20$ to $z \simeq 6$ (for other possible scenarios and details see e.g. [@haiman1], [@chui] and references therein).
Discussion
==========
The ionization history of the universe at high redshift as inferred by WMAP and GP observations is quite complicated: the universe reionized at $z \simeq 12\hbox{--}15$; the neutral fraction increased to better than one part in a thousand for $5.5 <z < 6.5$; and for $z \la 5$, the neutral fraction dropped to one part in a million. Theoretical analyses based on models of structure formation are not sophisticated enough to understand this ionization history. Future observations however might throw light on these observations. Firstly, future telescopes might be able to detect the sources of reionization upto $z \simeq 20$ ([@haiman1]). Future CMBR experiment Planck also is sensitive enough to disentangle the effects of complicated ionization histories [@kaplinghat].
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---
abstract: 'How long can a word be that avoids the unavoidable? Word $W$ encounters word $V$ provided there is a homomorphism $\phi$ defined by mapping letters to nonempty words such that $\phi(V)$ is a subword of $W$. Otherwise, $W$ is said to avoid $V$. If, on any arbitrary finite alphabet, there are finitely many words that avoid $V$, then we say $V$ is unavoidable. Zimin (1982) proved that every unavoidable word is encountered by some word $Z_n$, defined by: $Z_1 = x_1$ and $Z_{n+1} = Z_nx_{n+1}Z_n$. Here we explore bounds on how long words can be and still avoid the unavoidable Zimin words.'
author:
- 'Joshua Cooper\*'
- 'Danny Rorabaugh\*'
title: Bounds on Zimin Word Avoidance
---
[^1]
In 1929, Frank Ramsey proved that, for any fixed $r,n,\mu \in {{\mathbb{Z}}}^+$, every sufficiently large set $\Gamma$ with its $r$-subsets partitioned into $\mu$ classes is guaranteed to have a subset $\Delta_n \subseteq \Gamma$ such that all the $r$-subsets of $\Delta_n$ are in the same class [@Ramsey]. This was the advent of a major branch of combinatorics that became known as Ramsey theory. Often applied to graph theoretic structures, Ramsey theory looks at how large a random structure must be to guarantee that a given substructure exists or a given property is satisfied. Here we apply this paradigm to an existence result from the combinatorics of words.
A [*$q$-ary word*]{} is a string of characters, at most $q$ of them distinct.
Over a fixed $q$-letter alphabet, the set of all finite words forms a semigroup with concatenation as the binary operation (written multiplicatively) and the empty word $\varepsilon$ as the identity element. We also have a binary subword relation $\leq$ where $V \leq W$ when $W = UVU'$ for some words $U$, $V$, and $U'$. That is, $V$ appears contiguously in $W$.
We call word $W$ an [*instance*]{} of $V$ provided
- $V = x_0x_1 \cdots x_{m-1}$ where each $x_i$ is a letter;
- $W = A_0A_1 \cdots A_{m-1}$ with each $A_i \neq \varepsilon$ and $A_i = A_j$ whenever $x_i=x_j$.
Equivalently, $W$ is a [*$V$-instance*]{} provided there exists some semigroup homomorphism $\phi$ such that $\phi(x_i) = A_i \neq \varepsilon$ for each $i$.
$W = abbcabbxdc$ is an instance of $V = xyxzy$, with $\phi$ defined by $\phi(x) = abb$, $\phi(y)=c$, and $\phi(z)=xd$.
A word $U$ [*encounters*]{} word $V$ provided some subword $W \leq U$ is an instance of $V$. If $U$ fails to encounter $V$, then $U$ [*avoids*]{} $V$.
(0,0) node[$\varepsilon$]{}; (-1,.7) node\[above\][$a$]{}–(0,.3)–(1,.7) node\[above\][$b$]{}; (-1.5,1.7) node\[above\][$aa$]{}–(-1,1.3)–(-.6,1.7) node\[above\][$ab$]{}; (.6,1.7) node\[above\][$ba$]{}–(1,1.3)–(1.5,1.7) node\[above\][$bb$]{}; (-1.2,1.7)–(-1.8,2.2); (-1.2,2.2)–(-1.8,1.7); (1.2,1.7)–(1.8,2.2); (1.2,2.2)–(1.8,1.7); (-1.2,2.7) node\[above\][$aba$]{}–(-.6,2.3)–(-.4,2.7) node\[above\][$abb$]{}; (1.2,2.7) node\[above\][$bab$]{}–(.6,2.3)–(.4,2.7) node\[above\][$baa$]{}; (-.1,2.7)–(-.7,3.2); (-.1,3.2)–(-.7,2.7); (.1,2.7)–(.7,3.2); (.1,3.2)–(.7,2.7); (-1.7,3.7) node\[above\][$abaa$]{}–(-1.2,3.2)–(-.6,3.7) node\[above\][$abab$]{}; (1.7,3.7) node\[above\][$babb$]{}–(1.2,3.2)–(.6,3.7) node\[above\][$baba$]{}; (-2,3.7)–(-1.4,4.2); (-1.4,3.7)–(-2,4.2); (2,3.7)–(1.4,4.2); (1.4,3.7)–(2,4.2); (-.9,3.7)–(-.3,4.2); (-.3,3.7)–(-.9,4.2); (.9,3.7)–(.3,4.2); (.3,3.7)–(.9,4.2);
We see in Figure 1 that $xx$ is avoided by only finitely many words over a two-letter alphabet. However, it has been known for over a century [@Thue] that $xx$ can be avoided by arbitrarily long (even infinite) ternary words.
A word $V$ is [*unavoidable*]{} provided for any finite alphabet, there are only finitely many words that avoid $V$.
A. I. Zimin proved an elegant classification of all unavoidable words [@Zimin].
Define the [*$n^{th}$ Zimin word*]{} recursively by $Z_0 := \varepsilon$ and, for $n \in {{\mathbb{N}}}$, $Z_{n+1} := Z_nx_nZ_n$. Using the alphabet rather than indexed variables: $$Z_1 = a, \quad Z_2 = a{\bf b}a, \quad Z_3 = aba{\bf c}aba, \quad Z_4 = abacaba{\bf d}abacaba, \quad \ldots$$
Equivalently, $Z_n$ can be defined over the natural numbers as the word of length $2^n-1$ such that the $i^{\text{th}}$ letter is the 2-adic order of $i$ for $1 \leq i < 2^n$.
A word $V$ with $n$ distinct letters is unavoidable if and only if $Z_n$ encounters $V$.
Avoiding the Unavoidable
========================
From Zimin’s explicit classification of unavoidable words, a natural question arises in the Ramsey theory paradigm: for a fixed unavoidable word $V$, how long can a word be that avoids $V$? Our approach to this question is to start with avoiding the Zimin words, which gives upper bounds for all unavoidable words. Define $f(n,q)$ to be the smallest integer $M$ such that every $q$-ary word of length $M$ encounters $Z_n$.
\[upper\] For $n,q \in {{\mathbb{Z}}}^+$ and $Q := 2q+1$, $$f(n,q) \leq {}^{n-1}Q := Q^{Q^{^{\iddots^{{}_Q}}}},$$ with $Q$ occurring $n-1$ times in the exponential tower.
We proceed via induction on $n$. For the base case, set $n=1$. Every nonempty word is an instance of $Z_1$, so $f(1,q) = 1$.
For the inductive hypothesis, assume the claim is true for some positive $n$ and set $T:=f(n,q)$. That is, every $q$-ary word of length $T$ encounters $Z_n$. Concatenate any $q^T+1$ strings $W_0, W_1, \ldots, W_{q^T}$ of length $T$ with an arbitrary letter $a_i$ between $W_{i-1}$ and $W_{i}$ for each positive $i \leq q^T$: $$U := W_0 \; a_1\; W_1 \; a_2 \; W_2 \; a_3\; \cdots \; W_{q^T-1} \; a_{q^T} \; W_{q^T}.$$
By the pigeonhole principle, $W_i = W_j$ for some $i < j$. That string, being length $T$, encounters $Z_n$. Therefore, we have some word $W \leq W_i$ that is an instance of $Z_n$ and shows up twice, disjointly, in $U$. The extra letter $a_{i+1}$ guarantee that the two occurrences of $W$ are not consecutive. This proves that an arbitrary word of length $(T+1)(q^T + 1)-1$ witnesses $Z_{n+1}$, so $$f(n+1,q) \leq (T+1)(q^T + 1)-1 \leq (2q+1)^T = Q^T.$$
There is clearly a function $Q(n,q)$ such that $f(n+1,q) \leq {Q(n,q)}^{f(n,q)}$ and $Q(n,q)\rightarrow q$ as $n\rightarrow \infty$. No effort has been made to optimize the choice of function, as such does not decrease the tetration in the bound.
The technique used to prove Theorem \[upper\] is first found in Lothaire’s proof of unavoidability of $Z_n$ ([@Lothaire], 3.1.3). The technique in Zimin’s original proof [@Zimin] implicitly gives that for $n\geq 2$, $$f(n+1,q+1) \leq (f(n+1,q)+2|Z_{n+1}|)f(n,|Z_{n+1}|^2q^{f(n+1,q)}).$$ This is an Ackermann-type function for an upper bound, which is much larger than the primitive recursive bound from Theorem \[upper\].
Table 1 shows known values of $f(n,2)$. Supporting word-lists and Sage code are found in the Appendix.
$n$ $Z_n$ $f(n,2)$
----- ----------------- --------------
0 $\varepsilon$ 0
1 a 1
2 aba 5
3 abacaba 29
4 abacabadabacaba $\geq 10483$
: Values of $f(n,2)$ for $n\leq 4$.
\
Finding a Lower Bound with the First Moment Method
==================================================
Throughout this section, $q$ is a fixed integer greater than 1. Given a fixed alphabet of $q$ letters, $C(n,q,M)$ denotes the set of length-$M$ instances of $Z_n$. That is $$C(n,q,M) := \{W \mid W \in \{x_0,\ldots,x_{q-1}\}^M \text{ is a }Z_n\text{-instance}\}.$$
For all $n,M \in {{\mathbb{Z}}}^+$, $$|C(n,q,M+1)| \geq q\cdot|C(n,q,M)|.$$
Take arbitrary $W \in C(n,q,M)$. We can write $W = W_1W_0W_1$ with $W_1 \in C(n-1,q,N)$, where $2N < M$. Choose the decomposition of $W$ to minimize $|W_1|$. Then $W_1W_0x_iW_1 \in C(n,q,M+1)$ for each $i < q$.
The lemma follows, unless a $Z_n$-instance of length $M+1$ can be generated in two ways – that is, if $W_1W_0aW_1 = V_1V_0bV_1$ for some $V_1V_0V_1 = V$, where $|V_1|$ is also minimized. If $|V_1|<|W_1|$, then $V_1$ is a prefix and suffix of $W_1$, so $|W_1|$ was not minimized. But if $|V_1|>|W_1|$, then $W_1$ is a prefix and suffix of $V_1$, so $|V_1|$ was not minimized. Therefore, $|V_1|=|W_1|$, so $V_1=W_1$, which implies $a=b$ and $V=W$.
For all $n,M \in {{\mathbb{Z}}}^+$, $$\begin{aligned}
\Pr\left(W \in C(n,q,M+1) \mid W\in \{x_0,\ldots,x_{q-1}\}^{M+1}\right) \quad\\
\geq \Pr\left(W \in C(n,q,M) \mid W\in \{x_0,\ldots,x_{q-1}\}^M\right),
\end{aligned}$$ assuming uniform probability on words of a fixed length.
For all $n,M \in {{\mathbb{Z}}}^+$, $$|C(n,q,M)| \leq \left(\frac{q}{q-1}\right)^{n-1}q^{(M-2^n+n+1)}.$$
The proof proceeds by induction on $n$. For the base case, set $n=1$. Every non-empty word is an instance of $Z_1$, so $|C(1,q,M)| = q^M$.
For the inductive hypothesis, assume the claim is true for some positive $n$. The first inequality below derives from the following way to overcount the number of $Z_{n+1}$-instances of length $M$. Every such word can be written as $UVU$ where $U$ is a $Z_n$-instance of length $j<M/2$. Since an instance of $Z_n$ can be no shorter than $Z_n$, we have $2^n-1 \leq j < M/2$. For each possible $j$, there are $|C(n,q,j)|$ ways to choose $U$ and $q^{M-2j}$ ways to choose $V$. This is an overcount, since a Zimin-instance may have multiple decompositions.
$$\begin{aligned}
|C(n+1,q,M)| & \leq & \sum_{j = 2^n-1}^{{\left \lfloor{(M-1)/2}\right \rfloor}} |C(n,q,j)|q^{M-2j} \\
& \leq & \sum_{j = 2^n-1}^{{\left \lfloor{(M-1)/2}\right \rfloor}}\left(\frac{q}{q-1}\right)^{n-1}q^{(j-2^n+n+1)}q^{M-2j}\\
& = & \left(\frac{q}{q-1}\right)^{n-1}q^{(M-2^n+n+1)} \sum_{j = 2^n-1}^{{\left \lfloor{(M-1)/2}\right \rfloor}}q^{-j}\\
& < & \left(\frac{q}{q-1}\right)^{n-1}q^{(M-2^n+n+1)} \sum_{j = 2^n-1}^{\infty}q^{-j}\\
& = & \left(\frac{q}{q-1}\right)^{n-1}q^{(M-2^n+n+1)}\left(\frac{q^{-(2^n-1)+1}}{q-1}\right)\\
& = & \left(\frac{q}{q-1}\right)^{(n-1)+1}q^{(M -2^{n+1}+ (n+1)+1)}.
\end{aligned}$$
For all $n,M \in {{\mathbb{Z}}}^+$, $$\Pr\left(W \in C(n,q,M) \mid W\in \{x_0,\ldots,x_{q-1}\}^M\right) \leq \left(\frac{q}{q-1}\right)^{n-1}q^{(-2^n+n+1)},$$ assuming uniform probability on words of length $M$.
$$f(n,q) \geq q^{2^{(n-1)}(1+o(1))} \quad (q\rightarrow \infty, n \rightarrow \infty).$$
Let word $W$ consist of $M$ uniform, independent random selections from the alphabet $\{x_0, \ldots, x_{q-1}\}$. Define the random variable $X$ to count the number of subwords of $W$ that are instances of $Z_n$ (including repetition if a single subword occurs multiple times in $W$): $$X = |\{V \mid W \geq V \in C(n,q,|V|)\}|.$$ By monotonicity with respect to word length: $$\begin{aligned}
E(X) & \leq & |\{V \mid V \leq W\}| \cdot \Pr(W \in C(n,q,M))\\
& \leq & \binom{M+1}{2} \left(\frac{q}{q-1}\right)^{n-1}q^{(-2^n+n+1)}\\
& < & M^2 e^{(n-1)/(q-1)}q^{(-2^n+n+1)}.
\end{aligned}$$
There exists a word of length $M$ that avoids $Z_n$ when $E(X) < 1$. It suffices to show that: $$M^2 \left(e^{(n-1)/(q-1)}q^{(-2^n+n+1)}\right) \leq 1.$$
Solving for $M$: $$\begin{aligned}
M &\leq& \left(e^{(n-1)/(q-1)}q^{(-2^n+n+1)}\right)^{-1/2}\\
&=& q^{2^{(n-1)}}\left(e^{(n-1)/(q-1)}q^{(n+1)}\right)^{-1/2}\\
&=& q^{2^{(n-1)}(1+o(1))}.
\end{aligned}$$
Continuing work {#continuing-work .unnumbered}
===============
Current efforts to improve bounds on the probability that a word is an instance of $Z_n$ will help close the gap between the lower and upper bounds on $f(n,q).$
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to express our gratitude to Professor George McNulty for his algebraic contributions to the field of word avoidability [@McNulty], and for bringing this topic to our attention. Also, to the organizers of the [*45th Southeast International Conference on Combinatorics, Graph Theory, and Computing*]{}, who accepted this paper for presentation on 04 March 2014. And finally, to the participants of the Graduate Discrete Mathematics Seminar at the University of South Carolina, who formed a wonderful audience for early presentations of this work.
[5]{} M. Lothaire. [*Algebraic Combinatorics on Words.*]{} Cambridge University Press, Cambridge, 2002.
D.R. Bean, A. Ehrenfeucht, & G.F. McNulty. Avoidable Patterns in Strings of Symbols, [*Pac. J. of Math.*]{} [**85:2**]{}:261–294, 1979.
F.P. Ramsey. On a problem in formal logic, [*Proc. London Math. Soc.*]{} [**30**]{}:264–286, 1929.
W.A. Stein, et al. Sage Mathematics Software (Version 6.1.1), The Sage Development Team. (2014) http://www.sagemath.org.
A. Thue. [Ü]{}ber unendliche Zeichenreihen, [*Norske Vid. Skrifter I Mat.-Nat. Kl.*]{} Christiania 7:1–22, 1906.
A.I. Zimin. Blocking sets of terms, [*Mat. Sb.*]{} [**119**]{}, 1982; [*Math. USSR-Sb.*]{} [**47**]{}:353–364, 1984.
Appendix: Binary Words that Avoid $Z_n$ {#appendix-binary-words-that-avoid-z_n .unnumbered}
=======================================
All binary words that avoid $Z_2$. {#all-binary-words-that-avoid-z_2. .unnumbered}
----------------------------------
\
The following 13 words are the only words over the alphabet $\{0,1\}$ that avoid $Z_2 = aba$. $$\begin{matrix}
\varepsilon, & 0, & 00, & 001, & 0011, \\
& & 01, & 011, & \\
& 1, & 10, & 100, & \\
& & 11, & 110, & 1100.
\end{matrix}$$
Maximum-length binary words that avoid $Z_3$. {#maximum-length-binary-words-that-avoid-z_3. .unnumbered}
---------------------------------------------
\
The following 48 words are the only words of length $f(3,2)-1 = 28$ over the alphabet $\{0,1\}$ that avoid $Z_3= abacaba$. All binary words of length $f(3,2) = 29$ or longer encounter $Z_3$. This result is easily, computationally verified by constructing the binary tree of words on $\{0,1\}$, eliminating branches as you find words that encounter $Z_3$.\
[c c]{}
-------------------------------
0010010011011011111100000011,
0010010011111100000011011011,
0010010011111101101100000011,
0010101100110011111100000011,
0010101111110000001100110011,
0010101111110011001100000011,
0011001100101011111100000011,
0011001100111111000000101011,
0011001100111111010100000011,
0011011010010011111100000011,
0011011011111100000010010011,
0011011011111100100100000011,
0011111100000010010011011011,
0011111100000010101100110011,
0011111100000011001100101011,
0011111100000011011010010011,
0011111100100100000011011011,
0011111100100101101100000011,
0011111100110011000000101011,
0011111100110011010100000011,
0011111101010000001100110011,
0011111101010011001100000011,
0011111101101100000010010011,
0011111101101100100100000011,
-------------------------------
&
-------------------------------
1100000010010011011011111100,
1100000010010011111101101100,
1100000010101100110011111100,
1100000010101111110011001100,
1100000011001100101011111100,
1100000011001100111111010100,
1100000011011010010011111100,
1100000011011011111100100100,
1100000011111100100101101100,
1100000011111100110011010100,
1100000011111101010011001100,
1100000011111101101100100100,
1100100100000011011011111100,
1100100100000011111101101100,
1100100101101100000011111100,
1100110011000000101011111100,
1100110011000000111111010100,
1100110011010100000011111100,
1101010000001100110011111100,
1101010000001111110011001100,
1101010011001100000011111100,
1101101100000010010011111100,
1101101100000011111100100100,
1101101100100100000011111100.
-------------------------------
A long binary word that avoid $Z_4$: {#a-long-binary-word-that-avoid-z_4 .unnumbered}
------------------------------------
\
The following binary word of length 10482 avoids $Z_4 = abacabadabacaba$. This implies that $f(4,2)\geq 10483$. The word is presented here as an image with each row, consisting of 90 squares, read left to right. Each square, black or white, represents a bit. For example, the longest string of black in the first row is 14 bits long. We cannot have the same bit repeated $15 = |Z_4|$ times consecutively, as that would be a $Z_4$-instance. A string of 14 white bits is found in the 46th row.\
{width="270px"}
Verifying that a word avoids $Z_n$: {#verifying-that-a-word-avoids-z_n .unnumbered}
-----------------------------------
\
The code to generate a $Z_4$-avoiding word of length 10482 is messy. The following, easy-to-validate, inefficient, brute-force, Sage [@Sage] code was used for verification of the word above. It took about half a day, running on an IntelCorei5-2450M CPU $@$ 2.50GHz $\times$ 4.\
``` {frame="single"}
#Recursive function to test if V is an instance of Z_n
def inst(V,n):
if n==1:
if len(V)>0:
return 1
return 0
else:
top = ceil(len(V)/2)
for i in range(2^(n-1)-1,top):
if V[:i]==V[-i:]:
if inst(V[:i],n-1):
return 1
return 0
#Paste word here as a string
W =
L = len(W)
n = 4
#Check every subword V of length at least 2^n-1
for b in range(L+1):
for a in range(b-(2^n-1)):
if inst(W[a:b],n):
print a,b,W[a:b]
```
[^1]: \*University of South Carolina
|
---
bibliography:
- 'tGISguide.bib'
---
[**Why GPS makes distances bigger than they are** ]{}
Peter Ranacher $^{1\ast}$, Richard Brunauer$^{2}$, Wolfgang Trutschnig $^{3}$ Stefan Christiaan Van der Spek$^{4}$, and Siegfried Reich $^{2}$
[1]{} Department of Geoinformatics - Z\_GIS, University of Salzburg, Schillerstra[ß]{}e 30, 5020 Salzburg, Austria\
[2]{} Salzburg Research Forschungsgesellschaft mbH, Jakob Haringer Stra[ß]{}e 5/3, 5020 Salzburg, Austria\
[3]{} Department of Mathematics, University of Salzburg, Hellbrunner Stra[ß]{}e 34, 5020 Salzburg, Austria\
[4]{} Delft University of Technology, Faculty of Architecture, Department of Urbanism, Julianalaan 134, 2628BL Delft, The Netherlands
Abstract {#abstract .unnumbered}
========
Global Navigation Satellite Systems (GNSS), such as the Global Positioning System (GPS), are among the most important sensors for movement analysis. GPS is widely used to record the trajectories of vehicles, animals and human beings. However, all GPS movement data are affected by both measurement and interpolation error. In this article we show that measurement error causes a systematic bias in distances recorded with a GPS: the distance between two points recorded with a GPS is – on average – bigger than the true distance between these points. This systematic ‘overestimation of distance’ becomes relevant if the influence of interpolation error can be neglected, which is the case for movement sampled at high frequencies. We provide a mathematical explanation of this phenomenon and we illustrate that it functionally depends on the autocorrelation of GPS measurement error ($C$). We argue that $C$ can be interpreted as a quality measure for movement data recorded with a GPS. If there is strong autocorrelation any two consecutive position estimates have very similar error. This error cancels out when average speed, distance or direction are calculated along the trajectory.
Based on our theoretical findings we introduce a novel approach to determine $C$ in real-world GPS movement data sampled at high frequencies. We apply our approach to a set of pedestrian and a set of car trajectories. We find that the measurement error in the data is strongly spatially and temporally autocorrelated and give a quality estimate of the data. Finally, we want to emphasize that all our findings are not limited to GPS alone. The systematic bias and all its implications are bound to occur in any movement data collected with absolute positioning if interpolation error can be neglected.
Introduction
============
Global Navigation Satellite Systems (GNSSs), such as the Global Positioning System (GPS), have become essential sensors for collecting the movement of objects in geographical space. In movement ecology, GPS tracking is used to unveil the migratory paths of birds [@higuchi_pierre_2005], elephants [@douglas-hamilton_etal_2005] or roe deer [@andrienko_etal_2011a]. In urban studies, GPS movement data help detecting traffic flows [@zheng_etal_2011a] and human activity patterns in cities [@vanderspek_etal_2009]. In transportation research, GPS allows monitoring of intelligent vehicles [@zito_etal_1995] and mapping of transportation networks [@mintsis_etal_2004], to name but a few application examples.
Movement recorded with a GPS is commonly stored in form of a trajectory. A trajectory $\tau$ is an ordered sequence of spatio-temporal positions: $\tau ={< (\boldsymbol{P_1},t_1),...,(\boldsymbol {P_n},t_n)>}$, with $t_1 < ... < t_n$ [@gueting_schneider_2005]. The tuple $(\boldsymbol{P},t)$ indicates that the moving object was at a position $\boldsymbol{P}$ at time $t$. In order to represent the continuity of movement, consecutive positions $(\boldsymbol{P_i},t_i)$ and $(\boldsymbol{P_{j}},t_{j})$ along the trajectory are connected by an interpolation function [@macedo_etal_2008].
However, although satellite navigation provides global positioning at an unprecedented accuracy, GPS trajectories remain affected by errors. There are two types of error inherent in any kind of movement data, measurement error and interpolation error [@schneider_1999], and these errors inevitably also affect trajectories recorded with a GPS.
- Measurement error refers to the impossibility of determining the actual position $(\boldsymbol{P},t)$ of an object due to the limitations of the measurement system. In the case of satellite navigation, it reflects the spatial uncertainty associated with each position estimate.
- Interpolation error refers to the limitations on interpolation representing the actual motion between consecutive positions $(\boldsymbol{P_i},t_i)$ and $(\boldsymbol{P_{j}},t_{j})$. This error is influenced by the temporal sampling rate at which a GPS records the positions.
Measurement and interpolation errors cause the movement recorded with a GPS to differ from the actual movement of the object. This needs to be taken into account in order to achieve meaningful results from GPS data.
In this article we focus on GPS measurement error in movement data. We show that measurement error causes a systematic overestimation of distance. Distances recorded with a GPS are – on average – always bigger than the true distances travelled by a moving object, if the influence of interpolation error can be neglected. In practice, this is the case for movement recorded at high frequencies. We provide a rigorous mathematical explanation of this phenomenon. Moreover, we show that the overestimation of distance is functionally related to the spatio-temporal autocorrelation of GPS measurement error. We build on this relationship and provide a novel methodology to assess the quality of GPS movement data. Finally we demonstrate our method on two types of movement data, namely the trajectories of pedestrians and cars.
Section \[sec:related work\] introduces relevant work from previously published literature. Section \[sec:overestimation\] provides a mathematical explanation of why GPS measurement error causes a systematic overestimation of distance. Section \[sec:autocorr\] shows how this overestimation can be used to reason about the spatio-temporal auto-correlation of measurement error. Section \[sec:experiment\] describes the experiment and presents our experimental results, section \[sec:discussion\] discusses the results.\
Related work {#sec:related work}
============
Since GPS data have become a common component of scientific analyses its quality parameters have received considerable attention. These parameters include the accuracy of the estimated position, the availability and the update rate of the GPS signal, as well as the continuity, integrity, reliability and coverage of the service [@hofmann-wellenhof_etal_2003]. The accuracy of the estimated position (i.e. the expected conformance of a position provided with a GPS to the true position, or the anticipated measurement error) is clearly of utmost importance. Measurement error and its causes, influencing factors, and scale have been extensively discussed in published literature: measurement error has been shown to vary over time [@olynik_2002] and to be location-dependent. Shadowing effects, for example due to canopy cover, have a significant influence on its magnitude [@deon_etal_2002]. Measurement error is both random and caused by external influences, as well as systematic and caused by the system’s limitations [@parent_etal_2013].
Measurement error is the result of several influencing factors. According to [@langley_1997], these include:
- Propagation delay: atmospheric variations can affect the speed of the GPS signal and hence the time that it takes to reach the receiver;
- drift in the GPS clock: a drift in the on-board clocks of the different GPS satellites causes them to run asynchronously with respect to each other and to a reference clock;
- Ephemeris error: imprecise satellite data and incorrect physical models affect estimations of the true orbital position of each GPS satellite [@colombo_1986];
- Hardware error: the GPS receiver, being as fault-prone as any other measurement instrument, produces an error when processing the GPS signal;
- Multipath propagation: infrastructure close to the receiver can reflect the GPS signal and thus prolong its travel time from the satellite to the receiver;
- Satellite geometry: an unfavourable geometric constellation of the satellites reduces the accuracy of positioning results.
There are several quality measures to describe GPS measurement error, the most common being the *95 $\%$ radius* ($R95$), which is defined as the radius of the smallest circle that encompasses 95 $\%$ of all position estimates [@chin_1987]. The official GPS Performance Analysis Report for the Federal Aviation Administration issued by the [@gps_report_2013] states that the current set-up of the GPS allows to measure a spatial position with an average $R95$ of slightly over 3 meters using the Standard Positioning Service (SPS). The values in the report were, however, obtained from reference stations that were equipped with high quality receivers and had unobstructed views of the sky. It is reasonable to assume that the accuracy would be reduced in other recording environments, as measurement error depends to a considerable extent on the receiver, as well as on the geographic location [@gps_report_2013; @langley_1997]. This assumption is supported by published literature on GPS accuracy in forests [@sigrist_etal_1999] and on urban road networks [@modschnig_etal_2006], as well as on the accuracies of different GPS receivers [@wing_etal_2005; @zandbergen_2009]. On the other hand, the accuracy of GPS can be increased using differential global positioning systems (DGPS) such as the European Geostationary Navigation Overlay Service (EGNOS). DGPS estimate and correct the propagation delay in the ionosphere, thus yielding higher position accuracies [@hofmann-wellenhof_etal_2003].
A detailed overview of current GPS accuracy is provided in the quarterly GPS Performance Analysis Report for the Federal Aviation Administration. A good introduction to the GPS in general, and to its error sources and quality parameters in particular, has been provided by [@hofmann-wellenhof_etal_2003].\
The above-mentioned research has mainly focused on describing and understanding GPS measurement error. In addition to this, filtering and smoothing approaches have been proposed for recording movement data, in order to reduce the influence of errors on movement trajectories. A concise summary of these approaches can be found in [@parent_etal_2013]. [@jun_etal_2006] tested smoothing methods that best preserve travelled distance, speed, and acceleration. The authors found that Kalman filtering resulted in the least difference between the true movement and its representation.
GPS measurement error causes a systematic overestimation of distance {#sec:overestimation}
====================================================================
A GPS measurement consists of a spatial component (i.e. latitude $\phi$, longitude $\lambda$) and a temporal component (i.e. a time stamp $t$). In this article we will mainly focus on the spatial component.
The GPS uses the *World Geodetic System 1984 (WGS84)* as a coordinate reference system. For reasons of simplicity it is preferable to transform the GPS measurements to a Cartesian map projection, such as the Universal Transversal Mercator (UTM). A transformation from an ellipsoid (WGS84) to a Cartesian plane (UTM) leads to a distortion of the original trajectories [@hofmann-wellenhof_etal_2003]. For vehicle, pedestrian, or animal movements consecutive positions along a trajectory are usually sampled in intervals ranging from seconds to minutes. Thus, these positions are very close together in space so that the distortion is insignificant for most practical applications. Hence, for all the following consideration we assume that the movement is recorded in UTM.
Very generally, a spatial position in UTM is a two-dimensional coordinate
$${\boldsymbol{P}} = \left(
\begin{array}{c}
x\\
y\\
\end{array}
\right) ,$$
where $x$ is the metric distance of the position from a reference point in eastern direction and $y$ in northern direction. If a moving object is recorded at position ${\boldsymbol{P}}$ with a GPS, the position estimate $ \boldsymbol{P^m} = (x^m, y^m)$ is affected by measurement error. The relationship between the true position and its estimate is very trivially
$$\boldsymbol {P^m} = \boldsymbol{P}+\boldsymbol{{\varepsilon_P}} \text{,}$$
where $\boldsymbol{{\varepsilon_P}}$ is the horizontal measurement error expressed as a vector in the horizontal plane. $\boldsymbol{{\varepsilon_P}}$ is drawn from $\boldsymbol{{\mathcal{E}_P}}$, the distribution of measurement error at $\boldsymbol{{P}}$. We adopt the convention used by [@codling_etal_2008] to denote random variables with upper case letters and their numerical values with lower case letters.
We now provide a detailed mathematical explanation of why measurement error causes a systematic overestimation of distance in trajectories, if interpolation error can be neglected. Figure \[fig:thought\_experiment\] illustrates the problem statement in a simplified form. Consider a moving object equipped with a GPS. The moving object travels between two arbitrary positions $\boldsymbol{P}$ and $\boldsymbol{Q}$. Let $d_0 = d(\boldsymbol{P},\boldsymbol{Q})$ denote the Euclidean distance between these, henceforth referred to as reference distance. The object always moves along a straight line, consequently interpolation error can be neglected. The movement of the object can be described by the following five steps, which correspond to the subplots in Figure \[fig:thought\_experiment\].
1. The moving object starts at $\boldsymbol{P}$. The GPS obtains the position estimate $\boldsymbol{P^m}$ with measurement error $\boldsymbol{{\varepsilon_P}}$, which is drawn from $\boldsymbol{{\mathcal{E}_P}}$.
2. [ The moving object travels to $\boldsymbol{Q}$. The GPS obtains the position estimate $\boldsymbol{Q^m}$ with measurement error $\boldsymbol{{\varepsilon_Q}}$, which is drawn from $\boldsymbol{{\mathcal{E}_Q}}$. The distance between the two position estimates is calculated: ${d^m} = d(\boldsymbol{P^m},\boldsymbol{Q^m})$.]{}
3. [ The moving object returns to $\boldsymbol{P}$. The GPS obtains a position estimate and a new ${d^m}$ is calculated.]{}
4. [Steps 2 and 3 are repeated $n$ times, where $n$ is an infinitely large number.]{}
5. [After $n$ repetitions, the position estimates scatter around $\boldsymbol{P}$ and $\boldsymbol{Q}$ with measurement error $\boldsymbol{{\mathcal{E}_P}}$ and $\boldsymbol{{\mathcal{E}_Q}}$.]{}
![A moving object equipped with a GPS travels between two arbitrary positions[]{data-label="fig:thought_experiment"}](figures/thought_experiment_new.eps)
We claim that measurement error propagates to the expected measured distance $\mathbb{E}(d^m)$ and to the expected squared measured distance $\mathbb{E} (d_2^m)$ between the position estimates. More specifically, measurement error yields $\mathbb{E} (d^m) >d_0$ as well as $\mathbb{E}(d_2^m) >d_0^2$.
We are now going to rigorously prove this claim. To do so, we simplify notation, write $\boldsymbol{{\mathcal{E}_P}}=(X_1,Y_1)$ as well as $\boldsymbol{{\mathcal{E}_Q}}=(X_2,Y_2)$, and assume that there is no systematic bias, i.e. we have $\mathbb{E}(X_1)=\mathbb{E}(X_2)=\mathbb{E}(Y_1)=\mathbb{E}(Y_2)=0$. Since neither translations nor rotations affect distances between points we may, without loss of generality, consider $\boldsymbol{P}=(0,0)$ and $\boldsymbol{Q}=(d_0,0)$. Since linear transformations (like rotations) preserve expectation, rotating errors with expectation zero results in errors having expectation zero too. Having this we can now formulate the following first result for the expected squared distance $\mathbb{E}(d^2(\boldsymbol{P^{m}}, \boldsymbol{Q^{m}}))$. For mathematical background we refer to [@klenke_2013]. Notice that no assumptions (like absolute continuity or normality) about the underlying error distributions are needed, i.e. the result holds in full generality.
\[th:overestimation\] Suppose that $d_0>0$, that $\boldsymbol{P}=(0,0)$, and that $\boldsymbol{Q}=(d_0,0)$. Let $X_1,X_2$ both have distribution function $F$ and variance $\sigma_X^2$, and $Y_1,Y_2$ both have distribution function $G$ and variance $\sigma_Y^2$. Furthermore assume that $\mathbb{E}(X_1)=\mathbb{E}(X_2)=\mathbb{E}(Y_1)=\mathbb{E}(Y_2)=0$. Then the following two conditions are equivalent:
1. $\mathbb{E}(d_2^m)= \mathbb{E}(d^2(\boldsymbol{P^{m}}, \boldsymbol{Q^{m}})) > d_0^2$
2. $\min\{Cov(X_1,X_2), Cov(Y_1,Y_2)\} <1 $
In other words: the expected squared distance $\mathbb{E}(d_2^m)$ is strictly greater than $d_0^2$ unless the errors fulfil $X_1=X_2$ and $Y_1=Y_2$ with probability one (which describes the situation of always having identical errors in $\boldsymbol{P}$ and $\boldsymbol{Q}$).
**Proof:** Calculating $\mathbb{E}(d^2(\boldsymbol{P^{m}}, \boldsymbol{Q^{m}}))$ and using the fact that $\textit{Cov}(X_1,X_2)\leq \sigma_X^2$ and $\textit{Cov}(Y_1,Y_2)\leq \sigma_Y^2$ directly yields $$\begin{aligned}
\label{eq:overestimation}
\mathbb{E}(d^2(\boldsymbol{P^{m}},\boldsymbol{Q^{m}})) &=& \mathbb{E}(d_0+X_2-X_1)^2 + \mathbb{E}(Y_2-Y_1)^2 \nonumber \\
&=& d_0^2 + \mathbb{E}(X_2-X_1)^2 + \mathbb{E}(Y_2-Y_1)^2 = d_0^2 + \textit{Var}(X_2-X_1) + \textit{Var}(Y_2-Y_1) \nonumber \\
&=& d_0^2 + 2 \sigma_X^2 + 2 \sigma_Y^2 - 2 \textit{Cov}(X_1,X_2) - 2 \textit{Cov}(Y_1,Y_2) \\
&\geq& d_0^2.\nonumber\end{aligned}$$ Having this it follows immediately that $\mathbb{E}(d^2(\boldsymbol{P^{m}},\boldsymbol{Q^{m}}))=d_0^2$ if and only if $\textit{Cov}(X_1,X_2)= \sigma_X^2$ and $\textit{Cov}(Y_1,Y_2)=\sigma_Y^2$, which in turn is equivalent to the fact that $X_1=X_2$ and $Y_1=Y_2$ holds with probability one. $\blacksquare$
In general one is, however, interested in the expected distance $\mathbb{E} (d^m):= \mathbb{E}(d(\boldsymbol{P^{m}}, \boldsymbol{Q^{m}}))$ and not in the expected squared distance. Since, in general, $\mathbb{E}(Z^2)>d_0^2$ need not imply $\mathbb{E}(\vert Z \vert)>d_0$ for arbitrary random variables $Z$, a different method is used to prove the following main result:
\[th:overestimation2\] Suppose that the assumptions of Theorem \[th:overestimation\] hold. Then the following two conditions are equivalent:
1. $\mathbb{E}(d^m)= \mathbb{E}(d(\boldsymbol{P^{m}}, \boldsymbol{Q^{m}})) > d_0$
2. $\max\big\{\mathbb{P}(Y_1\not =Y_2), \mathbb{P}(X_2-X_1<-d_0)\big\} >0$
In other words: the expected distance $\mathbb{E}(d^m)$ is strictly greater than the true distance $d_0$ unless the errors fulfil $Y_1=Y_2$ with probability one and $\mathbb{P}(X_2-X_1<-d_0)=0$ holds.
**Proof:** Obviously we have $$\label{temp}
\sqrt{(d_0+X_2-X_1)^2 + (Y_2-Y_1)^2} \geq \vert d_0+X_2-X_1 \vert$$ Setting $Z:=X_2-X_1$ implies $\mathbb{E}(Z)=0$. Assume now that $\mathbb{P}(Z<-d_0)>0$ holds. Then the desired inequality follows immediately from $$\begin{aligned}
\label{temp2}
\mathbb{E}\vert Z + d_0 \vert &=& \int_{\mathbb{R}} \vert z+ d_0 \vert \, d \mathbb{P}^Z =
\int_{[-d_0,\infty]} (z+d_0) \,d \mathbb{P}^Z + \int_{(-\infty,-d_0)} -(z+d_0) \,d \mathbb{P}^Z \nonumber \\
&=& \underbrace{\int_{\mathbb{R}} (z+d_0) \,d \mathbb{P}^Z}_{=d_0} \,+ \,\,
\underbrace{(-2) \int_{(-\infty,-d_0)} (z+d_0) \,d \mathbb{P}^Z}_{>0}\\
&>& d_0. \nonumber\end{aligned}$$ In case we have $\mathbb{P}(Z<-d_0)=0$ but $\mathbb{P}(Y_1\not=Y_2)>0$ holds, then Inequality \[temp\] is strict with positive probability so we get $$\mathbb{E}(d^m)=\mathbb{E}\Big(\sqrt{(d_0+X_2-X_1)^2 + (Y_2-Y_1)^2} \Big)>\mathbb{E}(\vert d_0 +X_2-X_1 \vert)= \mathbb{E}(\vert Z + d_0 \vert) = d_0.$$ Altogether this shows that the second condition of Theorem \[th:overestimation2\] implies the first one.\
To prove the reverse implication, assume that $\max\big\{\mathbb{P}(Y_1\not =Y_2), \mathbb{P}(X_2-X_1<-d_0)\big\} =0$. Then, firstly, the left and the right hand-side of Inequality \[temp\] coincide with probability one, so $\mathbb{E}(d^m)=\mathbb{E}\big(\sqrt{(d_0+X_2-X_1)^2 + (Y_2-Y_1)^2} \big)=\mathbb{E}(\vert d_0 +X_2-X_1 \vert)$ holds. And secondly, directly applying Equality \[temp2\] yields $\mathbb{E}(\vert Z + d_0 \vert) = d_0$, which finally shows $\mathbb{E}(d^m)= d_0$. $\blacksquare$
\[corr:overestimation\] *It is worth mentioning that Theorem \[th:overestimation2\] has several interesting (and partially surprising) consequences: Whenever the errors in $x$-direction are unbounded (like in the case of normal distributions) the expected distance is always strictly greater than the true distance $d_0$. The same holds whenever the errors $Y_1$ and $Y_2$ in $y$-direction do not always coincide – a very realistic assumption for GPS trajectories.*
We want to underline that Theorem
\[th:overestimation\] and \[th:overestimation2\] hold in full generality for arbitrary distributions of GPS measurement error. Although GPS measurement error is often assumed to have a bivariate normal distribution and to be independent in both the $x$- and $y$-direction [@jerde_visscher_2005; @bos_etal_2007], [@chin_1987] puts forward convincing arguments why this is very likely not the case. Hence, the general validity of our findings is relevant.
For reasons of simplicity, we have assumed that $\boldsymbol{{\mathcal{E}_P}}$ and $\boldsymbol{{\mathcal{E}_Q}}$ follow the same distribution function and that there is no systematic bias, i.e. $\boldsymbol{{\mathcal{E}_P}}$ is centred around $\boldsymbol{P}$ and $\boldsymbol{{\mathcal{E}_Q}}$ around $\boldsymbol{Q}$. This assumption is generally acknowledged for in the literature. It builds, for example, the basis for algorithms to extract road maps from GPS tracking data [e.g. @wang_etal_2014]. Roads are assumed to be located where the density of the GPS measurements is the highest. Also Figure \[fig:collage\] shows that this assumption is indeed realistic for real-world GPS data. However, even a systematic bias does not necessary restrict the validity of our argument. Let us assume that $\mathbb{E}(X_1)=\mathbb{E}(X_2) \neq 0$ and $\mathbb{E}(Y_1)=\mathbb{E}(Y_2)\neq 0$, i.e. the mean of the error distribution has shifted away from $\boldsymbol{P}$ and $\boldsymbol{Q}$ respectively. As the shift is the same for $\boldsymbol{{\mathcal{E}_P}}$ and $\boldsymbol{{\mathcal{E}_Q}}$, the influence on distance calculations cancels out, Theorem
\[th:overestimation\] and \[th:overestimation2\] still hold. The validity of our proof is restricted only if $\mathbb{E}(X_1) \neq \mathbb{E}(X_2)$ or $\mathbb{E}(Y_1) \neq \mathbb{E}(Y_2)$. This implies that the mean of the error distribution changes abruptly between $\boldsymbol{P}$ and $\boldsymbol{Q}$. As – in practice – $\boldsymbol{P}$ and $\boldsymbol{Q}$ are very close in space, this scenario is not realistic for GPS measurement error.
How big is the overestimation of distance and why is this relevant? {#sec:autocorr}
===================================================================
In the previous Section we proved that distances recorded with a GPS are on average bigger than the distances travelled by a moving object, if interpolation error can be neglected. In this Section we provide an equation for $\textit{OD}$, the expected overestimation of distance. Moreover, we identify three parameters that influence the magnitude of $\textit{OD}$. First, let us define $\textit{OD}$ with the help of Equation \[eq:overestimation\]: $$\textit{OD} = \mathbb{E}(d^m_2)^{\frac{1}{2}} - d_0 = (d^2_0 + 2 \sigma_X^2 + 2 \sigma_Y^2 - 2 \textit{Cov} (X_1,X_2) - 2 \textit{Cov} (Y_1,Y_2)) ^{\frac{1}{2}} - d_0.$$ From this follows that $\textit{OD}$ is a function of three parameters:
1. [$d_0$, the reference distance between $\boldsymbol{P}$ and $\boldsymbol{Q}$]{}\
2. [$\textit{Var}_{\textit{gps}} = 2 \sigma_X^2 + 2 \sigma_Y^2$, a term for the variance of GPS measurement error]{}\
3. [$C = 2 \textit{Cov} (X_1,X_2) - 2 \textit{Cov} (Y_1,Y_2)$, a term for the auto-correlation of GPS measurement error. $C$ expresses the similarity of any two consecutive position estimates. If $C$ is big, consecutive position estimates are affected by similar GPS measurement error (see also Figure \[fig:influencing\_parameters\]]{}).
We can now simplify notation and write
$$\label{eq:simplified_OD}
\textit{OD} = (d^2_0 + \textit{Var}_{\textit{gps}} - C) ^{\frac{1}{2}} - d_0.$$
The influence of the three parameters on $\textit{OD}$ is further illustrated in Figure \[fig:influencing\_parameters\]. $\textit{OD}$ is small if the reference distance is big, the variance of GPS measurement error is small and the error has high positive autocorrelation. $\textit{OD}$ is big if the reference distance is small, the variance of GPS measurement error is big and the error has high negative autocorrelation.
![Overestimation of distance ($\textit{OD}$) and its influencing parameters.[]{data-label="fig:influencing_parameters"}](figures/od_function_new.eps)
To understand the magnitude of $\textit{OD}$ in real-world GPS data, let us assume for a moment that there is no autocorrelation of GPS measurement error, i.e. $C =0$. Moreover, let us assume that the variance of error is the same in $x$- and $y$- directions, i.e. $\sigma^2 = \sigma_X^2 = \sigma_Y^2$ and $\textit{Var}_{\textit{gps}} = 4 \sigma^2$. We can now visualize the relationship between $\textit{OD}$, $d_0$ and $\sigma$. Figure \[fig:OD\_compared\] a
shows that $\mathit{OD}$ increases as the spread of GPS measurement error ($\sigma$) increases; $d_0$ is assumed to be constant. For a constant $d_0$ of $5{m}$, for example, and $\sigma =2 \rm {m}$ the overestimation of distance roughly equals $2\rm{m}$ (yellow line). When $\sigma$ increases to $4 \rm {m}$ the overestimation of distance increases to $4 \rm {m}$. Figure \[fig:OD\_compared\] b shows that $\mathit{OD}$ decreases as $d_0$ increases; $\sigma$ is assumed to be constant. For a constant $\sigma$ of $3 \rm {m}$, for example, and $d_0 = 5{m}$ the overestimation of distance equals around $3\rm{m}$ (black line). When $d_0$ increases to $10{m}$ the overestimation of distance decreases to $2 \rm {m}$.
![The overestimation of distance $(\mathit{OD}$) increases as the spread of GPS measurement error ($\sigma$) increases, the reference distance ($d_0$) is constant (a); $\mathit{OD}$ decreases as $d_0$ increases $\sigma$ is constant (b).[]{data-label="fig:OD_compared"}](figures/od_compared_new.eps)
Remember that Figure \[fig:OD\_compared\]
shows the influence of $\textit{Var}_{\textit{gps}}$ if there is no autocorrelation of GPS measurement error. This is not very realistic for real world GPS data. In fact, [@el-rabbany_kleusberg_2003; @wang_etal_2002] and [@howind_etal_1999] show that GPS is affected by both spatial and temporal autocorrelation. This means that position estimates taken close in space and in time tend to have similar error.
How big is the autocorrelation of GPS measurement error? Let us reformulate Equation \[eq:simplified\_OD\] and solve for $C$:
$$\label{eq:C}
C = d_0^2 - \overbrace{•(\textit{OD} + d_0)^2} ^{\mathbb{E}(d^m_2)} + \textit{Var}_{\textit{gps}}.$$
This implies that we can calculate the autocorrelation of GPS measurement error if $\mathit{OD}$, $\textit{Var}_{\textit{gps}}$ and $d_0$ are known. Things become interesting if we consider what autocorrelation really means in the context of GPS positioning. In Figure \[fig:influencing\_parameters\], in the bottom left cell, the position estimates $\boldsymbol{P^m}$ and $\boldsymbol{Q^m}$ are highly auto-correlated and, hence, very similar. This leads to the effect that $d^m$ is very similar to $d_0$. In fact, this applies not only to distance, but to other movement parameters as well. Direction, speed, acceleration or turning angle must all be similar to the ‘true’ movement of the object if they are derived from highly auto-correlated GPS position estimates. Consequently, $C$ describes how well a GPS captures the movement of an object, if interpolation error can be neglected. Or in other words, $C$ is a quality measure for GPS movement data.
Assessing the quality of GPS movement data {#sec:experiment}
==========================================
Real world GPS data are affected by spatial as well as temporal autocorrelation [@el-rabbany_kleusberg_2003; @wang_etal_2002; @howind_etal_1999]. Spatial autocorrelation implies that GPS measurement error is not independent of space. Measurements obtained at similar locations will have similar error. Temporal autocorrelation implies that GPS measurement error is not independent of time. Measurements obtained at similar times will have a similar error due to similar atmospheric conditions and a similar satellite constellation [@bos_etal_2007]. We carried out a simple experiment to visualize temporal autocorrelation in real-world GPS data. We placed a GPS unit at a known position $\boldsymbol{P}$ and recorded about 720 position estimates over a period of about six hours at a sampling rate of $1/30~\mathrm{Hz}$. The resulting distribution is centred around $\boldsymbol{P}$ with an $\text{\textit{R95}}$ of about $3~\mathrm{m}$ (Figure \[fig:collage\] a). If only those position estimates are displayed that were recorded within a certain time interval, GPS measurement error reveals itself to be highly auto-correlated. Figure \[fig:collage\] b, for example, shows only those position estimates that were obtained within periods covering $5$ minutes before and after ${t_1,t_2,t_3}$.
In this Section we build on the relationship described in Equation \[eq:C\] and show the spatial and temporal autocorrelation in two sets of real-world GPS movement data. In the first experiment we identified to what degree a set of pedestrian movement data was affected by spatial and temporal autocorrelation. In the second experiment we derived the spatial autocorrelation in a set of car movement data. Based on this we tried to assess how well the GPS captured the movement of the car.
![The distribution of GPS measurement error at position $\boldsymbol{P}$ (a). Revealing the temporal autocorrelation of GPS measurement error (b). The movement of a pedestrian around a reference course (c).[]{data-label="fig:collage"}](figures/spatio_temporal_ac_theory_new.eps)
Experiment 1: Pedestrian trajectories
-------------------------------------
### Experimental setup {#experimental-setup .unnumbered}
For the first experiment, we equipped a pedestrian with a GPS. The pedestrian walked along a reference course with a well-established reference distances $d_0$. The movement of the pedestrian was recorded with a QSTARZ:BT-Q1000X GPS unit[^1] with ‘Assisted GPS’ activated.
Rather than using a high-quality GPS we collected all data with a low-budget GPS, a type of GPS common for recording movement data. We deliberately treated the GPS as a ‘black box’. This implies that the algorithm to calculate the position estimates from the raw GPS signal was not known. Moreover, we considered that it was sufficient to use only a single GPS unit, as the aim of the experiment was not to investigate the quality of the particular GPS, but to show the usefulness of our approach.
The reference course was located in an empty parking lot to avoid shadowing and multi-path effects. We staked out a square with sides that were $10~\mathrm{m}$ long and had markers at one meter intervals. A square was used in order to allow distance measurements to be collected in all four cardinal directions (approximately). The distance between the markers was used as a reference distance $d_0$.
The GPS position estimates were obtained by walking to the reference markers in turn and recording the position, moving around the square until all positions of the markers had been recorded. Position estimates were only taken at the reference markers, and only when the recording button was pushed manually. Two consecutive position estimates were taken within three to five seconds. A full circuit around the square took approximately between two and three minutes and resulted in $40$ positions being recorded. A total of $25$ circuits around the square were completed, without any breaks. This resulted in $1000$ GPS positions being collected in approximately one hour.
In pre-processing distance measurements $d^m$ were calculated between the position estimates and later compared to $d_0$ the reference distance between the markers. Then the average measured distance $\bar{d^m}$ was calculated and from this $\hat{\mathit{OD}} = \bar{d^m} -d_0$ and $\hat C = d^2_0 - \bar{d^m_2} +\textit{Var}_{\textit{gps}}$ were derived. $\mathit{\hat {OD}}$ and $\hat C$ are estimators for $\mathit{OD}$ and $C$.
We decided not to derive $\sigma_X$ and $\sigma_Y$ from observational data, but to set $\sigma_X = \sigma_Y = 3\mathrm{m}$. Hence, $\textit{Var}_{\textit{gps}}$ is not the observed variance of GPS measurement error, but a reference value to which $\mathit{OD}$ is later compared to. Consequently, our results do not show the exact value of $C$, but provide an estimate of $C$ with respect to $\textit{Var}_{\textit{gps}}$.
We increased the spatial separation between two position estimates of the pedestrian to illustrate the influence of spatial autocorrelation. Then we increased the temporal separation between two position estimates to illustrate the influence of temporal autocorrelation.
### Results {#results .unnumbered}
In contrast to the theoretical findings in Figure \[fig:OD\_compared\], overestimation of distance tended to increase as the reference distance $d_0$ increased. This was due to a decrease in the spatial autocorrelation of GPS measurement error. With increasing spatial separation of the position estimates, measurement error became less auto-correlated. Figure \[fig:spatial\_ac\] shows the relationship between the reference distance $d_0$ and $\hat{\mathit{OD}}$ (black dots) as well as $\hat{C}$ (black crosses).
![Overestimation of distance ($\hat{\mathit{OD}}$) and spatial autocorrelation of GPS measurement error ($\hat C$) in the pedestrian movement data.[]{data-label="fig:spatial_ac"}](figures/spatial_ac_new.eps)
We wanted to illustrate that the overestimation of distance was not caused by a small number of extreme outliers. Figure \[fig:histogram\] shows the histogram of $d^m - d_0$ for $d_0=1~\mathrm{m}$ (a), and for $d_0=5~\mathrm{m}$ (b) and their fit to a Gaussian distribution. Both histograms follow a Gaussian distribution $\mathcal{N} (\mu_{d} ,\sigma^2_{d}) $ rather well and outliers are almost non-existent. Note that $\mu_d$ and $\sigma^2_d$ in Figure \[fig:histogram\] refer to the values of the fitted Gaussian distribution and not to the empirically derived frequency.\
![Histogram of the difference between measured and reference distance ($d^m -d_0$) for $d_0=1~\mathrm{m}$ (a) and $d_0=5~\mathrm{m}$ (b)[]{data-label="fig:histogram"}](figures/spatial_ac_detail_new.eps)
In order to illustrate the temporal autocorrelation in GPS measurement error, we calculated the distance between non-consecutive position estimates around the square. One example is the distance between two position estimates, where the second one was obtained one circuit after the first later. The reference distance between the markers remained the same, e.g. $ d_0 = 1~\mathrm{m}$, but the position estimates were recorded within a longer time interval $\Delta t$. Figure \[fig:temporal\_ac\] shows the relationship between $\Delta t$ and $\hat{\mathit{OD}}$ (black dots) as well as $\hat{C}$ (black crosses) for a reference distance $ d_0= 1~\mathrm{m}$. $\hat{\mathit{OD}}$ increase with longer time intervals. The sharpest increase occurs between measurements that were taken promptly and those taken after about $2 {\frac{1}{2}}$ minutes. After $40$ minutes the curve levels out. This increase of $\hat{\mathit{OD}}$ was caused by the temporal auto-correlation of measurement error. For measurements taken within several seconds, measurement error appears to be strongly auto-correlated. However, auto-correlation falls sharply for measurements taken within $2 {\frac{1}{2}}$ minutes. From then on $\hat C$ gradually decreases as $\Delta t$ increases; again the curve levels out at about $40$ minutes.
![Overestimation of distance $\hat{\mathit{OD}}$ and temporal autocorrelation of GPS measurement error ($\hat C$) in the pedestrian movement data.[]{data-label="fig:temporal_ac"}](figures/temporal_ac_new.eps)
The data for the above experiment were calculated with a GPS for which the algorithm to calculate the position estimates from the raw GPS signal was not known. This raises the legitimate question, whether the results were produced by a smoothing algorithm rather than the behaviour of the GPS. Let us assume that the GPS used a smoothing algorithm. In simplified form, the current position estimate is then calculated from the last position estimate, the current GPS measurement and a movement model. For movement with constant speed and direction, smoothing yields trajectories that represent the true movement very accurately. However, sudden changes in movement, i.e. a sharp turn, are not followed by the trajectory. The current measurement implies a sharp turn, however, the movement model does not. Thus, the sharp turn becomes more elongated, the overestimation of distance increases. However, we did not find any support for an increase in the overestimation of distance after a sharp turn. This can also be seen in Figure \[fig:collage\] b.
Experiment 2: Car trajectories
------------------------------
In the first experiment the reference distance $d_0$ was staked out along a reference course. For obvious reasons this is not possible for recording the movement of a car. Hence we derived $d_0$ from speed measurements recorded with a car’s controller area network bus (CAN bus).
### Experimental Setup {#experimental-setup-1 .unnumbered}
We equipped a car with a GPS unit and tracked its movement for about 6 days. The car moved mostly in an urban road network at rather low speeds (average: $25~\mathrm{km/h}$). The temporal sampling rate of recording was $1~\mathrm{Hz}$. For the CAN bus measurements, a sensor recorded the rotation of the car’s drive axle, from which $d_0$ was inferred. Thus $d_0$ is the distance travelled by the car according to the CAN bus. For the same phases of movement we compared $d_0$ to $d^m$, the distance travelled by the car according to the GPS position estimates. As in the first experiment, we set $\sigma_X = \sigma_Y = 3\mathrm{m}$ and calculated $\textit{Var}_{\textit{gps}}$.
The data were first pre-processed and cleaned. Parts with very high speed (above $ 140~\mathrm{km/h}$) and very rapid acceleration (above $5~\mathrm{m/s^2
}$) were removed. Although the data consisted mostly of the car’s forward movements, there were also periods when it was either stationary or reversing in a parking lot. The data may also have included some periods during which shadowing caused a loss of the GPS signal (for example when driving in a tunnel). We therefore applied a simple mode detection algorithm to remove any such periods. The algorithm evaluates speed and acceleration along the trajectory and distinguishes segments that most probably reflect driving behaviour from those that are likely to reflect non-driving behaviour [@zheng_etal_2010]. Using the algorithm we were able to include only long phases of continuous driving, sampled at a continuous sampling frequency of $1~\rm{Hz}$. Following this pre-processing a total of about $ 195~\mathrm{km}$ of car trajectories remained for analysis.
### Results {#results-1 .unnumbered}
Figure \[fig:can\_bus\_autocorr\] shows that the autocorrelation of GPS measurement error decreased as the spatial separation between two consecutive position estimates increased. Nevertheless, $\hat C$ in Figure \[fig:can\_bus\_autocorr\] is always positive. This can be interpreted as a quality measure for the movement data. Consecutive position estimates were affected by less variance than initially suggested by $\textit{Var}_{\textit{gps}}$.
Although the results in Figure \[fig:can\_bus\_autocorr\] are similar to those obtained from the pedestrian movement data, they contain outliers. We believe that these outliers occur due to two reasons. First, the data comprise relatively few distance measurements for big $d_0$ because of the generally low speed of the car. Second, we could not guarantee a full temporal synchronization of both measurement systems (GPS and CAN bus). In other words, $d_0$ and $d^m$ might relate to slightly different time intervals. We found this lag to be around one second. We believe that this insight is important for the practical application of Equation \[eq:C\]. In order to provide valid results it requires both a significant number of distance measurements as well as a proper synchronisation of reference and measured distance.
![Overestimation of distance ($\hat{\mathit{OD}}$) and spatial autocorrelation of GPS measurement error ($\hat C$) in the car movement data.[]{data-label="fig:can_bus_autocorr"}](figures/can_bus_autocorr_new.eps)
Discussion and Outlook {#sec:discussion}
======================
In this paper we identified a systematic bias in GPS movement data. If interpolation error can be neglected GPS trajectories systematically overestimate distances travelled by a moving object. This overestimation of distance has previously been noted in the trajectories of fishing vessels [@palmer_2008]. For high sampling rates the distance travelled by the vessel was overestimated due to measurement error, while for lower sampling rates it was underestimated due to the influence of interpolation error. We provided a mathematical explanation for this phenomenon and showed that it functionally depends on three parameters, of which one is $C$, the spatio-temporal autocorrelation of GPS measurement error. We built on this relationship and introduced a novel approach to estimate $C$ in real-world GPS movement data. In this Section we want to discuss our findings and show their implications for movement analysis and beyond.
In the era of big data, more and more movement data are recorded at finer and finer intervals. For movement recorded at very high frequencies (e.g. $1 \rm {Hz}$) interpolation error can usually be neglected. Hence $\mathit{OD}$ is bound to occur in these data. However, this does not mean that high frequency movement data are of low quality, quite the opposite is true. Using the relationship between $C$ and $\mathit{OD}$ we showed experimentally that GPS measurement error in real world trajectories was affected by strong spatial and temporal autocorrelation. In other words, if the data were recorded close in space and time they captured the movement of the object better than if they were further apart.
Autocorrelation is important for movement analysis in many aspects. An appropriate sampling strategy for recording movement data, for example, should consider the influence of measurement error and address spatial and temporal autocorrelation. Since autocorrelation can be interpreted as a quality measure, it allows to reveal the performance of different GPS receivers in different recording environments. Moreover, autocorrelation has implications for simulation. [@laube_etal_2011] performed a simulation to reveal the complex interaction between measurement error and interpolation error and their effects on recording speed, turning angle and sinuosity. Their Monte Carlo simulation assumed GPS errors to scatter entirely randomly between each two consecutive positions. Our approach allows to verify whether this assumption is realistic.
### Where to find a reference distance? {#where-to-find-a-reference-distance .unnumbered}
For practical applications the biggest limitation of our experiments is their dependency on a valid reference distance. The moving object must traverse the reference distance along a straight line and without interpolation error, and at a precisely known time. Moreover, a large number of measurements has to be collected, since $C$ is derived from the expectation value of a random variable.
This limitation leads to a possibly interesting application of our findings, where the reference distance is derived from the GPS point speed measurements. Point speed measurements are calculated from the instantaneous derivative of the GPS signal using the Doppler effect. Point speed is very accurate [@bruton_etal_1999] and usually part of a GPS position estimate. Hence, for high sampling rates (e.g.$1$ [Hz]{}) point speed measurements can be used to infer the distance that a moving object has travelled between two position estimates. This distance is not affected by the overestimation of distance effect and could serve as a reference distance. Thus, GPS could be compared to itself to reveal the spatio-temporal autocorrelation of the position estimates. This approach would not require any other ground truth data. However, its feasibility and usefulness are yet to be tested.
Finally, we want to underline that our findings are not only relevant for GPS. The overestimation of distance is bound to occur in any type of movement data where distances are deduced from imprecise position estimates, of course only if interpolation error can be neglected.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was funded by the Austrian Science Fund (FWF) through the Doctoral College GIScience at the University of Salzburg (DK W 1237-N23). We thank Arne Bathke from the Department of Mathematics of the University of Salzburg for his invaluable help on quadratic forms.
[^1]: for specifications, please refer to: http://www.qstarz.com/Products/GPS/20Products/BT-Q1000.html
|
---
abstract: 'We study the effect of Hund’s splitting of repulsive interactions on electronic phase transitions in the multiorbital topological crystalline insulator Pb$_{1-x}$Sn$_{x}$Te, when the chemical potential is tuned to the vicinity of low-lying Type-II Van Hove singularities. Nontrivial Berry phases associated with the Bloch states impart momentum-dependence to electron interactions in the relevant band. We use a multipatch parquet renormalization group (RG) analysis for studying the competition of different electronic phases, and find that if the dominant fixed-point interactions correspond to antiparallel spin configurations, then a chiral $p$-wave Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) state is favored, otherwise, none of the commonly encountered electronic instabilities occur within the one-loop parquet RG approach.'
author:
- 'S.Kundu and V.Tripathi'
title: 'Role of Hund’s splitting in electronic phase competition in Pb$_{1-x}$Sn$_{x}$Te'
---
Topological crystalline insulators (TCIs) have low-energy surface states in certain high symmetry directions, protected by crystalline symmetry [@fu2011topological]. Unlike conventional Z$_{2}$ topological insulators [@hasan2010colloquium; @konig2008quantum; @moore2010birth; @qi2011topological], the nature of these low-energy states is sensitive to the surface orientation. In particular, it has been shown in the recently discovered TCI Pb$_{1-x}$Sn$_{x}$Te [@dziawa2012topological; @hsieh2012topological; @tanaka2012experimental; @xu2012observation] that the band structure of the (001) surface allows for the presence of Type-II Van Hove singularities [@PhysRevB.92.035132], with a diverging density of states, which opens up the possibility of a variety of competing Fermi-surface instabilities brought about by weak repulsive interparticle interactions [@dzyaloshinskii1987maximal; @schulz1987superconductivity; @lederer1987antiferromagnetism; @PhysRevLett.112.070403; @PhysRevB.89.144501]. In particular, the parquet approximation **** for studying competing phases in a system with multiple Fermi pockets has proved very useful in the context of unconventional superconductivity [@mineev1999introduction; @norman2011challenge; @sigrist1991phenomenological] in cuprates [@furukawa1998truncation], graphene [@nandkishore2012chiral] and semimetal thin films [@PhysRevB.93.155108]. However, in a multiorbital system like Pb$_{1-x}$Sn$_{x}$Te, phase competition needs to be studied taking into account the effect of Hund’s splitting of interactions. The importance of Hund’s coupling has generally been underemphasized in parquet renormalization group analyses of multiorbital systems for reasons of convenience, but recent developments show that Hund’s coupling may play an important role in electronic instabilities of multiorbital systems **** [@yuan2015triplet; @vafek2017hund].
In this paper, we employ a multipatch parquet renormalization group (RG) analysis including Hund’s splitting effects, and show that even relatively small amounts of Hund’s splitting can have a dramatic effect on the very existence of electronic instabilities on the surface of Pb$_{1-x}$Sn$_{x}$Te. Depending on the sign of the Hund’s splitting, we find that away from perfect nesting, either a chiral $p$-wave FFLO [@PhysRev.135.A550; @larkin1964nonuniform] state is stabilized or none of the commonly encountered electronic instabilities occur at the level of the one-loop parquet approach. A characteristic feature of Pb$_{1-x}$Sn$_{x}$Te is that the surface bands are effectively spinless, which rules out $s$-wave pairing, that would otherwise prevail over $p$-wave pairing in the presence of nonmagnetic disorder [@PhysRev.131.1553; @RevModPhys.75.657; @anderson1959theory].
The topological crystalline insulator surface that we consider offers certain natural advantages from an experimental point of view. It provides two-dimensional Van Hove singularities which are accessible through a small change in doping, unlike, say, graphene, where a very high level of doping is required. **** Interestingly, as we show below, the $p$-wave symmetry originates not from intrinsic Fermi surface deformations, but from the nontrivial Berry phases associated with the topological states. This is reminiscent of chiral $p$-wave superconductivity enabled by a topological Berry phase in fermionic cold atom systems with attractive momentum-independent interactions [@zhang2008p]. We argue that the $p$-wave superconductivity on the TCI surface is more robust against potential disorder [@PhysRevLett.109.187003; @nagai2015robust] than in, say, Sr$_{2}$RuO$_{4}$ [@PhysRevLett.80.161]. Moreover, the $p$-wave superconductivity here is intrinsic, unlike proximity-induced $p$-wave superconductivity on topological insulator surfaces where recently Majorana fermions have been detected [@he2017chiral]. Finally, such an FFLO state in a pure solid state system in the absence of an applied magnetic field is a rather unusual occurrence (see, e.g. Refs [@200815637] and [@hsu2017topological]). Ref. [@cho2012superconductivity] also discusses an intranode FFLO pairing in a doped Weyl semimetal, although the stability of such a state in this system is still a controversial issue [@bednik2015superconductivity; @wei2014odd; @zhou2016superconductivity].
![\[fig:The-RG-couplings\]The different types of Coulomb interaction processes in our low-energy model (Eq.\[eq:3\]). The solid lines and dashed lines denote two different patches $\overline{X_{1}}$ and $\overline{X_{2}}$ in momentum space, on the (001) surface. All the vertices have momentum-dependences as indicated for $h_{4}$. The $\sigma$’s refer to the particular spin components of the (spinor) wavefunctions associated with the bands under consideration (see text for more details). ](maintext_fig1){width="1.0\columnwidth"}
The band gap minima of IV-VI semiconductors are located at the four $L$ points in the FCC Brillouin zone. **** In [@liu2013two], the TCI surface states are classified into two types: *Type-I*, for which all four $L$-points are projected to the different time-reversal invariant momenta(TRIM) in the surface Brillouin zone, and *Type-II*, for which different $L$-points are projected to the same surface momentum. The (001) surface falls into the latter class of surfaces, for which the $L_{1}$ and $L_{2}$ points are projected to the $\overline{X_{1}}$ point on the surface, and the $L_{3}$ and $L_{4}$ points are projected to the symmetry-related $\overline{X_{2}}$ point. This leads to two coexisting massless Dirac fermions at $\overline{X_{1}}$ arising from the $L_{1}$ and the $L_{2}$ valley, respectively, and likewise at $\overline{X_{2}}$. The k.p Hamiltonian close to the point $\overline{X_{1}}$ on the (001) surface is derived on the basis of a symmetry analysis in [@liu2013two], and is given by $$H_{\overline{X_{1}}}(k)=(v_{x}k_{x}s_{y}-v_{y}k_{y}s_{x})+m\tau_{x}+\delta s_{x}\tau_{y},\label{eq:1}$$ where $k$ is measured with respect to $\overline{X_{1}}$, $\overrightarrow{s}$ is a set of Pauli matrices associated with the two spin components associated with each valley, $\tau$ operates in valley space, and the terms $m$ and $\delta$, which are off-diagonal in valley space, are added to describe intervalley scattering. The band dispersion and constant energy contours for the above surface Hamiltonian undergo a Lifshitz transition with increasing energy away from the Dirac point, and when the Fermi surface is at $\delta=26$ meV (as taken from [@liu2013two]) two saddle points $\overline{S_{1}}$ and $\overline{S_{2}}$ at momenta $(\pm\frac{m}{v_{x}},0)$ lead to a Van-Hove singularity in the density of states. A similar situation arises at the point $\overline{X_{2}}$.
In addition to the noninteracting part of the Hamiltonian described in Eq.1 above, we now consider interactions between surface electrons corresponding to different valleys and spins, which gives rise to the following terms in the Hamiltonian- $$H_{I}=\frac{1}{2}\sum_{a,b,c,d,\sigma,\sigma^{\prime}}U_{abcd}^{\sigma\sigma^{\prime}}c_{\sigma a}^{\dagger}c_{\sigma^{\prime}b}^{\dagger}c_{\sigma^{\prime}c}c_{\sigma d}\label{eq:2}$$ where $a,b,c,d$ refer to different valleys (which are either all the same, same in pairs or all different in the above sum) and $\sigma,\sigma^{\prime}$ refer to spins. Here, we consider $U_{abcd}^{\sigma\sigma^{\prime}}=U_{1}^{\sigma\sigma^{\prime}}$when $(a,c)$ belong to one $\overline{X}$-point (i.e. the L-valleys corresponding to $(a,c)$ are projected to one of the $\overline{X}$-points) and $(b,d)$ belong to the other $\overline{X}$-point. Similarly, $U_{abcd}^{\sigma\sigma^{\prime}}=U_{2}^{\sigma\sigma^{\prime}}$ when $(b,c)$ belong to one $\overline{X}$-point and $(a,d)$ belong to the other, $U_{3}^{\sigma\sigma^{\prime}}$ when $(a,b)$ belong to one $\overline{X}$-point and $(c,d)$ to the other, and $U_{4}^{\sigma\sigma^{\prime}}$when $a$,$b$,$c$ and $d$ all correspond to L-points projected to the same $\overline{X}$-point. The interactions depend only on the relative orientations of the spins, for example, $U^{\sigma\sigma^{\prime}}$ can be written as $U^{\sigma\sigma}\delta_{\sigma\sigma^{\prime}}+U^{\sigma\overline{\sigma}}(1-\delta_{\sigma\sigma^{\prime}})$. In our analysis, we have projected the interactions between electrons in the valley-spin picture to the positive-energy band lying closest to the Van-Hove singularities [@seesi]. The resulting multiplicative form factors $u_{\sigma ai}$(for a transformation from valley $a$, spin $\sigma$ to the $i$th band) lend a momentum dependence to the effective pairing interactions obtained upon projection. We find that the spin $\uparrow$ components of the form factors have an $\exp[i\theta_{k}]$ dependence in momentum space and transform as $\ell=1$ objects, whereas the phase of the spin $\downarrow$ components remains unchanged upon advancing by an angle of $2\pi$ around the $\overline{X_{r}}$($r=1,2$) points, and these show an $\ell=0$ angular dependence. These additional phase factors arise from the Berry phases associated with the surface states of the crystalline topological insulator. After projecting to the two bands intersecting with the Fermi level, we obtain the following low-energy theory $$\begin{aligned}
L & =\sum_{i}\psi_{i}^{\dagger}(\partial_{\tau}-\epsilon_{k}+\mu)\psi_{i}-\sum_{i,\sigma,\sigma^{\prime}}\frac{1}{2}h_{4}^{\sigma\sigma^{\prime}}\psi_{i}^{\dagger}\psi_{i}^{\dagger}\psi_{i}\psi_{i}\nonumber \\
& \qquad-\sum_{i\neq j,\sigma,\sigma^{\prime}}\frac{1}{2}(h_{1}^{\sigma\sigma^{\prime}}\psi_{i}^{\dagger}\psi_{j}^{\dagger}\psi_{i}\psi_{j}+h_{2}^{\sigma\sigma^{\prime}}\psi_{i}^{\dagger}\psi_{j}^{\dagger}\psi_{j}\psi_{i}\nonumber \\
& \qquad+h_{3}^{\sigma\sigma^{\prime}}\psi_{i}^{\dagger}\psi_{i}^{\dagger}\psi_{j}\psi_{j})\nonumber \\
& \qquad=\sum_{i}\psi_{i}^{\dagger}(\partial_{\tau}-\epsilon_{k}+\mu)\psi_{i}-(h_{4}^{0}+h_{4}^{1})\psi_{i}^{\dagger}\psi_{i}^{\dagger}\psi_{i}\psi_{i}\nonumber \\
& \qquad-\sum_{i\neq j}((h_{1}^{0}+h_{1}^{1})\psi_{i}^{\dagger}\psi_{j}^{\dagger}\psi_{i}\psi_{j}+(h_{2}^{0}+h_{2}^{1})\psi_{i}^{\dagger}\psi_{j}^{\dagger}\psi_{j}\psi_{i}\nonumber \\
& \qquad+(h_{3}^{0}+h_{3}^{1})\psi_{i}^{\dagger}\psi_{i}^{\dagger}\psi_{j}\psi_{j})\label{eq:3}\end{aligned}$$ with $h_{r}^{0}=\frac{1}{2}\sum_{\sigma}h_{r}^{\sigma\sigma}$ and $h_{r}^{1}=\frac{1}{2}\sum_{\sigma}h_{r}^{\sigma\overline{\sigma}}$ where the quadratic noninteracting part comes from the model in Eq.1. The chemical potential value $\mu$=0 corresponds to the system being doped to the Van Hove singularities. Here $h_{4}$ refers to different scattering processes within a band $i$, whereas $h_{1}$,$h_{2}$ and $h_{3}$ refer to exchange processes, Coulomb interactions and pair hopping between electrons corresponding to the two different bands under consideration (see Fig. \[fig:The-RG-couplings\]). Due to the distinct phase dependences associated with the form factors corresponding to spins $\uparrow$ and $\downarrow$, the effective interactions $h_{r}$ after projection to the low-energy bands also either have a phase factor of $\exp[i(\theta_{k}-\theta_{k^{\prime}})]$ (for spin-antiparallel configurations) and behave as $\ell=1$ objects, or have no additional phase factors (for spin-parallel configurations) and behave as $\ell=0$ objects. The coupling constants $h_{r}^{0}\propto h_{r}^{\sigma\sigma}$ and $h_{r}^{1}\propto h_{r}^{\sigma\overline{\sigma}}$ respectively correspond to $\ell=0$ and $\ell=1$ angular momentum components of the interaction in our simplified model in Eq.\[eq:3\] above. It is important to note that although the surface bands are effectively spinless, we associate spin indices $\sigma\sigma^{\prime}$(or equivalently the superscripts $0$ and $1$) with the interactions $h_{r}$ in the different scattering channels $r$, due to the phase dependences associated with interactions between electrons with different spin configurations. In doing so, we allow for the Coulomb interactions between electrons to depend on the spin configuration being considered, thereby incorporating the effects of Hund’s splitting of interactions in our treatment.
![\[fig:all\]Flow of couplings with renormalization group scale $y$, starting with repulsive interactions, where the couplings in different angular momentum channels($h_{r}^{0}$ and $h_{r}^{1}$) are assumed to be degenerate initially, at $(h_{r}^{0,1})_{initial}=0.1$. We find pair hopping between patches ($h_{3}$) and on-patch scattering ($h_{4}$) to be the dominant scattering channels. Here, the critical point $y_{c}\approx3.65$.\
The inset shows the evolution of the fixed-point couplings $g_{r}^{\ell}$($\ell=0,1$) as a function of $d_{1}(y_{c})$($=\frac{1}{\sqrt{1+y_{c}}}$), which is the ratio of the particle-hole to particle-particle susceptibilities at the fixed point $y_{c}$. ](maintext_fig2){width="1.0\columnwidth"}
To study the possible instabilities in this system, we construct a two-patch renormalization group for the interaction vertices. In the RG analysis, the instability is indicated in the form of a pole in the vertex function. We consider only the electrons near the saddle points at $\overline{X_{1}}$ and $\overline{X_{2}}$ on the (001) surface. In our RG analysis, we distinguish between coupling constants with different spin combinations ($h_{r}^{\sigma\sigma}$ and $h_{r}^{\sigma\overline{\sigma}}$, or equivalently $h_{r}^{0}$ and $h_{r}^{1}$ respectively) and write separate RG equations for the two kinds of interactions.
We perform RG analysis up to one-loop level, integrating out high-energy degrees of freedom gradually from an energy cutoff $\Lambda$, which is the bandwidth. The susceptibilities in the different channels schematically behave as $\chi_{0}^{pp}(\omega)\sim\ln[\Lambda/\omega]\ln[\Lambda/\text{max}(\omega,\mu)]$, $\chi_{Q}^{ph}(\omega)\sim\ln[\Lambda/\text{max}(\omega,\mu)]\ln[\Lambda/\text{max}(\omega,\mu,t)]$ and $\chi_{0}^{ph}(\omega),\chi_{Q}^{pp}(\omega)\sim\ln[\Lambda/\text{max}(\omega,\mu)]$, where $\omega$ denotes the energy away from the Van Hove singularities and $t$ represents terms in the Hamiltonian that destroy the perfect nesting.
![\[fig:g8\]Flow of couplings with renormalization group scale $y$, starting with repulsive interactions, where the $\ell=1$ components of all the couplings are chosen to larger than the $\ell=0$ components by $2\%$ initially, i.e. $\frac{|h_{r}^{1}-h_{r}^{0}|}{|h_{r}^{0}|}=0.02$, where $(h_{r}^{0})_{initial}=0.1$. We find the $\ell=1$ components of pair hopping between patches ($h_{3}$) and on-patch scattering ($h_{4}$) to be the most dominant couplings in this case. Here, the critical point $y_{c}\approx3.56$. ](maintext_fig3){width="0.9\columnwidth"}
We use $y\equiv\ln^{2}[\Lambda/\omega]\sim\chi_{0}^{pp}$ as the RG flow parameter, and describe the relative weight of the other channels as $d_{1}(y)=\frac{d\chi_{Q}^{ph}}{dy}$, $d_{2}(y)=\frac{d\chi_{0}^{ph}}{dy}$ and $d_{3}(y)=-\frac{d\chi_{Q}^{pp}}{dy}$, where $d_{1}(y)$ is taken to be a function $\frac{1}{\sqrt{1+y}}$[@nandkishore2012chiral], interpolating smoothly in between the limits $d_{1}(y=0)=1$ and $d_{1}(y\gg1)=\frac{1}{\sqrt{y}}$ , and $d_{2},d_{3}\ll d_{1}$. The multiplicative factor $d_{1}(y)$ essentially incorporates the effects of imperfect nesting in our analysis. The RG equations are obtained by evaluating second-order diagrams and collecting the respective combinatoric prefactors, for each of the interactions $h_{1}$,$h_{2}$,$h_{3}$ and $h_{4}$. The diagrams corresponding to the renormalization of the interaction $h_{2}$ are shown in Fig. 7 in the Supplementary as an illustrative example. The RG equations obtained are given by (where we have used the notation $\sigma\sigma\equiv0$ and $\sigma\overline{\sigma}\equiv1$ for each of the couplings) $$\begin{aligned}
\frac{dh_{1}^{0}}{dy} & =2d_{1}(-(h_{1}^{0})^{2}-(h_{3}^{1})^{2}-(h_{1}^{1})^{2}\nonumber \\
& +2h_{1}^{0}h_{2}^{0}+(h_{3}^{0})^{2}),\\
\frac{dh_{1}^{1}}{dy} & =2d_{1}(-2h_{1}^{0}h_{1}^{1}+2h_{1}^{1}h_{2}^{0}),\\
\frac{dh_{2}^{0}}{dy} & =2d_{1}((h_{2}^{0})^{2}+(h_{3}^{0})^{2}),\\
\frac{dh_{2}^{1}}{dy} & =2d_{1}((h_{2}^{1})^{2}+(h_{3}^{1})^{2}),\end{aligned}$$ $$\begin{aligned}
\frac{dh_{3}^{0}}{dy} & =-4h_{4}^{0}h_{3}^{0}+2d_{1}(4h_{2}^{0}h_{3}^{0}\nonumber \\
& -2h_{1}^{1}h_{3}^{1}),\\
\frac{dh_{3}^{1}}{dy} & =-4h_{4}^{1}h_{3}^{1}+2d_{1}(2h_{2}^{1}h_{3}^{1}\nonumber \\
& -2h_{1}^{0}h_{3}^{1}+2h_{2}^{0}h_{3}^{1}),\\
\frac{dh_{4}^{0}}{dy} & =-2((h_{4}^{0})^{2}+(h_{3}^{0})^{2}),\\
\frac{dh_{4}^{1}}{dy} & =-2((h_{4}^{1})^{2}+(h_{3}^{1})^{2}).\end{aligned}$$
These coupled differential equations are then solved, starting from initial values of interactions in the weak-coupling regime($h_{r}^{0}=h_{r}^{1}\sim0.1$). The results for the cases where (a) the couplings $h_{r}^{\ell}$ are degenerate for $\ell=0$ and $\ell=1$, (b) the couplings $h_{r}^{\ell}$ in the $\ell=1$ channel are chosen to dominate initially, (c) the couplings $h_{r}^{\ell}$ in the $\ell=0$ channel are chosen to dominate initially, are shown in the Figures \[fig:all\],\[fig:g8\] and \[fig:g7\] respectively. **** The figures show results for a Hund’s splitting of $2\%$, and we have verified that even for a splitting of $0.1\%$ introduced initially between the interactions in the $\ell=0$ and $\ell=1$ channels ($\frac{|h^{1}-h^{0}|}{|h^{1}+h^{0}|}\sim0.1\%$) the final set of dominant couplings $g_{r}^{\ell}$ near the critical point of the RG correspond to the value of $\ell$ which has been chosen to dominate initially. Thus, the results of our RG analysis are found to be extremely sensitive to the sign of the Hund’s splitting. In contrast, the results are remarkably insensitive to the magnitude as well as sign of an initial splitting introduced between the couplings $h_{r}$ corresponding to the different scattering channels $r=1-4$. This is graphically depicted in Fig.10 in the Supplementary.\
![\[fig:g7\]Flow of couplings with renormalization group scale $y$, starting with repulsive interactions, where the $\ell=0$ components of all the couplings are chosen to be larger than the $\ell=1$ components by $2\%$ initially, i.e. $\frac{|h_{r}^{0}-h_{r}^{1}|}{|h_{r}^{1}|}=0.02$, where $(h_{r}^{1})_{initial}=0.1$. We find the $\ell=0$ components of pair hopping between patches ($h_{3}$) and on-patch scattering ($h_{4}$) to be the most dominant couplings in this case. Here, the critical point $y_{c}\approx3.4$.\
The inset shows the behavior of $h_{r}(y)(y_{c}-y)$ as a function of $(y_{c}-y)$ close to the fixed point $y_{c}$. The $y$-intercepts of the different curves show the fixed-point values $g_{r}^{\ell}$ for the couplings $h_{r}^{\ell}(y)$. Evidently, the dominant couplings near the critical point correspond to the $\ell=0$ channel in this case.](maintext_fig4){width="1.0\columnwidth"}
![\[fig:exponents\]The exponents $\alpha$, which are negative, corresponding to the various susceptibilities: chiral $p$-wave superconductivity, CDW, SDW and uniform charge compressibility ($\kappa$), plotted as a function of $d_{1}(y_{c})$ for the case where each of the couplings $g_{r}^{\ell}$ for $r=1-4$ and $\ell=0,1$ are degenerate. The order of these exponents indicates that chiral $p$-wave superconductivity is the leading instability (with the most negative exponent $\alpha_{pw}$) throughout, and CDW and SDW have nearly the same values of exponents $\alpha$ in this case.](maintext_fig5){width="0.95\columnwidth"}
We now investigate the instabilities of the system by evaluating the susceptibilities $\chi$ for various types of order, introducing infinitesimal test vertices corresponding to different kinds of pairing into the action, such as $\triangle_{a}\psi_{a\sigma}^{\dagger}\psi_{a\sigma^{\prime}}^{\dagger}+\triangle_{a}^{*}\psi_{a\sigma}\psi_{a\sigma^{\prime}}$ for the patch $a=1,2$ (where the spin labels $\sigma,\sigma^{\prime}$ are meant to simply denote the presence or absence of the phase factors $\exp[i\theta_{k}]$) corresponding to particle-particle pairing on the patch [@nandkishore2012chiral].
The renormalization of the test vertex for particle-particle pairing on a patch is governed by the equation [@nandkishore2012chiral] $$\frac{\partial}{\partial y}\left(\begin{array}{c}
\Delta_{1}\\
\Delta_{2}
\end{array}\right)=2\left(\begin{array}{cc}
h_{4}^{1} & h_{3}^{1}\\
h_{3}^{1} & h_{4}^{1}
\end{array}\right)\left(\begin{array}{c}
\Delta_{1}\\
\Delta_{2}
\end{array}\right)\label{eq:13}$$ since we can only consider Cooper pairing in the $p$-wave channel for spinless electrons. By transforming to the eigenvector basis, we can obtain different possible order parameters, and choose the one corresponding to the most negative eigenvalue. The vertices with positive eigenvalues are suppressed under RG flow.
At an electronic instability, the most divergent susceptibility $\chi$ determines the nature of the ordered phase. Each of the couplings associated with the RG flow has an asymptotic form $h_{r}^{\ell}(y)=\frac{g_{r}^{\ell}}{y_{c}-y}$ near the instability threshold. The coefficients $g_{r}^{\ell}$ can be determined as a function of $d_{1}(y_{c})$ (the results for the case, where we start with identical initial values for each of the couplings, are shown in the inset in Fig. \[fig:all\]). We diagonalize the Eq. \[eq:13\] above and substitute the asymptodic form of the interactions in the most negative eigenvalue. This gives us the exponent $\alpha$ for the divergence of the susceptibility $\chi\propto(y_{c}-y)^{\alpha}$ for $p$-wave superconductivity. Likewise we can introduce test vertices for other possible instabilities and obtain the corresponding exponents for their susceptibilities [@seesi]. The exponents for intrapatch $p$-wave pairing, charge-density wave, spin-density wave, uniform spin, charge compressibility ($\kappa$) and finite-momentum $\pi$ pairing are given by- $$\begin{aligned}
\alpha_{\text{pw}} & =2(-g_{3}^{1}+g_{4}^{1}),\nonumber \\
\alpha_{\text{CDW}} & =-2(g_{3}^{1}-g_{1}^{0}-g_{1}^{1}+g_{2}^{0})d_{1}(y_{c}),\nonumber \\
\alpha_{\text{SDW}} & =-2(g_{3}^{1}+g_{2}^{1})d_{1}(y_{c}),\nonumber \\
\alpha_{\kappa} & =-2(-g_{4}^{1}-(g_{1}^{0}-g_{2}^{0}-g_{2}^{1}))d_{2}(y_{c}),\nonumber \\
\alpha_{s} & =-2(g_{4}^{1}+g_{1}^{1})d_{2}(y_{c}),\nonumber \\
\alpha_{\pi}^{0} & =2(g_{2}^{0}-g_{1}^{0})d_{3}(y_{c}),\nonumber \\
\alpha_{\pi}^{1} & =2(g_{2}^{1}-g_{1}^{1})d_{3}(y_{c}).\label{eq:14}\end{aligned}$$ The $p$-wave order here is chiral since its symmetry is dictated by the aforementioned $\exp[i\theta_{k}]$ dependence of the Berry phase factors in the wave functions. It is important to note that we have $p$-wave order on the patches, unlike [@PhysRevB.92.035132] and [@PhysRevB.93.155108]. Consequently, this is a finite-momentum pairing, with each patch $\overline{X_{i}}$ located at a finite momentum with respect to the $\overline{\Gamma}$ point on the surface. Furthermore, the relative phase of the $p$-wave order on different patches is $\pi$, which means that we have $d$-wave order between the patches [@seesi].
Figure \[fig:exponents\] shows the behavior of the exponents for $p$-wave pairing, SDW, CDW and charge compressibility as a function of $d_{1}(y_{c})$. Comparison between the values of these exponents shows that the most divergent susceptibility is $p$-wave superconductivity throughout the parameter range $0<d_{1}(y_{c})<1$. The CDW and SDW instabilities show a weaker divergence, and are followed by charge compressibility. The exponents for uniform spin susceptibility and $\pi$ pairing are always positive and hence, these orders are suppressed. In the case of perfect nesting, i.e $d_{1}=1$, the SDW and CDW instabilities become degenerate with $p$-wave superconductivity.
Now, if a finite Hund’s splitting is introduced initially such that $h_{r}^{1}>h_{r}^{0}$, the above analysis holds and $p$-wave superconductivity is still the dominant instability. However, for an initial Hund’s splitting of the opposite sign, i.e. $h_{r}^{0}>h_{r}^{1}$, we find that the dominant couplings $g_{r}^{\ell}$ at the instability threshold correspond to $\ell=0$. In this case, the exponents $\alpha$ for each of the susceptibilities $\chi$ considered in Eq.\[eq:14\] turn out to be either positive or numerically close to zero. This is due to subtle cancellations between contributions from the dominant couplings in different scattering channels. Thus, none of the instabilities considered above are found to occur in this case, within the one-loop approximation. Clearly, the nature of instabilities in this system is crucially dependent on the sign of the Hund’s splitting.
We now discuss the effects of weak disorder on superconductivity on our crystalline topological insulator surface. Since potential scattering of the electrons changes their momenta, we expect the $d$-wave pairing across the patches to be sensitive to such disorder. However, within a patch, the $p$-wave pairing is topologically protected. To see this, note that our order parameter $<\psi_{k}\psi_{-k}>\sim\Delta_{0}\exp[i\theta_{k}]$ ( where $\psi$ denotes the spinless fermion in the relevant band and $\theta_{k}$ arises from the nontrivial Berry phases). Translated to the valley-spin picture, this shows that the superconducting order parameter in terms of those fermions has no momentum dependence, and hence, cannot be degraded by weak potential disorder. The $p$-wave superconductivity is also found to survive in the presence of magnetic impurities for a finite Hund’s splitting of interactions [@Kundu2017Competing].
Finally, we discuss the experimental implications of our work. Recently, there have been reports of surface superconductivity induced on the surface of Pb$_{0.6}$Sn$_{0.4}$Te by forming a mesoscopic point contact using a nonsuperconducting metal [@das2016unexpected]. The observed transition temperature is in the range 3.7-6.5 K. We expect transition temperatures roughly an order of magnitude smaller than the bandwidth $\Lambda$, which is of the order of the band gap. However, the nature of the Cooper pair order in the experiment is not yet settled and further experimental work needs to be done in this direction to confirm our prediction of surface $p$-wave superconductivity in this material. Recently, we have come across a paper [@Mazur2017Majorana] which reports the detection of an electron-hole gap with a broad zero-bias conductance maximum at the topological surfaces of diamagnetic, paramagnetic, and ferromagnetic Pb$_{1-y-x}$Sn$_{y}$Mn$_{x}$Te (where $y\gtrsim0.67$ and $0\leq x<0.1$) using soft-contact spectroscopy. The MBS-like conductance spectra obtained with and without magnetic impurities are found to be intrinsic in origin, which we believe supports our claim. Our approach could also be useful for studying phase competition in other two-dimensional systems with multiple Fermi patches in the presence of Hund’s splitting. In particular, this could be relevant for Type-II Dirac surface states on certain surfaces of antiperovskites[@PhysRevB.95.035151], or for the bulk band structure of the Dirac semimetal Na$_{3}$Bi with multiple Dirac nodes connecting via a Lifshitz point[@PhysRevB.92.075115], in a quasi-2D approximation.
The authors gratefully acknowledge useful discussions with Kedar Damle and Rajdeep Sensarma. SK acknowledges Debjyoti Burdhan for his help with some of the figures. VT acknowledges DST for a Swarnajayanti grant (No. DST/SJF/PSA-0212012-13).
**Supplementary material for Role of Hund’s splitting in electronic phase competition in Pb$_{1-x}$Sn$_{x}$Te:**
Here we provide additional information on 1) electron interactions in the valley-spin picture and effective interactions when projected to a band and 2) RG equations for test vertices corresponding to different kinds of pairing, and 3) Fixed point values of different couplings as a function of $d_{1}(y_{c})$
**Interactions between electrons in the valley-spin basis:**
Here we derive the effective interaction model obtained upon projecting the interactions in the valley-spin basis to one of the surface bands (the positive energy band closest to the saddle points) for each of the $\overline{X}$ points. The interaction Hamiltonian for surface electrons with valley and spin labels is given by **$$H_{I}=\frac{1}{2}\sum_{a,b,c,d,\sigma,\sigma^{\prime}}U_{abcd}^{\sigma\sigma^{\prime}}c_{\sigma a}^{\dagger}c_{\sigma^{\prime}b}^{\dagger}c_{\sigma^{\prime}c}c_{\sigma d}\label{eq:1}$$** where $a,b,c,d$ refer to different valleys (which are either all the same, same in pairs or all different in the above sum) and $\sigma,\sigma^{\prime}$ refer to spins. Here, we consider $U_{abcd}^{\sigma\sigma^{\prime}}=U_{1}^{\sigma\sigma^{\prime}}$when $(a,c)$ belong to one $\overline{X}$-point (i.e. the L-valleys corresponding to $(a,c)$ are projected to one of the $\overline{X}$-points) and $(b,d)$ belong to the other $\overline{X}$-point. Similarly, $U_{abcd}^{\sigma\sigma^{\prime}}=U_{2}^{\sigma\sigma^{\prime}}$ when $(b,c)$ belong to one $\overline{X}$-point and $(a,d)$ belong to the other, $U_{3}^{\sigma\sigma^{\prime}}$when $(a,b)$ belong to one $\overline{X}$-point and $(c,d)$ to the other, and $U_{4}^{\sigma\sigma^{\prime}}$when $a$,$b$,$c$ and $d$ all correspond to L-points projected to the same $\overline{X}$-point. The interactions depend only on the relative orientations of the spins, for example, $U^{\sigma\sigma^{\prime}}$ can be written as $U^{\sigma\sigma}\delta_{\sigma\sigma^{\prime}}+U^{\sigma\overline{\sigma}}(1-\delta_{\sigma\sigma^{\prime}})$. For the k.p Hamiltonian $H_{\overline{X_{1}}}(k)$ and $H_{\overline{X_{2}}}(k)$ of the (001) surface, the operators corresponding to different bands can be rewritten in terms of the operators for different valley and spin combinations as follows $$\begin{aligned}
\psi_{1} & =A_{1}c_{\uparrow1}+B_{1}c_{\downarrow1}+C_{1}c_{\uparrow2}+D_{1}c_{\downarrow2},\nonumber \\
\psi_{2} & =A_{2}c_{\uparrow1}+B_{2}c_{\downarrow1}+C_{2}c_{\uparrow2}+D_{2}c_{\downarrow2},\nonumber \\
\psi_{3} & =A_{3}c_{\uparrow1}+B_{3}c_{\downarrow1}+C_{3}c_{\uparrow2}+D_{3}c_{\downarrow2},\nonumber \\
\psi_{4} & =A_{4}c_{\uparrow1}+B_{4}c_{\downarrow1}+C_{4}c_{\uparrow2}+D_{4}c_{\downarrow2},\nonumber \\
\psi_{5} & =A_{5}c_{\uparrow3}+B_{5}c_{\downarrow3}+C_{5}c_{\uparrow4}+D_{5}c_{\downarrow4},\nonumber \\
\psi_{6} & =A_{6}c_{\uparrow3}+B_{6}c_{\downarrow3}+C_{6}c_{\uparrow4}+D_{6}c_{\downarrow4},\nonumber \\
\psi_{7} & =A_{7}c_{\uparrow3}+B_{7}c_{\downarrow3}+C_{7}c_{\uparrow4}+D_{7}c_{\downarrow4},\nonumber \\
\psi_{8} & =A_{8}c_{\uparrow3}+B_{8}c_{\downarrow3}+C_{8}c_{\uparrow4}+D_{8}c_{\downarrow4},\label{eq:2}\end{aligned}$$ where $\{A_{i},B_{i},C_{i},D_{i},i=1$ to $8\}$ correspond to the complex conjugates of the nonzero components of the different normalized energy eigenvectors, and are functions of $k_{x}$ and $k_{y}$ in the two-dimensional momentum space. We denote the L-valleys projected to one of the $\overline{X}$-points by $1$ and $2$, and those projected to the other point by $3$ and $4$. Thus, the total number of bands is eight. Since the points $\overline{X}$ are decoupled from each other, four of these components for each eigenvector vanish, giving rise to the expression in Eq. \[eq:2\]. We can invert the above equations to write the $c_{\alpha a}'s$ in terms of $\psi_{i}'s$. Substituting all of these expressions into $H_{I}$ in Eq. \[eq:1\] above, and writing $c_{\alpha a}$ as $\sum_{i}u_{\alpha ai}\psi_{i}$, we have $$\begin{aligned}
H_{I} & =\frac{1}{2}(\sum_{a,b,c,d,\sigma,\sigma^{\prime}}\sum_{i,j,k,l}U_{abcd}^{\sigma\sigma^{\prime}}u_{\sigma ai}^{*}(k_{1}^{\prime})u_{\sigma^{\prime}bj}^{*}(k_{2}^{\prime})\nonumber \\
& \qquad\times u_{\sigma^{\prime}ck}(k_{1})u_{\sigma dl}(k_{2})\psi_{i}^{\dagger}\psi_{j}^{\dagger}\psi_{k}\psi_{l})\label{eq:3-1}\end{aligned}$$
where $k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}$ are constrained by momentum conservation, and $i$,$j$,$k$ and $l$ refer to the various bands, and $(a,b,c,d)$ are either all the same, same in pairs or all different in the above sum. Now, we are only interested in the two bands (for a given $\overline{X}$-point) which lie in the bulk band gap and are closer to the saddle points in energy. In particular, we shall concentrate on the positive energy bands lying closer to the saddle points for each of the $\overline{X}$ points, in which case we can drop all the terms from the above equations except those involving $\psi_{2}$ and $\psi_{6}$, the relevant bands in our case. We then have $c_{\uparrow1}=u_{\uparrow1}\psi_{2}$, $c_{\downarrow1}=u_{\downarrow1}\psi_{2}$, $c_{\uparrow2}=u_{\uparrow2}\psi_{2}$ and $c_{\downarrow2}=u_{\downarrow2}\psi_{2}$, and likewise for $\psi_{6}$ with the valleys $3$ and $4$, suppressing the contributions from the other bands. Considering only the contributions from the two lower positive energy bands (corresponding to the two $\overline{X}$ points) which are degenerate, the above Eq. \[eq:3-1\] can be rewritten as $$\begin{aligned}
H_{I} & =\sum_{i}\sum_{\sigma,\sigma^{\prime}}\frac{1}{2}h_{4}^{\sigma\sigma^{\prime}}\psi_{i}^{\dagger}\psi_{i}^{\dagger}\psi_{i}\psi_{i}\nonumber \\
& \qquad+\sum_{i\neq j}\sum_{\sigma,\sigma^{\prime}}\frac{1}{2}(h_{1}^{\sigma\sigma^{\prime}}\psi_{i}^{\dagger}\psi_{j}^{\dagger}\psi_{i}\psi_{j}\nonumber \\
& \qquad+h_{2}^{\sigma\sigma^{\prime}}\psi_{i}^{\dagger}\psi_{j}^{\dagger}\psi_{j}\psi_{i}+h_{3}^{\sigma\sigma^{\prime}}\psi_{i}^{\dagger}\psi_{i}^{\dagger}\psi_{j}\psi_{j})\label{eq:4}\end{aligned}$$ where the sum is over the two low-energy bands only, and $h_{1}^{\sigma\sigma^{\prime}}=\sum_{a,b,c,d}U_{1}^{\sigma\sigma^{\prime}}u_{\sigma ai}^{*}(k_{1}^{\prime})u_{\sigma^{\prime}bj}^{*}(k_{2}^{\prime})u_{\sigma^{\prime}ci}(k_{1})u_{\sigma dj}(k_{2})$ (where one of the low-energy bands denoted by $i$ has nonzero components for valleys $(a,c)$ and the other one denoted by $j$ for valleys $(b,d)$), and this gives us the corresponding coupling $h_{1}$ used in the low-energy theory in Eq.(3) of the main text, when scaled with respect to the number of such combinations of valleys. The rest of the couplings $h_{2}$, $h_{3}$ and $h_{4}$ can be similarly defined in terms of the interactions in the valley-spin picture and the form factors for the basis transformation. Thus, there are four kinds of allowed scattering terms between electrons belonging to the two bands under consideration. These correspond to exchange processes between electrons on the two different bands($h_{1}$), Coulomb interaction between electrons on different bands ($h_{2}$), pair hopping between the two bands ($h_{3}$) and scattering between different valleys within a band ($h_{4}$).
![\[fig:schematic\]A schematic representation of the (001) surface with two Dirac points and saddle points each at the $\overline{X_{1}}$and $\overline{X_{2}}$ points. The FFLO wave vector connecting the $\overline{X_{1}}$ point to the origin $\overline{\Gamma}$ is represented by $Q_{1}$. In our analysis, we obtain a phase factor of $\exp[i(\theta_{k}-\theta_{k^{\prime}})]$,with respect to the $\overline{X}$ points, associated with the effective interactions in the band picture, while the relative phases of the order parameter between the patches $\overline{X_{1}}$ and $\overline{X_{2}}$ is $\pi$ ($d$-wave)(with respect to the $\overline{\Gamma}$ point). The superconducting order parameter has the form $\Delta_{k}=\Delta_{0}\left(\protect\begin{array}{c}
1\protect\\
-1
\protect\end{array}\right)_{\overline{X}}\otimes\exp[i\theta_{k}]$. We have considered a situation where the Fermi surface of a patch encloses both the Van-Hove points. Electrons anywhere in the patch experience an enhanced density of states due to the proximity of one or more Van-Hove points. ](suppl_fig1){width="0.5\columnwidth"}
![\[fig:h4rg\]Diagrams for one-loop renormalization of the coupling $h_{2}$. The diagrams for $h_{1}$, $h_{3}$ and $h_{4}$ are similarly obtained.](suppl_fig2){width="0.8\columnwidth"}
![\[fig:Vertex-renormalization\]Test vertex renormalization corresponding to (a) particle-particle pairing on the patch, (b) particle-particle pairing between patches, (c) particle-hole pairing on the patch and (d) particle-hole pairing between patches, where $Q$ refers to the nesting vector between the patches $\overline{X_{1}}$and $\overline{X_{2}}$ in two dimensions. ](suppl_fig3){width="0.8\columnwidth"}
**Susceptibilities:**
The renormalization equations for the different kinds of ordering considered, in the particle-particle as well as particle-hole channel, are given as follows.
The renormalization of the test vertex corresponding to particle-hole pairing between the patches, in the $\ell=0$ channel is given by $$\frac{\partial}{\partial y}\left(\begin{array}{c}
\Delta_{12}\\
\Delta_{21}
\end{array}\right)=$$ $$-2d_{1}(y)\left(\begin{array}{cc}
h_{2}^{0}-h_{1}^{0}-h_{1}^{1} & -h_{3}^{1}\\
-h_{3}^{1} & h_{2}^{0}-h_{1}^{0}-h_{1}^{1}
\end{array}\right)\left(\begin{array}{c}
\Delta_{12}\\
\Delta_{21}
\end{array}\right)\label{eq:5}$$ and in the $\ell=1$ channel, by $$\frac{\partial}{\partial y}\left(\begin{array}{c}
\Delta_{12}\\
\Delta_{21}
\end{array}\right)=-2d_{1}(y)\left(\begin{array}{cc}
h_{2}^{1} & h_{3}^{1}\\
h_{3}^{1} & h_{2}^{1}
\end{array}\right)\left(\begin{array}{c}
\Delta_{12}\\
\Delta_{21}
\end{array}\right)\label{eq:6}$$ The renormalization of the test vertex corresponding to particle-particle pairing between the patches, in the $\ell=0$ channel, is given by $$\frac{\partial}{\partial y}\left(\begin{array}{c}
\Delta_{12}\\
\Delta_{21}
\end{array}\right)=2d_{3}(y)\left(\begin{array}{cc}
h_{2}^{0} & h_{1}^{0}\\
h_{1}^{0} & h_{2}^{0}
\end{array}\right)\left(\begin{array}{c}
\Delta_{12}\\
\Delta_{21}
\end{array}\right)\label{eq:7}$$ and in the $\ell=1$ channel, by $$\frac{\partial}{\partial y}\left(\begin{array}{c}
\Delta_{12}\\
\Delta_{21}
\end{array}\right)=2d_{3}(y)\left(\begin{array}{cc}
h_{2}^{1} & h_{1}^{1}\\
h_{1}^{1} & h_{2}^{1}
\end{array}\right)\left(\begin{array}{c}
\Delta_{12}\\
\Delta_{21}
\end{array}\right)\label{eq:8}$$ The renormalization of the test vertex corresponding to particle-hole pairing on a patch, in the $\ell=0$ channel, is given by $$\frac{\partial}{\partial y}\left(\begin{array}{c}
\Delta_{1}\\
\Delta_{2}
\end{array}\right)=$$ $$-2d_{2}(y)\left(\begin{array}{cc}
-h_{4}^{1} & h_{1}^{0}-h_{2}^{0}-h_{2}^{1}\\
h_{1}^{0}-h_{2}^{0}-h_{2}^{1} & -h_{4}^{1}
\end{array}\right)\left(\begin{array}{c}
\Delta_{1}\\
\Delta_{2}
\end{array}\right)\label{eq:9}$$ and in the $\ell=1$ channel, is given by $$\frac{\partial}{\partial y}\left(\begin{array}{c}
\Delta_{1}\\
\Delta_{2}
\end{array}\right)=-2d_{2}(y)\left(\begin{array}{cc}
h_{4}^{1} & h_{1}^{1}\\
h_{1}^{1} & h_{4}^{1}
\end{array}\right)\left(\begin{array}{c}
\Delta_{1}\\
\Delta_{2}
\end{array}\right)\label{eq:10}$$
The diagrams corresponding to the renormalization of the different kinds of pairing vertices are shown in Fig. \[fig:Vertex-renormalization\]. The most negative eigenvalue for Cooper pairing on the patch is given by $2(-h_{3}^{1}+h_{4}^{1})$ which corresponds to the eigenvector $\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
-1 & 1\end{array}\right)$, competing with those for CDW and SDW order, given by $-2(h_{3}^{1}-h_{1}^{0}-h_{1}^{1}+h_{2}^{0})d_{1}(y)$ (corresponding to the eigenvector $\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
-1 & 1\end{array}\right)$) and $-2(h_{3}^{1}+h_{2}^{1})d_{1}(y)$ (corresponding to the eigenvector $\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
1 & 1\end{array}\right)$) respectively. This is followed by particle-hole pairing on a patch in the $\ell$=0 channel, with the more negative eigenvalue given by $-2(-h_{4}^{1}-(h_{1}^{0}-h_{2}^{0}-h_{2}^{1}))d_{2}(y)$ (corresponding to the eigenvector $\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
-1 & 1\end{array}\right)$ ). Thus, the dominant instability of our system, namely $p$-wave superconductivity, appears in the $\ell=1$ channel.
![\[fig:gi0l=00003D1\]The fixed point values for $g_{r}^{0}$ as a function of $d_{1}(y_{c})$ for the case where the $\ell$=0 components of all the couplings dominate initially. Note that the fixed point values $g_{2}^{0}$ and $g_{1}^{0}$ turn out to be identical. ](suppl_fig4){width="0.45\columnwidth"}
(a)Flow of the couplings with the RG scale $y$, with an initial splitting in the different scattering channels $r$. Here we have chosen the initial value of $h_{1}$ to be greater than all the other $h_{r}$ by $10\%$, i.e. $\frac{|h_{1}^{\ell}-h_{r}^{\ell}|}{|h_{r}^{\ell}|}=0.1$($r\protect\neq1$) for $\ell=0,1$, where $(h_{r}^{\ell})_{initial}=0.1$ for $r\protect\neq1$. The resulting order of the couplings at the fixed-point $y_{c}$ is identical to the case where all the couplings are chosen to be degenerate initially (see Fig.2 in main text). This illustrates that our RG flows are insensitive to the initial order of the couplings in different scattering channels $r=1-4$ , as long as $h_{r}^{0}=h_{r}^{1}$ for all $r$. Here, the critical point $y_{c}\approx3.8$.\
(b)Flow of the couplings with the RG scale $y$, with $h_{1}^{0}>h_{1}^{1}$ by $10\%$ initially, i.e. $\frac{|h_{1}^{0}-h_{1}^{1}|}{|h_{1}^{1}|}=0.1$, where $(h_{r}^{0})_{initial}=0.1$ for $r\protect\neq1$ and $(h_{r}^{1})_{initial}=0.1$ for all $r$. This changes the order of the couplings at the fixed point drastically, and the couplings $h_{3}^{0}$ and $(-h_{4}^{0})$ now dominate near the fixed point of RG flow. Here, the critical point $y_{c}\approx3.5$. ](suppl_fig5a "fig:"){width="0.45\columnwidth"}
(b)Flow of the couplings with the RG scale $y$, with an initial splitting in the different scattering channels $r$. Here we have chosen the initial value of $h_{1}$ to be greater than all the other $h_{r}$ by $10\%$, i.e. $\frac{|h_{1}^{\ell}-h_{r}^{\ell}|}{|h_{r}^{\ell}|}=0.1$($r\protect\neq1$) for $\ell=0,1$, where $(h_{r}^{\ell})_{initial}=0.1$ for $r\protect\neq1$. The resulting order of the couplings at the fixed-point $y_{c}$ is identical to the case where all the couplings are chosen to be degenerate initially (see Fig.2 in main text). This illustrates that our RG flows are insensitive to the initial order of the couplings in different scattering channels $r=1-4$ , as long as $h_{r}^{0}=h_{r}^{1}$ for all $r$. Here, the critical point $y_{c}\approx3.8$.\
(b)Flow of the couplings with the RG scale $y$, with $h_{1}^{0}>h_{1}^{1}$ by $10\%$ initially, i.e. $\frac{|h_{1}^{0}-h_{1}^{1}|}{|h_{1}^{1}|}=0.1$, where $(h_{r}^{0})_{initial}=0.1$ for $r\protect\neq1$ and $(h_{r}^{1})_{initial}=0.1$ for all $r$. This changes the order of the couplings at the fixed point drastically, and the couplings $h_{3}^{0}$ and $(-h_{4}^{0})$ now dominate near the fixed point of RG flow. Here, the critical point $y_{c}\approx3.5$. ](suppl_fig5b "fig:"){width="0.45\columnwidth"}
**Fixed point values of couplings as a function of $d_{1}(y_{c})$:**
As discussed in the main text, the different couplings $h_{r}^{\ell}(y)$ have an asymptotic form $\frac{g_{r}^{\ell}}{y_{c}-y}$ near the critical point $y_{c}$ of the RG flow. In order to determine the behavior of the fixed point values $g_{r}^{\ell}$ for the different couplings as a function of $d_{1}(y_{c})$, we substitute this asymptotic form into the RG equations (Eq.4-11 of the main text) to obtain the polynomial equations
$$\begin{aligned}
g_{1}^{0} & =2d_{1}(y_{c})(-(g_{1}^{0})^{2}-(g_{3}^{1})^{2}-(g_{1}^{1})^{2}\nonumber \\
& +2g_{1}^{0}g_{2}^{0}+(g_{3}^{0})^{2}),\nonumber \\
g_{1}^{1} & =2d_{1}(y_{c})(-2g_{1}^{0}g_{1}^{1}+2g_{1}^{1}g_{2}^{0}),\nonumber \\
g_{2}^{0} & =2d_{1}(y_{c})((g_{2}^{0})^{2}+(g_{3}^{0})^{2}),\nonumber \\
g_{2}^{1} & =2d_{1}(y_{c})((g_{2}^{1})^{2}+(g_{3}^{1})^{2}),\nonumber \\
g_{3}^{0} & =-4g_{4}^{0}g_{3}^{0}+2d_{1}(y_{c})(4g_{2}^{0}g_{3}^{0}-2g_{1}^{1}g_{3}^{1}),\nonumber \\
g_{3}^{1} & =-4g_{4}^{1}g_{3}^{1}+2d_{1}(y_{c})(2g_{2}^{1}g_{3}^{1}\nonumber \\
& -2g_{1}^{0}g_{3}^{1}+2g_{2}^{0}g_{3}^{1}),\nonumber \\
g_{4}^{0} & =-2(g_{4}^{0})-2(g_{3}^{0})^{2},\nonumber \\
g_{4}^{1} & =-2(g_{4}^{1})^{2}-2(g_{3}^{1})^{2}.\label{eq:11}\end{aligned}$$
These coupled equations are then solved with appropriate initial conditions, to determine $g_{r}^{\ell}$($\ell=0,1$) as a function of $d_{1}(y_{c})$, which is the ratio of the particle-hole and particle-particle susceptibilities at the fixed point $y_{c}$. The behaviour of $g_{r}^{\ell}$ as a function of $d_{1}(y_{c})$ when all the couplings are chosen to be degenerate initially, is shown in the inset in Fig.2 of the main text. The corresponding behavior when the degeneracy between the couplings in the $\ell=0$ and $\ell=1$ channels is lifted (such that $g_{r}^{0}>g_{r}^{1}$ for all $r$) is shown in Fig. \[fig:gi0l=00003D1\] (here we have only shown the behavior of the couplings $g_{r}^{0}$, as the fixed-point values $g_{r}^{1}$ turn out to be very small in this case).
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abstract: 'Susceptibility of deep neural networks to adversarial attacks poses a major theoretical and practical challenge. All efforts to harden classifiers against such attacks have seen limited success till now. Two distinct categories of samples against which deep neural networks are vulnerable, “adversarial samples" and “fooling samples", have been tackled separately so far due to the difficulty posed when considered together. In this work, we show how one can defend against them both under a unified framework. Our model has the form of a variational autoencoder with a Gaussian mixture prior on the latent variable, such that each mixture component corresponds to a single class. We show how selective classification can be performed using this model, thereby causing the adversarial objective to entail a conflict. The proposed method leads to the rejection of adversarial samples instead of misclassification, while maintaining high precision and recall on test data. It also inherently provides a way of learning a selective classifier in a semi-supervised scenario, which can similarly resist adversarial attacks. We further show how one can reclassify the detected adversarial samples by iterative optimization.[^1]'
author:
- |
Partha Ghosh[^2]\
Max Planck Institute of Intelligent Systems\
Tübingen, Germany\
Arpan Losalka$^*$\
IBM Research AI\
New Delhi, India\
Michael J Black\
Max Planck Institute of Intelligent Systems\
Tübingen, Germany\
bibliography:
- 'ResistingAdvAtack.bib'
title: Resisting Adversarial Attacks Using Gaussian Mixture Variational Autoencoders
---
Introduction {#intro}
============
The vulnerability of deep neural networks to adversarial attacks has generated a lot of interest and concern in the past few years. The fact that these networks can be easily fooled by adding specially crafted noise to the input, such that the original and modified inputs are indistinguishable to humans , clearly suggests that they fail to mimic the human learning process. Even though these networks achieve state-of-the-art performance, often surpassing human level performance on the test data used for different tasks, their vulnerability is a cause of concern when deploying them in real life applications, especially in domains such as health care , autonomous vehicles and defense, etc.
Adversarial Attacks and Defenses
--------------------------------
Adversarially crafted samples can be classified into two broad categories, namely (i) adversarial samples and (ii) fooling samples as defined by . Existence of adversarial samples was first shown by Szegedy et al. , while fooling samples , which are closely related to the idea of “rubbish class” images were introduced by Nguyen et al. . Evolutionary algorithms were applied to inputs drawn from a uniform distribution, using the predicted probability corresponding to the targeted class as the fitness function to craft such fooling samples. It has also been shown that Gaussian noise can be directly used to trick classifiers into predicting one of the output classes with very high probability .\
Adversarial attack methods can be classified into (i) white box attacks , which use knowledge of the machine learning model (such as model architecture, loss function used during training, etc.) for crafting adversarial samples, and (ii) black box attacks , which only require the model for obtaining labels corresponding to input samples. Both these kinds of attacks can be further split into two sub categories, (i) targeted attacks, which trick the model into producing a chosen output, and (ii) non-targeted attacks, which cause the model to produce any undesired output . The majority of attacks and defenses have dealt with adversarial samples so far , while a relatively smaller literature deals with fooling samples . However, to the best of our knowledge, no prior method tries to defend against both kinds of samples simultaneously under a unified framework. State-of-the-art defense mechanisms have tried to harden a classifier by one or more of the following techniques: adversarial retraining , preprocessing inputs , deploying auxiliary detection networks or obfuscating gradients . One common drawback of these defense mechanisms is that they do not eliminate the vulnerability of deep networks altogether, but only try to defend against previously proposed attack methods. Hence, they have been easily broken by stronger attacks, which are specifically designed to overcome their defense strategies .
Szegedy et al. argue that the primary reason for the existence of adversarial samples is the presence of small “pockets” in the data manifold, which are rarely sampled in the training or test set. On the other hand, Goodfellow et al. have proposed the “linearity hypothesis” to explain the presence of adversarial samples. Under our approach as detailed in Sec. \[res\_adv\_smpls\], the adversarial objective poses a fundamental conflict of interest, and inherently addresses both these possible explanations.
Approach
--------
We design a generative model that finds a latent random variable $\mathbf{z}$ such that data label $\mathbf{y}$ and the data $\mathbf{x}$ become conditionally independent given $\mathbf{z}$, i.e., $P(\mathbf{x}, \mathbf{y} | \mathbf{z}) = P(\mathbf{x}|\mathbf{z})P(\mathbf{y}|\mathbf{z})$. We base our generative model on VAEs , and obtain an inference model that represents $P(\mathbf{z}|\mathbf{x})$ and a generative model that represents $P(\mathbf{x}|\mathbf{z})$. We perform label inference $P(\mathbf{y}|\mathbf{x})$ by computing $\arg\max_{\mathbf{y}} P(\mathbf{y}|\mathbf{z}= \mathbb{E}_{P(\mathbf{z}|\mathbf{x})}[\mathbf{z}])$. We choose the latent space distribution $P(\mathbf{z})$ to be a mixture of Gaussians, such that each mixture component represents one of the classes in the data. Under this construct, inferring the label given latent encoding, i.e., $P(\mathbf{y}|\mathbf{z})$ becomes trivial by computing the contribution of the mixture components. Adversarial samples are dealt with by thresholding in the latent and output spaces of the generative model and rejecting the inputs for which $P(\mathbf{x}) \approx 0$. In Figure \[model\_pipeline\], we describe our network at test and train time.
Our contributions can be summarized as follows.
- We show how VAE’s can be trained with labeled data, using a Gaussian mixture prior on the latent variable in order to perform classification.
- We perform selective classification using this framework, thereby rejecting adversarial and fooling samples.
- We propose a method to learn a classifier in a semi-supervised scenario using the same framework, and show that this classifier is also resistant against adversarial attacks.
- We also show how the detected adversarial samples can be reclassified into the correct class by iterative optimization.
- We verify our claims through experimentation on 3 publicly available datasets: MNIST , SVHN and COIL-100 .
Related Work
============
A few pieces of work in the existing literature on defense against adversarial attacks have attempted to use generative models in different ways.
Samangouei et al. propose training a Generative Adversarial Network (GAN) on the training data of a classifier, and use this network to project every test sample on to the data manifold by iterative optimization. This method does not try to detect adversarial samples, and does not tackle “fooling images”. Further, this defense technique has been recently shown to be ineffective . Other pieces of work have also shown that adversarial samples can lie on the output manifold of generative models trained on the training data for a classifier .
PixelDefend, proposed by Song et al. also uses a generative model to detect adversarial samples, and then rectifies the classifier output by projecting the adversarial input back to the data manifold. However, Athalye et al. have shown that this method can also be broken by bypassing the exploding/vanishing gradient problem introduced by the defense mechanism.
MagNet uses autoencoders to detect adversarial inputs, and is similar to our detection mechanism in the way reconstruction threshold is used for detecting adversarial inputs. This defense method does not claim security in the white box setting. Further, the technique has also been broken in the grey box setting by recently proposed attack methods .
Traditional autoencoders do not constrain the latent representation to have a specific distribution like variational autoencoders. Our use of variational autoencoders allows us to defend against adversarial and fooling inputs simultaneously, by using thresholds in the latent and output spaces of the model in conjunction. This makes the method secure to white box attacks as well, which is not the case with MagNet.
Further, even state of the art defense mechanisms and certified defenses have been shown to be ineffective for simple datasets such as MNIST . We show via extensive experimentation on different datasets how our method is able to defend against strong adversarial attacks, as well as end to end white box attacks.
Method
======
Variational Autoencoders
------------------------
We consider the dataset $\mathbf{X}=\{\mathbf{x}^{(i)}\}_{i=1}^{N}$ consisting of $N$ i.i.d. samples of a random variable $\mathbf{x}$ in the space $\mathcal{X}$. Let $\mathbf{z}$ be the latent representation from which the data is assumed to have been generated. Similar to Kingma et al. , we assume that the data generation process consists of two steps: (i) a value $\mathbf{z}^{(i)}$ is sampled from a prior distribution $P_{\theta^{*}}(\mathbf{z})$; (ii) a value $\mathbf{x}^{(i)}$ is generated from a conditional distribution $P_{\theta^{*}}(\mathbf{x|z})$. We also assume that the prior $P_{\theta^{*}}(\mathbf{z})$ and likelihood $P_{\theta^{*}}(\mathbf{x|z})$ come from parametric families of distributions $P_{\theta}(\mathbf{z})$ and $P_{\theta}(\mathbf{x|z})$ respectively. In order to maximize the data likelihood $P_{\theta}(\mathbf{x}) = \int P_{\theta}(\mathbf{z})P_{\theta}(\mathbf{x}|\mathbf{z})d\mathbf{z}$, VAEs use an encoder network $Q_\phi(\mathbf{z}|\mathbf{x})$, that approximates $P_\theta(\mathbf{z}|\mathbf{x})$. The evidence lower bound (ELBO) for VAE is given by $$\begin{split}
ELBO(\mathbf{x},\theta,\phi) = \mathbb{E}_{\mathbf{z}\sim Q_{\phi}(\mathbf{z}|\mathbf{x})}[\log P_\theta(\mathbf{x}|\mathbf{z})]\\
- D_{\mathit{KL}}[Q_{\phi}(\mathbf{z}|\mathbf{x}) || P(\mathbf{z})]
\end{split}
\label{vae_loss}$$ where $D_{\mathit{\mathit{KL}}}$ represents the KL divergence measure. Using a Gaussian prior $P_{\theta}(\mathbf{z})$ and a Gaussian posterior $Q_{\phi}(\mathbf{z}|\mathbf{x})$, variational autoencoders maximize this lower bound deriving a closed form expression for the KL divergence term.
Modifying the Evidence Lower Bound
----------------------------------
VAEs do not enforce any lower or upper bound on encoder entropy $H(Q_\phi(\mathbf{z}|\mathbf{x}))$. This can result in blurry reconstruction due to sample averaging in case of overlap in the latent space. On the other hand, unbounded decrease in $H(Q_\phi(\mathbf{z}|\mathbf{x}))$ is not desirable either, as in that case the VAE can degenerate to a deterministic autoencoder leading to holes in the latent space. Hence, we seek an alternative design in which we fix this quantity to a constant value. In order to do so, we express the KL divergence in terms of entropy. $$\begin{split}
&D_{\mathit{KL}}[Q_{\phi}(\mathbf{z}|\mathbf{x}) \, \Vert \, P_{\theta}(\mathbf{z})] \\
&= -\mathbb{E}_{\mathbf{z} \sim Q_{\phi}(\mathbf{z} \vert \mathbf{x})}\left[\log \, P_{\theta}(\mathbf{z}) - \log \, Q_{\phi}(\mathbf{z} \vert \mathbf{x}) \right] \\
&= -\mathbb{E}_{\mathbf{z} \sim Q_{\phi}(\mathbf{z} \vert \mathbf{x})}\left[\log \, P_{\theta}(\mathbf{z}) \right] + \mathbb{E}_{\mathbf{z} \sim Q_{\phi}(\mathbf{z} \vert \mathbf{x})}\left[\log \, Q_{\phi}(\mathbf{z} \vert \mathbf{x}) \right] \\
&= H(Q_{\phi}(\mathbf{z} \vert \mathbf{x}), P_{\theta}(\mathbf{z})) - H(Q_{\phi}(\mathbf{z} \vert \mathbf{x}))
\end{split}
\label{kl_div_eq}$$ where $H(Q_{\phi}(\mathbf{z} \vert \mathbf{X}), P_{\theta}(\mathbf{z}))$ represents the cross entropy between $Q_{\phi}(\mathbf{z}|\mathbf{X})$ and $P_{\theta}(\mathbf{z})$. It can be noted that we need to minimize the KL divergence term. Hence, if we assume that $H(Q_{\phi}(\mathbf{z} \vert \mathbf{x}))$ is constant, then we can drop this term during optimization (please refer to the next section for details of how $H(Q_{\phi}(\mathbf{z} \vert \mathbf{x}))$ is enforced to be constant). This lets us replace the KL divergence $D_{\mathit{KL}}[Q_{\phi}(\mathbf{z}|\mathbf{X}) \, \Vert \, P_{\theta}(\mathbf{z})]$ in the loss function with $H(Q_{\phi}(\mathbf{z} \vert \mathbf{X}), P_{\theta}(\mathbf{z}))$. $$\begin{split}
&ELBO(\mathbf{x},\theta,\phi) \\
&= \mathbb{E}_{\mathbf{z}\sim Q_{\phi}(\mathbf{z}|\mathbf{x})}[\log P_\theta(\mathbf{x}|\mathbf{z})]
- H(Q_{\phi}(\mathbf{z} \vert \mathbf{x}), P_{\theta}(\mathbf{z})) \\
&= \mathbb{E}_{\mathbf{z}\sim Q_{\phi}(\mathbf{z}|\mathbf{x})}[\log P_\theta(\mathbf{x}|\mathbf{z})]
+ \mathbb{E}_{\mathbf{z} \sim Q_{\phi}(\mathbf{z} \vert \mathbf{x})}\left[\log \, P_{\theta}(\mathbf{z}) \right]
\end{split}
\label{kl_div_eqival}$$ The choice of fixing the entropy of $Q_{\phi}(\mathbf{z} \vert \mathbf{x})$ is further justified via experiments in section \[expts\].
{width="80.00000%"}
Supervision using a Gaussian Mixture Prior {#sup_gmm_prior}
------------------------------------------
In this section, we modify the above ELBO term for supervised learning by including the random variable $\mathbf{y}$ denoting labels. The following expression can be derived for the log-likelihood of the data. $$\begin{split}
&\log (P_{\theta}(\mathbf{x},\mathbf{y})) = \mathbb{E}_{\mathbf{z}\sim Q_{\phi}(\mathbf{z}|\mathbf{x})}
[\log(P_{\theta}(\mathbf{x},\mathbf{y}|\mathbf{z}))] \\ &- D_{\mathit{KL}}[Q_{\phi}(\mathbf{z}|\mathbf{x}) || P_{\theta}(\mathbf{z})] + D_{\mathit{KL}}[Q_{\phi}(\mathbf{z}|\mathbf{x}) || P_{\theta}(\mathbf{z}|\mathbf{x},\mathbf{y})]
\end{split}
\label{elbo_with_y_pre}$$ Noting that $D_{\mathit{KL}}[Q_{\phi}(\mathbf{z}|\mathbf{x}) || P_{\theta}(\mathbf{z}|\mathbf{x},\mathbf{y})]\geq0$, and replacing $D_{\mathit{KL}}[Q_{\phi}(\mathbf{z}|\mathbf{x}) || P_{\theta}(\mathbf{z})]$ with $\mathbb{E}_{\mathbf{z} \sim Q_{\phi}(\mathbf{z} \vert \mathbf{x})}\left[\log \, P_{\theta}(\mathbf{z}) \right]$ by assuming $H(Q_\phi(\mathbf{z}|\mathbf{x}))$ to be constant (as shown in Eqn. \[kl\_div\_eqival\]), we get the following lower bound on the data likelihood. $$\begin{split}
ELBO(\mathbf{x},\mathbf{y},\theta,\phi) = \mathbb{E}_{\mathbf{z}\sim Q_{\phi}(\mathbf{z}|\mathbf{x})} [\log(P_{\theta}(\mathbf{x},\mathbf{y}|\mathbf{z}))] \\
+ \mathbb{E}_{\mathbf{z} \sim Q_{\phi}(\mathbf{z} \vert \mathbf{x})}\left[\log \, P_{\theta}(\mathbf{z}) \right]
\end{split}
\label{elbo_with_y}$$ We choose our VAE to use a Gaussian mixture prior for the latent variable $\mathbf{z}$. We further choose the number of mixture components to be equal to the number of classes $k$ in the training data. The means of each of these components, $\boldsymbol{\mu}_{1},\boldsymbol{\mu}_{2},...,\boldsymbol{\mu}_{k}$ are assumed to be the one-hot encodings of the class labels in the latent space. It can be noted here that although this choice enforces the latent dimensionality to be $k$, it can be easily altered by choosing the means in a different manner. For example, means of all the mixture components can lie on a single axis in the latent space. Unlike usual VAEs, our encoder network outputs only the mean of $Q_\phi(\mathbf{z}|\mathbf{x})$. We use the reparameterization trick introduced by Kingma et al. , but sample the input $\boldsymbol{\epsilon}$ from $N(0,\Sigma_{constant})$ in order to enforce the entropy of $Q_\phi(\mathbf{z}|\mathbf{x})$ to be constant. Here, each mixture component corresponds to one class and $\mathbf{x}$ is assumed to be generated from the latent space according to $P_{\theta}(\mathbf{x}|\mathbf{z})$ irrespective of $\mathbf{y}$. Therefore, $\mathbf{x}$ and $\mathbf{y}$ become conditionally independent given $\mathbf{z}$, i.e. $\log(P_{\theta}(\mathbf{x},\mathbf{y}|\mathbf{z})) = \log(P_{\theta}(\mathbf{x}|\mathbf{z})) + \log(P_{\theta}(\mathbf{y}|\mathbf{z}))$. $$\begin{split}
&ELBO(\mathbf{x},\mathbf{y},\theta,\phi) \\ &= \mathbb{E}_{\mathbf{z}\sim Q_{\phi}(\mathbf{z}|\mathbf{x})} \left[\log(P_{\theta}(\mathbf{x}|\mathbf{z})) + \log(P_{\theta}(\mathbf{y}|\mathbf{z})) + \log \, P_{\theta}(\mathbf{z}) \right] \\
&= \mathbb{E}_{\mathbf{z}\sim Q_{\phi}(\mathbf{z}|\mathbf{x})} \left[\log(P_{\theta}(\mathbf{x}|\mathbf{z})) + \log(P_{\theta}(\mathbf{z}|\mathbf{y})) + \log \, P_{\theta}(\mathbf{y}) \right]
\end{split}
\label{elbo_with_y}$$ Assuming the the classes to be equally likely, the final loss function for an input $\mathbf{x}^{(i)}$ with label $\mathbf{y}^{(i)}$ becomes the following. $$\begin{split}
\mathcal{L}(\mathbf{x^{(i)}}, \mathbf{y}^{(i)}, \boldsymbol{\epsilon}) = ||\mathbf{x}^{(i)} &- g(f(\mathbf{x}^{(i)})+\epsilon) ||^2 \\
&+ \alpha|| f(\mathbf{x}^{(i)}) - \boldsymbol{\mu}_{\mathbf{y}^{(i)}} ||^2 \end{split}
\label{final_loss}$$ where the encoder is represented by $f$, the decoder is represented by $g$ and $\boldsymbol{\mu}_{\mathbf{y}^{(i)}}$ represents the mean of the mixture component corresponding to $\mathbf{y}^{(i)}$. $\alpha$ is a hyper-parameter that trades off between reconstruction fidelity, latent space prior and classification accuracy. The label $\mathbf{y}$ for an input sample $\mathbf{x}$ can be obtained following the Bayes Decision rule. $$\begin{split}
\arg\max_{\mathbf{y}}\,\, & P_{\theta}(\mathbf{y}|\mathbf{x}) = \arg\max_{\mathbf{y}} P_{\theta}(\mathbf{x,y})\\
&= \arg\max_{\mathbf{y}} \int_{\mathbf{z}} P_{\theta}(\mathbf{x,y}|\mathbf{z})P_{\theta}(\mathbf{z})d\mathbf{z} \\
&= \arg\max_{\mathbf{y}} \int_{\mathbf{z}} P_{\theta}(\mathbf{x}|\mathbf{z})P_{\theta}(\mathbf{y}|\mathbf{z})P_{\theta}(\mathbf{z})d\mathbf{z} \\
&= \arg\max_{\mathbf{y}} \int_{\mathbf{z}} P_{\theta}(\mathbf{z}|\mathbf{x})P_{\theta}(\mathbf{y}|\mathbf{z})P_{\theta}(\mathbf{x})d\mathbf{z} \\
&= \arg\max_{\mathbf{y}} \int_{\mathbf{z}} P_{\theta}(\mathbf{z}|\mathbf{x})P_{\theta}(\mathbf{y}|\mathbf{z})d\mathbf{z}
\end{split}
\label{classification_rule}$$ $P_{\theta}(\mathbf{z}|\mathbf{x})$ can be approximated by $Q_{\phi}(\mathbf{z}|\mathbf{x})$, i.e., the encoder distribution. This corresponds to the Bayes decision rule, in the scenario where there is no overlap among the classes in the input space, $\phi$ has enough variability and $Q_{\phi^*}(\mathbf{z}|\mathbf{x})$ is able to match $P_{\theta^*}(\mathbf{z}|\mathbf{x})$ exactly. Semi-supervised learning follows automatically, by using the loss function in Eqn. \[final\_loss\] for labeled samples, and the loss corresponding to Eqn. \[kl\_div\_eqival\] for unlabeled samples.
In order to compute the class label as defined in equation \[classification\_rule\], we use a single sample estimate of the integration by simply using the mean of $Q_\phi(\mathbf{z}|\mathbf{x})$ as the $\mathbf{z}$ value in our experiments. This choice does not affect the accuracy as long as the mixture components representing the classes are well separated in the latent space.
Resisting adversarial attacks {#res_adv_smpls}
-----------------------------
In order to successfully reject adversarial samples irrespective of the method of its generation, we use thresholding at the encoder and decoder outputs. This allows us to reject any sample $\mathbf{x}$ whose encoding $\mathbf{z}$ has low probability under $P_\theta(\mathbf{z})$, i.e., if the distance between its encoding and the encoding of the predicted class label in the latent space exceeds a threshold value, $\tau_{enc}$ (since $P_\theta(\mathbf{z})$ is a mixture of Gaussians). We further reject those input samples which have low probability under $P_\theta(\mathbf{x}|\mathbf{z})$, i.e., if the reconstruction error exceeds a certain threshold, $\tau_{dec}$ (since $P_\theta(\mathbf{x}|\mathbf{z})$ is Gaussian). Essentially, a combination of these two thresholds ensures that $P_\theta(\mathbf{x}) = \int_\mathbf{z}P_\theta(\mathbf{x}|\mathbf{z})P_\theta(\mathbf{z})d\mathbf{z}$ is not low.
Both $\tau_{enc}$ and $\tau_{dec}$ can be determined based on statistics obtained while training the model. In our experiments, we implement thresholding in the latent space as follows: we calculate the Mahalanobis distance between the encoding of the input and the encoding of the corresponding mixture component mean, and reject the sample if it exceeds the critical chi-square value ($3\sigma$ rule in the univariate case). Similarly, for $\tau_{dec}$, we use the corresponding value for the reconstructions errors. However, in general, any value can be assigned to these two thresholds, and they determine the risk to coverage trade-off for this selective classifier.
If the maximum allowed $L_{p}$ norm of the perturbation $\boldsymbol{\eta}$ is $\gamma$, then the adversary, trying to modify an input $\mathbf{x}$ from class $c_{1}$, must satisfy the following criteria.
1. $\operatorname*{arg\,min}_{c_i}|| f(\mathbf{x}+\boldsymbol{\eta}) - \boldsymbol{\mu}_{c_i} ||_{2} = c_{2}$ where $ c_{2}\neq c_{1}$
2. $ || f(\mathbf{x}+\boldsymbol{\eta}) - \boldsymbol{\mu}_{c_{2}} ||_{2} \leq \tau_{enc} $
3. $||\boldsymbol{\eta}||_{p} \leq \gamma$
4. $|| (\mathbf{x}+\boldsymbol{\eta}) - g(f(\mathbf{x}+\boldsymbol{\eta}) + \boldsymbol{\epsilon}) ||_{2} \leq \tau_{dec}$ where $\boldsymbol{\epsilon} \sim N(0, \Sigma_{constant})$
By the first three constraints, the encoding of $\mathbf{x}$ and $\mathbf{x}+\boldsymbol{\eta}$ must belong to different Gaussian mixture components in the latent space. However, constraint $4$ requires the distance between the reconstruction obtained from the encoding of $\mathbf{x}+\boldsymbol{\eta}$ to be close to $\mathbf{x}+\boldsymbol{\eta}$, i.e., close to $\mathbf{x}$ in the pixel space. This is extremely hard to satisfy because of the low probability of occurrence of holes in the latent space within $\tau_{enc}$ distance from the means.
Similarly, for the case of fooling samples, it can be argued that even if an attacker manages to generate a fooling sample which tricks the encoder, it will be very hard to simultaneously trick the decoder to reconstruct a similar image belonging to the rubbish class.
Reclassification
----------------
Once a sample is detected as adversarial by either or both the thresholds discussed above, we attempt to find its true label using the decoder only. By definition of adversarial images, $\mathbf{x}_{adv} = \mathbf{x}_{org} + \boldsymbol{\eta}$, where $\mathbf{x}_{adv}$ is the adversarial image corresponding to the original image $\mathbf{x}_{org}$, and $||\boldsymbol{\eta}||_{p}$ is small. Hence, we can conclude that for any given image $\mathbf{x}$, $||\mathbf{x} - \mathbf{x}_{adv}||_p \approx ||\mathbf{x} - \mathbf{x}_{org}||_p$. Suppose $\mathbf{z}^*$ is given by Eqn. \[finding\_z\]. $$\mathbf{z}^* = \arg\min_{\mathbf{z}}{||g(\mathbf{z}) - \mathbf{x}_{org}||_p}
\label{finding_z}$$ Following the argument stated above, we can approximate $\mathbf{z}^*\approx \mathbf{z}^*_{adv} = \operatorname*{arg\,min}_\mathbf{z}{||g(\mathbf{z}) - \mathbf{x}_{adv}||_p}$. We can now find the label of the adversarial sample as $\arg\min_{c_i}{||\boldsymbol{\mu}_{c_i} - \mathbf{z}^*_{adv}||_2}$. Essentially, for reclassification, we try to find the $\mathbf{z}$ in the latent space, which, when decoded, gives the minimum reconstruction error from the adversarial input. However, if Eqn. \[finding\_z\] returns a $\mathbf{z}$ that lies beyond $\tau_{enc}$ from the corresponding mean, or if the reconstruction error exceeds $\tau_{dec}$, we conclude that the sample is a fooling sample and reject the sample. It can be noted here that if this network is deployed in a scenario where fooling samples are not expected to be encountered, one can choose not to reject samples during reclassification, thereby increasing coverage. Also, starting from a single value of $\mathbf{z}$ can cause the optimization process to get stuck at a local minimum. A better alternative is to run $k$ different optimization processes with $\mathbf{z} = \boldsymbol{\mu}_{1}, \boldsymbol{\mu}_{2}, \dots, \boldsymbol{\mu}_{k}$ as the initial values, and choose the $\mathbf{z}$ which gives minimum reconstruction error as $\mathbf{z}^*_{adv}$. Given enough compute power is available, these $k$ processes can be run in parallel. In our experiments, we follow these two strategies while reclassifying adversarial samples.
Experiments {#expts}
===========
We verify the effectiveness of our network through numerical results and visual analysis on three different datasets - MNIST, SVHN and COIL-100. For different datasets, we make minimal changes to the hyper-parameters of our network, partly due to the difference in the image size and image type (grayscale/colored) in each dataset.
{width="0.75\linewidth"}
### Implementation details.
We use an encoder network with convolution, max-pooling and dense layers to parameterize $Q_\phi(\mathbf{z}|\mathbf{x})$, and a decoder network with convolution, up-sampling and dense layers to parameterize $P_\theta(\mathbf{x}|\mathbf{z})$. We choose the dimensionality of the latent space to be the same as the number of classes for MNIST and COIL-100. However, noting that the size of images is larger for SVHN compared to MNIST, and also, because the dataset contains colored images, we choose the dimensionality of the latent space for SVHN as $32$ instead of $10$. The choice of means also varies slightly for this dataset, as we pad zeros to the one-hot encodings of the class labels to allow for the extra latent dimensions. The standard deviation of the encoder distribution is chosen such that the chance of overlap of the mixture components in the latent space is negligible and the classes are well separated. We use $1/3000$ as the variance for the MNIST dataset, and reduce this value as the latent dimensionality increases for the other datasets. We use the ReLU nonlinearity in our network, and sigmoid activation in the final layer so that the output lies in the allowed range $[0,1]$. We use the Adam optimizer for training.
[cccc c c c c c c c]{} & &
------------------------------------------------------------------------
\
& & & &
------------------------------------------------------------------------
\
& & & &
------------------------------------------------------------------------
\
Dataset & SOTA & Accuracy & Accuracy & Error & Rejection & Accuracy & Accuracy & Error & Rejection
------------------------------------------------------------------------
\
MNIST & 99.79% & 99.67% & 97.97% & 0.22% & 1.81% & 99.1% & 98.17% & 0.52% & 1.31%\
SVHN & 98.31% & 95.06% & 92.80% & 4.58% & 2.62% & 86.42% & 83.54% & 13.64% & 2.82%\
COIL-100 & 99.11% & 99.89% & 98.40% & 0% & 1.60% & - & - & - & -\
### Qualitative evaluation.
Since our algorithm relies upon the reconstruction error between the generated and the original samples, we first show a few randomly chosen images generated by the network (for both supervised ad semi-supervised scenarios) corresponding to test samples of different classes from the three datasets in Figure \[generated\_coil\_data\].
### Numerical results.
In Table \[Comp\_normal\_classifier\], we present the accuracy, error and rejection percentages obtained by our method with and without thresholding. For semi-supervised learning, we have taken $100$ randomly chosen labeled samples from each class for both MNIST and SVHN during training. It is important to note here that the SOTA for COIL-100 was obtained on a random train-test split of the dataset, and hence, the accuracy values are not directly comparable.
### Adversarial attacks on encoder.
We use the encoder part of the network trained on the MNIST dataset to generate adversarial samples using the *Fast Gradient Sign Method (FGSM)* with varying $\epsilon$ values . The corresponding results are shown in Figure \[fgsm\_fig\]. The behavior is as desired, i.e., with increasing $\epsilon$, percentage of misclassified samples rises to a maximum value of only $3.89\%$ and then decreases, while the accuracy decreases monotonically and the rejection percentage increases monotonically. Similar results are obtained for the semi-supervised model, as shown in Figure \[fgsm\_fig\], although the maximum error rate is higher in this case. We further tried the FGSM attack from the Cleverhans library with the default parameters on the SVHN and COIL-100 datasets, and all the generated samples were rejected by the models after thresholding. Similarly, we generated adversarial samples for all three datasets using stronger attacks from Cleverhans with default parameter settings, including the Momentum Iterative Method and Projected Gradient Descent . In these cases as well, all generated adversarial samples were successfully rejected by thresholding.
This indicates that since all these attacks lack knowledge of the decoder network, they only manage to produce samples which fool the encoder network, but are easily detected at the decoder output. From this set of experiments, we conclude that the only effective method of attacking our model would be to design a complete white-box attack that has knowledge of the decoder loss as well, as well as the two thresholds. Further, since we do not use any form of gradient obfuscation in our defense mechanism, a complete white-box attacker would represent a strong adversary.
![We run FGSM with varying $\epsilon$ on the models trained on MNIST data in both supervised and semi-supervised scenarios. Although the error rate is higher for the semi-supervised network, the rejection ratio rises monotonically for both networks with increasing $\epsilon$, and the error rate for the supervised model stays below 5%.[]{data-label="fgsm_fig"}](fgsm_plot.pdf){width="45.00000%"}
### White-box adversarial attack. {#whitebox_attc_sec}
We present the results for completely white-box targeted attack on our model for the COIL-100 and MNIST datasets in figures \[fooling\_and\_adv\_samples\]a and \[fooling\_and\_adv\_samples\]b. Here, the adversary has complete knowledge of the encoder, the decoder, as well as the rejection thresholds. The results shown correspond to random samples from the first two classes of objects for the COIL-100 dataset, and the classes $2$ and $5$ for MNIST dataset. We perform gradient descent on the adversarial objective as given in Eqn. \[ad\_obj\]. The target class is set to $6$ for MNIST images from class $2$, $9$ for MNIST images from class $5$, and the class other than that of the source image for the COIL-100 images. $$\begin{split}
&\operatorname*{arg\,min}_{\boldsymbol{\eta}}{\mathcal{L}_{adv}} = \\
& \qquad \qquad \operatorname*{arg\,min}_\mathbf{\eta} [((||\mathbf{x_o} + \boldsymbol{\eta} - g(f(\mathbf{x_o} + \boldsymbol{\eta}))||^2)/\tau_{dec})^a \\
& + ((\mathbf{\mu_t} - f(\mathbf{x_o} + \mathbf{\eta}))\Sigma_t(\mathbf{\mu_t} - f(\mathbf{x_o} + \boldsymbol{\eta}))/\tau_{enc})^b + ||\boldsymbol{\eta}||^2]
\label{ad_obj}
\end{split}$$ where $\mathbf{x_o}$ is the original image we wish to corrupt, $\boldsymbol{\mu_t}$ is the mean of target class, $\boldsymbol{\eta}$ is the noise added, $f, g$ are the encoder and decoder respectively, and $\Sigma_t$ denotes target class covariance in latent space. $a>1$ and $b>1$ represent constant exponents which ensure that the adversarial loss grows steeply when the two threshold values are exceeded. Essentially, we aim for low reconstruction error and small change in the adversarial image while moving its embedding close to the target class mean. $\boldsymbol{\eta}$ is initialized with zeros.
We also ran the white box attack on $100$ randomly sampled images from each of the $10$ classes for MNIST and SVHN, by setting each of the $9$ other classes as the target class. The samples generated by optimizing the adversarial objective in each of these cases were either correctly classified or rejected.
-------------------------------------------------- ------------------------------------------------- ---------------------------------------------------
Adversarial Samples (MNIST) Adversarial Samples (COIL) Fooling Samples
{width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"}
(a) (b) (c)
-------------------------------------------------- ------------------------------------------------- ---------------------------------------------------
### Fooling images.
We take $100$ images sampled from the uniform distribution as inputs and optimize the white-box fooling attack objective given by Eqn. \[fooling\_obj\], with each of the classes from the MNIST and SVHN datasets as the target classes. In Figure \[fooling\_and\_adv\_samples\]c, we visualize some of the images to which the attack converged and their reconstructions for the MNIST dataset, with the target classes $1, 2, \dots, 6$. $$\begin{split}
\operatorname*{arg\,min}_{\boldsymbol{\eta}}&{\mathcal{L}_{fool}} = \operatorname*{arg\,min}_{\boldsymbol{\eta}}[(||\boldsymbol{\eta} - g(f(\boldsymbol{\eta}))||^2/\tau_{dec})^a \\
&+ ((\boldsymbol{\mu_t} - f(\boldsymbol{\eta}))\Sigma_t(\boldsymbol{\mu_t} - f(\boldsymbol{\eta}))/\tau_{enc})^b]
\label{fooling_obj}
\end{split}$$ Here, $\boldsymbol{\eta}, a, b, f$, $g$, $\Sigma_t$ and $\boldsymbol{\mu_t}$ are as described in sec. \[whitebox\_attc\_sec\].
It has been shown that fooling samples are extremely easy to generate for state-of-the-art classifier networks . Our technique, by design, gains resilience against such attacks as well. Since by definition, a fooling sample cannot look like a legitimate sample, it can not have small pixel space distance with any real image. This is exactly what can be noticed in the results in Figure \[fooling\_and\_adv\_samples\]c, where reconstruction errors are very high. Hence, most of the images to which this attack converges are rejected at the decoder, although they had managed to fool the encoder when considered in isolation. For the few cases where the images are not rejected, we observe that the attack method actually converged to a legitimate image of the target class.
### Reclassifying Adversarial samples.
In this section we present the performance of our reclassification technique. Although one could have used our decoder network to perform both “ordinary” and “adversarial” sample classification using Eqn. \[finding\_z\], but this process involves an iterative optimization. Hence, we only use it for the detected adversarial samples. The results are summarized in Table \[reclassification\_acc\].
------------ ------ ------ ------ ------ ------ -- -- -- --
$\epsilon$ 0.06 0.12 0.18 0.24 0.30
Accuracy 97% 93% 91% 87% 87%
------------ ------ ------ ------ ------ ------ -- -- -- --
: We present the reclassification accuracy for samples generated using FGSM on the MNIST dataset.[]{data-label="reclassification_acc"}
Following the same reclassification scheme, we also find that the method is able to correctly classify rejected test samples, thereby improving the overall accuracy achieved by the proposed method. For example, among the 181 samples rejected by the supervised model for the MNIST test dataset (as per Table \[Comp\_normal\_classifier\]), 110 samples are now correctly classified, improving the accuracy to 99.07%.
### Entropy of $Q_{\phi}(\mathbf{z} \vert \mathbf{x})$. {#const_entropy}
To compare the performance of the proposed network with the corresponding network with variable entropy of $Q_{\phi}(\mathbf{z} \vert \mathbf{x})$, we ran experiments by letting $H(Q_{\phi}(\mathbf{z} \vert \mathbf{x}))$ to be variable, and keeping all other parameters same. We tried the FGSM attack against the encoder of the model thus obtained, and observed that the adversarial sample detection capability of the network reduces drastically. This is justified by the fact that the reconstructions tend to be blurry in this case, thereby leading to a high reconstruction threshold. The results are shown in figure \[fgsm\_variable\].
![We run FGSM with varying $\epsilon$ on the model with variable encoder distribution entropy, trained on MNIST data. The rejection rate stays low in this case, while the error rate increases with increasing $\epsilon$.[]{data-label="fgsm_variable"}](vanilla_plot){width="30.00000%"}
In order to further study the difference between the two cases, we train both variants of the network on the CelebA dataset, and observe that the “Fréchet Inception Distance (FID) score is significantly better for the model with a constant $H(Q_{\phi}(\mathbf{z} \vert \mathbf{x}))$ (50.4) than the one with variable $H(Q_{\phi}(\mathbf{z} \vert \mathbf{x}))$ (58.3). The FID scores are obtained by randomly sampling $10,000$ points from the latent distribution, and comparing the distribution of the images generated from the these points with the training image distribution.
Discussion
==========
In this work, we have successfully demonstrated how a generative model can be used to gain defensive strength against adversarial attacks on images of relatively high resolution (128x128 for the COIL-100 dataset for example). However, the proposed network is limited by the generative capability of VAE based architectures, and thus, might not scale effectively to ImageNet scale datasets . In spite of this fact, keeping the underlying principles for adversarial sample detection and reclassification as described in this work, recent advances in invertible generative models such as Glow can be exploited to scale to more complex datasets. Further, as discussed earlier, the problem of defending against adversarial attacks still remains an unsolved problem even for datasets with more structured images. Hence our method can be used for practical applications such as secure medical image classification , biometrics identification, etc.
Human perception involves both discriminative and generative capabilities. Similarly, our work proposes a modification to VAEs to incorporate discriminative ability, besides using its generative ability to gain robustness against adversarial samples. The input space dimensionality (to the decoder) is drastically smaller compared to the input space dimensionality of image classifiers. Hence, it is much easier to attain dense coverage in the latent space, thereby minimizing the possibility of the occurrence of holes, leading to defensive capability against both adversarial and fooling images. With our construct, selective classification and semi-supervised learning become feasible under the same framework. A possible direction of future research would be to study how effectively the proposed approach can be scaled to more complex datasets by using recently proposed invertible generative modeling techniques.
Acknowledgement
===============
We are extremely grateful to Mr. Arnav Acharyya for his invaluable contribution to the discussions that helped shape this work.
[^1]: Accepted for publication in the Proceedings of the Thirty-Third AAAI Conference on Aritificial Intelligence (AAAI 2019)
[^2]: Equal contribution
|
[**On the Properties of Konishi-Kaneko Map**]{}
V.G. Gurzadyan, A. Melkonyan, and K.Oganessyan
Astronomy Centre, University of Sussex, UK and Department of Theoretical Physics, Yerevan Physics Institute, Yerevan 375036, Armenia.
An interesting possibility of study of the dynamics of $N$-body gravitating systems is provided by the Konishi-Kaneko iterated map \[\[KonK\]-\[KIn\]\]. Particular interest has the demonstrated by Inagaki \[\[Inag\]\] agreement of Konishi-Kaneko map with the thermodynamic considerations and the results of numerical simulations.
At least two points outline the importance of iterated maps:
1\. Iterated maps can enable one to avoid in certain way the principal difficulties associated with $N$-body systems – the non-compactness of the phase space and singularity of Newtonian interaction.
2\. Iterated maps can be rather informative in revealing the mechanisms of developing of chaos.
The Konishi-Kaneko map is defined as follows \[\[KonK\],\[KonKErg\]\]: $$\label{R1}
p^{n+1}_i=p^n_i + k\sum_{j=1}^{N}\sin 2\pi(x^n_j-x^n_i),$$ $$\label{R2}
x^{n+1}_i=x^n_i+p^{n+1}_i; (mod1) .$$ This system describes 1-dimensional $N$-body system with a potential of interaction which is free of singularity and is attractive if $k > 0$. It represents a symplectic 2D-map of interval $(0,1)$ and should have some common properties with maps known few decades ago, such as the Ulam map \[\[Ulam\]\] proposed to describe the Fermi mechanism of acceleration of cosmic rays, and the map of Zaslavsky and Sinakh \[\[Zasl\]\]. Among the recent interesting studies in this area we mention the papers by Kim (see e.g.\[\[Kim\]\]), where the symmetrically coupled two 1D systems are studied and the existence of Feigenbaum bifurcations is shown, with the value of Feigenbaum constant $\delta=8.72..$. Chaotic properties of the present non-symmentrical, i.e. Konishi-Kaneko system were observed in \[\[KonKErg\]\], where the Lyapunov numbers of the system were calculated for clustered and non-clustered states. Note, that at $N=2$ we have an integrable system.
The Jacobian of Konishi-Kaneko system is: $$\frac{\partial(y^{n+1}, x^{n+1})}{\partial(y^n, x^n)}=1.$$ The corresponding Hamiltonian system was studied in \[\[Inag\]\] with respect the thermodynamic instability.
Obviously not every map defined on $(0,1)$ interval can possess Feigenbaum bifurcations. Therefore first we have to check the necessary condition of existing of period-doubling bifurcations, i.e. the negativity of the Schwartzian derivative: $$Sf\equiv f'''/f''-3/2(f''/f')^2 < 0.$$ This condition is fulfilled for Eqs.(\[R1\]),(\[R2\]) since $f'''/f' < 0$ for any value of $k$.
To obtain the bifurcation scale ${\delta}$ we have to find out the values of period-doubling bifurcation points which must satisfy the conditions $$\sum_{j}^{N} |x_{j}^{n+1} - x_{j}^n| < \epsilon, (2^1=2);\\
\sum_{j}^{N} |x_{j}^{n+2} - x_j^n| < \epsilon, \sum_{j}^{N} |x_j^{n+3} -
x_j^{n+1}| < \epsilon, (2^2=4);\\$$ for each $k_n, n=1,2,...$, respectively. The ${\epsilon}$ is the accuracy of the obtained values of $k_n$. The accuracy of calculation of $k_1$ e.g. for $N=10$ was $\epsilon\approx
10^{-5}$, and $10^{-4}$ for $k_2$ and $k_3$. These calculations were enough to find out the Feigenbaum universal number $\delta=8.72...$. The results of calculations for $N=10$ are given in Table 1.
Table 1
$k_2-k_1 $ $k_3-k_2$ $\delta $
------------------- -------------------- ---------------- -- -- --
$0.000196\dots $ $0.00002222\dots $ $8.82\dots$
$0.000194\dots$ $0.00002222\dots$ $8.78\dots$
$0.00019368\dots$ $0.00002222\dots$ $8.71584\dots$
$0.00019368\dots$ $0.00002210\dots$ $8.76359\dots$
$0.00019368\dots$ $0.00002225\dots$ $8.70490\dots$
$0.00019368\dots$ $0.00002223\dots$ $8.71219\dots$
$0.00019368\dots$ $0.00002221\dots$ $8.71950\dots$
$0.00019368\dots$ $0.00002219\dots$ $8.72682\dots$
$0.00019368\dots$ $0.000022205\dots$ $8.72315\dots$
$0.00019368\dots$ $0.000022209\dots$ $8.71950\dots$
$0.00019368\dots$ $0.000022207\dots$ $8.72315\dots$
The results e.g. for $N = 3,5,7$ were absolutely identical with those for $N=10$, though with different accuracy $\epsilon$. We have noticed a clear decrease in the accuracy with the increase of the number of particles.
The estimation of the values of $k_n$ requires careful procedure of calculations because of the complicated character of the system and of the sensitivity on the iterations of $k_n$ and the accuracy $\epsilon$.
Using the obtained values of $k_n$ and the formula $$k_{\infty} = {(\delta k_{n+1} - k_n)} / {(\delta-1)},$$ we also estimate the $k_{\infty}$, from which the chaotic behavior of the system is established and the map never repeats itself: $$k_{\infty} = 0.1307\dots .$$ At $k>k_{\infty}$ the system should have positive Lyapunov numbers as shown in \[\[KonKErg\]\].
The period-doubling points correspond to the phase transitions of second order \[\[AO\]\], and can enable the study of such systems via the methods of thermodynamic formalism \[\[BT\]\].
[88]{}
\[KonK\] Konishi T., Kaneko K.: 1992, J.Phys. A, [**25**]{}, 6283.
\[KonKErg\] Konishi T., Kaneko K.: 1994, in: ‘[*Ergodic Concepts in Stellar Dynamics*]{}’, Eds: V.G.Gurzadyan, D.Pfenniger, Springer-Verlag, p.95.
\[Inag1\] Inagaki S.: 1993, Prog.Theor.Phys., [**90**]{}, 577.
\[Inag\] Inagaki S.: 1994, in: ‘[*Ergodic Concepts in Stellar Dynamics*]{}’, Eds: V.G.Gurzadyan, D.Pfenniger, Springer-Verlag, p.105.
\[KIn\] Konishi T., Inagaki S.: 1993, PASJ, [**45**]{}, 733.
\[Ulam\] Ulam S.M.: 1963, in: Proceed. IV Berkeley Symp. on Math.Stat. and Probability, Univ. California Press.
\[Zasl\] Zaslavsky G.M., Sinakh V.N.: 1970, Radiofizika, [**13**]{}, 604.
\[Kim\] Kim S-Y.: 1994, Phys. Rev. E. B [**50**]{}, 1922.
\[AO\] Ananikian N.S., Oganessyan K.A.: 1995, Phys. Lett., A, [**200**]{}, 205.
\[BT\] Bohr T., Tel T.: 1988, in [*Directions in Chaos*]{}, Ed.U.Bai-lin Hao, vol.II, p.195. World Sci.
|
---
abstract: 'In this work, we develop multivariate functional singular spectrum analysis (MFSSA) over different dimensional domains which is the functional extension of multivariate singular spectrum analysis (MSSA). In the following, we provide all of the necessary theoretical details supporting the work as well as the implementation strategy that contains the recipes needed for the algorithm. We provide a simulation study showcasing the better performance in reconstruction accuracy of a multivariate functional time series (MFTS) signal found using MFSSA as compared to other approaches and we give a real data study showing how MFSSA enriches analysis using intraday temperature curves and remote sensing images of vegetation. MFSSA is available for use through the [[Rfssa]{}]{} R package.'
author:
- |
Jordan Trinka\
Department of Mathematical and Statistical Sciences,\
Marquette University, USA\
and\
Hossein Haghbin\
Department of Statistics,\
Persian Gulf University, Iran\
and\
Mehdi Maadooliat\
Department of Mathematical and Statistical Sciences,\
Marquette University, USA
bibliography:
- 'Mybib.bib'
title: '**Multivariate Functional Singular Spectrum Analysis Over Different Dimensional Domains**'
---
=4
\#1
[*Keywords:*]{} Multivariate Singular Spectrum Analysis, Functional Time Series, Hilbert Space, Functional SVD, Remote Sensing Data
Introduction {#sec:int}
============
A common problem in time series analysis is detection, extraction, and exploration of mean, seasonal, trend, and noise components in time series data. A technique known as singular spectrum analysis (SSA) has been developed as a nonparametric, exploratory method which can be used to identify such interesting components in ordinary time series where observations are scalars [@golyandina2001]. Often times, many variables are observed as a result of a single stochastic process and investigation of time series components can be made richer by performing a multivariate analysis of these vector observations. The MSSA algorithm is a technique that has seen success over its univariate SSA counterpart in decomposing a multidimensional time series into components if the covariates are moderately correlated [@golyandina2012]. MSSA also has been broken up into two approaches of vertical MSSA (VMSSA) and horizontal MSSA (HMSSA) where VMSSA involves the vertical stacking of univariate Hankel trajectory matrices while HMSSA works with the horizontal stacking of the same elements [@hassani2018]. Over the course of the last 15 years, MSSA has seen significant success in various areas of application see [@groth2011; @golyandina2012; @silva2018; @hassani2019].
Functional data analysis embodies the evaluation and exploration of data that is comprised of functions such as curves or surfaces [@ramsay2005]. Functional PCA (FPCA) is a technique that is used to find the most informative directions in a time-independent collection of functional subjects [@ramsay2005]. Univariate Functional Singular Spectrum Analysis (FSSA) was developed by @haghbin2019 as a novel technique that is used to decompose a time-dependent collection of functional subjects, known as a functional time series (FTS), into mean, seasonal, trend, and noise components. FSSA works to decompose a FTS in a similar fashion as SSA using a functional singular value decomposition (fSVD). This method was compared with other techniques of dimension reduction of a FTS including dynamic functional principal component analysis (DFPCA) [@hormann2012] and it was found that FSSA is the ideal approach in terms of reconstruction accuracy.
Multivariate functional data are observed when a stochastic process gives rise to multiple different functions over possibly different dimensional domains. Multivariate FPCA (MFPCA) was developed so that more than one variable of functional subjects could be included in the analysis. @chiou2014 extended MFPCA to include a normalized approach which accounts for differences in degrees of variability in the covariates as well as differences in units. MFPCA was further extended by @happ2018 to account for different dimensional domains so that one could perform dimension reduction on multivariate functional data that might be comprised of curves, surfaces, or any other finite dimensional domain altogether. A primary assumption of MFPCA is that the functional data are independent of time. With the goal of performing dimension reduction on a MFTS, one might conjecture to use FSSA on the covariates independently of one another but this fails to capture any cross-correlations between variables. MFSSA provides us a way to perform dimension reduction of a MFTS while capturing these cross-correlations to further enrich analysis and strengthen reconstruction accuracy of the true signal. In addition, MFSSA is developed, in the following, to handle functions taken over any finite dimensional domain. This can allow the user to explore relationships between time dependent curves, images, or any other hyperplane.
The rest of the paper is organized to first introduce the reader to MSSA, we then discuss the functional extension of MSSA known as MFSSA and how one can generalize MSSA into MFSSA by developing both horizontal MFSSA (HMFSSA) and vertical MFSSA (VMFSSA). We also show that VMFSSA solves the same problem as MFSSA using a unitary operator. We finish the paper by discussing a simulation study illustrating when MFSSA outperforms all other known methods in terms of reconstruction accuracy and a real data study where we use weather station intraday temperature curves and remote sensing images in a bivariate analysis to explore some of the more interesting qualities of MFTS data through the use of MFSSA. In supplementary material, we provide further interesting plots and animations for our real data study, we provide another real data study that uses surface reflectance density curves, we develop HMFSSA fully, and we provide proofs of all lemmas and propositions. In addition to all of this work, the MFSSA algorithm has been implemented in the [[Rfssa]{}]{} package and we also include a shiny app that can be launched from within the package allowing the user to explore the work with already loaded data or their own data.
General Scheme of MSSA {#mssa}
======================
MSSA is a type of SSA developed to analyze multivariate time series. The algorithm is broken up into two different approaches known as VMSSA and HMSSA. The MSSA algorithm consists of the following four steps:
### MSSA I. Embedding {#emmssa .unnumbered}
Given $p$ univariate time series of length $N$, $\{y_i^{(j)}\}_{i=1,\dots,N}^{j=1,\dots,p}$, a multivariate time series can be considered as a series of length $N$ of $p$-tuples, $\vec{y}_{i}:=\left(y_i^{(1)},\dots,y_i^{(p)}\right) \in \mathbb{H}:=\mathbb{R}^{p}$, in the form of $\mathbf{y}_{N}:=(\vec{y}_{1},\dots,\vec{y}_{N})$. One may choose an integer $L$, where $L < \frac{N}{2}$, set $K=N-L+1$, and create the set of $L \times K$, univariate trajectory matrices, $\{\mathbf{X}^{\left(j\right)}\}_{j=1}^{p}$. These trajectory matrices have the form $$\label{unitrajmat}
\mathbf{X}^{\left(j\right)}:=\left[\mathbf{x}^{\left(j\right)}_{1},\dots,\mathbf{x}^{\left(j\right)}_{K}\right],$$ where $\mathbf{x}_{k}^{\left(j\right)}:=\left[y_{k}^{\left(j\right)},\dots,y_{k+L-1}^{\left(j\right)}\right]^{\top}$ is referred as $k^{th}$ lagged vector associated with variable $j$. In the HMSSA, we concatenate the univariate trajectory matrices horizontally to obtain an $L \times pK$ multivariate trajectory matrix $$\label{hmssaem}
\mathbf{X}:= \left[\mathbf{X}^{\left(1\right)}, \dots, \mathbf{X}^{\left(p\right)}\right],$$ where as in the VMSSA, we concatenate those univariate trajectory matrices vertically to obtain the associated $pL \times K$ multivariate trajectory matrix $$\label{vmssaem}
\mathbf{X}:= \begin{bmatrix}\mathbf{X}^{\left(1\right)}\\ \vdots\\ \mathbf{X}^{\left(p\right)}\end{bmatrix}.$$ A Hankel matrix is defined as that whose antidiagonal elements are equivalent. One may note that since each univariate trajectory matrix, $\mathbf{X}^{(j)}$, is Hankel, therefore the multivariate trajectory matrix, $\mathbf{X}$, is block Hankel.
As we shall see in Section \[vmfssa\], there would be an interchangeable relationship between the extension of VMSSA and MFSSA. Without loss of generality, in the remaining of this section we focus on the VMSSA. Therefore, we have that $\mathbf{X}:\mathbb{R}^{K} \rightarrow \mathbb{R}^{pL}$. Often times, this embedding step is viewed as applying an invertible transformation $\mathcal{T}:\mathbb{R}^{N} \rightarrow \mathbb{R}^{pL \times K}$ such that $$\label{Tmssa}
\mathbf{X}=\mathcal{T}(\mathbf{y}_{N}).$$
### MSSA II. Decomposition {#decmssa .unnumbered}
In the decomposition step we perform an SVD of the rank $r$ trajectory matrix, $\mathbf{X}$. The formulation for the SVD is given as $$\mathbf{X}=\sum_{i=1}^{r}\sigma_{i}\mathbf{u}_{i}\mathbf{v}_{i}^{\top}=\sum_{i=1}^{r}\mathbf{X}_{i}, \nonumber$$ where $\{\sigma_{i}\}_{i=1}^{r}$ are the singular values, $\{\mathbf{v}_{i}\}_{i=1}^{r}$ forms an orthonormal basis for the domain of $\mathbf{X}$, $\{\mathbf{u}_{i}\}_{i=1}^{r}$ forms an orthonormal basis for the range of $\mathbf{X}$, and the set of rank one matrices, $\{\mathbf{X}_{i}\}_{i=1}^{r}$, are known as elementary matrices.
### MSSA III. Grouping {#mssa-iii.-grouping .unnumbered}
For grouping, we partition the set of indices of $\{1,2,\dots,r\}$ into $m$ disjoint subsets $\{I_{1}, I_{2},\dots,I_{m} \}$ such that for any positive integer $q=1,\cdots,m$, the matrix $\mathbf{X}_{I_{q}}$ is defined as $\mathbf{X}_{I_{q}}:=\sum_{i \in I_{q}} \mathbf{X}_{i}$. This allows us to write the original trajectory matrix, $\mathbf{X}$, as $$\label{mssaexpansion}
\mathbf{X}=\mathbf{X}_{I_{1}}+\mathbf{X}_{I_{2}}+\cdots+\mathbf{X}_{I_{m}}.$$ The grouping should be done so that each $\mathbf{X}_{I_{q}}$ describes a different feature of the original time series such as trend or seasonality which can be achieved by looking at exploratory plots like paired-plots or scree plots [@golyandina2001; @hassani2018].
### MSSA IV. Reconstruction {#mssa-iv.-reconstruction .unnumbered}
For any $pL\times K$ block Hankel matrix, one may use $\mathcal{T}^{-1}$ to obtain the associated multivariate time series. Note that the matrices $\mathbf{X}_{I_{q}}$’s ($q=1,\cdots,m$), given in , are not necessary block Hankel, and therefore we can not use $\mathcal{T}^{-1}$ transformation. A popular remedy in the literature is to use orthogonal projection approach and approximate $\mathbf{X}_{I_{q}}$’s with appropriate block Hankel matrices.
The matrix $\mathbf{X}_{I_{q}}$ can be written in the block form: $$\label{mssablockapprox}
\mathbf{X}_{I_{q}}= \begin{bmatrix}\mathbf{X}_{I_{q}}^{\left(1\right)}\\ \vdots\\ \mathbf{X}_{I_{q}}^{\left(p\right)}\end{bmatrix},\nonumber$$ where $\mathbf{X}_{I_{q}}^{\left(j\right)}$ is an $L \times K$ matrix for $j=1,\dots,p$. The orthogonal projection of the $\mathbf{X}_{I_q}$ onto the space of the block Hankel matrices can be done by averaging the antidiagonal elements of each $\mathbf{X}_{I_{q}}^{\left(j\right)}$. We denote this approximated block Hankel matrix as $\widetilde{\mathbf{X}}_{I_{q}}$, and use the inverse transformation, $\mathcal{T}^{-1}$, to obtain $$\tilde{\mathbf{y}}_{N}^{q}:=\mathcal{T}^{-1}(\widetilde{\mathbf{X}}_{I_{q}}), \nonumber$$ and as such, we have $\mathbf{y}_{N}\approx\tilde{\mathbf{y}}_{N}^{1}+\dots+\tilde{\mathbf{y}}_{N}^{m}$.
Separability
------------
Let $\mathbf{y}_{N}$ and $\mathbf{z}_{N}$ be two multivariate time series of length $N$. The weighted-correlation (w-correlation) between $y_{N}$ and $z_{N}$ is defined as $$\label{w-corr}
\rho_{1,2}^{\left(w\right)}:= \frac{{\left\langle \mathbf{y}_{N}, \mathbf{z}_{N} \right \rangle}_{w}}{{\| \mathbf{y}_{N} \|}_{w}{\| \mathbf{z}_{N} \|}_{w}}, \nonumber$$ where ${\left\langle \mathbf{y}_{N}, \mathbf{z}_{N} \right \rangle}_{w} := \sum_{j=1}^{p}\sum_{i=1}^{N}w_{i}y_{i}^{(j)}z_{i}^{(j)}$, $w_{i}:=\text{min}\{i,L,N-i+1\}$, and ${\| \mathbf{y} \|}_{w}:=\sqrt{{\left\langle \mathbf{y}, \mathbf{y} \right \rangle}_{w}}$. Like in all types of SSA, a correlation close to zero is desired for reconstructed time series.
Parameter Selection
-------------------
The two parameters of SSA are the window length, $L$, and how one does the grouping. Since every type of SSA is a nonparametric, data-driven approach to analysis, differing choices of $L$ will give different results. A rule of thumb is that $L$ should be chosen to be a multiple of a periodicity that is present in the data but no greater than $N/2$ [@golyandina2001; @golyandina2013]. As stated earlier, it is ideal to perform the grouping such that there is no correlation between reconstructions.
Theoretical Foundations of MFSSA {#fssa-method}
================================
The mathematical foundations in the following subsection are used throughout the paper and form the theoretical backbone of the MFSSA algorithm.
Preliminaries and Notations
---------------------------
For each $j=1,\cdots,p$, consider an $m_j$-dimensional domain, ${{T}}_{j}$, to be a compact subset of $\mathbb{R}^{m_{j}}$, and let $\mathbb{F}_j:=L^{2}\left({{T}}_{j}\right)$ to be the Hilbert space of square integrable real functions defined on ${T}_{j}$. We define the Cartesian product space $\mathbb{H}:= \mathbb{F}_1 \times \cdots \times \mathbb{F}_{p}$, where each $\vec{x}\in \mathbb{H}$, can be denoted by the $p$-tuple $\left(x^{(1)},\dots,x^{(p)}\right)$. Note that $\mathbb{H}$ is a Hilbert space equipped with inner product $$\label{observations inner product}
{\left\langle \vec{x}, \vec{y} \right \rangle}_{\mathbb{H}}:=\sum_{j=1}^{p}{\left\langle x^{(j)}, y^{(j)} \right \rangle}_{\mathbb{F}_j}=\sum_{j=1}^{p}\int_{{{T}}_{j}}x^{\left(j\right)}\left(s_{j}\right)y^{\left(j\right)}\left(s_{j}\right)ds_{j}, \quad s_{j} \in {{T}}_{j}, \nonumber$$ for some $\vec{x},\vec{y} \in \mathbb{H}$. We specify a MFTS of length $N$ as $\mathbf{y}_{N}:=(\vec{y}_{1},\dots,\vec{y}_{N})$, where $\vec{y}_{i} \in \mathbb{H}$.
Similarly, for a given $L\in\mathbb{N}$, $\mathbb{H}^L$ stands for the Cartesian product of $L$ copies of $\mathbb{H}$, and each $\mathbf{x}\in\mathbb{H}^L$ can be denoted by the $L$-tuple $\left(\vec{x}_{1},...,\vec{x}_{L}\right)$. Clearly $\mathbb{H}^L$ is a Hilbert space with respect to the inner product $${\left\langle \mathbf{x}, \mathbf{y} \right \rangle}_{\mathbb{H}^{L}}:=\sum_{i=1}^{L} {\left\langle \vec{x}_{i}, \vec{y}_{i} \right \rangle}_{\mathbb{H}},\quad \mathrm{for}\ \mathbf{x},\mathbf{y} \in \mathbb{H}^{L}.$$
Next we define $\mathbb{H}^{L \times K}$ to be the space spanned by linear operators $\mathbfcal{V}:\mathbb{R}^{K} \rightarrow \mathbb{H}^{L}$, specified by $\left[\vec{v}_{i,k}\right]_{i=1,\dots,L}^{k=1,\dots,K}$, as $$\label{rangeofy}
\mathbfcal{V}(\pmb{a}):=\left(\sum_{k=1}^{K}a_{k}\vec{v}_{1,k}, \dots,\sum_{k=1}^{K}a_{k}\vec{v}_{L,k}\right),\qquad \pmb{a}:=(a_1, a_2, \dots, a_K) \in \mathbb{R}^{K}, \nonumber$$ where $\vec{v}_{i,k}\in\mathbb{H}$. Now for two operators $\mathbfcal{V}$, $\mathbfcal{Z} \in \mathbb{H}^{L \times K}$, the Frobenius inner product can be defined as $$\label{Frobenius Norm}
{\left\langle \mathbfcal{V}, \mathbfcal{Z} \right \rangle}_{F}:=\sum_{i=1}^{L}\sum_{k=1}^{K}{\left\langle \vec{v}_{i,k}, \vec{z}_{i,k} \right \rangle}_{\mathbb{H}},\nonumber$$ which induces the Frobenius norm given by ${\| \mathbfcal{V} \|}_{F}:=\sqrt{{\left\langle \mathbfcal{V}, \mathbfcal{V} \right \rangle}_{F}}$. We denote by $\mathbb{H}_{H}^{L \times K}$ the Hankel subspace of $\mathbb{H}^{L \times K}$ such that for any $\widetilde{\mathbfcal{V}} = \left[\vec{\tilde{v}}_{i,k}\right] \in \mathbb{H}_{H}^{L \times K}$ there exists a $\vec{g}_{u} \in \mathbb{H}$ such that ${\| \vec{\tilde{v}}_{i,k}-\vec{g}_{u} \|}_\mathbb{H}=0$ where $u=i+k$.
MFSSA Algorithm {#mfssa.algorithm}
---------------
Similar to other SSA algorithms, MFSSA consists of four steps: Embedding, Decomposition, Grouping, and Reconstruction.
### MFSSA I. Embedding {#mfssa-i.-embedding .unnumbered}
As one may note the columns of a univariate trajectory matrix, as given in , are the corresponding lagged vectors. Therefore a trajectory matrix can be seen as a linear operator from $\mathbb{R}^K$ to the space of linear combinations of the lagged vectors. @haghbin2019 used this as a motivation to introduce the trajectory operator for FSSA.
In a similar fashion, we define multivariate functional lagged vectors in $\mathbb{H}^{L}$ of the form $$\label{mlagvec}
\pmb{x}_{k}:=\left(\vec{y}_{k}, \vec{y}_{k+1},\dots,\vec{y}_{k+L-1}\right), \quad k=1,\dots,K.$$ One may define a linear operator, specified with $\pmb{x}_k$’s, to obtain the trajectory operator, $\mathbfcal{X}:\mathbb{R}^{K} \rightarrow \mathbb{H}^{L}$. As such, for some $\pmb{a}=(a_1, a_2, \dots, a_K) \in \mathbb{R}^{K}$, we have $$\label{mfssa x op}
\mathbfcal{X}(\pmb{a}):=\sum_{k=1}^{K}a_{k}\pmb{x}_{k}.$$ Notice that $\text{R}\left(\mathbfcal{X}\right)=\text{sp}\{\pmb{x}_{j}\}_{j=1}^{K}$ is the range of the operator $\mathbfcal{X}$ with rank $r$, where $1\leq r \leq \min(pL,K)$. This step of embedding can also be viewed as applying the invertible transformation, $\mathcal{T}:\mathbb{H}^{N} \rightarrow \mathbb{H}_{H}^{L \times K}$, such that $$\label{B operator}
\mathbfcal{X}=\mathcal{T}(\mathbf{y}_{N}).$$
\[prop:traj\] The operator given in is a bounded and linear operator with adjoint $\mathbfcal{X}^{*}:\mathbb{H}^{L} \rightarrow \mathbb{R}^{K}$ $$\mathbfcal{X}^{*}\mathbf{z}:=\left({\left\langle \pmb{x}_{1}, \mathbf{z} \right \rangle}_{\mathbb{H}^{L}}, {\left\langle \pmb{x}_{2}, \mathbf{z} \right \rangle}_{\mathbb{H}^{L}}, \dots, {\left\langle \pmb{x}_{K}, \mathbf{z} \right \rangle}_{\mathbb{H}^{L}}\right)^{\top} \in \mathbb{R}^{K}.$$
### MFSSA II. Decomposition {#mfssadecomp .unnumbered}
Notice that the compact operator $\mathbfcal{X}$, is of rank $r$. Therefore one may employ Theorem 7.6 of [@weidmann1980] and obtain the SVD for the operator $\mathbfcal{X}$ as $$\label{fsvd}
\mathbfcal{X}(\pmb{a})=\sum_{i=1}^{r}\sigma_{i}\langle \mathbf{v}_{i},\pmb{a}\rangle_{\mathbb{R}^K}\boldsymbol{\psi}_{i}=\sum_{i=1}^{r}\sigma_{i}\mathbf{v}_{i}\otimes \boldsymbol{\psi}_{i}\left(\pmb{a}\right)=\sum_{i=1}^{r} \mathbfcal{X}_{i}(\pmb{a}).$$ Here, $\{\sigma_{i}\}_{i=1}^{r}$ are the singular values, $\{\mathbf{v}_{i}\}_{i=1}^{r}$ are the orthonormal right singular vectors spanning $\mathbb{R}^{r}$, $\{\boldsymbol{\psi}_{i}\}_{i=1}^{r}$ are the orthonormal left singular functions spanning an $r$-dimensional subspace of $\mathbb{H}^{L}$. Now we define the rank one elementary operators $\mathbfcal{X}_{i}:=\sigma_{i}\mathbf{v}_{i}\otimes \boldsymbol{\psi}_{i}$, where $\otimes$ stands for the tensor(outer) product. It is easy to see that $\mathbfcal{X}=\sum_i\mathbfcal{X}_i$. We call the result of the multivariate fSVD (mfSVD) of $\mathbfcal{X}$ and we call the set $(\sigma_{i},\boldsymbol{\psi}_{i},\mathbf{v}_{i})$ to be the $i^{th}$ eigentriple of $\mathbfcal{X}$.
\[prop:mfssadecomp\] Let $(\sigma_{i},\boldsymbol{\psi}_{i},\mathbf{v}_{i})$ be the $i^{th}$ eigentriple of $\mathbfcal{X}$, $i=1,\dots,r$. The following hold: $$\boldsymbol{\psi}_{i}=\sigma_{i}^{-1}\mathbfcal{X}\mathbf{v}_{i}, \quad \mathbf{v}_{i}=\sigma_{i}^{-1}\mathbfcal{X}^{*}\boldsymbol{\psi}_{i}. \nonumber$$
### MFSSA III. Grouping {#mfssa-iii.-grouping .unnumbered}
The grouping step of MFSSA follows the same flavor as the grouping step of MSSA. We partition the set of indices $\{1,2,\dots,r\}$ into $m$ disjoint subsets $\{I_{1},I_{2},\dots,I_{m}\}$ such that for any $q$, the operator $\mathbfcal{X}_{I_{q}}$ is defined as $\mathbfcal{X}_{I_{q}}:=\sum_{i \in I_{q}} \mathbfcal{X}_{i}$. As such, we write $$\label{mfssa traj op}
\mathbfcal{X} = \mathbfcal{X}_{I_{1}} + \mathbfcal{X}_{I_{2}} + \cdots + \mathbfcal{X}_{I_{m}}.\nonumber$$ Similar to [@haghbin2019], exploratory plots, such as scree plots, paired-plots, w-correlation plots, and others can be developed to determine how to obtain the $m$ disjoint groups.
### MFSSA IV. Reconstruction {#mfssa:recon .unnumbered}
Let $\mathbfcal{Y} \in \mathbb{H}^{L \times K}$, then since $\mathbb{H}_{H}^{L \times K}$ is a closed subspace of $\mathbb{H}^{L \times K}$, we have by the Projection Theorem that there exists a unique $\widetilde{\mathbfcal{Y}} \in \mathbb{H}_{H}^{L \times K}$ such that $$\label{reconstruct XIq}
{\| \mathbfcal{Y}-\widetilde{\mathbfcal{Y}} \|}_{F}^{2} \leq {\| \mathbfcal{Y}-\widetilde{\mathbfcal{Z}} \|}_{F}^{2}, \nonumber$$ for any $\widetilde{\mathbfcal{Z}} \in \mathbb{H}_{H}^{L \times K}$. Define the projector $\Pi:\mathbb{H}^{L \times K} \rightarrow \mathbb{H}_{H}^{L \times K}$ such that we have $\Pi \mathbfcal{Y}=\widetilde{\mathbfcal{Y}}$. We achieve this projection by using Lemma 3.1 of [@haghbin2019] and the resulting diagonal averaging technique that $$\label{fssadiagavg}
\vec{\tilde{y}}_{i,k}:=\frac{1}{n_{u}}\sum_{\left(n,m\right):n+m=u}\vec{y}_{n,m},$$ where $n_{u}$ is the number of $\left(n,m\right)$ pairs such that $n+m=u$. With this projection, we have that $\Pi\mathbfcal{X}_{I_{q}}=\widetilde{\mathbfcal{X}}_{I_{q}}$ for $q=1,\dots,m$. We then employ the inverse of $\mathcal{T}$ from to obtain the following formula for the reconstruction $$\label{mfssarecon}
\tilde{\mathbf{y}}^{q}_{N}:=\mathcal{T}^{-1}\widetilde{\mathbfcal{X}}_{I_{q}},\nonumber$$ where $\mathbf{y}_{N} = \sum_{q=1}^{m}\tilde{\mathbf{y}}^{q}_{N}$.
Separability
------------
Let $\mathbf{x}_{N} = \mathbf{y}_{N}+\mathbf{z}_{N}$ where each $\mathbf{y}_{N}$ and $\mathbf{z}_{N}$ are multivariate functional time series. We define the weighted-covariance between multivariate functional time series as $$\label{w-corr}
{\left\langle \mathbf{y}_{N}, \mathbf{z}_{N} \right \rangle}_{w} := \sum_{j=1}^{p}\sum_{i=1}^{N}w_{i}{\left\langle y_{i}^{(j)}, z_{i}^{(j)} \right \rangle}_{\mathbb{F}_j},\nonumber$$ where $w_{i} := \text{min}\{i, L, N-i+1\}$. We call $\mathbf{x}_{N}$ separable if ${\left\langle \mathbf{y}_{N}, \mathbf{z}_{N} \right \rangle}_{w} = 0$. The weighted-covariance measure shown here can also be used to form a so-called $w$-correlation between MFTS.
MFSSA Implementation {#mfssaimp}
====================
Similar to the discussion of implementation in [@haghbin2019], we observe discrete samples of functional data that are then converted into functional objects using smoothing methods. Techniques that are used to form the functional data observations can be found in [@ramsay2007]. Let $\{\nu_{i}^{(j)}\}_{i \in \mathbb{N}}$ be the collection of basis functions in $\mathbb{F}_j$ for $j = 1,...,p$. Each observation in $\mathbb{F}_j$ can be projected onto the subspace $F_{j}:=\text{sp}\{\nu_{i}^{(j)}\}_{i=1}^{d_{j}}$ where $d_{j}$ can be determined by a variety of techniques like cross-validation. To this end, each $y_{i}^{(j)} \in \mathbb{F}_{j}$ can be projected to $F_{j}$ as $$\label{basis expansion}
\hat{y}_{i}^{(j)}:=\sum_{k=1}^{d_{j}}c_{i,k}^{(j)}\nu_{k}^{(j)}, \quad i=1,...,N, \quad c_{i,k}^{(j)} \in \mathbb{R}.\nonumber$$ where $\hat{y}_{i}^{(j)} \in F_{j}$. Now we set $d_0:=0$, $d:=\sum_{j=0}^{p}d_{j}$, and $\mathbb{H}_d:=F_1\times \ldots \times F_p \subseteq \mathbb{H}.$ For the rest of this section we provide the implementation of the MFSSA on MFTS $\mathbf{y}_{N}:=(\vec{\hat{y}}_{1},\dots,\vec{\hat{y}}_{N})$, where $\vec{\hat{y}}_{i}:=(\hat{y}_{i}^{(1)},\ldots, \hat{y}_{i}^{(p)})\in \mathbb{H}_d$.
For each $q\in\{1,\ldots, d\}$, there exist a unique $j_q\in\{1,\ldots,p\}$ such that $\sum_{i=0}^{j_q-1}d_i +1 \leq q \leq \sum_{i=0}^{j_q}d_i$. Now consider $\vec{\nu}_q \in \mathbb{H}_d$, as a multivariate functional object of length $p$ with all zero functions, except $j_q$-th element, which is $\nu_{\ell_q}^{(j_q)}$, where $\ell_q:={q-\sum_{i=0}^{j_q-1}d_i}$.
\[lem:nu\] The following holds:
- Each multivariate functional object $\vec{\hat{y}}_{i}$ can be uniquely represented as a linear combination of $\vec{\nu}_q$’s $$\label{vbasis expansion}
\vec{\hat{y}}_{i}:=\sum_{q=1}^{d}c_{i,\ell_q}^{(j_q)}\vec{\nu}_q, \quad i=1,...,N.\nonumber$$
- The set $\{\vec{\nu}_q\}_{q=1}^{d}$ is a basis system of $\mathbb{H}_{d}$.
Now for each $k\in\{1,\ldots, Ld\}$, one can see that there exist unique $q_k\in\{1,\ldots,d\}$ and $r_k\in\{1,\ldots,L\}$ such that $k=(q_k-1)L+r_k$. Consider ${\pmb \phi}_{k}$ as a functional vector of length $L$ with all zero functions, except $r_k$-th element, which is $\vec{\nu}_{q_k}$.
\[lem:phi\] The sequence $\{{\pmb \phi}_{k}\}_{k=1}^{Ld}$ is a basis system for $\mathbb{H}_d^L$, where $\mathbb{H}_d^L$ is the Cartesian product of $L$ copies of $\mathbb{H}_d$.
Using the Lemma \[lem:phi\], one may define a linear operator $\mathbfcal{P}:\mathbb{R}^{Ld}\rightarrow \mathbb{H}_d^L$, specified with ${\pmb \phi}_{k}$’s, where each $\pmb{x}\in \mathbb{H}_d^L$ can be written as $$\label{A}
\pmb{x} = \sum_{i=1}^{Ld} b_{i}{\pmb \phi}_{i}=\mathbfcal{P}(\mathbf{b}).$$ We call ${\mathbf{b}}=(b_1, \ldots,b_{Ld})\in\mathbb{R}^{Ld},$ the corresponding coefficient vector of $\pmb{x}$ with respect to the operator $\mathbfcal{P}$. Similar to one may define the functional lagged vectors for the MFTS $\mathbf{y}_{N}$ as $\pmb{x}_{k}=\left(\vec{\hat{y}}_{k},\vec{\hat{y}}_{k+1},\dots,\vec{\hat{y}}_{k+L-1}\right)\in \mathbb{H}_d^L$, where $k=1,\dots,K$. Therefore the associated trajectory operator, given in , would be $\mathbfcal{X}:\mathbb{R}^{K} \rightarrow \mathbb{H}_d^{L}$.
\[lem:p\] The following holds:
- The corresponding coefficient vector of the functional lagged vector $\pmb{x}_k$ with respect to the operator $\mathbfcal{P}$ is $$\label{lem:phi_b}
{\bf b}_k:=\left[c_{k,\ell_1}^{(j_1)},\ldots, c_{k+L-1,\ell_1}^{(j_1)}, c_{k,\ell_2}^{(j_2)},\ldots, c_{k+L-1,\ell_2}^{(j_2)},\ldots, c_{k+L-1,\ell_d}^{(j_d)} \right]^\top\in \mathbb{R}^{Ld}.$$
- For any $\pmb{a} \in \mathbb{R}^{K}$, we have $\mathbfcal{X}(\pmb{a})=\mathbfcal{P}(\mathbf{B}\pmb{a}),$ where ${\mathbf B}:=\left[b_{k,i}\right]_{i=1,\dots,Ld}^{k=1,\dots,K}=\left[\mathbf{b}_{1}, \mathbf{b}_{2}, \dots, \mathbf{b}_{K}\right]_{Ld \times K}$, and $b_{k,i}$ is the $i^{th}$ element of ${\mathbf b}_{k}$.
The following theorem gives us the recipes necessary to obtain the eigentriples of $\mathbfcal{X}$.
\[thm:recipe\] Suppose ${\mathbf X}:={\mathbf G}^{1/2}{\mathbf B}$ where ${\mathbf G}:=\left[{\left\langle \boldsymbol{\phi}_{i}, \boldsymbol{\phi}_{j} \right \rangle}_{\mathbb{H}^{L}}\right]_{i,j=1}^{Ld}$ is the Gram matrix. Denote the collection $\left({\sigma}_{i},{\mathbf{v}}_{i},{\pmb{u}}_{i}\right)$ as the $i^{\text{th}}$ eigentriple of $\mathbf{X}$. Now define ${\boldsymbol{\psi}}_{i}:=\mathbfcal{P} ({\mathbf G}^{-1/2}{\pmb{u}}_{i})$. The following holds:
- $\mathbfcal{X}^{*}{\boldsymbol{\psi}}_{i}={\sigma}_{i}{\mathbf{v}}_{i}$
- $\mathbfcal{X}{\mathbf{v}}_{i}={\sigma}_{i}{\boldsymbol{\psi}}_{i}$
- The collection $\{{\boldsymbol{\psi}}_{i}\}_{i=1}^{r}$ form an orthonormal basis for $R(\mathbfcal{X})$.
The collection of triples $({\sigma}_{i},{\mathbf{v}}_{i},{\boldsymbol{\psi}}_{i})_{i=1}^{r}$ defines the mfSVD of $\mathbfcal{X}$.
Generalizing MSSA to MFSSA
==========================
One may note that a key step in extending different SSA approaches, is how to obtain the trajectory matrix (operator) in the embedding step (see e.g., Sections \[emmssa\] and \[mfssa.algorithm\]). Despite the fact in SSA, where the trajectory matrix is a linear combination of the associated lagged vectors, that is not the case for MSSA.
In Section 3, we obtain MFSSA by generalizing FSSA, where we introduce the trajectory operator as a linear combination of multivariate lagged vectors. Alternatively, one may mimic the approach of MSSA algorithms (HMSSA or VMSSA) and develop new trajectory operators that are not necessarily based on lagged vectors. The following subsections would extend HMSSA and VMSSA to obtain the functional versions respectively.
From HMSSA to HMFSSA {#HMSSA2MMFSSA}
--------------------
As one may see the columns of $\mathbf{X}^{(j)}$ in , $\mathbf{x}_k^{(j)}$’s, are the univariate lagged vectors for the $j^{th}$ variable. Therefore one can see the $\mathbf{X}^{(j)}$ as an operator from $\mathbb{R}^K \rightarrow \mathbb{R}^L$, which can be seen as a linear combination of these lagged vectors: $$\mathbf{X}^{(j)}\pmb{a}^{(j)}=\sum_{k=1}^{K}a_k^{(j)}\mathbf{x}_k^{(j)},\qquad \pmb{a}^{(j)}:=(a_1^{(j)},\dots,a_K^{(j)})\in \mathbb{R}^K.$$ In the embedding step of HMSSA, the trajectory matrix, given in , can be seen as a linear operator, $\mathbf{X}:\mathbb{R}^{pK}\rightarrow \mathbb{R}^L$, where $$\label{hmssa:linear comb}
\mathbf{X}\pmb{a}=\sum_{j=1}^{p}\sum_{k=1}^{K}a_k^{(j)}\mathbf{x}_k^{(j)},\qquad \pmb{a}:=(\pmb{a}^{(1)},\dots,\pmb{a}^{(p)})\in \mathbb{R}^{pK}.$$ In order to extend to the functional space, we need to assume that the lag vectors in HMFSSA, denoted with $\pmb{x}_k^{(j)}$, are in the space $\mathbb{F}_j^L$, for $j=1,\dots,p$. But the linear combination of $\pmb{x}_k^{(j)}$’s are well-defined if and only if $\mathbb{F}_1^L=\cdots=\mathbb{F}_p^L$, or equivalently $T_1=\dots=T_p$. We shall call the extension of this special case as HMFSSA and we present it in the supplementary material.
From VMSSA to VMFSSA {#vmfssa}
--------------------
In the embedding step of VMSSA, the trajectory matrix, given in , can be seen as a linear operator, $\mathbf{X}:\mathbb{R}^{K}\rightarrow \mathbb{R}^{pL}$, with $$\mathbf{X}\pmb{a}=\sum_{k=1}^{K}a_j\underline{\mathbf{x}}_k,\qquad \pmb{a}:=({a}_{1},\dots,{a}_{K})\in \mathbb{R}^{K}\quad\mathrm{and\ } \underline{\mathbf{x}}_k:=
\begin{bmatrix}\mathbf{x}_k^{(1)}\\ \vdots\\ \mathbf{x}_k^{(p)}\end{bmatrix}\in \mathbb{R}^{pL}.$$ To develop VMFSSA, we need to extend this operator to the functional space, i.e., $\underline{\mathbf{x}}_k$ should belong to a new unfolded Hilbert space, $\mathbb{H}^{p,L}:=\underbrace{\mathbb{F}_{1}\times \cdots \times \mathbb{F}_{1}}_{L\text{ times}}\times \ldots \times \underbrace{\mathbb{F}_{p} \times \cdots \times \mathbb{F}_{p}}_{L\text{ times}}$. Here, each $\underline{\mathbf{x}}\in \mathbb{H}^{p,L}$ is denoted by $\underline{\mathbf{x}}:=\left(x_{1}^{\left(1\right)}, \cdots, x_{L}^{\left(1\right)}, \ldots, x_{1}^{\left(p\right)}, \cdots, x_{L}^{\left(p\right)}\right)$. It is easy to see that $\mathbb{H}^{p,L}$ is a Hilbert space equipped with inner product $$\label{eqinnp}
{\left\langle \underline{\mathbf{x}}, \underline{\mathbf{y}} \right \rangle}_{\mathbb{H}^{p,L}}:=\sum_{i=1}^{L}\sum_{j=1}^{p}{\left\langle x_{i}^{\left(j\right)}, y_{i}^{\left(j\right)} \right \rangle}_{\mathbb{F}_{j}}=\sum_{i=1}^{L} {\left\langle \vec{x}_{i}, \vec{y}_{i} \right \rangle}_{\mathbb{H}}={\left\langle \mathbf{x}, \mathbf{y} \right \rangle}_{\mathbb{H}^{L}}.\nonumber$$ Therefore, there exists a unitary operator $\mathcal{U}:\mathbb{H}^{L} \rightarrow \mathbb{H}^{p,L}$ where $\mathcal{U}(\mathbf{x})=\underline{\mathbf{x}}$, and we have an isomorphism between $\mathbb{H}^{L}$ and $\mathbb{H}^{p,L}$. Now one may define the linear operator $\underline{\mathcal{X}}:\mathbb{R}^K\rightarrow\mathbb{H}^{p,L}$, specified with $\underline{\mathbf{x}}_k$’s, as $$\underline{\mathcal{X}}\pmb{a}:=\sum_{k=1}^{K}a_j\underline{\mathbf{x}}_k,\qquad \pmb{a}\in \mathbb{R}^{K}\quad\mathrm{and}\quad \underline{\mathbf{x}}_k\in\mathbb{H}^{p,L}.$$
The following theorem illustrates the equivalency between the MFSSA and VMFSSA results.
\[thm:vmfssa\] Let $({\sigma}_{i},{\mathbf{v}}_{i},\boldsymbol{\psi}_{i})_{i=1}^r$ to be the eigentriples of ${\mathbfcal{X}}$. The following holds:
- $\underline{\mathcal{X}}=\mathcal{U}\mathbfcal{X}$.
- Furthermore, $\underline{\mathcal{X}}$ is a rank $r$ operator with the eigentriples $({\sigma}_{i},{\mathbf{v}}_{i},\underline{\boldsymbol{\psi}}_{i})_{i=1}^r$, where $\underline{\boldsymbol{\psi}}_{i}=\mathcal{U}\boldsymbol{\psi}_{i}$.
Therefore the decompositions obtained via MFSSA and VMFSSA are interchangeable and subsequently the respective groupings and reconstructions are equivalent.
Numerical Studies
=================
In order to explore the capabilities of MFSSA and HMFSSA we implement a simulation study where we compare our two novel algorithms to other approaches of MFTS reconstruction of the true signal. We also present an application to remote sensing data which is used to further illustrate the interesting qualities of MFTS data that are discovered by MFSSA.
Simulation Study {#simstudy}
----------------
For the simulation, we generate a bivariate FTS of lengths $N=\{100,200\}$ by projecting the following discrete observations sampled in equidistance on the unit interval onto a B-spline basis with 15 degrees of freedom $$\begin{aligned}
Y_{t}^{\left(1\right)}\left(s_{i}\right)&:=y_{t}^{\left(1\right)}+X_{t}^{\left(1\right)}\\
Y_{t}^{\left(2\right)}\left(s_{i}\right)&:=y_{t}^{\left(2\right)}+X_{t}^{\left(2\right)},\quad s_{i} \in \left[0,1\right],\quad i=1,\dots,100,\quad t=1,\dots,N.\end{aligned}$$ where $y_{t}^{\left(1\right)}:=\mu_{t}+\delta_{t}^{\left(1\right)}$ and $y_{t}^{\left(2\right)}:=\delta_{t}^{\left(2\right)}$ are nonrandom, true signal terms. We take $\mu_{t}:=kt$ as an increasing trend component with $k=\{0.00,0.02\}$, $\delta_{t}^{\left(j\right)}$ are taken as seasonal components with expressions given as $$\begin{aligned}
\delta_{t}^{\left(1\right)}&:=&e^{s_{i}^{2}}\cos\left(2\pi\omega_{1}t\right)-e^{1-s_{i}^{2}}\cos\left(2\pi\omega_{2}t\right)-\sin\left(2\pi\omega_{1}t\right)\cos\left(4\pi s_{i}\right)\\ &&+\sin\left(2\pi\omega_{2}t\right)\sin\left(\pi s_{i}\right)\\
\delta_{t}^{\left(2\right)}&:=&e^{s_{i}^{2}}\sin\left(2\pi\omega_{1}t\right)+\cos\left(2\pi\omega_{1}t\right)\cos\left(4\pi s_{i}\right),\end{aligned}$$ where $\omega_{1}=\{0.1, 0.5\}$, $\omega_{2}=\{0,0.25\}$, and $X_{t}^{\left(j\right)}$ are error terms for $j=1,2$. The error terms follow four models drawn directly from [@haghbin2019], one being a Gaussian white noise and the other three coming from a functional autoregressive model of order 1 (FAR1) given by $$\label{FAR1}
X_{t}\left(s\right):=\Psi X_{t-1}\left(s\right)+\epsilon_{t}\left(s\right),\nonumber$$ where the collection $\{\epsilon_{t}\left(s\right)\}_{t=1}^{N}$ are taken as independent functions of Brownian motion over the unit interval and $\Psi$ is an integral operator with kernel $$\label{psi}
\psi\left(s,u\right):=\gamma_{0}\left(2-(2s-1)^{2}-(2u-1)^{2}\right). \nonumber$$ We choose $\gamma_{0}$ such that the norm of $\Psi$, given as $$\label{hibsch norm}
{\| \Psi \|}^{2}:=\int_{0}^{1}\int_{0}^{1}{\lvert \psi\left(s,u\right) \rvert}^{2}dsdu, \nonumber$$ takes on values of $0$, $0.5$, or $0.9$ in order to obtain our autoregressive models. Due to the presence of a trend component and two frequencies, we require five components to reconstruct the true structures which is due to the fact that each of the two frequencies is expressed in a sine and a cosine term. We compare reconstruction results of MFSSA, HMFSSA, FSSA performed on each covariate independently of one another, MSSA (HMSSA), and DFPCA ran on each covariate independently of one another. For MSSA we specify that the data matrix, $Q$, follows the form $$\label{Q mssa sim}
Q:=\left[Q_{1}, Q_{2}\right]^{\top},\nonumber$$ such that $Q_{j}:=\left[Y_{t}^{\left(j\right)}\left(s_{i}\right)\right]_{i=1,\dots,100}^{t=1,\dots,N}$ for $j=1,2$ with $i$ being representative of rows of $Q_{j}$ and $t$ of columns. For all of the SSA-based algorithms we set $L=\{20,40\}$ and for all algorithms, we measure the error of each reconstruction with the following root mean square error (RMSE) $$\label{RMSE}
\text{RMSE}:=\sqrt{\frac{1}{N\times n \times p}\sum_{j=1}^{p}\sum_{t=1}^{N}\sum_{i=1}^{n}\left(y_{t}^{\left(j\right)}\left(s_{i}\right)-\hat{y}_{t}^{\left(j\right)}\left(s_{i}\right)\right)^{2}}, \nonumber$$ where $\hat{y}_{t}^{\left(j\right)}\left(s_{i}\right)$ is the reconstruction of covariate $j$, at time point $t$, evaluated at point $s_{i}$. For every unique combination of parameters and error terms, we repeat $100$ times and report the mean of the RMSE’s in the following plots whose vertical axes are taken over a log scale.
![Simulation Study[]{data-label="fig:sim_study"}](graphs.pdf "fig:"){width=".84\textwidth"} \[fig:sim-A\]
![Simulation Study[]{data-label="fig:sim_study"}](graphs.pdf "fig:"){width=".84\textwidth"} \[fig:sim-B\]
We see in the top plot that $L=20$ and in the bottom plot, $L=40$, while the vertical lines separate out the simulated data by noise models and in addition, each tick mark on the horizontal should be read as $[N,\omega_{1},\omega_{2},k]$. From these two subfigures, we find that MFSSA almost always outperforms other techniques of dimension reduction for a MFTS while HMFSSA also outperforms other techniques occasionally.
Application to Remote Sensing and Weather Station Data
------------------------------------------------------
It is well known that the amount of vegetation present in a region is closely related to the temperature of that same area. Researchers can use this correlation to get a better understanding of how the vegetation and temperature in a region changes over time together through use of multivariate analysis techniques. Data that tracks the intraday hourly mean temperature, in celsius, for a variety of United States weather stations is available for download from [@tempdata]. In addition, Satellite images of varying resolutions, regions, time periods, spectral bands, and their variants have been made available for download and analyzed using various techniques [@tuck2014]. The normalized difference vegetation index (NDVI) measure, which is bounded between zero and one, is used to track the amount of vegetation, is computed as the difference of the near-infrared and red bands which is then divided by the sum of the same spectral quantities [@lambin1999]. NDVI values closer to one are indicative of more vegetation being present while values closer to zero are indicative of less vegetation. It is common practice to average the NDVI measures of each image to form a time series and then analyze it with techniques such as X12-ARIMA [@panuju2012]. The issue with this approach is that two different densities that correspond to two different NDVI images might have similar sample means and to this end, more informative approaches should be used. The work of [@haghbin2019] estimated a density for each NDVI image taken of a region of Jambi, Indonesia in 16 day increments between February 18, 2002, and July 28, 2019. They then applied FSSA to the time series of densities and discovered a trend component indicating a loss of vegetation over the course of a decade that was not detected by other techniques.
It was determined that using MSSA over SSA can lead to richer analysis of correlated data [@golyandina2012]. If a variable with strong seasonality components and another variable with strong mean components are included together in an MSSA analysis, we expect to find strong seasonality and mean component reflected in the singular values and singular vectors. To illustrate this concept continues into the functional realm, we use a bivariate example of intraday hourly mean temperature curves and NDVI images of a parallelogram shaped region just east of Glacier National Park in Montana, U.S.A. located between longitudes of $113.30^{\circ} \text{W} - 113.56^{\circ} \text{W}$ and latitudes of $48.71^{\circ} \text{N} - 48.78^{\circ} \text{N}$ starting January 1, 2008 and ending September 30, 2013 every 16 days. We start by applying FSSA with a lag of $45$ to the functional curves and images separately from one another, where this choice of lag captures annual behavior in the MFTS, and we obtain the following plots of the singular vectors.
\[fssatempndvi\]
[.49]{} ![FSSA on Intraday Temperature Curves and FSSA on NDVI Functional Images[]{data-label="fig:fssa_temp_ndvi"}](temp_curves.pdf "fig:"){width=".8\textwidth"} \[fig:temp\_A\]
[.5]{} \[fig:temp\_vecs\_B\]
[.49]{} \[fig:NDVI\_C\]
[.5]{} \[fig:NDVI\_vecs\_D\]
It is clear from plot (B) of Figure \[fig:fssa\_temp\_ndvi\] that there exists a strong seasonality component in the intraday temperature curves of plot (A) accounting for $54.72\%$ of the variation in the data while a mean behavior component accounts for $15.38\%$ of the variation in the data. We also see from plot (D) of Figure \[fig:fssa\_temp\_ndvi\] that the mean component captures $65.07\%$ of the variation of the NDVI images data where plot (C) is one such observations while the seasonality components only account for $28.56\%$ of the variation of the data. We normalize the intraday temperature curves by dividing each sampling point by the standard deviation of all the sampling points since the NDVI images have values that are significantly smaller. We now apply MFSSA with a lag of $45$ to the normalized intraday temperature curves and NDVI images in a bivariate analysis to obtain the following plots.
[0.32]{} ![MFSSA Exploratory Plots[]{data-label="fig:mfssa_temp_NDVI"}](mfssa_sing_vals.pdf "fig:"){width=".98\textwidth"} \[fig:mfssa\_sing\_vals\_A\]
[0.33]{} \[fig:mfssa\_wcor\_B\]
[0.33]{} ![MFSSA Exploratory Plots[]{data-label="fig:mfssa_temp_NDVI"}](mfssa_vecs.pdf "fig:"){width=".98\textwidth"} \[fig:mfssa\_sing\_vecs\]
Plots (A) and (B) of Figure \[fig:mfssa\_temp\_NDVI\] show that component one should be grouped by itself, two should be grouped with three, and four with five. Plot (C) of Figure \[fig:mfssa\_temp\_NDVI\] shows that in the bivariate analysis, the mean component becomes dominant with the seasonal components taking on the second and third main sources of variation. This shows that combining the temperature curves and NDVI images functional data into a bivariate analysis reveals a stronger mean component as opposed to the weaker mean component seen in plot (E) of Figure \[fig:fssa\_temp\_ndvi\].
Discussion
==========
Throughout this paper, we presented MFSSA as a novel technique of dimension reduction of a MFTS. We found that the MFSSA problem is solved by performing VMFSSA and we also developed HMFSSA, presented in supplementary material, as another approach but found that it was more restrictive and not as informative as MFSSA. We also developed MFSSA to be able to handle functions taken over different dimensional domains to uncover a more dominant mean component for the intraday temperature curves/NDVI images bivariate analysis. The MFSSA algorithm is available for use in the [[Rfssa]{}]{} package [@rfssapackage], available through CRAN.
Supplementary Materials {#supplementary-materials .unnumbered}
=======================
The supplementary material includes plots and animations of the left singular functions of our real data study in the manuscript, another remote sensing real data study example, and the full development of HMFSSA. We also include proofs of the lemmas and propositions of the manuscript.
Left Singular Functions of MFSSA
================================
In this section, we build on the real data study presented in the manuscript by presenting the left singular functions. As mentioned, we apply FSSA to the temperature curves and NDVI images separately, We also implement MFSSA to the temperature curves and NDVI images together, both with a lag of $45$, to obtain the following.
\[fssatempfuns\]
[.49]{} ![Subfigures (A), (C): FSSA Left Singular Functions. Subfigures (B), (D): MFSSA Left Singular Functions[]{data-label="fig:fssa_funs_temp"}](temp_funs.pdf "fig:"){width=".85\textwidth"} \[fig:curves\_A\]
[0.50]{} ![Subfigures (A), (C): FSSA Left Singular Functions. Subfigures (B), (D): MFSSA Left Singular Functions[]{data-label="fig:fssa_funs_temp"}](mfssa_temp_funs.pdf "fig:"){width=".85\textwidth"} \[fig:mfssa\_curves\_B\]
[.49]{} \[fig:ani\_curves\_C\]
[0.50]{} \[fig:mfssa\_ani\_curves\_D\]
Plot (A) of Figure \[fig:fssa\_funs\_temp\] shows all $L=45$ functions of the first four left singular functions of FSSA for the temperature data while plot (C) steps through each function in an animation. Plot (B) of Figure \[fig:fssa\_funs\_temp\] shows all $L=45$ functions of the first four left singular functions of MFSSA for the temperature data while plot (D) steps through each function in an animation. We see in the temperature data, that when MFSSA is applied, the mean component becomes stronger. We apply FSSA to the images with a lag of $45$ and compare the resulting left singular functions for the NDVI images to those we obtain via the MFSSA analysis in the following animations.
\[mfssatempNDVIfuns\]
[.49]{} \[fig:images\_B\]
[0.50]{} \[fig:images\_B\]
Here, we see little difference between the animations.
MFSSA Applied to Remote Sensing Density Curves
==============================================
To further show that MFSSA enriches data analysis of correlated variables, we use a bivariate example of near-infrared (NIR) and shortwave infrared (SWIR) images taken every eight days of a region just outside of the city of Jambi, Indonesia between $103.61^{\circ} \text{E} - 103.68^{\circ} \text{E}$ and $1.67^{\circ} \text{S} - 1.60^{\circ} \text{S}$ over the timeline of February 18, 2000 and November 25, 2019. The wavelength of the NIR images range from 841-876 nanometers (nm) and the wavelength of the SWIR images are within the values of 2105-2155 nm. NIR light can be used for imaging vegetation as it is used in the calculation of the NDVI measure [@lambin1999] while shortwave infrared is often used for imaging the moisture content in soil where a lower surface reflectance (SR) corresponds to higher moisture content [@shin2017]. As mentioned in [@prasetyo2016], it appears that this particular part of the Jambi province was a hot spot for controlled fires between 2001 and 2015 and this loss of vegetation over the course of about a decade will be reflected in lower NIR and higher SWIR SR values as time moves on. We obtain the KDEs of both the NIR and the SWIR SR images using Silverman’s rule of thumb [@silverman1986] which we then project onto a cubic B-spline basis where the degrees of freedom are chosen using the GCV criterion. In addition, we replaced outliers in the SWIR densities with the average of densities from the preceding and proceeding days. Similar results, as compared to the following, still hold even if the outliers are not removed. Applying FSSA with a lag of $45$ to the NIR and SWIR densities separately, where this choice of lag approximately captures annual behavior, gives the following exploratory plots.
\[fssanirswir\]
[.49]{} ![FSSA on NIR and SWIR Densities[]{data-label="fig:swi_fssa"}](swir.pdf "fig:"){width=".8\textwidth"} \[fig:NDVI-A\]
[.5]{} \[fig:NDVI-B\]
[.49]{} \[fig:NDVI-C\]
[.5]{} \[fig:NDVI-D\]
Figure \[fig:swi\_fssa\] subfigures (A) and (B) give us the right singular vectors and left singular functions of the NIR densities while Figure \[fig:swi\_fssa\] subfigures (C) and (D) are the right singular vectors and left singular functions of the SWIR densities. We find that applying FSSA to the NIR densities captures seasonality in the second and third components while trend is present in the fourth component similar to the NDVI results of [@haghbin2019]. Applying FSSA to the SWIR densities shows that trend is a more dominant behavior captured in the second component as compared to the seasonal behaviors captured in components three and four. Applying MFSSA decomposition with a lag of $45$ to the bivariate NIR/SWIR example, where this lag is chosen to capture annual behavior, gives the following exploratory plots.
[0.32]{} ![KDEs of NIR and SWIR Images as well as MFSSA Exploratory Plots[]{data-label="fig:mfssa_nir_swir"}](bivariate.pdf "fig:"){width=".98\textwidth"} \[fig:swir\_nr-A\]
[0.33]{} \[fig:swir\_nr-B\]
[0.33]{} ![KDEs of NIR and SWIR Images as well as MFSSA Exploratory Plots[]{data-label="fig:mfssa_nir_swir"}](swir_nr.pdf "fig:"){width=".98\textwidth"} \[fig:swir\_nr-C\]
[0.33]{} ![KDEs of NIR and SWIR Images as well as MFSSA Exploratory Plots[]{data-label="fig:mfssa_nir_swir"}](swir_nr.pdf "fig:"){width=".98\textwidth"} \[fig:swir\_nr-D\]
[0.33]{} ![KDEs of NIR and SWIR Images as well as MFSSA Exploratory Plots[]{data-label="fig:mfssa_nir_swir"}](swir_nr.pdf "fig:"){width=".93\textwidth"} \[fig:swir\_nr-E\]
[0.32]{} ![KDEs of NIR and SWIR Images as well as MFSSA Exploratory Plots[]{data-label="fig:mfssa_nir_swir"}](swir_nr.pdf "fig:"){width=".93\textwidth"} \[fig:swir\_nr-F\]
The bivariate FTS can be found in Figure \[fig:mfssa\_nir\_swir\] subfigure (A) while Figure \[fig:mfssa\_nir\_swir\] subfigures (B) and (C) are plots of singular values and w-correlation respectively. See that Figure \[fig:mfssa\_nir\_swir\] subfigure (D) gives us our MFSSA right singular vectors which showcases the weights that are multiplied by the left singular functions shown in Figure \[fig:mfssa\_nir\_swir\] subfigures (E) and (F). Since we are performing MFSSA, we obtain $45$ left eigenfunctions that correspond to the NIR densities as well as another set of $45$ left eigenfunctions that correspond to the SWIR densities. Notice the trend behavior for the NIR densities is present in component two as according to Figure \[fig:mfssa\_nir\_swir\] subfigure (E) which indicates that adding SWIR densities into the analysis with the NIR densities created a more pronounced trend result as compared with Figure \[fig:swi\_fssa\] subfigure (B). To this end, we find that performing a bivariate analysis on the NIR/SWIR densities enriched our data analysis as expected.
HMFSSA
======
We begin this section with our discussion of moving from HMSSA to HMFSSA. As we clarified in subsection 5.1 of the manuscript, we need to assume $T:=T_{1}=\cdots=T_{p}$, $\mathbb{F}:=L^2(T)$, and $\pmb{x}_k^{(j)}$’s belong to a common space $\mathbb{F}^L$, for $j=1,\cdots,p$. Notice that while the domain for each variable is the same, one may evaluate each variable at different points along $T$. We present the four main steps of the HMFSSA algorithm in the following subsection.
Embedding, Decomposition, Grouping, and Reconstruction {#embdechmfssa}
------------------------------------------------------
We choose $0 < L < \frac{N}{2}$, set $K=N-L+1$, and we define the linear operator $\utilde{\mathcal{X}}:\mathbb{R}^{pK} \rightarrow \mathbb{F}^{L}$ given by $$\utilde{\mathcal{X}}\left(\pmb{a}\right)=\sum_{j=1}^{p}\sum_{k=1}^{K}a_{k}^{\left(j\right)}\pmb{x}_k^{(j)}, \qquad \pmb{a}\in\mathbb{R}^{pK} \nonumber$$ which follows a similar form as compared to equation (5.1) of the manuscript. The operator, $\utilde{\mathcal{X}}$, is block Hankel, has rank $1\leq \tilde{r} \leq pK$, and we have that $\text{R}\left(\utilde{\mathcal{X}}\right) =\text{sp}\{\pmb{x}_k^{(j)}\}_{k=1,\dots,K}^{j=1,\dots,p}$. It is easy to see from the range of $\utilde{\mathcal{X}}$ why all variables must share a common domain $T$.
Since $\utilde{\mathcal{X}}$ is a finite rank operator and thus compact, we utilize Theorem 7.6 from [@weidmann1980] to obtain the following fSVD for HMFSSA $$\label{hmfssa:decomp}
\utilde{\mathcal{X}}\left(\pmb{a}\right)=\sum_{i=1}^{\tilde{r}}\utilde{\sigma}_{i}{\left\langle \utilde{\mathbf{v}}_{i}, \pmb{a} \right \rangle}_{\mathbb{R}^{pK}}\utilde{\boldsymbol{\psi}}_{i}=\sum_{i=1}^{\tilde{r}}\utilde{\sigma}_{i}\utilde{\mathbf{v}}_{i}\otimes\utilde{\boldsymbol{\psi}}_{i}\left(\pmb{a}\right)=\sum_{i=1}^{\tilde{r}}\utilde{\mathcal{X}}_{i}\left(\pmb{a}\right), \nonumber$$ where $\{\utilde{\sigma}_{i}\}_{i=1}^{\tilde{r}}$ are the singular values, $\{\utilde{\mathbf{v}}_{i}\}_{i=1}^{\tilde{r}}$ are the orthonormal right singular vectors that span $\mathbb{R}^{\tilde{r}}$, and $\{\utilde{\boldsymbol{\psi}}_{i}\}_{i=1}^{\tilde{r}}$ are the orthonormal left singular functions that span an ${\tilde{r}}$-dimensional subspace of $\mathbb{F}^{L}$. Also notice that $\{\utilde{\mathcal{X}}_{i}\}_{i=1}^{\tilde{r}}$ are rank one elementary operators similar to those seen in equation (3.4) of the manuscript.
The grouping stage of HMFSSA is similar to the grouping stage of other types of SSA where we form operators $\utilde{\mathcal{X}}_{I_{q}}:\mathbb{R}^{pK} \rightarrow \mathbb{F}^{L}$ for $1\leq q \leq m$. We finish by projecting each $\utilde{\mathcal{X}}_{I_{q}}$ onto the subspace of block Hankel operators that map from $\mathbb{R}^{pK}$ to $\mathbb{F}^{L}$ to form a collection of $q$ reconstructed MFTS where the projection is completed blockwise using the diagonal averaging technique of [@haghbin2019].
HMFSSA Implementation
---------------------
Implementation of HMFSSA is similar to that of [@haghbin2019] since $\utilde{\mathcal{X}}$ maps to $\mathbb{F}^{L}$. Let $\{\nu_{k}\}_{k \in \mathbb{N}}$ be a known basis of the space $\mathbb{F}$ such that any $x \in \mathbb{F}$ can be projected onto the subspace $\mathbb{H}_d:=\text{sp}\{\nu_{i}\}_{i=1}^{d}$. As such, each $\hat{x} \in \mathbb{H}_d$ can be represented as $$\label{hmfssaimpexp}
\hat{x}=\sum_{k=1}^{d}c_{k,i}\nu_{k}, \quad i=1,\dots,N, \quad c_{k,i} \in \mathbb{R}, \nonumber$$ Let $\mathbb{H}_{d}^{L}$ be the $d$-dimensional subspace formed from the Cartesian product of $L$ copies of $\mathbb{H}_d$, then the rest of the work in defining basis elements of $\mathbb{H}_{d}^{L}$ follows directly from [@haghbin2019]. The work involving the expansion of the lagged vectors, the range of $\utilde{\mathcal{X}}$, the definition of the coefficient matrix $\mathbf{B}:=\left[b_{k,i}\right]_{i=1,\dots,Ld}^{k=1,\dots,pK}=\left[\mathbf{b}_{1}, \mathbf{b}_{2}, \dots, \mathbf{b}_{pK}\right]_{Ld \times pK}$, and the HMFSSA version of Theorem 4.1 seen in the manuscript, also follows from [@haghbin2019] except for the fact that we replace $K$ with $pK$.
HMFSSA SWIR/NIR Study
---------------------
To show that HMFSSA separates out MFTS behavior based on the covariate, we apply HMFSSA with a lag of $45$ to the NIR/SWIR example and obtain the following plots.
[0.33]{} \[fig:swir\_nr-A\]
[0.33]{} \[fig:swir\_nr-B\]
[0.32]{} \[fig:swir\_nr-C\]
[0.32]{} \[fig:swir\_nr-D\]
[0.33]{} \[fig:swir\_nr-E\]
[0.33]{} \[fig:swir\_nr-F\]
In this case, we have $K$ right singular vectors that correspond to NIR densities and $K$ right singular vectors that correspond to SWIR densities. It appears that the first component captures mean behavior of SWIR densities while the second component captures mean behavior of the NIR densities seen in Figure \[fig:hmfssa\_nir\_swir\] subfigures (E) and (F) which is confirmed when we compare with Figure \[fig:swi\_fssa\] subfigures (A) and (C). Rather than combining information to create a more pronounced mean component, HMFSSA works to separate out these behaviors by variable which is expected due to the similarity between HMFSSA and FSSA.
Proofs
======
Notice that since $\text{R}\left(\mathbfcal{X}\right)=\text{sp}\{\pmb{x}_{k}\}_{k=1}^{K}$, then $\mathbfcal{X}$ is a rank $1\leq r \leq K$ operator and thus compact. As such, we have that $\mathbfcal{X}$ is bounded. Let $\pmb{a},\pmb{b} \in \mathbb{R}^{K}$ and $c \in \mathbb{R}$, then we have that $$\mathbfcal{X}\left(\pmb{a}+c\pmb{b}\right)=\sum_{k=1}^{K}\left(a_{k}+cb_{k}\right)\pmb{x}_{k}=\sum_{k=1}^{K}a_{k}\pmb{x}_{k}+c\sum_{k=1}^{K}b_{k}\pmb{x}_{k}=\mathbfcal{X}\pmb{a}+c\mathbfcal{X}\pmb{b} \nonumber$$ which implies that $\mathbfcal{X}$ is a linear operator. Now let $\mathbf{z} \in \mathbb{H}^{L}$, then $${\left\langle \mathbfcal{X}\pmb{a}, \mathbf{z} \right \rangle}_{\mathbb{H}^{L}}=\sum_{k=1}^{K}a_{k}{\left\langle \pmb{x}_{k}, \mathbf{z} \right \rangle}_{\mathbb{H}^{L}}=\pmb{a}^{\top}\mathbfcal{X}^{*}\mathbf{z}={\left\langle \pmb{a}, \mathbfcal{X}^{*}\mathbf{z} \right \rangle}_{\mathbb{R}^{K}},\quad \pmb{a}\in\mathbb{R}^{K} \nonumber$$ and we have that $\mathbfcal{X}^{*}$ is the adjoint of $\mathbfcal{X}$.
Let $\mathbf{S}=\mathbfcal{X}^{*}\mathbfcal{X}$ be the $K \times K$ variance/covariance matrix for the $L$-lagged vectors of $\mathbfcal{X}$. Since $\mathbf{S}$ is a rank $1\leq r\leq K$ matrix, the eigendecomposition of $\mathbf{S}$ gives a set of orthonormal vectors, $\{\mathbf{v}_{i}\}_{i=1}^{r}$, such that for any $\pmb{a} \in \mathbb{R}^{r}$ we have the expansion $\pmb{a}=\sum_{i=1}^{r}\left(\pmb{a}^{\top}\mathbf{v}_{i}\right)\mathbf{v}_{i}$. Notice that the set $\{\mathbf{v}_{i}\}_{i=1}^{r}$ are the right singular vectors of $\mathbfcal{X}$, then it is true that $$\mathbfcal{X}\left(\pmb{a}\right)=\sum_{i=1}^{r}\left(\pmb{a}^{\top}\mathbf{v}_{i}\right)\mathbfcal{X}\mathbf{v}_{i}=\sum_{i=1}^{r}\sigma_{i}\left(\pmb{a}^{\top}\mathbf{v}_{i}\right)\boldsymbol{\psi}_{i}.\nonumber$$ This implies that $\mathbfcal{X}\mathbf{v}_{i}=\sigma_{i}\boldsymbol{\psi}_{i}$ and we have $\boldsymbol{\psi}_{i}=\sigma_{i}^{-1}\mathbfcal{X}\mathbf{v}_{i}$. Now, suppose that we have some $\mathbf{z}\in \text{sp}\{\boldsymbol{\psi}_{i}\}_{i=1}^{r}$. Then we have the expansion given by $\mathbf{z}=\sum_{i=1}^{r}{\left\langle \mathbf{z}, \boldsymbol{\psi}_{i} \right \rangle}_{\mathbb{H}^{L}}\boldsymbol{\psi}_{i}$. By Theorem 7.6 of [@weidmann1980], we have that $\mathbfcal{X}^{*}$ has an SVD with the same eigentriples of $\mathbfcal{X}$ and we obtain the following $$\mathbfcal{X}^{*}\mathbf{z}=\sum_{i=1}^{r}{\left\langle \mathbf{z}, \boldsymbol{\psi}_{i} \right \rangle}_{\mathbb{H}^{L}}\mathbfcal{X}^{*}\boldsymbol{\psi}_{i}=\sum_{i=1}^{r}\sigma_{i}{\left\langle \mathbf{z}, \boldsymbol{\psi}_{i} \right \rangle}_{\mathbb{H}^{L}}\mathbf{v}_{i} \nonumber$$ which implies that $\mathbfcal{X}^{*}\boldsymbol{\psi}_{i}=\sigma_{i}\mathbf{v}_{i}$ and we have that $\mathbf{v}_{i}=\sigma_{i}^{-1}\mathbfcal{X}^{*}\boldsymbol{\psi}_{i}$
\
- Let $M_{j_{q}}=\sum_{i=0}^{j_{q}}d_{i}$, then we obtain the following elements of $\mathbb{H}_{d}$ $$\begin{aligned}
\vec{\hat{y}}_i^{\left(1\right)}&=\begin{pmatrix}\hat{y}_{i}^{\left(1\right)}& 0 & \cdots & 0 \end{pmatrix}=\sum_{q=1}^{d_{1}}c_{i,\ell_{q}}^{\left(1\right)}\vec{\nu}_{q}\\
\vec{\hat{y}}_i^{\left(2\right)}&=\begin{pmatrix}0 & \hat{y}_{i}^{\left(2\right)} & 0 & \cdots & 0 \end{pmatrix}=\sum_{q=d_1+1}^{d_{1}+d_2}c_{i,\ell_{q}}^{\left(2\right)}\vec{\nu}_{q}\\
&\vdots \\
\vec{\hat{y}}_i^{\left(j_q\right)}&=\begin{pmatrix}0 & \cdots & 0 & \hat{y}_{i}^{\left(j_{q}\right)} & 0 & \cdots & 0 \end{pmatrix}=\sum_{q=M_{j_{q}-1}+1}^{M_{j_{q}}}c_{i,\ell_{q}}^{\left(j_q\right)}\vec{\nu}_{q}\\
&\vdots \\
\vec{\hat{y}}_i^{\left(p\right)}&=\begin{pmatrix}0& \cdots & 0 & \hat{y}_{i}^{\left(p\right)} \end{pmatrix}=\sum_{q=M_{j_{p}-1}+1}^{d}c_{i,\ell_{q}}^{\left(p\right)}\vec{\nu}_{q}.\end{aligned}$$ From this, we find that any $\vec{\hat{y}}_{i} \in \mathbb{H}_{d}$ can be expressed as $$\begin{aligned}
\vec{\hat{y}}_{i}&=\begin{pmatrix}\hat{y}_{i}^{\left(1\right)}& \hat{y}_{i}^{\left(2\right)}& \cdots & \hat{y}_{i}^{\left(j_q\right)} & \cdots & \hat{y}_{i}^{\left(p\right)} \end{pmatrix}\\ &=\vec{\hat{y}}_i^{\left(1\right)}+\vec{\hat{y}}_i^{\left(2\right)}+\cdots+\vec{\hat{y}}_i^{\left(j_{q}\right)}+\cdots+\vec{\hat{y}}_i^{\left(p\right)}=\sum_{q=1}^{d}c_{i,\ell_{q}}^{\left(j_{q}\right)}\vec{\nu}_{q}.\end{aligned}$$
- This part of the proof is a direct consequence of the proof of part *i)*
The proof of this Lemma is almost identical to the proof of Lemma 4.1 of [@haghbin2019] and holds without loss of generality.
\
- Let $M_{j_{q}}=\sum_{i=0}^{j_q} d_{i}$ and denote the $i^{\text{th}}$ element of $\mathbf{b}_{k}$ with $b_{k,i}$, then we obtain the following elements of $\mathbb{H}_{d}^{L}$
$$\begin{aligned}
\pmb{x}_{k}^{\left(1\right)}&=\begin{pmatrix}
\vec{\hat{y}}_{k}^{\left(1\right)}\\
\vec{\hat{y}}_{k+1}^{\left(1\right)}\\
\vdots\\
\vec{\hat{y}}_{k+L-1}^{\left(1\right)}
\end{pmatrix}=\begin{pmatrix}
\vec{\hat{y}}_{k}^{\left(1\right)}\\
0\\
\vdots\\
0
\end{pmatrix}+\begin{pmatrix}
0\\
\vec{\hat{y}}_{k+1}^{\left(1\right)}\\
0\\
\vdots\\
0
\end{pmatrix}+\begin{pmatrix}
0\\
0\\
\vdots\\
0\\
\vec{\hat{y}}_{k+L-1}^{\left(1\right)}
\end{pmatrix}=\sum_{i=1}^{Ld_{1}}b_{k,i}\boldsymbol{\phi}_{i}\\
&=\sum_{q_{i}=1}^{d_{1}}\sum_{r_{i}=1}^{L}c_{k+r_{i}-1,\ell_{q_{i}}}^{\left(1\right)}\boldsymbol{\phi}_{i}\\
\pmb{x}_{k}^{\left(2\right)}&=\begin{pmatrix}
\vec{\hat{y}}_{k}^{\left(2\right)}\\
\vec{\hat{y}}_{k+1}^{\left(2\right)}\\
\vdots\\
\vec{\hat{y}}_{k+L-1}^{\left(2\right)}
\end{pmatrix}=\begin{pmatrix}
\vec{\hat{y}}_{k}^{\left(2\right)}\\
0\\
\vdots\\
0
\end{pmatrix}+\begin{pmatrix}
0\\
\vec{\hat{y}}_{k+1}^{\left(2\right)}\\
0\\
\vdots\\
0
\end{pmatrix}+\begin{pmatrix}
0\\ 0\\ \vdots\\ 0\\
\vec{\hat{y}}_{k+L-1}^{\left(2\right)}
\end{pmatrix}=\sum_{i=Ld_{1}+1}^{L\left(d_{1}+d_{2}\right)}b_{k,i}\boldsymbol{\phi}_{i}\\
&=\sum_{q_{i}=d_{1}+1}^{d_{1}+d_{2}}\sum_{r_{i}=1}^{L}c_{k+r_{i}-1,\ell_{q_{i}}}^{\left(2\right)}\boldsymbol{\phi}_{i}\\
&\vdots\end{aligned}$$
$$\begin{aligned}
\pmb{x}_{k}^{\left(p\right)}&=\begin{pmatrix}
\vec{\hat{y}}_{k}^{\left(p\right)}\\
\vec{\hat{y}}_{k+1}^{\left(p\right)}\\
\vdots\\
\vec{\hat{y}}_{k+L-1}^{\left(p\right)}
\end{pmatrix}=\begin{pmatrix}
\vec{\hat{y}}_{k}^{\left(p\right)}\\
0\\ \vdots\\ 0
\end{pmatrix}+\begin{pmatrix}
0\\
\vec{\hat{y}}_{k+1}^{\left(p\right)}\\
0\\ \vdots\\ 0
\end{pmatrix}+\begin{pmatrix}
0\\ 0\\ \vdots\\ 0\\
\vec{\hat{y}}_{k+L-1}^{\left(p\right)}
\end{pmatrix}=\sum_{i=LM_{p-1}+1}^{Ld}b_{k,i}\boldsymbol{\phi}_{i}\\
&=\sum_{q_{i}=M_{p-1}+1}^{d}\sum_{r_{i}=1}^{L}c_{k+r_{i}-1,\ell_{q_{i}}}^{\left(p\right)}\boldsymbol{\phi}_{i}.\end{aligned}$$
As a result, we find that $\pmb{x}_{k}=\pmb{x}_{k}^{\left(1\right)}+\pmb{x}_{k}^{\left(2\right)}+\cdots+\pmb{x}_{k}^{\left(p\right)}=\sum_{i=1}^{Ld}b_{k,i}\boldsymbol{\phi}_{i}$ and the coefficients found in $\mathbf{b}_{k}$ are found in equation (4.3) of the manuscript.
- $$\label{expansion of X}
\mathbfcal{X}(\pmb{a})=\sum_{k=1}^{K}a_{k}\pmb{x}_k
=\sum_{i=1}^{Ld}\left(\sum_{k=1}^{K} b_{k,i}a_{k}\right)\pmb {\phi}_{i}
=\mathbfcal{P}(\mathbf{B}\pmb{a}).$$
This proof is a direct consequence of Theorem 4.1 of [@haghbin2019]
\
- Let $\pmb{a} \in \mathbb{R}^{K}$. Then we have that $$\mathcal{U}\mathbfcal{X}\left(\pmb{a}\right)=\sum_{k=1}^{K}a_{k}\mathcal{U}\pmb{x}_{k}=\sum_{k=1}^{K}a_{k}\underline{\mathbf{x}}_{k}=\underline{\mathcal{X}}\left(\pmb{a}\right) \nonumber$$ and as such, we have that $\underline{\mathcal{X}}=\mathcal{U}\mathbfcal{X}$.\
- Again, let $\pmb{a} \in \mathbb{R}^{K}$, then we have $$\mathcal{U}\mathbfcal{X}\left(\pmb{a}\right)=\sum_{i=1}^{r}\sigma_{i}\pmb{a}^{\top}\mathbf{v}_{i}\mathcal{U}\boldsymbol{\psi}_{i}=\sum_{i=1}^{r}\sigma_{i}\pmb{a}^{\top}\mathbf{v}_{i}\underline{\boldsymbol{\psi}}_{i}=\underline{\mathcal{X}}\left(\pmb{a}\right). \nonumber$$ This implies that the $i^{\text{th}}$ eigentriple of $\underline{\mathcal{X}}$ is $\left(\sigma_{i},\mathbf{v}_{i},\underline{\boldsymbol{\psi}}_{i}\right)$ and that $\underline{\boldsymbol{\psi}}_{i}=\mathcal{U}\boldsymbol{\psi}_{i}$.
|
---
abstract: 'We use an improved version of the SU(3) flavour parity-doublet quark-hadron model to investigate the higher order baryon number susceptibilities near the chiral and the nuclear liquid-gas transitions. The parity-doublet model has been improved by adding higher-order interaction terms of the scalar fields in the effective mean field Lagrangian, resulting in a much-improved description of nuclear ground-state properties, in particular the nuclear compressibility. The resulting phase diagram of the model agrees qualitatively with expectations from lattice QCD, i.e., it shows a crossover at zero net baryo-chemical potential and a critical point at finite density. Using this model, we investigate the dependence of the higher-order baryon number susceptibilities as function of temperature and chemical potential. We observe a strong interplay between the chiral and liquid-gas transition at intermediate baryo chemical potentials. Due to this interplay between the chiral and the nuclear liquid-gas transitions, the experimentally measured cumulants of the net baryon number may show very different beam energy dependence, subject to the actual freeze-out temperature.'
author:
- 'A. Mukherjee$^{1,2}$, J. Steinheimer$^1$ and S. Schramm$^{1,2}$'
bibliography:
- 'bibnew.bib'
title: |
Higher-order baryon number susceptibilities:\
interplay between the chiral and the nuclear liquid-gas transitions
---
Introduction
============
The theory of quantum chromodynamics (QCD) is expected to have a rich phase structure at finite chemical potential and temperature [@Stephanov:1998dy; @McLerran:2007qj; @Alford:2007xm]. Its study is a central topic of high energy nuclear physics. Experimental programs at the Large Hadron Collider (LHC) and the Relativistic Heavy Ion Collider (RHIC) are currently investigating the properties of hot and dense QCD matter. Future programs at RHIC, the Facility for Anti-Proton and Ion Research (FAIR) and the NICA facility are aimed at a better understanding of the phase transition from hadronic to de-confined quark matter and the transition from a phase where chiral symmetry is broken to one where it is restored.\
Theoretical studies employing lattice QCD methods have already established that the transition from hadrons to quarks proceeds as a smooth crossover in the case of vanishing net baryon number density [@Borsanyi:2010cj; @Bazavov:2010sb]. For finite net baryon density, the use of standard lattice QCD methods is limited by the so-called fermion sign problem. Some conclusions can be drawn by extending the lattice thermodynamic quantities, via a Taylor expansion around $\mu_\text{B}=0$ [@Allton:2002zi; @Gunther:2016vcp], for values of $\mu_{\rm B}/T<2$. However, to go to even higher densities, higher orders of the expansion coefficients need to be known to a very good accuracy, a requirement which cannot be met with the current computational possibilities. Thus, the current conclusions are that a first order phase transition seems very unlikely for chemical potentials smaller than $\mu_\text{B}/T \approx 2-3$ (i.e., for $\mu_{\rm B} < 200-300$ MeV).\
At very large net baryon densities and low temperatures, astrophysical observations may also help to constrain the QCD Equation of State. Nuclear matter ground-state properties have been derived from measurements to a high accuracy. On the other hand, properties of compact stars like their masses and eventually, their radii, can serve as important information for determining the equation of state at several times nuclear ground-state density (c.f., e.g., [@Steiner:2010fz]).\
With the currently available information, we can theorise that, in the temperature ($T$) and baryo-chemical potential ($\mu_\text{B}$) plane, the conjectured QCD phase diagram consists, primarily, of three parts:
1. a high $\mu_\text{B}$ and temperature region, where chiral symmetry is restored, containing a de-confined quark-gluon plasma (QGP),
2. a low temperature and moderate $\mu_\text{B}$ region containing dense, strongly interacting nuclear matter and nuclei, and
3. a low temperature and low chemical potential region made up of purely hadronic matter described by a hadron resonance gas.
The transition from a phase where chiral symmetry is broken to one where it is restored, for lower temperatures, is conjectured to be a first-order phase transition, which switches to a continuous transition at a point known as the ’Critical Point’ ($T_\text{C}$). At values of $\mu_\text{B}$ lower than that at $T_\text{C}$, the transition is termed a ’Crossover Transition’. The transition from a dilute gas of hadrons to bound nuclear matter is also a first-order phase transition, generally called the nuclear liquid-gas phase transition.\
In order to verify this suggested phase structure experimentally, heavy-ion collision experiments are conducted at various beam energies. Some particularly interesting observables in these experiments are the moments of the multiplicity distributions, the susceptibilities of the different conserved net charges (baryon number, strangeness and electric charge) [@Aggarwal:2010wy; @PhysRevLett.112.032302]. The reason behind this interest is that one proposed universal characteristic of the critical point of QCD is the divergence of the correlation length, $\xi \rightarrow \infty$, of the order-parameter ($\sigma$ and $\zeta$) fields. As a consequence, higher-order fluctuation moments of observables coupled to these fields also diverge, at least for an infinite system size and relaxation time [@Stephanov:2008qz; @Stephanov:1998dy].\
The direct comparison of these measured susceptibilities with lattice QCD results is neither straightforward, nor entirely unambiguous. Currently known complications in the interpretation of such measurements include: corrections of experimental efficiency and acceptance effects [@Bzdak:2012ab; @Bzdak:2016qdc; @kitazawa2016efficient], cluster formation [@Feckova:2015qza], conservation laws [@Begun:2004gs; @Bzdak:2012an], corrections due to the finite size of the system [@Gorenstein:2008et], fluctuations of the system volume[@Gorenstein:2011vq; @Sangaline:2015bma] and fluctuations present in the initial state of the collision [@Spieles:1996is]. Moreover, it has been pointed out that the hadronic decoupling phase, which occurs after hadronisation, can have a strong influence on the observed multiplicity distributions [@Kitazawa:2012at; @Asakawa:2000wh; @Jeon:2000wg; @Steinheimer:2016cir].\
On the theoretical side, the measured susceptibilities, at least for large values of the chemical potential, are usually compared to effective models, which normally do not include a de-confinement or chiral transition and a nuclear liquid-gas transition simultaneously.
Current heavy-ion experiments create systems with widely varying $\mu_{\rm B}$ and high $T$ values. In fact only the experiments at the LHC and high-energy RHIC can be directly connected to LQCD results. Most interesting results on the baryon number susceptibilities, which is the central topic of the manuscript, have been obtained at rather low beam energies e.g. at the RHIC beam energy scan and the HADES experiment at GSI. Here the values of chemical potential are large and temperatures are moderate. In our paper we focus on the fact that at such high values of chemical potential ($\mu_{\rm B} > 400$ MeV) the effect of nuclear interactions can and should not be neglected. In this paper, we will discuss how the observed susceptibilities may change if one takes into account an equation of state that includes a nuclear liquid-gas transition, as well as a first order chiral transition at high baryon densities. We will also show, how the observed susceptibilities change with beam energy for different freeze-out lines in the phase diagram and how the interplay between the liquid-gas transition and the chiral transition manifests itself in the beam-energy dependence.
Model Description
=================
The Parity-Doublet Model
------------------------
The parity-doublet model, as used in this paper, serves as an effective approach to describe the strongly interacting hadronic, and in extension, quark matter. In this approach, an explicit mass term for baryons in the Lagrangian is possible, which preserves chiral symmetry. Here, the signature for chiral symmetry restoration is the degeneracy of the usual baryons and their respective negative-parity partner states. In the following, we outline the basic SU(3) parity model and determine nuclear matter saturation properties with this ansatz. Subsequently, we calculate the phase diagram of isospin-symmetric matter by varying the baryo-chemical potential and temperature of the system.
In the model approach, positive and negative parity states of the baryons are grouped in doublets $N = (N^+,N^-)$ as discussed in [@PhysRevD.39.2805; @Hatsuda:1988mv]. The flavour SU(3) extension of the approach, using the non-linear representation of the fields, is quite straightforward, as shown in [@Nemoto:1998um] and details can be found in [@Steinheimer:2011ea]. In addition, as outlined in [@Papazoglou:1997uw], one constructs SU(3)-invariant terms in the Lagrangian including the meson-baryon and meson-meson self-interaction terms.
Taking into account the scalar and vector condensates in mean-field approximation, the resulting Lagrangian ${\cal L_{\rm B}}$ reads as [@Steinheimer:2011ea] $$\begin{aligned}
{\cal L_{\rm B}} &=& \sum_i (\bar{B_\text{i}} i {\partial\!\!\!/} B_\text{i})
+ \sum_i \left(\bar{B_\text{i}} m^*_\text{i} B_\text{i} \right) \nonumber \\ &+&
\sum_i \left(\bar{B_\text{i}} \gamma_\mu (g_{\omega \text{i}} \omega^\mu +
g_{\rho \text{i}} \rho^\mu + g_{\phi \text{i}} \phi^\mu) B_\text{i} \right) ~,
\label{lagrangian2}\end{aligned}$$ summing over the states of the baryon octet. The effective masses of the baryons (assuming isospin-symmetric matter) are $$\begin{aligned}
m^*_{\text{i}\pm} = \sqrt{ \left[ (g^{(1)}_{\sigma \text{i}} \sigma + g^{(1)}_{\zeta \text{i}} \zeta )^2 + (m_0+n_\text{s} m_\text{s})^2 \right]}\nonumber \\
\pm g^{(2)}_{\sigma \text{i}} \sigma \pm g^{(2)}_{\zeta \text{i}} \zeta ~,
\label{effmass}\end{aligned}$$ with the $g^\text{(j)}_\text{i}$ as the coupling constants of the baryons with the scalar fields. In addition, there is an SU(3) symmetry-breaking mass term proportional to the strangeness, $n_\text{s}$, of the respective baryon. Note that the parity-doublet models allow for two different scalar coupling terms $i=1,2$.
The scalar meson interaction, driving the spontaneous breaking of the chiral symmetry, can be written in terms of SU(3) invariants $I_2 = (\sigma^2+\zeta^2) ,~ I_4 = -(\sigma^4/2+\zeta^4) $ and $I_6 = (\sigma^6 + 4\zeta^6) $ as: $$V = V_0 + \frac{1}{2} k_0 I_2 - k_1 I_2^2 - k_2 I_4 + k_6 I_6 ~,
\label{veff}$$ where $V_0$ is fixed by demanding a vanishing potential in the vacuum.\
In this work, the last term, $k_6 I_6$, has been introduced following Ref. [@Motohiro:2015taa], which results in an improved lowering of the calculated nuclear matter compressibility value to 267 MeV, which is now in reasonable agreement with the phenomenologically obtained range of about $200-280$ MeV.
The set of scalar coupling constants are fitted in order to reproduce the vacuum masses of the nucleon, and the $\Lambda$, $\Sigma$, and $\Xi$ hyperons , whereas the vector couplings are chosen to reproduce reasonable values for nuclear ground-state properties (see Ref. [@Steinheimer:2011ea]).\
As a likely parity partner of the nucleon we choose the N(1535) resonance with its correspondent mass. In order to keep the number of parameters small, we assume equal mass splitting of the baryons with their respective parity partners, therefore setting $g^{(2)}_{\zeta \text{i}} = 0$. An SU(3) description, in addition to enhancing the number of degrees of freedom, also necessarily increases the number of parameters. In order to avoid being overwhelmed by too many new parameters, we assume, for simplicity, that the splitting of the various baryon species and their respective parity partners is of the same value for all baryons, which is achieved by setting $g^{(2)}_{\sigma \text{i}} \equiv g^{(2)}$ and $g^{(2)}_{\zeta \text{i}} = 0$.\
The hyperonic vector interactions were tuned to generate phenomenologically acceptable optical potentials of the hyperons in ground-state nuclear matter, with $U_\Lambda(\rho_o) = -28\,{\rm MeV}$ and $U_\Xi(\rho_o) = -18\,{\rm MeV}$. The mass difference due to the strange quark was fixed at $m_\text{s} = 150$ MeV. All parameter values used are summarized in Table \[modpar\].
------------------------------ ------------------------- ----------------------
$k_0$ $k_1$ $k_2$
$(242.61 \text{ MeV})^2$ 4.818 -23.357
$k_6$ $\epsilon$ $g_\sigma^1$
$(0.276)^6 \text{ MeV}^{-2}$ $(75.98 \text{ MeV})^4$ -8.239296
$g_\sigma^8$ $\alpha_\sigma$ $g_{\text{N}\omega}$
-0.936200 2.435059 5.45
------------------------------ ------------------------- ----------------------
: Model Parameters: the F, D, and S-type couplings $\alpha_\sigma$, $g_\sigma^8$ and $g_\sigma^1$ determine the couplings of the various baryons.[]{data-label="modpar"}
Mesons and Quarks
-----------------
At some temperature, QCD exhibits a transition from a hadronic to a de-confined phase, at which point, the quarks become the dominant degrees of freedom. This transition occurs as a smooth crossover, at least for $\mu_{\mathrm{B}}=0$. Consequently there has been discussion about the actual temperature up to which a hadronic description is still valid [@Aoki:2006we; @Bazavov:2010bx; @Vovchenko:2015cbk]. We can only say for sure that the order parameter of the chiral transition, the chiral condensate, has an inflection point at the pseudo-critical temperature $T_{\mathrm{PS}} \approx 155$ MeV, and that de-confinement occurs in a temperature region of $T_\text{dec}\approx 150 - 400$ MeV. Nevertheless, at some point, the hadronic parity-doublet model will not be the appropriate effective description of QCD and one needs to introduce a de-confinement mechanism in the model. In this work, we will apply a mechanism that has been introduced in [@Steinheimer:2010ib], to add a de-confinement transition in a chiral hadronic model. This is done by adding an effective quark and gluon contribution, as done in the PNJL approach [@Fukushima:2003fw; @Ratti:2005jh]. This model uses the Polyakov loop $\Phi$ as the order parameter for de-confinement. $\Phi$ is defined via $$\Phi=\frac{1}{3}\text{Tr}[\exp{(i\int d\tau A_4)}],
\label{phidef}$$ where $A_4=iA_0$ is the temporal component of the SU(3) gauge field, distinguishing $\Phi$ and its conjugate $\Phi^{*}$, at finite baryon densities [@Fukushima:2006uv; @Allton:2002zi; @Dumitru:2005ng].\
The effective masses of the quarks are generated by the scalar mesons, except for a small explicit mass term ($\delta m_\text{q}=5$ MeV and $\delta m_\text{s}=150$ MeV, for the strange quark) and $m_0$: $$\begin{aligned}
&m_\text{q}^*=g_{\text{q}\sigma}\sigma+\delta m_\text{q} + m_{0\text{q}}&\nonumber\\
&m_\text{s}^*=g_{\text{s}\zeta}\zeta+\delta m_\text{s} + m_{0\text{q}},&\end{aligned}$$ with values of $g_{\text{q}\sigma}=g_{\text{s}\zeta}= 4.0$. Similar to the case of the baryons, we also introduced a mass parameter $m_{0\text{q}}= 165$ MeV for the quarks. Again, this additional mass term can be due to a coupling of the quarks to the dilaton field (gluon condensate). Given such a mass term, the quarks do not appear in the nuclear ground-state, which would be a clearly non-physical result. This also permits us to set the vector type repulsive interaction strength of the quarks, to zero. A non-zero vector interaction strength would lead to a massive deviation of the quark number susceptibilities from lattice data, as has been indicated in different mean field studies [@kunihiro1991quark; @Ferroni:2010xf; @Steinheimer:2010sp; @steinheimer2014lattice].\
A coupling of the quarks to the Polyakov loop is introduced in the thermal energy of the quarks. Their thermal contribution to the grand-canonical potential $\Omega$, can be written as: $$\Omega_\text{q}=-T \sum_{{\rm i}\in Q}{\frac{\gamma_\text{i}}{(2 \pi)^3}\int{d^3k \ln\left(1+\Phi \exp{\frac{E_\text{i}^*-\mu_\text{i}}{T}}\right)}}$$ and $$\noindent
\Omega_{\overline{\text{q}}}=-T \sum_{\text{i}\in Q}{\frac{\gamma_\text{i}}{(2 \pi)^3}\int{d^3k \ln\left(1+\Phi^* \exp{\frac{E_\text{i}^*+\mu_\text{i}}{T}}\right)}}~.$$ The sums run over all quark flavours, where $\gamma_\text{i}$ is the corresponding degeneracy factor, $E_\text{i}^*=\sqrt{m_\text{i}^{*2}+p^2}$ the energy and $\mu_\text{i}^*$ the chemical potential of the quark.\
All thermodynamic quantities: energy density $e$, entropy density $s$, as well as the densities of the different particle species $\rho_\text{i}$, are derived from the grand-canonical potential. It includes the effective potential $U(\Phi,\Phi^*,T)$, which controls the dynamics of the Polyakov loop. For simplicity, in our approach we adopt the ansatz proposed in [@Ratti:2005jh]: $$\begin{aligned}
U&=&-\frac12 a(T)\Phi\Phi^*\nonumber\\
&+&b(T)\ln[1-6\Phi\Phi^*+4(\Phi^3\Phi^{*3})-3(\Phi\Phi^*)^2],\end{aligned}$$ with $a(T)=a_0 T^4+a_1 T_0 T^3+a_2 T_0^2 T^2$, $b(T)=b_3 T_0^3 T$.\
The parameters $a_0, a_1, a_2$ and $b_3$ are initially fixed, as in [@Ratti:2005jh], by demanding a first order phase transition in the pure gauge sector at $T_0=270$ MeV, and that the Stefan-Boltzmann limit of a gas of gluons is reached for $T\rightarrow\infty$. In general, of course, the presence of quarks may have a significant influence on the Polyakov potential [@Schaefer:2007pw], and in order to obtain a crossover transition at $\mu_\text{B}=0$, we change $T_0$ to 200 MeV.\
In the following, as a way to remove the hadrons once quarks are de-confined, we introduce excluded volumes for the hadrons in the system. Including effects of finite-volume particles in a thermodynamic model for hadronic matter was proposed long ago [@Hagedorn:1980kb; @Baacke:1976jv; @Gorenstein:1981fa; @Hagedorn:1982qh; @Rischke:1991ke; @Cleymans:1992jz; @Kapusta:1982qd; @Bugaev:2000wz; @Bugaev:2006pt; @Satarov:2009zx]. In recent publications [@Steinheimer:2010ib; @Steinheimer:2010sp], we adopted this ansatz to successfully describe a smooth transition from a hadronic to a quark dominated system (see also [@Sakai:2011fa]).\
In particular, we introduce the quantity $v_\text{i}$, which is the volume excluded by a particle of species $i$, where we only distinguish between baryons, mesons and quarks. Consequently, $v_\text{i}$ can assume three values: $$\begin{aligned}
v_\text{quark}&=&0 \nonumber \\
v_\text{baryon}&=&v \nonumber \\
v_\text{meson}&=&v/a, \nonumber\end{aligned}$$ where $a$ is a number larger than one. In our calculations, we choose a value of $a=8$, which assumes that the radius $r$ of a meson is half of the radius of a baryon. Note that at this point, we neglect any possible density-dependent and Lorentz contraction effects on the excluded volumes as introduced in [@Bugaev:2000wz; @Bugaev:2006pt]. The modified chemical potential $\widetilde{\mu}_\text{i}$, which is connected to the real chemical potential $\mu_\text{i}$ of the $i$-th particle species, is obtained by the following relation: $$\widetilde{\mu}_\text{i}=\mu_\text{i}-v_\text{i} \ P,$$ where $P$ is the sum over all partial pressures. To be thermodynamically consistent, all densities ($\widetilde{e_\text{i}}$, $\widetilde{\rho_\text{i}}$ and $\widetilde{s_\text{i}}$) have to be multiplied by a volume correction factor $f$, which is the ratio of the total volume $V$ and the reduced volume $V'$, not being occupied: $$f=\frac{V'}{V}=(1+\sum_{\rm i}v_\text{i}\rho_\text{i})^{-1}$$ $$e=\sum_{\rm i} f \ \widetilde{e_\text{i}}, \quad
\rho_\text{i}=f \ \widetilde{\rho_\text{i}}, \quad
s=\sum_{\rm i} f \ \widetilde{s_\text{i}}~.$$ As a consequence, the chemical potentials of the hadrons are decreased by the quarks, but not vice-versa. In other words, as the quarks start appearing, they effectively suppress the hadrons by changing their chemical potential, while the quarks are only affected through the volume correction factor $f$.
Results
=======
Comparison with Lattice QCD {#latsec}
---------------------------
A comparison with data obtained from lattice QCD calculations is necessary in order to benchmark the model results and their subsequent modifications. To that end, we determine the interaction measure $I$, defined as: $$I = \frac{\varepsilon-3P}{T^4} ~,
\label{intmes}$$ with $\varepsilon, P$ and $T$ as the energy density, pressure and temperature, respectively. The model result for $I$ at $\mu_\text{B}=0$ as function of temperature, in comparison to available lattice data [@Borsanyi:2013bia], is shown in Fig. \[ept\].
We observe that, indeed, the model gives a good description of LQCD thermodynamics below the pseudo-critical temperature $T_{\rm C}$. But, although the shapes obtained from both sets of data are similar, the peak value of the interaction measure is much higher in case of the parity-doublet model than that obtained from lattice QCD. This is likely a result of our use of the standard Polyakov loop potential for the description of the quark and gluon deconfinement. For future investigations, it is therefore interesting to implement an improved version of the Polyakov potential which better describes the thermodynamics at $\mu_{\rm B}=0$. At this point, we want to clarify the intent of this paper again: instead of constraining our model parameters by a fit to lattice QCD results at $\mu_{\rm B}=0$, we constrain our model parameters by actual observables at large baryon densities and low temperatures, e.g., nuclear ground-state properties and neutron star observations. Starting from these parameters, we then extend the model to low densities where the remaining free parameters (mainly those of the Polyakov loop) are subsequently used, to get at least a reasonable description of low$-\mu_{\rm B}$ lattice results. Within the current set up of the model, a perfect description of lattice QCD results appears to be unachievable.
Checking model results for nuclear model ground-state properties, we obtain phenomenologically acceptable values of a nucleonic binding energy of $-16.00$ MeV and a compressibility ($\kappa$) of 267.12 MeV, for a saturation density $\rho_0 = 0.142 \text{ fm}^{-3}$. Note that the compressibility, which in general tends to be very high in parity-doublet models, has a reasonable value (see also [@Motohiro:2015taa]).
Pressure and Quark Fraction {#pqf}
---------------------------
Some interesting characteristics of the system may be revealed by observing the pressure of the system, along the transition lines, as a function of temperature.
For the nuclear liquid-gas transition we define this line as the maximum of the derivative of the net-baryon density with respect to the baryo-chemical potential. Similarly, for the chiral transition, we define it as the maximum of the derivative of the $\sigma$ field (chiral condensate) with respect to the baryo-chemical potential (or the temperature, for baryo-chemical potential values of $\mu_{\rm B} < 400$ MeV, i.e. beyond the merger of the two transition lines). Note that both criteria can be used equivalently for either transition, as the net-baryon density and value of the sigma field are intimately related (c.f. [@Walecka:1974qa; @Bender:2003jk; @Cohen:1991nk] and references therein). This means that when we observe a rapid change in the net baryon number density, we will also observe a rapid change in the chiral condensate and vice-versa. Thus, both criteria can be used to identify the crossover lines of the chiral and LG transition. Note that, if there was an additional separation of the chiral and deconfinement line (e.g., as discussed in [@Ferreira:2013tba]), the situation we try to describe would be even more complicated.
In the region where both first-order transitions switch to crossovers, we fit a double-Gaussian function to the derivative of the net-baryon density with respect to the baryo-chemical potential, assigning each peak to one transition line. One should, of course, note that the two transitions show clear differences. Even though the value of the chiral condensate changes slightly at the liquid gas transition, chiral symmetry is restored much later; after the chiral transition; where the chiral condensate, essentially, drops to zero.
The model results for the pressure along the transition and crossover lines, are shown in Fig. \[pplot\], where the baryo-chemical potential increases with decreasing temperature along both lines (cf. Fig. \[crit\]). The behaviour of the pressure along the transition line for the liquid-gas transition is as expected. Since the baryon density along the liquid-gas transition does not change considerably with increasing temperature, the change in the pressure is driven, primarily, by the increase in the entropy caused by the increasing temperature. Such a behaviour can be observed when the specific entropy in the gas phase is larger than that in the liquid phase, as derived from the Clapeyron equation [@Hempel:2013tfa].
For the chiral transition, the change of the pressure along the transition line is more complicated. At large temperatures and small chemical potentials, the pressure essentially follows the trend of the nuclear liquid-gas transition, as the meson-dominated system transitions smoothly into a system dominated by quark and gluons. As the chemical potential increases, however, the change in degrees of freedom is manifested more strongly in a change of net baryon number, as the system transitions from heavy baryons to light baryons. Consequently, the change in net baryon number dominates the change of pressure and thus, the pressure along the transition line shows a behaviour opposite to that observed during the liquid-gas transition. As the transition line goes to even lower temperatures, the behaviour of the pressure changes direction again. This time, however, it is a result of the change in curvature of the transition line in the $T$-$\mu_{\text{B}}$ diagram (cf. Fig. \[crit\]). This is, most likely, an artefact of the Polyakov model, which is not very reliable at low values of temperature and large values of baryo-chemical potential. In any case, it is important to note that the pressure at zero temperature for the de-confinement transition takes a finite value, which is an important property of a “realistic" model for the QCD EoS.
In order to illustrate the change in degrees of freedom at the transition lines, one can determine the so-called quark fraction $q_{\rm f}$, defined as: $$q_{\rm f} = \frac{\varepsilon_{\text{quark}}+\varepsilon_{\text{Polyakov}}}{\varepsilon_{\text{baryon}}+\varepsilon_{\text{meson}}+\varepsilon_{\text{Polyakov}}}~;
\label{qf}$$ with $\varepsilon_{\text{quark}}$, $\varepsilon_{\text{baryon}}$, $\varepsilon_{\text{meson}}$ and $\varepsilon_{\text{Polyakov}}$ denoting the energy density contributions from the quarks, baryons (including quarks), mesons and the Polyakov loop contribution from the gluons, respectively. The variation in this quantity, as a function of temperature, is shown in Fig. \[qfrac\] along both transition lines.
Along the LG transition line, the quark fraction is essentially zero for temperatures below 100 MeV (where the interplay between the two crossover transitions is negligible). Above this value it gradually rises (cf. Fig. \[qfrac\]) as the LG crossover line approaches the de-confinement crossover (cf. Fig. \[crit\]), thereby introducing an increasing number of quarks in the system.\
For the chiral transition, the quark fraction starts to increase, quite sharply, at around 100 MeV (cf. Fig. \[qfrac\]). Below that temperature, the transition is, apparently, a dominantly chiral one, with only a slow change in degrees of freedom. At very low temperatures, however, a slow change in the quark fraction is observed once again. This is because, at these temperatures, quarks can be introduced into the system due to the large chemical potential and due to the quark-suppressing effect of the Polyakov potential disappearing at low temperatures.
Susceptibilities and the QCD Phase Diagram {#suscsec}
------------------------------------------
The thermodynamics of QCD at small values of $\mu_\text{B}/T$ can be obtained by a Taylor expansion of lattice results at $\mu_\text{B} = 0$, in terms of the baryo-chemical potential. Expanding the pressure, $$P = -\Omega = \frac{T\ln\mathcal{Z}}{V}~,
\label{pressold}$$ where $\Omega$ is the grand-canonical potential, $V$ the volume and $\mathcal{Z}$ the grand-canonical partition function, the corresponding expansion coefficients $c_\text{n}^\text{B}$, or alternatively, the baryon number susceptibilities $\chi_\text{n}^\text{B}$, result as: $$\frac{\chi_\text{n}^\text{B}}{T^{4-n}} = n! \ c_\text{n}^\text{B} (T) = \frac{\partial^\text{n}(P(T,\mu_\text{B})/T^4)}{\partial(\mu_\text{B}/T)^\text{n}}~.
\label{susc}$$ The behaviour of these coefficients - or susceptibilities - especially, the third-order $\chi_3^{\rm B}$ (skewness) and fourth-order $\chi_4^{\rm B}$ (kurtosis), in and around the phase transitions, are expected to provide stronger signals of criticality, as compared to the second-order $\chi_2^{\rm B}$, because they diverge with a higher power of the correlation length of the order parameter, close to a second-order-type phase transition.
In experiment, usually the normalized ratios $\chi_3^{\rm B}/\chi_2^{\rm B}$ and $\chi_4^{\rm B}/\chi_2^{\rm B}$ are extracted from data, in order to remove the volume dependence of the susceptibilities (N. B.: this does not remove their dependence on volume fluctuations). Before calculating the susceptibilities, it is useful to clearly identify the crossover and first-order phase transition lines of the QCD phase diagram, within this model; as discussed in section \[pqf\] and as shown in Fig. \[crit\]. We observe that both critical end-points occur at a very low temperature. We also observe that the associated crossover lines, while first separated, merge at an intermediate chemical potential $\mu_{\rm B} \approx 400$ MeV. The figure also shows lines of constant entropy-per-baryon (isentropes) for various values of entropy-per-baryon. The isentropes show a distinct structure, a bending over at the crossover, as the dominant degrees of freedom change from hadrons to quarks. At the junction of the liquid-gas and chiral crossover transitions, the isentropes signal a sharpening of the transition generated by the interplay of the two crossovers.
To calculate the susceptibilities, the equations of motion, following from Eqs. (\[lagrangian2\]) and (\[veff\]), are solved self-consistently in mean field approximation, by minimising the grand-canonical potential as a function of the baryo-chemical potential and the temperature, as before. Then, the second-, third- and fourth-order derivatives of Eqn. (\[susc\]) are numerically calculated using a five-point formula. For all temperatures ranging from 15 MeV to 180 MeV, and all baryo-chemical potential values from 0 MeV to 1200 MeV, the results with the previously discussed ratios of susceptibilities are shown in Figs. \[tt\] and \[tt2\].
The figures illustrate the effect that the two phase transitions have on the susceptibility values. In the de-confinement phase the susceptibilities have values smaller than 1, as expected for a gas of low-mass fermions. In the region below the LG phase transition (at values of $\mu_\text{B} < 600 \ \text{MeV}$), they are consistently close to 1, since the system is composed of bound hadrons, where a value of 1 for the cumulants of conserved charges is expected. For the region between the crossover transitions from liquid (bound hadrons) to gas (of hadron resonances) and from a hadron resonant gas (HRG) to the QGP, an interplay between the two phase transitions can be observed. This results in the cumulants sometimes having values below 1, or even below zero, and sometimes having values greater than 1, in this intermittent region.
In order to give a rough estimate of the susceptibility ratios that could be expected from experiment, one has to define the point in the phase diagram at which the fluctuations are, essentially, frozen out. This point will be different for each beam energy and system size, and in general, is not trivially defined. However, it has been found that the measured mean multiplicities of stable hadrons can be nicely described by a thermal fit, with a single value of $T$ and $\mu_\text{B}$, for a specific beam energy. For different beam energies, different $T$ and $\mu_\text{B}$ values are obtained, thus producing the so-called ’Freeze-out Line’ [@Andronic:2008gu]. By fitting experimental data, the equation of a freeze-out line can be obtained as:
![\[colour online\] Susceptibility ratio $\chi_3^{\rm B}/\chi_2^{\rm B}$ as a function of temperature and baryo-chemical Potential. The solid black lines denote the $1^{st}$-order LG and chiral transitions, the dashed black lines denote the crossovers and the green (solid and dashed) lines denote the freeze-out curves for $T_{\rm lim}$ values 165 MeV and 120 MeV, respectively.[]{data-label="tt"}](chi3_chi2.eps){width="51.00000%"}
$$T \text{ (MeV)} = \frac{T_{\text{lim}}}{1+ \exp \left[2.60 - \frac{\ln \left(\sqrt{s_{\text{NN}} \text{ (GeV)}}\right)}{0.45}\right]}~,
\label{tfrz}$$
where $\mu_\text{B}$ and $\sqrt{s_{\text{NN}}}$ are related as
$$\mu_\text{B} \text{ (MeV)} = \frac{1303}{1+0.286\sqrt{s_{\text{NN}} \text{ (GeV)}}}~;
\label{mfrz}$$
with $\sqrt{s_{\text{NN}}}$ being the beam energy in GeV and $T_{\text{lim}}$ being a parameter. Again, one must keep in mind that Eqs. (\[tfrz\]) and (\[mfrz\]) represent a mere approximation, and the true freeze-out process is much more complicated than is assumed in this study [@Steinheimer:2016cir]. Nevertheless, it is worthwhile to study the behaviour of the normalized cumulants along different possible freeze-out lines.
In this study, two different freeze-out lines, obtained by using two different values of $T_{\text{lim}}$ (165 MeV and 120 MeV) in Eqs. (\[tfrz\]) and (\[mfrz\]) as shown in Figs. \[tt\] and \[tt2\], are used. Here, the higher value corresponds to the expected latest point of chemical equilibrium while the lower value is closer to the kinetic freeze-out point. For an ideal Boltzmann gas, the susceptibility ratio $\chi_4^{\rm B}/\chi_2^{\rm B}$ along these freeze-out lines can been shown to be equal to 1.
![\[colour online\] Same as Fig. \[tt\] for the susceptibility ratio $\chi_4^{\rm B}/\chi_2^{\rm B}$.[]{data-label="tt2"}](chi4_chi2.eps){width="51.00000%"}
The extracted values of the normalized cumulants are displayed in Figs. \[low1\] and \[high1\], as functions of the beam energy $\sqrt{s_\text{NN}}$. In the case of the low freeze-out temperature, the measured cumulants essentially resemble those of an ideal HRG, down to beam energies $\sqrt{s_\text{NN}}\le 10$ GeV. Below that energy, the measured susceptibilities actually probe the critical behaviour of the nuclear liquid-gas transition and not that of the QCD chiral transition, as already found in [@Fukushima:2014lfa; @Vovchenko:2015pya; @Vovchenko:2016rkn]. If, however, the higher freeze-out temperature is realized, one can observe a different dependence of the measured cumulants on the beam energy. A peak in the susceptibility ratio is then observed, at a beam energy of $\sqrt{s_\text{NN}}\approx 20$ GeV, due to the steepening of the chiral crossover with respect to chemical potential, at finite $\mu_\text{B}$ (Note: not due to the appearance of a critical point). At lower beam energies, the critical behaviour of the nuclear liquid-gas transition can be observed again.
In Fig. \[high1\] we also compare our results with the value of $\chi_4^{\rm B}/\chi_2^{\rm B}$ which has been extracted from lattice QCD calculations at $\mu_{\rm B}=0$ and $T \approx 150 $ MeV [@Borsanyi:2013hza]. One can already see, that the lattice data slightly below $T_{\rm PC}$ still has a significant uncertainty, and a quantitative comparison with our results is difficult for low temperatures.
At this point it would be interesting to directly compare our susceptibility ratios with experimental data. As has been shown in, e.g., [@Luo:2017faz]; the values of the cumulant ratios extracted from experiment depend strongly on the selected acceptance, as well as the centrality. Furthermore, experiments only measure net-protons; not net-baryons. It is therefore not clear what we should compare our grand canonical values to. One should also keep in mind, that a direct comparison of our grand canonical results with experimental data is not possible due to the many effects discussed in [@Bzdak:2012ab; @Bzdak:2016qdc; @kitazawa2016efficient; @Feckova:2015qza; @Begun:2004gs; @Bzdak:2012an; @Gorenstein:2008et; @Gorenstein:2011vq; @Sangaline:2015bma; @Spieles:1996is; @Kitazawa:2012at; @Asakawa:2000wh; @Jeon:2000wg; @Steinheimer:2016cir]. The point of our paper is, rather, to discuss the effects of including realistic nuclear matter; in a model with hadron-quark phase transition; on the baryon-number susceptibilities. The eventual comparison of the cumulants to experimental observables has to be determined in a dynamical approach to heavy-ion collisions, which may use our model EoS as an input.
It was pointed out in [@Chen:2016sxn] that, given a critical point of a particular universality class (and only one critical point!), the dependence of the normalized cumulants, as functions of one another, should show a particular universal banana-type shape.
Figs. \[low\] and \[high\] show the shapes obtained from the parity-doublet model calculations. Due to the fact that this model actually has two separate transitions, which are difficult to disentangle, the resulting shapes do not resemble a banana, but are more complicated. In general, when there is an interplay between two phase transitions the relationship between the skewness and the kurtosis is affected by the remnants of the crossover regions related to both the LG and chiral transition, as shown in Fig. \[high\]. Even for the $T_{\text{lim}}$ = 120 MeV freeze-out line (cf. Figs. \[tt\] and \[tt2\]), the aforementioned interactions, for $\sqrt{s_\text{NN}} \ge 2$ GeV, give results considerably different from those which are obtained using universality arguments (cf. Fig. \[low\]), as only the liquid-gas transition is observed.
An important result of this work is the strong dependence of the range of values for the ratios, at large beam energies, on the choice of the freeze-out point. Since both transitions can have an impact on the observed cumulant ratios, it is therefore important to understand the point of origin, during the system’s evolution, of the measured fluctuations, a problem which cannot be solved within the bounds of the present model, as it requires a dynamical description of the nuclear collisions, including the propagation of critical fluctuations.
Conclusion
==========
We have presented an improved version of the hadronic, three-flavour, parity-doublet model including a de-confinement transition to quarks and gluons. The main modification is the inclusion of a six-point interaction term, which significantly improves the nuclear matter saturation properties of the model. With this, we have constructed a model which gives a satisfactory description of nuclear matter, as well as qualitatively describes lattice QCD thermodynamics at $\mu_{\rm B}=0$.\
We have employed the model to study the interplay between the nuclear liquid-gas transition and the chiral transition at large temperatures. We find that this interplay does have an effect on the equation of state and the extracted susceptibilities in a significant range of the phase diagram. This means that the influence of dense nuclear matter on the phase structure, even at large temperatures and moderate chemical potentials, cannot be neglected.\
Furthermore, we have studied the beam energy dependence of the normalized cumulants from our model for different freeze-out conditions. Again, we observe a strong influence of nuclear matter interactions on the observed fluctuations, particularly for low beam energies.\
Our work highlights the fundamental importance of consistently including the properties of interacting nuclear matter in an effective model of the QCD Equation of State for interpreting experimental data of particle fluctuations in heavy-ion collisions.\
Acknowledgements
================
This work was supported by BMBF. The computational resources were provided by the LOEWE Frankfurt Centre for Scientific Computing (LOEWE-CSC).
|
---
abstract: 'The use of weak measurements for performing quantum tomography is enjoying increased attention due to several recent proposals. The advertised merits of using weak measurements in this context are varied, but are generally represented by novelty, increased efficacy, and foundational significance. We critically evaluate two proposals that make such claims and find that weak measurements are not an essential ingredient for most of their advertised features.'
author:
- 'Jonathan A. Gross'
- Ninnat Dangniam
- Christopher Ferrie
- 'Carlton M. Caves'
title: 'On the novelty, efficacy, and significance of weak measurements for quantum tomography'
---
Introduction {#sec:intro}
============
The business of quantum state tomography is converting multiple copies of an unknown quantum state into an estimate of that state by performing measurements on the copies. The naïve approach to the problem involves measuring different observables (represented by Hermitian operators) on each copy of the state and constructing the estimate as a function of the measurement outcomes (corresponding to different eigenvalues of the observables). Though tomography can be performed in such a way, there are more general ways of interrogating the ensemble; indeed, generalizations such as ancilla-coupled [@Dar2002] and joint [@Mas1993] measurements lead one to evaluate the problem of tomography from the perspective of *generalized measurements* [@Nie2010], an approach which has yielded many optimal tomographic strategies [@Hol1982; @Hra1997; @BagBalGil2006; @Blu2010; @Gro2010].
An interesting subclass of generalized measurements is the class of *weak measurements* [@BarLanPro1982; @CavMil1987; @DarYue1996; @FucJac2001; @OreBru2005]. [Figure \[fig:weak-measurement\]]{} gives a quantum-circuit description of a weak measurement. Weak measurements are often the only means by which an experimentalist can probe her system, thus making them of practical interest [@ChuGerKub1998; @SmiSilDeu2006; @GillettDaltonLanyon2010; @SayDotZho2011; @VijMacSli2012; @CamFluRoc2013; @CooRioDeu2014]. Sequential weak measurements are also useful for describing continuous measurements [@WisemanMilburn2010].
Weak measurements are also central in the more contentious formalism of *weak values* [@AAV1988]. In particular, the technique of *weak-value amplification* [@HostenKwiat2008] has generated much debate over its metrological utility [@StrubiBruder2013; @Knee2013a; @TanYam2013; @FerCom2013; @Knee2013b; @ComFerJia13a; @ZhaDatWam13; @DresselMalikMiatto2014; @JordanMartinez-RinconHowell2014; @LeeTsutsui2014; @KneComFerGau14].
The two proposals we review in this paper assert that it is useful to approach the problem of tomography with weak measurements holding a prominent place in one’s thinking. Some care needs to be taken in identifying whether a particular emphasis has the potential to be useful when thinking about tomography, given the large body of work already devoted to the subject. Since weak measurements are included in the framework of generalized measurements, none of the known results for optimal measurements in particular scenarios are going to be affected by shifting our focus to weak measurements. In [Sec. \[sec:eval-princ\]]{} we outline criteria for evaluating this shift of focus.
![A circuit depicting an ancilla-coupled measurement. Here $A$ is a two-system Hermitian operator, ${\left\vert{\psi}\right\rangle}$ is the state of the system being measured, ${\left\vert{\phi}\right\rangle}$ is the initial state of the meter, $\epsilon$ is a real number parameterizing the strength of the measurement, and $O$ is a standard observable with outcomes $o_j$. If ${\left\vert\epsilon\right\vert}\ll1$ the measurement is weak, $U(\epsilon)\simeq{\mathds{1}}$, and very little is learned or disturbed about the system by measuring the meter.[]{data-label="fig:weak-measurement"}](weak-measurement.pdf){width="0.95\linewidth"}
We apply these criteria to two specific tomographic schemes that advocate the use of weak measurements. *Direct state tomography* (Sec. \[sec:dst\]) utilizes a procedure of weak measurement and postselection, motivated by weak-value protocols, in an attempt to give an operational interpretation to wavefunction amplitudes [@LunSutPat2011]. *Weak-measurement tomography* (Sec. \[sec:weak-tomo\]) seeks to outperform so-called “standard” tomography by exploiting the small system disturbance caused by weak measurements to recycle the system for further measurement [@DasArv2014].
Evaluation principles {#sec:eval-princ}
=====================
Here we present our criteria for evaluating claims about the importance of weak measurements for quantum state tomography. The primary tool we utilize is generalized measurement theory, specifically, describing a measurement by a positive-operator-valued measure (POVM). A POVM assigns a positive operator $E_F$ to every measurable subset $F$ of the set $\Omega$ of measurement outcomes $\chi\in\Omega$. For countable sets of outcomes, this means the measurement is described by the countable set of positive operators, $$\begin{aligned}
\left\{ E_{\chi} \right\}_{\chi\in\Omega}\,.
\label{eqn:countable-povm}\end{aligned}$$ The positive operators $E_F$ are then given by the sums $$\begin{aligned}
E_F&=\sum_{\chi\in F}E_{\chi}\,.
\label{eqn:povm-sums}\end{aligned}$$ For continuous sets of outcomes the positive operator associated with a particular measurable subset $F$ is given by the integral $$\begin{aligned}
E_F&=\int_FdE_\chi\,.
\label{eqn:continuous-povm}\end{aligned}$$ These positive operators capture all the statistical properties of a given measurement, in that the probability of obtaining a measurement result $\chi$ within a measurable subset $F\subseteq\Omega$ for a particular state $\rho$ is given by the formula $$\begin{aligned}
\Pr(\chi\in F|\rho)&=\operatorname{Tr}\big(\rho E_F \big)\,.
\label{eqn:meas-stats}\end{aligned}$$ That each measurement yields some result is equivalent to the completeness condition, $$\begin{aligned}
E_\Omega&={\mathds{1}}\,.
\label{eqn:povm-completeness}\end{aligned}$$
POVMs are ideal representations of tomographic measurements because they contain all the information relevant for tomography, i.e., measurement statistics, while removing many irrelevant implementation details. If two wildly different measurement protocols reduce to the same POVM, their tomographic performances are .
Novelty {#sec:eval-novel}
-------
The authors of both schemes we evaluate make claims about the novelty of their approach. These claims seem difficult to substantiate, since no tomographic protocol within the framework of quantum theory falls outside the well-studied set of tomographic protocols employing generalized measurements. To avoid trivially dismissing claims in this way, however, we define a relatively conservative subset of measurements that might be considered “basic” and ask if the proposed schemes fall outside of this category.
The subset of measurements we choose is composed of randomly chosen one-dimensional orthogonal projective measurements \[hereafter referred to as *random ODOPs*; see [Fig. \[fig:random-odop-limits\]]{}(a)\]. These are the measurements that can be performed using only traditional von Neumann measurements, given that the experimenter is allowed to choose randomly the observable he wants to measure. This is quite a restriction on the full set of measurements allowed by quantum mechanics. Many interesting measurements, such as symmetric informationally complete POVMs, like the tetrahedron measurement shown in [Fig. \[fig:random-odop-limits\]]{}(b), cannot be realized in such a way. With ODOPs assumed as basic, however, if the POVM generated by a particular weak-measurement scheme is a random ODOP, we conclude that weak measurements should not be thought of as an essential ingredient for the scheme.
Identifying other subsets of POVMs as “basic” might yield other interesting lines of inquiry. For example, when doing tomography on ensembles of atoms, weak collective measurements might be compared with nonadaptive separable projective measurements [@SmiSilDeu2006; @CooRioDeu2014].
Efficacy
--------
Users of tomographic schemes are arguably less interested in the novelty of a particular approach than they are in its performance. There is a variety of performance metrics available for state estimates, some of which have operational interpretations relevant for particular applications. Given that we have no particular application in mind, we adopt a reasonable figure of merit, Haar-invariant average fidelity, which fortuitously is the figure of merit already used to analyze the scheme we consider in [Sec. \[sec:weak-tomo\]]{}. This is the fidelity, $f\big(\rho,\hat\rho(\chi)\big)$, of the estimated state $\hat{\rho}(\chi)$ with the true state $\rho$, averaged over possible measurement records $\chi$ and further averaged over the unitarily invariant (maximally uninformed) prior distribution over pure true states. For the case of discrete measurement outcomes, this quantity is written as $$\begin{aligned}
F\big( \hat{\rho},E \big)
&{\mathrel{\mathop:}=}\int d\rho\sum_\chi\Pr(\chi|\rho)
f\big( \rho,\hat{\rho}(\chi) \big)\,.
\label{eqn:avg-fidelity}\end{aligned}$$
An obvious problem with this figure of merit is its dependence on the estimator $\hat{\rho}$. We want to compare measurement schemes directly, not measurement–estimator pairs. To remove this dependence we should calculate the average fidelity with the optimal estimator for each measurement, expressed as $$\begin{aligned}
F(E)&{\mathrel{\mathop:}=}\max_{\hat{\rho}}F\big(\hat{\rho},E\big)\,.
\label{eqn:opt-est-avg-fidelity}\end{aligned}$$
To avoid straw-man arguments, it is also important to compare the performance of a particular tomographic protocol to the optimal protocol, or at least the best known protocol. Both proposals we review in this paper are nonadaptive measurements on each copy of the system individually. Since there are practical reasons for restricting to this class of measurements, we compare to the optimal protocol subject to this constraint.
This brings up an interesting point that can be made before looking at any of the details of the weak-measurement proposals. For our chosen figure of merit, the optimal individual nonadaptive measurement is a random ODOP (specifically the Haar-invariant measurement, which samples a measurement basis from a uniform distribution of bases according to the Haar measure). Therefore, weak-measurement schemes cannot hope to do better than random ODOPs, and even if they are able to attain optimal performance, weak measurements are clearly not an essential ingredient for attaining that performance.
Foundational significance
-------------------------
Many proposals for weak-measurement tomography are motivated not by efficacy, but rather by a desire to address some foundational aspect of quantum mechanics. This desire offers an explanation for the attention these proposals receive in spite of the disappointing performance we find when they are compared to random ODOPs.
There are two prominent claims of foundational significance. The first is that a measurement provides an operational interpretation of wavefunction amplitudes more satisfying than traditional interpretations. This is the motivation behind the direct state tomography of Sec. \[sec:dst\], where the measurement allegedly yields expectation values directly proportional to wavefunction amplitudes rather than their squares.
The second claim is that weak measurement finds a clever way to get around the uncertainty–disturbance relations in quantum mechanics. The intuition behind using weak measurements in this pursuit is that, since weak measurements minimally disturb the system being probed, they might leave the system available for further use; the information obtained from a subsequent measurement, together with the information acquired from the preceding weak measurements, might be more information in total than can be obtained with traditional approaches. Of course, generalized measurement theory sets limits on the amount of information that can be extracted from a system, suggesting that such a foundational claim is unfounded. We more fully evaluate this claim in Sec. \[sec:weak-tomo\].
Direct state tomography {#sec:dst}
=======================
In [@LunSutPat2011] and [@LunBab2012] Lundeen *et al.* propose a measurement technique designed to provide an operational interpretation of wavefunction amplitudes. They make various claims about the measurement, including its ability to make “the real and imaginary components of the wavefunction appear directly” on their measurement device, the absence of a requirement for global reconstruction since “states can be determined locally, point by point,” and the potential to “characterize quantum states *in situ* …without disturbing the process in which they feature.” The protocol is thus often characterized as [*direct state tomography*]{} (DST).
To evaluate these claims, we apply the principles discussed in Sec. \[sec:eval-princ\]. Lundeen *et al.* have outlined procedures for both pure and mixed states. We focus on the pure-state problem for simplicity, although much of what we identify is directly applicable to mixed-state . To construct the POVM, we need to describe the measurement in detail. The original proposal for DST of Lundeen *et al.* calls for a continuous meter for performing the weak measurements. As shown by Maccone and Rusconi [@MacRus2014], the continuous meter can be replaced by a qubit meter prepared in the positive $\sigma_x$ eigenstate ${\left\vert{+}\right\rangle}$, a replacement we adopt to simplify the analysis. Since wavefunction amplitudes are basis-dependent quantities, it is necessary to specify the basis in which we want to reconstruct the wavefunction. We call this the *reconstruction basis* and denote it by $\left\{ {\left\vert{n}\right\rangle}
\right\}_{0\leq n<d}$, where $d$ is the dimension of the system we are reconstructing.
The meter is coupled to the system via one of a collection of interaction unitaries $\left\{ U_{\varphi,n} \right\}_{0\leq n<d}$, where $$\begin{aligned}
U_{\varphi,n}&{\mathrel{\mathop:}=}e^{-i\varphi{\left\vert{n}\middle\rangle\middle\langle{n}\right\vert}\otimes\sigma_z}\,.
\label{eqn:dst-unitary}\end{aligned}$$ The strength of the interaction is parametrized by $\varphi$. A weak interaction, i.e., one for which $|\varphi|\ll1$, followed by measuring either $\sigma_y$ or $\sigma_z$ on the meter, effects a weak measurement of the system. In addition, after the interaction, there is a strong (projective) measurement directly on the system in the *conjugate basis* $\left\{{\left\vert{c_j}\right\rangle}\right\}_{0\leq j<d}$, which is defined by $$\begin{aligned}
{\left\langle{n}\middle\vert{c_j}\right\rangle}=\omega^{nj}/\sqrt d\;,\qquad
\omega{\mathrel{\mathop:}=}e^{2\pi i/d}\,.
\label{eqn:conjugate-basis}\end{aligned}$$
The protocol for DST of Lundeen *et al.*, motivated by thinking in terms of weak values, discards all the data except for the case when the outcome of the projective measurement is $c_0$. This protocol is depicted as a quantum circuit in [Fig. \[fig:dst-protocol\]]{}.
![Quantum circuit depicting direct state tomography. The meter is coupled to the system via one of a family of unitaries, $\left\{ U_{\varphi,n} \right\}_{0\leq n<d}$, each of which corresponds to a reconstruction-basis element. The meter is then measured in either the $y$ or $z$ basis to obtain information about either the real or imaginary part of the wavefunction amplitude of the selected reconstruction-basis element. This procedure is postselected on obtaining the $c_0$ outcome from the measurement of the system in the conjugate basis. While the postselection is often described as producing an effect on the meter, the circuit makes clear that the measurements can be performed in either order, so it is equally valid to say the measurement of the meter produces an effect on the system.[]{data-label="fig:dst-protocol"}](dst-protocol.pdf){width=".9\linewidth"}
For each $n$, the expectation values of $\sigma_y$ and $\sigma_z$, conditioned on obtaining the outcome $c_0$ from the projective measurement, are given by $$\begin{aligned}
\label{eqn:sigmay}
\begin{split}
{\left.{{\left\langle{\sigma_y}\right\rangle}}\right\vert_{n,c_0}}
&=\frac{2\sin\varphi}{d\,\Pr\!\big(c_0|U_{n,\varphi},\psi\big)}\operatorname{Re}(\psi_n\Upsilon^*)\\
&\qquad+\frac{\sin2\varphi-2\sin\varphi}{d\,\Pr\!\big(c_0|U_{n,\varphi},\psi\big)}
|\psi_n|^2\,,
\end{split}\\
{\left.{{\left\langle{\sigma_z}\right\rangle}}\right\vert_{n,c_0}}
&=\frac{2\sin\varphi}{d\,\Pr\!\big(c_0|U_{n,\varphi},\psi\big)}
\operatorname{Im}(\psi_n\Upsilon^*)\,,
\label{eqn:sigmaz}\end{aligned}$$ where $\psi_n{\mathrel{\mathop:}=}{\left\langle{n}\middle\vert{\psi}\right\rangle}$. The probability for obtaining outcome $c_0$ is $$\begin{aligned}
\begin{split}
&\Pr\!\big(c_0|U_{n,\varphi},\psi\big)\\
&\;=\frac{1}{d}\Big(|\Upsilon|^2+
2(\cos\varphi-1)\big[\operatorname{Re}(\psi_n\Upsilon^*)-|\psi_n|^2\big]\Big)\,,
\end{split}\end{aligned}$$ and $$\begin{aligned}
\label{eqn:Upsilon}
\Upsilon&{\mathrel{\mathop:}=}\sum_n\psi_n\,.\end{aligned}$$ We can always choose the unobservable global phase of ${\left\vert{\psi}\right\rangle}$ to make $\Upsilon$ real and positive. With this choice, which we adhere to going forward, ${\left.{{\left\langle{\sigma_y}\right\rangle}}\right\vert_{n,c_0}}$ provides information about the real part of $\psi_n$, and ${\left.{{\left\langle{\sigma_z}\right\rangle}}\right\vert_{n,c_0}}$ provides information about the imaginary part of $\psi_n$.
Specializing these results to weak measurements gives $$\begin{aligned}
\psi_n
&=\frac{\Upsilon}{2\varphi}\!\left({\left.{{\left\langle{\sigma_y}\right\rangle}}\right\vert_{c_0,n}}+
i\!{\left.{{\left\langle{\sigma_z}\right\rangle}}\right\vert_{c_0,n}} \right)+\mathcal{O}(\varphi^2)\,.
\label{eqn:reconstruction-formula}\end{aligned}$$ This is a remarkably simple formula for estimating the state ${\left\vert{\psi}\right\rangle}$! There is, however, an important detail that should temper our enthusiasm. Contrary to the claim in [@LunBab2012], this formula does not allow one to reconstruct the wavefunction point-by-point (amplitude-by-amplitude in this case of a finite-dimensional system), because one has no idea of the value of the “normalization constant” $\Upsilon$ until *all* the wavefunction amplitudes have been measured. This means that while ratios of wavefunction amplitudes can be reconstructed point-by-point, reconstructing the amplitudes themselves requires a global reconstruction. Admittedly, this reconstruction is simpler than commonly used linear-inversion techniques, but it comes at the price of an inherent bias in the estimator, arising from the weak-measurement approximation, as was discussed in [@MacRus2014].
The scheme as it currently stands relies heavily on postselection, a procedure that often discards relevant data. To determine what information is being discarded and whether it is useful, we consider the measurement statistics of $\sigma_y$ and $\sigma_z$ conditioned on an arbitrary outcome $c_m$ of the strong measurement. To do that, we first introduce a unitary operator $Z$, diagonal in the reconstruction basis, which cyclically permutes conjugate-basis elements and puts phases on reconstruction-basis elements: $$\begin{aligned}
Z{\left\vert{c_j}\right\rangle}={\left\vert{c_{j+1}}\right\rangle}\,,\qquad
Z{\left\vert{n}\right\rangle}=\omega^n{\left\vert{n}\right\rangle}\,.
\label{eqn:z-unitary}\end{aligned}$$ As is illustrated in [Fig. \[fig:postselect\]]{}, postselecting on outcome $c_m$ with input state ${\left\vert{\psi}\right\rangle}$ is equivalent to postselecting on $c_0$ with input state $Z^{-m}{\left\vert{\psi}\right\rangle}=\sum_n\omega^{-mn}\psi_n{\left\vert{n}\right\rangle}$.
Armed with this realization, we can write reconstruction formulae for all postselection outcomes, $$\begin{aligned}
\psi_n=\omega^{mn}\frac{\Upsilon}{2\varphi}\!
\left( {\left.{{\left\langle{\sigma_y}\right\rangle}}\right\vert_{c_m,n}}+
i\!{\left.{{\left\langle{\sigma_z}\right\rangle}}\right\vert_{c_m,n}} \right)+\mathcal{O}(\varphi^2)\,.
\label{eqn:alt-reconstruction}\end{aligned}$$ This makes it obvious that all the measurement outcomes in the conjugate basis give “direct” readings of the wavefunction in the weak-measurement limit. Postselection in this case is not only harmful to the performance of the estimator, it is not even necessary for the interpretational claims of . Henceforth, we drop the postselection and include all the data produced by the strong measurement.
The uselessness of postselection is not a byproduct of the use of a qubit meter. In the continuous-meter case, the conditional expectation values in the weak-measurement limit are given as weak values $$\begin{aligned}
{\left.{{\left\langle{{\left\vert{x}\middle\rangle\middle\langle{x}\right\vert}}\right\rangle}}\right\vert_{p}}&=\frac{{\left\langle{p}\middle\vert{x}\right\rangle}{\left\langle{x}\middle\vert{\psi}\right\rangle}}
{{\left\langle{p}\middle\vert{\psi}\right\rangle}}\,.
\label{eqn:weak-value}\end{aligned}$$ Weak-value-motivated DST postselects on meter outcome $p=0$ to hold the amplitude ${\left\langle{p}\middle\vert{x}\right\rangle}$ constant and thus make the expectation value proportional to the wavefunction ${\left\langle{x}\middle\vert{\psi}\right\rangle}$. Since ${\left\langle{p}\middle\vert{x}\right\rangle}$ is only a phase, however, it is again obvious that postselecting on any value of $p$ gives a “direct” reconstruction of a rephased wavefunction. Shi *et al.* [@Shi2015] have developed a variation on Lundeen’s protocol that requires measuring weak values of only one meter observable. This is made possible by keeping data that is discarded in the original postselection process.
We now consider whether the weak measurements in DST contribute anything new to tomography. It is already clear from Eqs. (\[eqn:sigmay\]) and (\[eqn:sigmaz\]) that for this protocol to provide data that is proportional to amplitudes in the reconstruction basis, the weakness of the interaction is only important for the measurement of $\sigma_y$. We are after something deeper than this, however, and to get at it, we change perspective on the protocol of [Fig. \[fig:dst-protocol\]]{}, asking not how postselection on the result of the strong measurement affects the measurement of $\sigma_y$ or $\sigma_z$, but rather how those measurements change the description of the strong measurement. As is discussed in [Fig. \[fig:dst-protocol\]]{}, this puts the protocol on a footing that resembles that of the random ODOPs in [Fig. \[fig:random-odop-limits\]]{}(a).
The measurement of $\sigma_z$, which provides the imaginary-part information, is trivial to analyze, because the analysis can be reduced to drawing more circuits. In [Fig. \[fig:imaginary-measurement\]]{}(a), the interaction unitary is written in terms of system unitaries $U_{n,\pm}{\mathrel{\mathop:}=}e^{\mp i\varphi\otimes{\left\vert{n}\middle\rangle\middle\langle{n}\right\vert}}$ that are controlled in the $z$-basis of the qubit. The $\sigma_z$ measurement on the meter commutes with the interaction unitary, so using the principle of deferred measurement, we can move this measurement through the controls, which become classical controls that use the results of the measurement. The resulting circuit, depicted in [Fig. \[fig:imaginary-measurement\]]{}(b), shows that the imaginary part of each wavefunction amplitude can be measured by adding a phase to that amplitude, with the sign of the phase shift determined by a coin flip. This is a particular example of the random ODOP described by [Fig. \[fig:random-odop-limits\]]{}(a). We conclude that weak measurements are not an essential ingredient for determining the imaginary parts of the wavefunction amplitudes.
Measuring the real parts is more interesting, since the $\sigma_y$ measurement does not commute with the interaction unitary. We proceed by finding the Kraus operators that describe the post-measurement state of the system. The strong, projective measurement in the conjugate basis has Kraus operators $K_m={\left\vert{c_m}\middle\rangle\middle\langle{c_m}\right\vert}$, whereas the unitary interaction $U_{\varphi,n}$, followed by the measurement of $\sigma_y$ with outcome $\pm$, has (Hermitian) Kraus operator $$\begin{aligned}
\begin{split}
K_{\pm}^{(y,n)}&{\mathrel{\mathop:}=}{\left\langle{\pm y}\right\vert}U_{\varphi,n}{\left\vert{+}\right\rangle}\\
&=\frac{1}{\sqrt{2}}\left({\mathds{1}}+\big(\sqrt2s_\pm-1\big){\left\vert{n}\middle\rangle\middle\langle{n}\right\vert}\right)\,,\\
s_\pm&{\mathrel{\mathop:}=}\pm\sin(\varphi\pm\pi/4)\,,
\end{split}
\label{eqn:dst-kraus-ops}\end{aligned}$$ where the eigenstates of $\sigma_y$ are ${\left\vert{\pm y}\right\rangle}{\mathrel{\mathop:}=}\big(e^{\mp i\pi/4}{\left\vert{0}\right\rangle}+e^{\pm i\pi/4}{\left\vert{1}\right\rangle}\big)/\sqrt2$. The composite Kraus operators, $K_{\pm,m}^{(y,n)}=K_mK_\pm^{(y,n)}$, yield POVM elements $E_{\pm,m}^{(y,n)}{\mathrel{\mathop:}=}K_{\pm,m}^{(y,n)\dagger}K_{\pm,m}^{(y,n)}=
K_\pm^{(y,n)}{\left\vert{c_m}\middle\rangle\middle\langle{c_m}\right\vert}K_\pm^{(y,n)}$. For each $n$, these POVM elements make up a rank-one POVM with $2d$ outcomes.
The POVM elements can productively be written as $$\begin{aligned}
\label{eqn:dst-povm}
E_{\pm,m}^{(y,n)}&=\alpha^{(y)}_\pm{\left\vert{b_{\pm,m}^{(y,n)}}\middle\rangle\middle\langle{b_{\pm,m}^{(y,n)}}\right\vert}\,,\\
\begin{split}
{\left\vert{b_{\pm,m}^{(y,n)}}\right\rangle}&{\mathrel{\mathop:}=}K_\pm^{(y,n)}{\left\vert{c_m}\right\rangle}\Big/\sqrt{\alpha^{(y)}_\pm}\\
&=\left({\left\vert{c_m}\right\rangle}+\big(\sqrt2 s_\pm-1\big)\omega^{mn}{\left\vert{n}\right\rangle}\right)\Big/
\sqrt{2\alpha^{(y)}_\pm}\,,
\end{split}\\
\alpha^{(y)}_\pm&{\mathrel{\mathop:}=}\frac12\!\left(1-
\frac{1}{d}+\frac{2}{d}s_\pm^2\right)\,.\end{aligned}$$ The POVM for each $n$ does not fit into our framework of random ODOPs, but can be thought of as within a wider framework of random POVMs. Indeed, the Neumark extension [@Neumark1940; @Peres1993] teaches us how to turn any rank-one POVM into an ODOP in a higher-dimensional Hilbert space, where the dimension matches the number of outcomes of the rank-one POVM.
Vallone and Dequal [@ValDeq2015] have proposed an augmentation of the original DST to obtain a “direct” wavefunction measurement without the need for the weak-measurement approximation. The essence of their protocol is to perform an additional $\sigma_x$ measurement on the meter. The statistics of this measurement allow the second-order term in $\varphi$ to be eliminated from the real-part calculation, giving a reconstruction formula that is exact for all values of $\varphi$. Of course, the claim that the original DST protocol “directly” measures the wavefunction is misleading, and directness claims for Vallone and Dequal’s modifications are necessarily more misleading. Even ratios of real parts of wavefunction amplitudes no longer can be obtained by ratios of simple expectation values, since these calculations rely on both $\sigma_x$ *and* $\sigma_y$ expectation values for different measurement settings.
We analyze this additional meter measurement in the same way we analyzed the $\sigma_y$ measurement. The Kraus operators corresponding to the meter measurements are $$\begin{aligned}
K_{+}^{(x,n)}&{\mathrel{\mathop:}=}{\left\langle{+}\right\vert}U_{\varphi,n}{\left\vert{+}\right\rangle}
={\mathds{1}}+\big(\cos\varphi-1\big){\left\vert{n}\middle\rangle\middle\langle{n}\right\vert}\,,\\
K_{-}^{(x,n)}&{\mathrel{\mathop:}=}{\left\langle{-}\right\vert}U_{\varphi,n}{\left\vert{+}\right\rangle}
=\sin\varphi{\left\vert{n}\middle\rangle\middle\langle{n}\right\vert}\,.
\label{eqn:dst-x-kraus-ops}\end{aligned}$$ The composite Kraus operators, $K_{\pm,m}^{(x,n)}=K_mK_\pm^{(x,n)}$, yield POVM elements $E_{\pm,m}^{(x,n)}$ that can be written as $$\begin{aligned}
\label{eqn:dst-x-povm}
E_{\pm,m}^{(x,n)}&=\alpha^{(x)}_\pm{\left\vert{b_{\pm,m}^{(x,n)}}\middle\rangle\middle\langle{b_{\pm,m}^{(x,n)}}\right\vert}\,,\\
\begin{split}
{\left\vert{b_{+,m}^{(x,n)}}\right\rangle}
&{\mathrel{\mathop:}=}K_+^{(x,n)}{\left\vert{c_m}\right\rangle}\Big/\sqrt{\alpha^{(x)}_+}\\
&=\Big({\left\vert{c_m}\right\rangle}+\big(\cos\varphi-1\big)\omega^{mn}{\left\vert{n}\right\rangle}\Big)\Big/\sqrt{\alpha^{(x)}_+}\,,
\end{split}\\\label{eqn:bxnminusm}
{\left\vert{b_{-,m}^{(x,n)}}\right\rangle}
&{\mathrel{\mathop:}=}K_-^{(x,n)}{\left\vert{c_m}\right\rangle}\Big/\sqrt{\alpha^{(x)}_-}
={\left\vert{n}\right\rangle}\,,\\
\alpha^{(x)}_+&{\mathrel{\mathop:}=}1-\frac{\sin^2\!\varphi}{d}\,,\qquad
\alpha^{(x)}_-{\mathrel{\mathop:}=}\frac{\sin^2\!\varphi}{d}\,.\end{aligned}$$
It is useful to ponder the form of the POVM elements for the $\sigma_y$ and $\sigma_x$ measurements of the DST protocols. For the original DST protocol of [Fig. \[fig:dst-protocol\]]{}, without postselection, the only equatorial measurement on the meter is of $\sigma_y$; the corresponding POVM elements, given by Eq. (\[eqn:dst-povm\]), are nearly measurements in the conjugate basis, except that the $n$-component of the conjugate basis vector is changed in magnitude by an amount that depends on the result of the $\sigma_y$ measurement. For the augmented DST protocol of Vallone and Dequal, the additional POVM elements (\[eqn:dst-x-povm\]), which come from the measurement of $\sigma_x$ on the meter, are quite different depending on the result of the $\sigma_x$ measurement. For the result $+$, the POVM element is similar to the POVM elements for the measurement of $\sigma_y$, but with a different modification of the $n$-component of the conjugate vector. For the result $-$, the POVM element is simply a measurement in the reconstruction basis; as we see below, the addition of the measurement in the reconstruction basis has a profound effect on the performance of the augmented DST protocol outside the region of weak measurements, an effect unanticipated by the weak-value motivation.
Although claims regarding the efficacy of DST are rather nebulous, we consider the negative impact of the weak-measurement limit on tomographic performance. In doing so, we assume for simplicity that the system is a qubit, in some unknown pure state that is specified by polar and azimuthal Bloch-sphere angles, $\theta$ and $\phi$. In this case we assume that the reconstruction basis is the eigenbasis of $\sigma_z$; the conjugate basis is the eigenbasis of $\sigma_x$.
The method we use to evaluate the effect of variations in $\varphi$ is taken from the work of de Burgh *et al.* [@deBurgh2008], which uses the *Cramér–Rao bound* (CRB) to establish an asymptotic (in number of copies) form of the average fidelity.
![(Color online) CRB $C(\varphi)$ of Eq. (\[eqn:CRB\]) for original DST (solid black) and augmented DST (dotted green for probability $1/4$ for $\sigma_x$ and $\sigma_y$ measurements and $1/2$ probability for a $\sigma_z$ measurement; dashed red for equal probabilities for all three measurements). As the plot makes clear, the optimal values for $\varphi$ are far from the weak-measurement limit. Values of $\varphi$ for which $\varphi^2\simeq0$ give exceptionally large CRBs, confirming the intuition that weak measurements learn about the true state very slowly. The CRB for original DST also grows without bound as $\varphi$ approaches $\pi/2$, since that measurement strength leads to degenerate Kraus operators and a POVM that, not informationally complete, consists only of projectors onto $\sigma_x$ and $\sigma_y$ eigenstates of the system. The CRB remains finite when the meter measurements are augmented with $\sigma_x$, since the resultant POVM at $\varphi=\pi/2$ then includes $\sigma_z$ projectors on the system \[i.e., projectors in the reconstructions basis; see Eq. (\[eqn:bxnminusm\])\], giving an informationally complete overall .[]{data-label="fig:fisher-term"}](DST_yz_vs_xyz_CR-bound.pdf){width="1\linewidth"}
In analyzing the two DST protocols, original and augmented, we assume that the two values of $n$ are chosen randomly with probability $1/2$. For the original protocol, we choose the $\sigma_z$ and $\sigma_y$ measurements with probability $1/2$. For the augmented protocol, we make one of two choices: equal probabilities for the $\sigma_x$, $\sigma_y$, and $\sigma_z$ measurements or probabilities of $1/2$ for the $\sigma_z$ measurement and $1/4$ for the $\sigma_x$ and $\sigma_y$ measurements. Formally, these assumed probabilities scale the POVM elements when all of them are combined into a single overall .
The asymptotic form involves the Fisher informations, $J_\theta$ and $J_\phi$, for the two Bloch-sphere state parameters, calculated from the statistics of whatever measurement we are making on the qubit. The CRB already assumes the use of an optimal estimator. When the number of copies, $N$, is large, the average fidelity takes the simple form $$\begin{aligned}
\label{eqn:asympt-avg-fid}
F(\varphi)&\simeq 1-\frac{1}{N}C(\varphi)\,,\\
\begin{split}
C(\varphi)
&=\int_{0}^{\pi}d\theta\,\sin\theta\\
&\quad\times\int_{0}^{2\pi}d\phi\,\frac{1}{4}\!\!\left(
\frac{1}{J_\theta(\theta,\phi,\varphi)}+
\frac{\sin^2\theta}{J_\phi(\theta,\phi,\varphi)}\!\right)\,.
\end{split}\label{eqn:CRB}\end{aligned}$$ Though we have derived analytic expressions for the Fisher informations, it is more illuminating to plot the CRB $C(\varphi)$, obtained by numerical integration (see [Fig. \[fig:fisher-term\]]{}). For original DST, the optimal value of $\varphi$ is just beyond $\pi/4$, invalidating all qualities of “directness” that come from assuming $\varphi\ll1$. For the augmented DST of Vallone and Dequal, the optimal value of $\varphi$ moves toward $\pi/2$, even further outside the region of weak measurements. In both cases, $C(\varphi)$ blows up at $\varphi=0$; for weak measurements, $C(\varphi)$ is so large that the information gain is glacial.
We visualize this asymptotic behavior by estimating the average fidelity over pure states as a function of $N$ using the sequential Monte Carlo technique [@smc_foot], for various protocols and values of $\varphi$. [Figure \[fig:optimal-dst-sim\]]{} plots these results and shows how the average fidelity, for the optimal value of $\varphi$, approaches the asymptotic form (\[eqn:asympt-avg-fid\]) as $N$ increases. We note that the estimator used in these simulations is the estimator optimized for average fidelity discussed in [@BagBalGil2006]. If we were to use the reconstruction formula proposed by Lundeen *et al.*, the performance would be worse.
![(Color online) Average infidelity $1-F$ as a function of the number $N$ of system copies for three measurements. The dashed red curve is for augmented DST with equal probabilities for the three meter measurements; the value $\varphi=1.25$ is close to the optimal value from [Fig. \[fig:fisher-term\]]{}. The other two curves are for original DST: the solid black curve is for $\varphi=0.89$, which is close to optimal (this curve nearly coincides with the dashed red curve for augmented DST); the dashed-dotted purple curve is for a small value $\varphi=0.1$, where the weak-measurement approximation is reasonable. The three dotted curves give the corresponding asymptotic behavior $C(\varphi)/N$. The two weak-measurement curves illustrate the glacial information acquisition when weak measurements are used; the dashed-dotted curve hasn’t begun to approach the dotted asymptotic form for $N=10^3$.[]{data-label="fig:optimal-dst-sim"}](DST_yz_vs_xyz_F.pdf){width="1\linewidth"}
Our conclusions are the following. First, postselection contributes nothing to . Its use comes from attention to weak values, but postselection is actually a negative for tomography because it discards data that are just as cogent as the data that are retained in the weak-value scenario. Second, weak measurements in this context add very little to a tomographic framework based on random ODOPs. Finally, the “direct” in DST is a misnomer [@misnomer] because the protocol does not provide point-by-point reconstruction of the wavefunction.
The inability to provide point-by-point reconstruction is a symptom of a general difficulty. Any procedure, classical or quantum, for detecting a complex amplitude when only absolute squares of amplitudes are measurable involves interference between two amplitudes, say, $A$ and $B$, so that some observed quantity involves a product of two amplitudes, say, ${\rm Re}(A^*B)$. If one regards $A$ as “known” and chooses it to be real, then ${\rm Re}(B)$ can be said to be observed directly. This is the way amplitudes and phases of classical fields are determined using interferometry and square-law detectors.
Of course, quantum amplitudes are not classical fields. One loses the ability to say that one amplitude is known and objective, with the other to be determined relative to the known amplitude. Indeed, if one starts from the tomographic premise that nothing is known and everything is to be estimated from measurement statistics, then $A$ cannot be regarded as “known.” DST fits into this description, with the sum of amplitudes, $\Upsilon$ of Eq. (\[eqn:Upsilon\]), made real by convention, playing the role of $A$. The achievement of DST is that this single quantity is the only “known” quantity needed to construct all the amplitudes $\psi_n$ from measurement statistics. Single quantity or not, however, $\Upsilon$ must be determined from the entire tomographic procedure before any of the amplitudes $\psi_n$ can be estimated.
Weak-measurement tomography {#sec:weak-tomo}
===========================
The second scheme we consider is a proposal for qubit tomography by Das and Arvind [@DasArv2014]. This protocol was advertised as opening up “new ways of extracting information from quantum ensembles” and outperforming, in terms of fidelity, tomography performed using projective measurements on the system. The optimality claim cannot be true, of course, since a random ODOP based on the Haar invariant measure for selecting the ODOP basis is optimal when average fidelity is the figure of merit, but the novelty of the information extraction remains to be evaluated.
![Quantum circuit depicting the weak-measurement tomography protocol of Das and Arvind. Two identical meters are used as ancillas to perform the weak $z$ and $x$ measurements. The circuit makes clear that there is nothing important in the order the measurements are performed after the interactions have taken place, so we consider the protocol as a single ancilla-coupled measurement.[]{data-label="fig:weak-protocol"}](weak-protocol.pdf){width=".95\linewidth"}
The weak measurements in this proposal are measurements of Pauli components of the qubit. These measurements are performed by coupling the qubit system via an interaction unitary, $$\begin{aligned}
U^{(j)}=e^{-i\sigma_j\otimes P}\,,
\label{eqn:weak-meas-unitary}\end{aligned}$$ to a continuous meter, which has position $Q$ and momentum $P$ and is prepared in the Gaussian state $$\begin{aligned}
{\left\vert{\phi}\right\rangle}=\sqrt[4]{\frac{\epsilon}{2\pi}}\int_{-\infty}^{\infty}dq\,e^{-\epsilon q^2/4}{\left\vert{q}\right\rangle}\,.
\label{eqn:weak-meas-Gaussian}\end{aligned}$$ The position of the meter is measured to complete the weak measurement. The weakness of the measurement is parametrized by $\epsilon=1/\Delta q^2$.
{width="90.00000%"}
The Das-Arvind protocol involves weakly measuring the $z$ and $x$ Pauli components and then performing a projective measurement of $\sigma_y$. We depict this protocol as a circuit in [Fig. \[fig:weak-protocol\]]{}. Das and Arvind view this protocol as providing more information than is available from the projective $\sigma_y$ measurement because the weak measurements extract a little extra information about the $z$ and $x$ Pauli components without appreciably disturbing the state of the system before it is slammed by the projective $\sigma_y$ measurement. Again, we turn the tables on this point of view, with its notion of a little information flowing out to the two meters, to a perspective akin to that of the random ODOP of [Fig. \[fig:random-odop-limits\]]{}(a). We ask how the weak measurements modify the description of the final projective measurement. For this purpose, we again need Kraus operators to calculate the POVM of the overall .
The Kraus operators for the projective measurement are $K^{(y)}_\pm={\left\vert{\pm
y}\middle\rangle\middle\langle{\pm y}\right\vert}$, and the (Hermitian) Kraus operator for a weak measurement with outcome $q$ on the meter is $$\begin{aligned}
\begin{split}
K^{(j)}(q)&{\mathrel{\mathop:}=}{\left\langle{q}\right\vert}\,U^{(j)}{\left\vert{\phi}\right\rangle}\sqrt{dq}\\
&=\sqrt[4]{\frac{\epsilon}{2\pi}}\exp\!\left(-\frac{\epsilon(q^2+1)}{4}\right)\\
&\quad\times\Big({\mathds{1}}\cosh(\epsilon q/2)+\sigma_j\sinh(\epsilon q/2)\Big)
\sqrt{dq}\,.
\end{split}
\label{eqn:weak-kraus-ops}\end{aligned}$$ The Kraus operators for the whole measurement procedure are $K_\pm(q_1,q_2){\mathrel{\mathop:}=}K^{(y)}_\pm K^{(x)}(q_2)K^{(z)}(q_1)$. From these come the infinitesimal POVM elements for outcomes $q_1$, $q_2$, and $\pm$: $$\begin{aligned}
\begin{split}
d&E_\pm(q_1,q_2)\\
&\,{\mathrel{\mathop:}=}K^\dagger_\pm(q_1,q_2)K_\pm(q_1,q_2)\\
&\,=K^{(z)}(q_1)K^{(x)}(q_2){\left\vert{\pm y}\middle\rangle\middle\langle{\pm y}\right\vert}K^{(x)}(q_2)K^{(z)}(q_1)\,.
\end{split}\end{aligned}$$ These POVM elements are clearly rank-one.
Using the Pauli algebra, we can bring the POVM elements into the explicit form, $$\begin{aligned}
\begin{split}
dE_\pm(q_1,q_2)&=K^\dagger_\pm(q_1,q_2)K_\pm(q_1,q_2)\\
&=dq_1\,dq_2\,G(q_1,q_2)
\frac{1}{2}({\mathds{1}}+\hat{{\mathbf{n}}}_\pm(q_1,q_2)\cdot\boldsymbol{\sigma})\,,
\end{split}
\label{eqn:weak-continuous-povm}\end{aligned}$$ where we have introduced a probability density and unit vectors, $$\begin{aligned}
\label{eqn:prob-dens}
\begin{split}
G(q_1,q_2)&{\mathrel{\mathop:}=}\frac{\epsilon}{2\pi}\exp\!\left(-\frac{\epsilon(q_1^2+q_2^2+2)}{2}\right)\\
&\qquad\times\cosh\epsilon q_1\cosh\epsilon q_2\,,
\end{split}\\
\hat{{\mathbf{n}}}_\pm(q_1,q_2)&{\mathrel{\mathop:}=}\frac{\hat{{\mathbf{x}}}\sinh\epsilon q_2
\pm\hat{{\mathbf{y}}}
+\hat{{\mathbf{z}}}\sinh\epsilon q_1\cosh\epsilon q_2}
{\cosh\epsilon q_1\cosh\epsilon q_2}\,.
\label{eqn:unit-vec}\end{aligned}$$ We note that $G(q_1,q_2)=G(-q_1,-q_2)$ and $\hat{{\mathbf{n}}}_\pm(q_1,q_2)=-\hat{{\mathbf{n}}}_\mp(-q_1,-q_2)$. This means that the overall POVM is made up of a convex combination of equally weighted pairs of orthogonal projectors and is therefore a random . From this perspective, the weak measurements are a mechanism for generating a particular distribution from which different projective measurements are sampled; i.e., they are a particular way of generating a distribution $P(\lambda)$ in [Fig. \[fig:random-odop-limits\]]{}. Several of these distributions are plotted in [Fig. \[fig:bloch-plots\]]{}.
![(Color online) Average infidelity $1-F$ as a function of the number $N$ of system copies for three measurements: Das and Arvind’s measurement protocol (dotted-dashed blue) with $\epsilon=0.575$; MUB consisting of Pauli $\sigma_x$, $\sigma_y$, and $\sigma_z$ measurements (solid black); random ODOP consisting of projective measurements sampled from the Haar-uniform distribution (dashed red). The dotted lines, $1/N$ and $13/12N$, are the CRBs defined by Eq. (\[eqn:CRB\]) for the optimal generalized tomographic protocol and MUB measurements, respectively.[]{data-label="fig:weak-mub-haar-sim"}](Haar_MUB_Weak_F.pdf){width="1\linewidth"}
It is interesting to note that the value of $\epsilon$ that Das and Arvind identified as optimal (about $0.575$) produces a distribution that is nearly uniform over the Bloch sphere. This matches our intuition when thinking of the measurement as a random ODOP, since the optimal random ODOP samples from the uniform distribution
To visualize the performance of this protocol, we again use sequential Monte Carlo simulations of the average fidelity. Das and Arvind compare their protocol to a measurement of $\sigma_x$, $\sigma_y$, and $\sigma_z$, whose eigenstates are *mutually unbiased bases* (MUB). In [Fig. \[fig:weak-mub-haar-sim\]]{}, we compare Das and Arvind’s protocol for $\epsilon=0.575$ to a MUB measurement and to the optimal projective-measurement-based tomography scheme, i.e., the Haar-uniform random .
We don’t discuss the process of binning the position-measurement results that Das and Arvind engage in, since such a process produces a rank-2 POVM that is equivalent to sampling from a discrete distribution over projective measurements and then adding noise, a practice that necessarily degrades tomographic performance.
We conclude that the protocol does not offer anything beyond that offered by random ODOPs and that its claim of extracting information about the system without disturbance is not supported by our analysis. In particular, when operated optimally, it is essentially the same as the strong projective measurements of a Haar-uniform random . It is true that the presence of the $z$ and $x$ measurements provides more information than a projective $y$ measurement by itself, but this is not because the $z$ and $x$ measurements extract information without disturbing the system.
Summary and conclusion
======================
Our analysis of weak-measurement tomographic schemes gives us guidance for future forays into tomography.
POVMs contain the necessary and sufficient information for comparing the performance of tomographic techniques. Specific realizations of a POVM might provide pleasing narratives, but these narratives are irrelevant for calculating figures of merit. Optimal POVMs for many figures of merit and technical limitations are known. A new tomographic proposal should identify restrictions on the set of available POVMs that come about from practical considerations and compare itself to the best known POVM in that set. The question of the optimality of Das and Arvind’s tomographic scheme is easily answered by identifying what POVMs arise from “projective measurement-based tomography” and realizing these POVMs are optimal even in the generalized nonadaptive, individual-measurement scenario.
Claims about novel properties of a state-reconstruction technique should be evaluated as a comparison with a motivated restriction on the set of POVMs. The false dichotomy between “tomographic methods” and whatever new method is being proposed obfuscates that all new methods implement a POVM and that reconstructing a state from POVM statistics is nothing but tomography. Our analysis shows that even the relatively bland and conceptually simple set of random ODOPs captures most of the behavior exhibited by more exotic protocols.
It is appropriate to move beyond the minimal, platform-independent POVM description when considering ease of implementation or when trying to provide a helpful conceptual framework. Nonetheless, a pleasing conceptual framework should not be confused with an optimal experimental arrangement. If the experimental setup described by Lundeen *et al.* happens to be the easiest to implement in one’s lab, the state should still be reconstructed using techniques developed in the general POVM setting rather than the perturbative reconstruction formula presented in work on DST, regardless of how attractive one finds the wavefunction-amplitude analogy.
We thank Josh Combes for helpful discussions. This work was support by National Science Foundation Grant No. PHY-1212445. CF was also supported by the Canadian Government through the NSERC PDF program, the IARPA MQCO program, the ARC via EQuS Project No. CE11001013, and by the U.S. Army Research Office Grant Nos. W911NF-14-1-0098 and W911NF-14-1-0103.
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In the long tradition of institutions abandoning a name in favor of initials, to divorce from some original product or purpose, we recommend the use of DST in the hope that the “direct” can be forgotten.
|
---
abstract: 'We present clustering results from the 2dF QSO Redshift Survey (2QZ) which currently contains over 20,000 QSOs at $z<3$. The two-point correlation function of QSOs averaged over the entire survey ($\bar{z}\simeq1.5$) is found to be similar to that of local galaxies. When sub-dividing the sample as a function of redshift, we find that for an Einstein-de Sitter universe QSO clustering is constant (in comoving coordinates) over the entire redshift range probed by the 2QZ, while in a universe with $\Om=0.3$ and $\lo=0.7$ there is a marginal increase in clustering with redshift. Sub-dividing the 2QZ on the basis of apparent magnitude we find only a slight difference between the clustering of QSOs of different apparent brightness, with the brightest QSOs having marginally stronger clustering. We have made a first measurement of the redshift space distortion of QSO clustering, with the goal of determining the value of cosmological parameters (in partcular $\lo$) from geometric distortions. The current data do not allow us to discriminate between models, however, in combination with constraints from the evolution of mass clustering we find $\Omega_{\rm m}=1-\lo=0.23^{+0.44}_{-0.13}$ and $\beta(z\sim1.4)=0.39^{+0.18}_{-0.17}$. The full 2QZ data set will provide further cosmological constraints.'
author:
- 'S.M. Croom$^1$, B.J. Boyle$^1$, N.S. Loaring$^2$, L. Miller$^2$, P. Outram$^3$, T. Shanks$^3$, R.J. Smith$^4$, F. Hoyle$^5$'
title: Clustering in the 2dF QSO Redshift Survey
---
Introduction
============
The 2dF QSO Redshift Survey (2QZ) aims to compile a homogeneous catalogue of $\sim25000$ QSOs using the Anglo-Australian Telescope (AAT) 2-degree Field facility (2dF) [@2dfpaper]. This catalogue will constitute a factor of $\ga50$ increase in numbers to a equivalent flux limit over previous data sets [@bsfp90]. The main science goal of the 2QZ is to use QSOs as a probe of large-scale structure in the Universe over a range of scales from 1 to $1000\Mpc$, out to $z\simeq3$. These measurements have the potential to help determine the fundamental cosmological parameters governing the Universe. The 2QZ will also significantly advance our understanding of the AGN/QSO phenomenon, by allowing us to carry out statistical studies of large numbers of QSO spectra, and also discovering rare and extreme types of AGN, for example broad-absorption-line QSOs or post-starburst QSOs [@brother99]. As well as QSOs, there are a large number of other interesting sources being discovered including compact narrow-emission-line galaxies, white dwarfs and cataclysmic variables.
In this paper we will concentrate on the statistical measurements of QSO clustering. The basic properties of the 2QZ are outlined in Section \[sec:survey\]. We then discuss various measurements of large-scale structure including the correlation function (Section \[sec:xir\]) and redshift-space distortions (Section \[sec:zspace\]).
The 2dF QSO Redshift Survey {#sec:survey}
===========================
The 2QZ currently (September 2001) contains 20573 QSOs below a redshift of $z\simeq3$. Observations will be completed in January 2002, by which time close to 25000 QSOs will have been observed. A large fraction of the data is already publicly available; the 2QZ “10k catalogue” [@2qzpaper5] contains spectra of 11000 QSOs, and almost 10000 other sources. The spectra and catalogue are available electronically at [http://www.2dfquasar.org]{}. The analysis below mostly concerns this 10k catalogue.
QSO candidates were selected as being point sources and bluer than the stellar locus, based on broad band $u\bj r$ magnitudes from Automatic Plate Measuring (APM) facility measurements of UK Schmidt Telescope (UKST) photographic plates. The magnitude limits are $18.25<\bj\leq20.85$. The survey comprises 30 UKST fields, arranged in two $75^{\circ}\times5^{\circ}$ declination strips centred in the South Galactic Cap (SGC) at $\delta=-30^{\circ}$ and the North Galactic Cap (NGC) at $\delta=0^{\circ}$ with RA ranges $\alpha=21^h40$ to $3^h15$ and $\alpha=9^h50$ to $14^h50$ respectively. The completed survey will cover approximately 740 deg$^2$ [@croom; @smith; @2qzpaper3].
Spectroscopic observations have been carried out using the 2dF instrument at the AAT in conjunction with the 2dF Galaxy Redshift Survey (2dFGRS) [@2dfgrs], as the 2QZ and 2dFGRS areas cover the same regions of sky. Spectroscopic data are reduced using the 2dF pipeline reduction system [@2dfman]. The identification of QSO spectra and redshift estimation was carried out using the [AUTOZ]{} code written specifically for this project. This program compares template spectra of QSOs, stars and galaxies to the observed spectra. Identifications are then confirmed by eye for all spectra. The mean spectroscopic completeness for the sample is 89 per cent.
The spatial and redshift distribution of QSOs is shown in Fig \[fig:wedge\]. At redshifts $z\ga2$ the decline in QSO numbers is due to the increased reddening of QSO colours by absorption in the Ly$\alpha$ forest. QSOs in this region are therefore missed by our colour selection. At low redshifts we will miss objects in which the host galaxy contributes significantly to the flux, due to both the extended nature of the sources and their redder colours. An estimate of the survey incompleteness due to the colour selection is given by Boyle et al. (2000) [@2qzpaper1].
The QSO correlation function {#sec:xir}
============================
We have measured the redshift space correlation function, $\xi\qso(s)$, of 2QZ QSOs, both averaged over the entire sample, and sub-divided into redshift or magnitude intervals. $\xi\qso(s)$ has been estimated assuming two representative cosmological models; the $\Om=1$ Einstein-de Sitter model (EdS) and a model with $\Om=0.3$ and $\lo=0.7$ ($\Lambda$). We use the minimum variance [@ls93] correlation function estimator, and details of our method can be found in Croom et al. (2001) [@2qzpaper2].
In Fig. \[fig:xi\]a shows the QSO correlation function for EdS and $\Lambda$ universes averaged over $0.3<z\leq2.9$, based on the QSOs in the 10k catalogue. The amplitude and shape of $\xi\qso(s)$ is comparable to that of local galaxies (Fig. \[fig:xi\]b). Fitting a standard power law of the form $\xi\qso(s)=(s/s_0)^{-\gamma}$ we find that $s_0=3.99^{+0.28}_{-0.34}\Mpc$ and $\gamma=1.55^{+0.10}_{-0.09}$ for an EdS cosmology. The effect of a significant cosmological constant is to increase the separation of QSOs, so that in the $\Lambda$ model the best fit is $s_0=5.69^{+0.42}_{-0.50}\Mpc$ and $\gamma=1.56^{+0.10}_{-0.09}$. Both these best fit lines are shown in Fig \[fig:xi\]a.
We then sub-divide the 2QZ into five redshift intervals containing approximately equal number of QSOs. The correlation function is measured in each redshift interval separately, and a power law is fit to the result (assuming the same slope as found from the full sample). The resulting clustering scale lengths are shown in Fig. \[fig:xi\]c. The clustering of QSOs is constant as a function of redshift over the entire range probed by the 2QZ (EdS). In the $\Lambda$ case there is a marginal increase of clustering with increasing redshift, but a constant value cannot be ruled out. Making comparisions to the clustering of high redshift Ly-break galaxies [@adelberger98], at $z\sim3$, we see that these also have a similar clustering strength to that of the 2QZ QSOs. The solid lines in Fig. \[fig:xi\]c denote the linear theory evolution of clustering. The data disagrees with the linear theory prediction, implying that QSOs must have a redshift dependent bias factor, $b\qso(z)$. We derive an empirical fit to the bias of the QSOs assuming $b(z)=1+(b(0)-1)G(\Om,\lo,z)^\beta$ where $G(\Om,\lo,z)$ is the linear growth factor (dot-dashed lines in Fig. \[fig:xi\]c). The best fit values are $b(0)=1.45^{+0.21}_{-0.16}$ and $\beta=1.68^{+0.44}_{-0.40}$ (EdS) and $b(0)=1.28^{+0.16}_{-0.11}$ and $\beta=1.89^{+0.49}_{-0.46}$ ($\Lambda$). A more detailed analysis is carried out by Croom et al. (2001) [@2qzpaper2].
We lastly sub-divide our sample into 3 apparent magnitude slices with equal number of QSOs in each. The measured values of $s_0$ as a function of mean apparent magnitude are shown in Fig. \[fig:xi\]d. There is no significant difference between the different magnitude slices, although the brightest QSOs do have marginally stronger clustering. We note that sub-dividing on the basis of apparent magnitude is approximately equvalent to selection relative to $M^*$ due to the extreme luminosity evolution of QSOs over the redshift range of our sample [@2qzpaper1].
Cosmological parameters from redshift-space distortions {#sec:zspace}
=======================================================
By comparing clustering along and across the line of sight, and modelling the effects of peculiar velocities (both linear and non-linear) it is possible to detect the geometric distortions present if the wrong cosmological model is assumed when determining the clustering [@ap79; @bph96]. At high redshift, the geometric distortion is particularly sensitive to the cosmological constant $\lo$. In practice, a model which also takes into account linear infall is fit to provide a constraint in the $\lo$ vs. $\beta=\Om^{0.6}/b$ plane.
The QSO power spectrum from the 2QZ 10k sample, measured along and across the line of sight, $P(k_\parallel,k_\perp)$, is shown in Fig. \[fig:zspace\]. Unfortunately the current 2QZ data only provides a joint constraint on $\Omega_{\rm m}=1-\lo$ and $\beta$ (shading in Fig. \[fig:zspace\]b). Infact there is also some dependence on the value of the small scale non-linear pair-wise velocity dispersion, $\sigp$. Because we are looking at $P(k_\parallel,k_\perp)$ on large scales this is not a major issue, but the effects of varying the assumed velocity dispersion is shown in Fig. \[fig:zspace\]b.
However, we can obtain a useful constraint on the cosmological world model by combining the above approach with our knowledge of the linear growth of clustering. By using the value of $\beta$ and the two-point correlation function for nearby galaxies derived from the 2dfGRS [@peacock01] we can determine the clustering of matter at $z=0$ and for a given cosmology we can then derive the clustering of mass as a function of $z$. Comparison to the amplitude of QSO clustering then gives the mean value of the QSO bias and hence $\beta$. This approach gives a differing relation between $\Omega_{\rm m}$ and $\beta$ (dotted and dot-dashed lines in Fig. \[fig:zspace\]b), and thus breaks the degeneracy. The current best fit derived from this method is $\Omm=0.23^{+0.44}_{-0.13}$ and $\beta=0.39^{+0.18}_{-0.17}$. Ths analysis is discussed further in Outram et al., (2001) [@2qzpaper6].
With the completion of the 2QZ at the end of 2001, analysis of the complete data set will produce further constraints on cosmological world models. This will include different types of analysis than those discussed here, including the gravitational lensing properties of the 2QZ QSOs. It is hoped that the 2QZ will become a major resource for the astronomical community, particularly when the final data becomes public at the end of 2002.
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---
abstract: |
[**Abstract**]{} For a semisimple complex Lie algebra $\mathfrak g$, the BGG category $\CatO$ is of particular interest in representation theory. It is known that Irving’s shuffling functors $\Sh_w$, indexed by elements $w\in W$ of the Weyl group, induce an action of the braid group $B_W$ associated to $W$ on the derived categories $\Db(\CatO_\lambda)$ of blocks of $\CatO$.
We show that for maximal parabolic subalgebras $\mathfrak p$ of $\SL_n$ corresponding to the parabolic subgroup $W_\mathfrak p=S_{n-1}\times S_1$ of $S_n$, the derived shuffling functors $\mathbf L\Sh_{s_i}$ are instances of Seidel and Thomas’ spherical twist functors. Namely, we show that certain parabolic indecomposable projectives $P^\mathfrak p(w)$ are spherical objects, and the associated twist functors are naturally isomorphic to $\mathbf L\Sh_w[-1]$ as auto-equivalences of $\Db(\CatO^\mathfrak p)$.
We give a short and accessible overview of the main properties of the BGG category $\CatO$, the construction of shuffling- and spherical twist functors, and show some examples how to determine images of these. To this end, we employ the equivalence of blocks of $\CatO$ and the module categories of a certain path algebras.
author:
- Fabian Lenzen
bibliography:
- '../sources.bib'
title: 'Shuffling functors and spherical twists on $\Db(\CatO_0)$'
---
=1
Contents {#contents .unnumbered}
========
Introduction
============
Consider a finite dimensional semisimple complex Lie algebra $\mathfrak g$ with respective Cartan and Borel subalgebras $\mathfrak h$ and $\mathfrak b$. Representations of $\mathfrak g$ are equivalent to modules over the universal enveloping algebra $U(\mathfrak g)$ [@Humphreys:Lie §V]. The *BGG category* $\CatO$ of $\mathfrak g$ is the full subcategory of $\Mod{U(\mathfrak g)}$ consisting of modules that
are finitely generated,
have a weight space decomposition $M=⨁_{𝜆∈𝔥^*} M_𝜆$ and
are locally $𝔫$-finite; , for every $v∈M$, the orbit $U(𝔫^+)⋅v$ is finite dimensional.
Category 𝓞, blocks and shuffling functors
-----------------------------------------
The category $\CatO$ has a decomposition $\CatO=\bigoplus_{\lambda}\CatO_\lambda$ into $\CatO_\lambda$, indexed by dominant weights $\lambda$. Denote the Weyl group of $\mathfrak g$ by $W$ and let $W_\lambda\leq W$ be its stabiliser subgroup of a weight $\lambda$. Each block $\CatO_\lambda$ contains the simple modules $L(w\cdot \lambda)$, the indecomposable projectives $P(w\cdot\lambda)$ and the Verma modules $M(w\cdot\lambda)$ indexed by $w\in W/W_\lambda$ with $W_\lambda$ the stabiliser subgroup $\lambda$. If a block $\CatO_\lambda$ is fixed, we just write $L(w)$, $P(w)$ and $M(w)$ for the respective objects therein. Each block $\CatO_\lambda$ is Morita equivalent to modules over a quasi-hereditary algebra [@BGG].
A weight $\lambda$ is called if $W_\lambda$ is trivial; , if $\lambda$ does not lie on any reflection plane. All blocks $\CatO_\lambda$ associated to regular weights are equivalent as categories; in the following we shall thus work in the $\CatO_0$ which contains the trivial $\mathfrak g$-representation $L(e\cdot 0)=\mathbf C$.
A *Coxeter system* consists of a group $W$, a set $S$ of generators and a presentation $W=⟨s∈S \mid s^2 = e, sts\dotsm = tst\dotsm ⟩$ with $m_{st}$ factors $s,t$ on both sides. The $s\in S$ are called . The matrix $(m_{st})_{s,t\in W}$ is called the of $W$. To $W$, there is the associated $B_W=⟨s∈S \mid sts\dotsm = tst\dotsm⟩$, such that there is a natural quotient map $B_W\onto W$.
The Weyl group of $\mathfrak g$ is a Coxeter group. In particular, the symmetric group $S_n$, which is the Weyl group of $\SL_n$, is a Coxeter group, generated by the simple reflections $s_1,\dotsc,s_{n-1}$, the Coxter matrix has entries $m_{s_i,s_j} = 2\ \text{if $i=j$}, 1\ \text{if $|i-j|=1$,}\ 0 \ \text{otherwise}$, and $B_n≔B_{S_n}$ is the well-known Artin braid group.
Let $s\in W$ be a simple reflection and $\mu$ be a weight with $W^\mu=\{e,s\}$. The is the composition $$\Theta_s\colon\CatO_0\xrightarrow{T_0^\mu}\CatO_\mu\xrightarrow{T_\mu^0}\CatO_0$$ of the two bi-adjoint translation functors $T_0^\mu$ and $T_\mu^0$. $\Theta_s$ is independent from the choice of $\mu$, and is an exact self-adjoint auto-equivalence of the block $\CatO_0$ [@Jantzen:Moduln-mit-hoechstem-Gewicht §2.10]. It is uniquely characterised by the existence of short exact sequences $$\label{eqn:translation-defining-ses}
0\to M(w)\to \Theta_s M(w) \to M(ws) \to 0
\quad\text{and by}\quad
\Theta_s M(w)\isom\Theta_s M(ws)$$ for $w<ws$ [@Jantzen:Moduln-mit-hoechstem-Gewicht Satz 2.10]. Furthermore, $\Theta_s^2 = \Theta_s ⊕ \Theta_s$ [@Humphreys:CatO cf. §7.14].
From the adjunctions $T_\mu^0\dashv T_0^\mu$ and $T_0^\mu\dashv T_\mu^0$ we we get adjunction maps $\unit_s\colon\id\Rightarrow\Theta_s$ and $\counit_s\colon\Theta_s\Rightarrow\id$. The and the $\Sh_s$ and $\Ch_s$ are the respective cokernel and kernel $$\Sh_s\coloneqq\coker(\unit_s)\colon\CatO_0\to\CatO_0,
\qquad
\Csh_s≔\ker(\counit_s):\CatO_0\to\CatO_0,$$ see [@Carlin:Extensions-of-Vermas §2; @Irving:Shuffled-Verma-Modules §3]. By the snake lemma, $\Sh_s$ is right- and $\Csh_s$ is left-exact, and we consider the derived functors $\mathbf L\Sh_s \isom\cone(\unit)$ and $\mathbf R\Csh_s\isom\cocone(\counit)$.
Note that for $w<ws$ we have $$\begin{aligned}
\mathbf L^i\Sh_s &= 0\ \text{for $i\neq 0,1$}, & \mathbf R^j \Csh_s &= 0\ \text{for $j\neq -1,0$},\\
\mathbf L^1\Sh_s M(w) &= 0, & \mathbf L^1\Sh_s M(ws) &= 0.\end{aligned}$$
Actions of 𝑊 on 𝐾₀(𝓞₀) and Dᵇ(𝓞₀)
---------------------------------
The $K_0(\mathcal C)$ of an abelian category $\mathcal C$ is the abelian group generated by $[M]$ of objects $M\in\mathcal C$, subject to the relation $[E]=[A]+[B]$ whenever $E$ is an extension of $A$ by $B$. For $\mathcal T$ a triangulated category, $K_0(\mathcal T)$ is defined in the same way, with “extension” replaced by “distinguished triangle” in the obvious way.
Any exact ( triangulated) functor induces a group homomorphism on $K_0(\mathcal C)$ ( $K_0(\mathcal T)$). Both definitions of $K_0$ are compatible in the sense that the inclusion $\mathcal C \into \Db(\mathcal C)$ induces an isomorphism $K_0(\mathcal C)\isom K_0(\Db(\mathcal C))$ of abelian groups [@Grothendieck:Tohoku].
Each of the collections $\{L(w)\}_{w\in W}$, $\{M(w)\}_{w\in W}$ and $\{P(w)\}_{w\in W}$ is a $\mathbf Z$-base of $K_0(\CatO_0)$. The shuffling functors thus induce a right action of $W$ on $K_0(\CatO_0)$, defined for simple reflections $s$ by the assignment $$W \to\Aut\bigl(K_0(\CatO_0)\bigr),\quad
s \mapsto [\Sh_s]\colon [M(w)] \mapsto [M(ws)].$$
The assignment $$B_W\to\Aut\bigl(\Db(\CatO_0)\bigr),\quad
s\mapsto\mathbf{L}\Sh_s,\quad
s^{-1}\mapsto\mathbf{R}\Csh_s$$ defines an action of $B_W$ on the derived category $\Db(\CatO_0)$, and the following square commutes: $$\begin{tikzcd}
B_W \ar[d, "\mathbf L\Sh_{(-)}"'] \ar[r, "\can", two heads] & W \dar["{[\Sh_{(-)}]}"] \\
\Aut(\Db(\CatO_0)) \ar[r, "{[-]}"'] & \Aut(K_0(\CatO_0)).
\end{tikzcd}$$
Seidel and Thomas’ spherical twist functors
-------------------------------------------
Let $\mathcal C$ be a $k$-linear abelian category $\mathcal C$ of finite global dimension. Seidel and Thomas have constructed an action of the braid group $B_n\coloneqq B_{S_n}$ of the symmetric group $S_n$ on $\mathcal C$ in terms of [@seidel-thomas:braid-group-actions]. We give a short summary of their construction.
We denote the Hom-space in $\Db(\mathcal C)$ by $\Hom^*_{\Db(\mathcal C)}$; it is a graded $k$-vector space whose degree $d$-part $\Hom^d_{\Db(\mathcal C)}$ consists of chain maps of homological degree $d$.
In contrast, for chain complexes $X, Y\in\Ch(\mathcal C)$ we denote by $\hom^\bullet_\mathcal C(X, Y)$ the chain complex of arbitrary graded morphisms $X\to Y$ (, not necessarily chain maps); it is a chain complex of $𝐂$-vector spaces with differential $\mathrm d_{\hom^k}(f)=\mathrm d_Y f-(-1)^k f d_X$ Note that $H^* \hom^\bullet_\mathcal C(-,-) = \Hom^*_{K^\mathrm b}(-, -)$ computes the Hom-space in the bounded homotopy category [@Weibel:Hom-Alg §2.7.5].
Finally, for $V$ a complex of $𝐂$-vector spaces and $X\in\Db(\mathcal C)$, we denote by $\lin^\bullet(V, X)$ the complex of $𝐂$-linear maps from $V$ to $X$, which is a chain complex in $\Db(\mathcal C)$ with differential $\mathrm d_{\lin}(f) v \coloneqq (-1)^{\deg v}[\mathrm dfv - f\mathrm dv]$. For details, consider [@seidel-thomas:braid-group-actions].
An object $E\in\Db(\mathcal C)$ is called for $d\geq 0$ if
\[def:spherical-object:finite-total-dimension\] for any $F∈\Db(𝓒)$, the spaces $\Hom^*_{\Db(𝓒)}(E, F)$ and $\Hom^*_{\Db(𝓒)}(F, E)$ are of finite total dimension, and
\[def:spherical-object:spherical-cohomology\] $\Hom^*_{\Db(𝓒)}(E, E) ≅ 𝐂[x]/(x^2)$ as algebras, where $x$ is a morphism of degree $d$.
It is called if it has in addition the :
\[def:spherical-object:calabi-yau\] For all $F$ and $i$, composition $\circ$ of morphisms is a non/degenerate pairing\
$\Hom^i_{\Db}(F, E) ⊗ \Hom^{d-i}_{\Db}(F, E) \to \Hom^d_{\Db}(E, E) ≅ 𝐂$. [^1]
Consider the evaluation and coevaluation maps $$\begin{aligned}
\ev\colon \Xi_E F≔\hom^\bullet(E, F)⊗F & \to F, & \ev'\colon F & \to\lin^\bullet(\hom^\bullet(F, E), E)\eqqcolon \Xi'_E, \\
\phi⊗f & \mapsto\phi(f), & f & \mapsto\phi\mapsto\phi(f).
\end{aligned}$$ If $E$ is $d$-spherelike, one can associate the and *-cotwist functor* $$\begin{aligned}
T_E &≔ \cone(\ev\colon\Xi_E \Rightarrow\id_\mathcal C), &
T'_E &≔ \cocone(\ev'\colon\id_\mathcal C\Rightarrow\Xi'_E).
\end{aligned}$$ There is an adjunction $T'_E\dashv T_E$
If $E$ is $d$-spherical, then $T_E$ and $T'_E$ are mutually inverse auto-equivalences of $\Db(\mathcal C)$.
\[notation:total-complexes\] Given a double (or triple) complex $X^{\bullet\bullet}$, we denote its $⊕$-total complex by $\{X^{\bullet\bullet}\}$. In particular, since we can regard a morphism $f\colon X\to Y$ of chain complexes trivially as a double complex with $Y$ laced in degree zero, the mapping cone can be written as $\cone(f)=\{f\colon X\to Y\}$.
\[rmk:twisting-E-by-itself\] Since for every $d$-spherical object $E$, $\hom^\bullet(E, E) \isom ⟨\id⟩_\mathbf C⊕⟨x⟩_\mathbf C[-d]$ as complexes vector spaces, we have $$T'_E E = \{\begin{psmallmatrix}1\\x\end{psmallmatrix}\colon \degZero{E}\to E⊕E[d]\} \qis E[1-d],$$ where the left $E$ is in degree zero.
A collection $\{E_1,\dotsc, E_n\}$ of $d$-spherical objects is an if $\dim\Hom^*_{\Db(\mathcal C)}(E_i, E_j) = \begin{smallcases}1 & \text{if $|i-j|=1$},\\0 & \text{otherwise}\end{smallcases}$.
Given an $\mathrm{A}_n$-configuration $\{E_1,\dotsc, E_n\}$ of $d$-spherical objects, the assignment $$B_n\to\Aut(\Db(\mathcal C)),\quad s_i \mapsto T_{E_i}$$ defines an action of the braid group.
Objective {#objective .unnumbered}
---------
The category $\CatO_0$ is a $\mathbf C$-linear category of finite global dimension and hence satisfies the requirements of [@seidel-thomas:braid-group-actions]. For $\mathfrak g=\SL_n$, we want to understand whether there is an $\mathrm{A}_n$-configuration in $\Db(\CatO_0)$ such that the associated twist functors relate to the shuffling functors. We shall prove the following:
For a maximal parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{sl}_n$ corresponding to the parabolic subgroup $S_{n-1}×S_1\leq S_n$, there is an $\mathrm{A}_{n-1}$-configuration of $0$-spherical objects in $\Db(\CatO^\mathfrak p_0)$ such that the associated spherical twist functor and the restriction $\restrict{𝐋\Sh_{s_i}}{\smash{\Db(𝓞^𝔭_0)}}$ are naturally isomorphic auto/equivalences of $\Db(𝓞^𝔭_0)$.
Outline
-------
We first introduce some machinery. gives an overview of some of the most important properties of blocks of $\CatO$. We do not require any prior knowledge about $\CatO$. We include a short refresher on Kazhdan-Lusztig theory, quivers and graded algebras.
In we explain how to compute images of the shuffling functors for the special case $\mathfrak g=\SL_2$. Our proof for the respective version of \[thm:main-result\] serves as a model for the general case, which is worked out in \[sec:sln-case\].
Tools for 𝓞 {#sec:tools-for-O}
===========
In this section, we collect the most important properties of $\CatO_\lambda$. For the following, is it not necessary to assume that $\lambda$ be a regular weight.
Composition series {#sec:composition-series}
------------------
Every module $M\in\CatO_\lambda$ admits a composition series $$0\subsetneq M_1\subsetneq\cdots\subsetneq M_\ell=M$$ with subquotients $M_i/M_{i-1}$ isomorphic to the simple modules $L(w\cdot \lambda)$ [@Jantzen:Moduln-mit-hoechstem-Gewicht Satz 1.13]. By the Jordan-Hölder theorem, any composition series for $M$ involves the same isomorphism classes of the simple factors, up to their order of appearance. One writes $[M:L(w\cdot\lambda)]$ for the the multiplicity of $L(w\cdot \lambda)$ in any composition series for $M$.
We write a composition composition factors from the simple submodules at the bottom to the simple quotients at the top. Any permutation of factors on the same horizontal level occurs in some composition series for $M$.
1. For $\mathfrak g=\SL_2$, the non-simple Verma module has a composition series $M(e)=\begin{psmallmatrix}L(e)\\L(s)\end{psmallmatrix}$, which is just another way to say that there is a short exact sequence $0\to\nolinebreak[2] L(s)\to M(e)\to L(e)\to 0$.
2. For $\mathfrak g=\SL_3$, the Verma module $M(s)=\begin{psmallmatrix}L(s)\\L(st)\quad L(ts)\\L(w_0)\end{psmallmatrix}$ has the simple quotient $L(s)$ and the simple submodule $L(w_0)$. Its non-trivial submodules are $L(w_0)$, $\begin{psmallmatrix}L(st)\\L(w_0)\end{psmallmatrix}$, $\begin{psmallmatrix}L(ts)\\L(w_0)\end{psmallmatrix}$ and $\begin{psmallmatrix}L(st)\quad L(ts)\\L(w_0)\end{psmallmatrix}$.
A morphism in $\CatO$ is compatible with composition series, , it takes factors from the top of its domain to the bottom of its codomain, preserving the order. The cokernel (resp. kernel) of a morphism consists of the simple factors that are not mapped onto (from). Since $\dim\Hom_\CatO(L(v), L(w))=\delta_{vw}$, a morphism is determined by a scalar for each composition factor of the domain.
\[caveat:restrictions-of-composition-series\] Compatibility with the composition series is necessary but not sufficient for the existence of a morphism in $\CatO_0$; , not every composition series diagram describes an actual morphism. Moreover, it is hard to tell which other factors of $M$ are contained in the submodule generated by some simple composition factor of $M$.
Certain modules, such as the indecomposable projectives $P(w\cdot\lambda)$, additionally admit a whose subquotients are isomorphic to Verma modules $M(w\cdot\lambda)$; one writes $(M:M(w\cdot\lambda))$ for the respective (unique) multiplicity of $M(w\cdot\lambda)$ in any standard filtration of $M$.
$(P(v\cdot\lambda):M(w\cdot\lambda))=[M(w\cdot\lambda):L(v\cdot\lambda)]$.
Kazhdan-Lusztig-theory
----------------------
There is a $(W/W_\lambda)^2$-parametrised collection $\{p_{vw}\}$ of polynomials in $\mathbf Z[q^{\pm 1}]$, called , which occur as base change coefficients between the standard- and the Kazhdan-Lusztig basis of the Iwahori-Hecke algebra $\mathsf{H}_q(W)$ [@KL:Representations-of-Coxeter-groups].
The composition factor multiplicities in $\CatO$ are given by $$\label{eqn:BGG-reciprocity}
(P(v\cdot\lambda):M(w\cdot\lambda)) = [M(w\cdot\lambda):L(v\cdot\lambda)] = p_{vw}(1).$$
We employ the convention from [@Soergel:KL-polynomials; @Lusztig:hecke-algebras] for the bases of $\mathsf{H}_q(W)$ and thus for the normalisation of the $p_{vw}$’s. Another widespread convention yields the formula $[M(w\cdot\lambda):L(v\cdot\lambda)] = p_{w_0v,w_0w}(1)$.
Gradings {#sec:gradings}
--------
The following paragraphs summarise how to pass from $\CatO_\lambda$ to a graded category $\gCatO_\lambda$; see [@stroppel:gradings-and-translation] for details. The following results are of fundamental importance:
The functor $\mathbf V_\lambda\colon\CatO_\lambda \to\Mod{}[\End_\CatO(P(w_0\cdot\lambda))]$, called the *combinatoric functor*, is fully faithful on projectives.
Let $C$ be the *coinvariant algebra* $C≔\mathbf C[\mathfrak h^*]/(\mathbf C[\mathfrak h^*]_+^W)$ of $\mathfrak g$, with the ideal $(\mathbf C[\mathfrak h^*]_+^W)$ generated by strictly positively graded $W$-invariant polynomials. Then there is an isomorphism $\End_\CatO(P(w_0\cdot\lambda)) \isom C^{W_\lambda}$.
The category $\CatO_\lambda$ has a projective generator $P_\lambda ≔ \bigoplus_{w\in W/W_\lambda} P(w\cdot\lambda)$. By Morita’s theorem [@Bass:K-theory Thm. II.1.3], there is an equivalence of categories $$\label{eqn:morita-eq}
\CatO_\lambda\xrightarrow{\simeq}\Mod{}[\underbrace{\End_\CatO(P_\lambda)}_{{}\eqqcolon A_\lambda}], M \mapsto\Hom_\CatO(P_\lambda, M).$$
The algebra $\End_\CatO(P(w_0\cdot\lambda))$ carries a natural non-negative grading exhibited by the Endomorphismensatz, so we can consider its category of graded modules. Fully-faithfulness of $\mathbf V_\lambda$ thus endows $A_\lambda$ with a grading. The equivalence in motivates:
Let $\gCatO_\lambda ≔ \Mod[g]{}[A_\lambda]$ be the category of graded $A_\lambda$-modules.
We denote the grading shift on $\Mod[g]{}[A_\lambda]$ by $\langle-\rangle$, where $M⟨1⟩_i ≔ M_{i-1}$.
A module $M\in \CatO_\lambda$ is if there is a graded module $\tilde M\in\gCatO_\lambda$ such that forgetting the grading yields $\mathrm f\tilde M=M$. In particular, simple, Verma- and indecomposable projective modules are gradable [@stroppel:gradings-and-translation §§2f.]. It will turn out to be natural to define these modules to have lowest non-trivial degree $0$.. In the following, we shall not distinguish notationally between these modules and their graded lifts.
Since $A_\lambda$ is non-negatively graded, the grading of modules reflects their submodule structure. The same data is encoded in the exponents of Kazhdan-Lusztig polynomials:
A for a module $M\in\CatO_0$ is a filtration of minimal length with semisimple subquotients $M_i$. A module $M$ is called *rigid* if it has a unique Loewy filtration; for instance, Verma modules are rigid.
The semisimple quotients are precisely the horizontal layers in the composition series diagrams.
The composition factor multiplicities in the degrees $M(v)_i$ of a Verma module satisfy $$p_{vw}=\sum_k\bigl[M(v)_{\ell(v)-\ell(w)+2k}:L(w)\bigr] q^k,$$ , any summand $q^k$ of $p_{vw}$ corresponds to a factor $L(w)$ in the $k$-th layer of the Loewy filtration for $M(v)$, with the zeroth layer at the top.
A graded analogue of the BGG reciprocity theorem holds.
\[def:graded-translation\] On $\gCatO_\lambda$, the graded $\Theta_s$ is uniquely defined by short exact sequences $$0\to M(w)⟨1⟩ \xto{\unit}\Theta_s M(w) \to M(ws)\to 0
\quad\text{and}\quad
\Theta_s M(ws) = \Theta_s M(w)⟨-1⟩$$ for $w<ws$. The adjunction $\unit\colon M\to\Theta_sM$ is a degree-$1$-map and $\Theta_s^2 \isom \Theta_s⟨-1⟩ ⊕ \Theta_s⟨1⟩$. The graded (co)shuffling functors are defined analogously to the non-graded case.
The Grothendieck group $K_0(\CatO^\mathbf Z_0)$ becomes a $\mathbf Z[q^{\pm 1}]$-module by $q[M]≔[M⟨1⟩]$. There is an isomorphism of $𝐙[q^{±1}]$-modules $$K_0\bigl(𝓞^\mathbf{Z}_0(\SL_n)\bigr) \to \mathsf{H}_q(S_n),\quad
[M(w)⟨q⟩] \mapsto qH_w,\quad
[P(w)⟨q⟩] \mapsto qC_w,$$ where $\mathsf{H}_q(S_n)$ denotes the Iwahori-Hecke algebra of $S_n$ with standard basis $\{H_w\}_{w\in W}$ and Kazhdan-Lusztig-basis $\{C_w\}_{w\in W}$. Under this isomorphism, the action via $[\Sh_s]$ corresponds to the right multiplication $(\cdot H_s)$.
Quivers {#sec:quivers}
-------
The $Q_\lambda$ associated to$A_\lambda$ has isoclasses of simple $A_\lambda$-modules as vertices their extensions as edges.
Let $\mathfrak a_\lambda$ be the ideal of the path algebra $\mathbf CQ_\lambda$ generated by the relations of extensions of the simple $A_\lambda$-modules. Then $\mathbf CQ_\lambda/\mathfrak a_\lambda \isom A_\lambda$ by Gabriel’s theorem [@Gabriel:Unzerlegbare-Darstellungen-I; @Gabriel:Unzerlegbare-Darstellungen-II].
We denote the composition of morphisms $v_2\xleftarrow{a}v_1$ and $v_3\xleftarrow{b}v_2$ of a quiver by $v_3\xleftarrow{ba}v_1$. We denote trivial path associated to a vertex $v$ by $\epsilon_v$.
By the equivalence \[eqn:morita-eq\], $Q_\lambda$ is the quiver with vertices indexed by $W$ and edges $w\from v$ given by the irreducible morphisms from $\Hom_\CatO(P(v\cdot\lambda), P(w\cdot\lambda))$, , morphisms that cannot be factored non-trivially. The ideal $\mathfrak a_\lambda$ is generated by relations of these morphisms.
The path algebra of any quiver $Q_\lambda$ is non-negatively graded by the length of a path in terms of arrows. One can show that $\mathfrak a_\lambda$ is a homogeneous ideal; hence $A_\lambda$ is graded as well. This grading coincides with the one induces via $\mathbf V_\lambda$ from \[sec:gradings\] [@stroppel:gradings-and-translation].
To summarise, we have that $\CatO_\lambda\simeq\Mod{}[A_\lambda]$ and $\CatO_\lambda^\mathbf Z\simeq\Mod[g]{}[A_\lambda]$. The algebra $A_\lambda$ has a complete set of primitive idempotents $\{\epsilon_w\}_{w\in W}$, given by the trivial paths in $Q_\lambda$, which correspond to the identities of the $P(w)$’s. This equivalence maps indecomposable projectives $P(w)$ in $\CatO_\lambda$ to the indecomposable projectives in $\Mod{}[A_\lambda]$; which are precisely the right ideals $\epsilon_w A_\lambda$ of all paths ending in $w$ [@Barot:Rep-Theory cor. 4.18, rmk. 4.20].
A canonical $\mathbf C$-basis of $\epsilon_w A_\lambda$ is given by paths that are in one-to-one correspondence with the simple composition factors of $P(w)$. Explicitly, each composition factor $L(v)⟨i⟩$ (, a factor $L(v)$ residing in the $i$-th layer) corresponds to a basis vector of $\epsilon_w A_\lambda$ from the degree $i$-part $(\epsilon_w A_\lambda e_v)_i$.
From know on, we shall stick to the principal block $\CatO_0$ and omit all the subscript-$\lambda$’s from $A$ and $Q$.
Parabolic subalgebras {#sec:parabolic-subalgebras}
---------------------
Let $(W,S)$ be a Coxeter system and $S_\mathfrak p\subseteq S$ be any subset of the simple reflections of $W$.
The associated $W_\mathfrak p\leq W$ is the subgroup $W_\mathfrak p=⟨s_i⟩_{i\in S_\mathfrak p}$ of $W$. Every left coset in $W_\mathfrak p\backslash W$ has a unique representative of minimal length [@Humphreys:Coxeter-Groups §1.10]. We denote the set of such representatives by $W^\mathfrak p$.
To the quiver $Q$ we associate the full subquiver $Q^\mathfrak p$ of $Q$ with vertex set $Q^\mathfrak p=W^\mathfrak p$ and define the respective algebra $A^\mathfrak p≔A/(e_v)_{v\in W_\mathfrak p}$. The category $\CatO^\mathfrak p_0 ≔ \Mod{}[A^\mathfrak p]$. is equivalent to the smallest Serre subcategory of $\CatO^\mathfrak p_0$ containing all simple modules $L(w)$ for $w\in W^\mathfrak p$ [@kildetoft-mazorchuk:parabolic-projective-functors §2]. The quotient map $A\onto A^\mathfrak p$ induces an induction-restriction-adjunction $$\Ind^{\mathfrak p}\colon \CatO_0 \xtofrom{\dashv} \CatO^\mathfrak p_0 \noloc \Res_{\mathfrak p},$$ where $\Res_\mathfrak p$ is fully faithful and thus turns $\CatO^\mathfrak p_0$ into a subcategory of $\CatO_0$. The functor $\Res_\mathfrak p$ has both left and right adjoints, where its left adjoint $Z^\mathfrak p≔\Ind^\mathfrak p=-⊗_A A^\mathfrak p$ ( right adjoint $Z_\mathfrak p$), called ( ), assigns to a module $M$ its largest quotient ( submodule) which only has simple composition factors corresponding to words from $W^\mathfrak p$ [@mazorchuk:categorification Thm. 6.1].
Let $P^\mathfrak p(w)≔Z^\mathfrak p(P(w))=\epsilon_wA^\mathfrak p$ and $M^\mathfrak p(v) ≔ Z^\mathfrak p(M(v))$. As notation suggests, the $P^\mathfrak p(w)$’s are the indecomposable projectives in $\CatO^\mathfrak p_0$ [@mazorchuk:categorification §4.6].
The constructions and statements from have projective analogues; namely:
The category $\CatO^\mathfrak p_0$ has a projective generator $P^\mathfrak p=\bigoplus_{w\in W^\mathfrak p}P^\mathfrak p(w)$.
The analogously constructed graded version $\CatO_0^{\mathbf Z,\mathfrak p}$ of $\CatO_0^\mathfrak p$ contains simples, parabolic Vermas and indecomposable projectives.
The $P^\mathfrak p(w)$’s have standard filtrations with subquotients isomorphic to $M^\mathfrak p(v)$’s.
The respective multiplicities satisfy a parabolic BGG reciprocity theorem [@RC:splitting.criteria Prop. 4.5, Thm. 6.1].
Parabolic Kazhdan-Lusztig polynomials $p^\mathfrak p_{vw}\in\mathbf Z[q^{\pm 1}]$ [@Deodhar:parabolic-KL §3; @Soergel:KL-polynomials] occur as base change coefficients for parabolic Hecke algebras.
The composition factor multiplicities can be computed using the parabolic generalised Kazhdan-Lusztig theorem [@Irving:filtered-category-OS Cor. 7.1.3; @CC:parabolic-KL Thm. 1.3]: $$p^\mathfrak p_{vw} = \sum_{k\geq 0}(P^\mathfrak p(w) : M^\mathfrak p(v)⟨k⟩)q^k = \sum_{k\geq 0} [M^\mathfrak p(v) : L(w)⟨k⟩]q^k,$$
Explicitly, this means that a summand $q^k$ in $p^\mathfrak p_{vw}$ means that the factor $M^\mathfrak p(v)$ occurs in the $k$-th layer of a Loewy series of $P^\mathfrak p(w)$, counted from the top [@BGS:Koszul-Duality Thm. 3.11.4].
The functor $Z^\mathfrak p$ commutes with projective functors, in particular with $\Theta_s$ [@mazorchuk:categorification Thm. 6.1]. This implies that $\Theta_s$ as well as $\Sh_s$ restrict to $\CatO^\mathfrak p_0$, and $\Theta_s$ is uniquely characterised by short exact sequences $$0\to M^\mathfrak p(w)⟨1⟩\xto{\unit}P^\mathfrak p(w)\to M^\mathfrak p(ws)\to 0
\quad\text{and}\quad
\Theta_s M(ws) = \Theta_s M(w)⟨-1⟩$$ for $w<ws$.
\[caveat:parabolic-projectives-not-projective\] The inclusion $\CatO^\mathfrak p_0\subseteq \CatO_0$ does not preserve projectives.
𝐵n-actions for 𝔰𝔩₂ {#sec:sl2-case}
==================
Consider the Lie algebra $\mathfrak{g}=\SL_2$ and its Weyl group $S_2=\{e, s\}$. The Verma modules and the indecomposable projectives have composition series $$P(e)
= M(e)
= \begin{psmallmatrix}L(e)\\L(s)\end{psmallmatrix},
\qquad
P(s)
= \begin{psmallmatrix}M(s)\\M(e)\end{psmallmatrix}
= \begin{psmallmatrix} L(s)\\L(e)\\L(s)\end{psmallmatrix},
\qquad M(s) = L(s);$$ these can be computed using Kazhdan-Lusztig polynomials. Therefore, $\CatO_0$ is equivalent to $\Mod{}[A]$ for the path algebra $A=\mathbf CQ/(ba)$ of the quiver $Q=e\xtofrom[b]{a}s$. The arrows $a,b$ of $Q$ and their relations correspond to the unique (up to scalars) non-trivial morphisms $a\colon P(e)\into P(s)$ and $b\colon P(s)\to P(e)$ [@stroppel:gradings-and-translation].
The action of Shₛ {#sec:shuffling-sl2}
-----------------
We are interested in the images $\Sh_s M$ for the following modules $M$:
1. $M(e)=P(e)$: from \[eqn:translation-defining-ses\] we get $\Theta_s M(e) = \begin{psmallmatrix}M(s)\\M(e)\end{psmallmatrix}=P(s)$ and $\Sh_s M(e)= M(s)$.
2. $M(s)$: Up to scalars, there is a unique morphism $M(s) \into M(e)$; hence we obtain that $\Sh_s M(s)\isom \begin{psmallmatrix}M(s)\\M(e)/M(s)\end{psmallmatrix} = \begin{psmallmatrix}L(s)\\L(e)\end{psmallmatrix}=M^\vee(s)$.
3. $P(s)$: using $\Theta_s^2\isom\Theta_s\oplus\Theta_s$, it follows from $P(s)=\Theta_s M(e)$ that $\Sh_s P(s) \isom P(s)$.
Similar arguments show that $\mathbf R\Csh_s P(s)\qis P(s)$ and $\mathbf R\Csh_s P(e)$ is the mapping cone $\mathbf R\Csh_s P(e)\qis\{P(s)\to P(e)\}$ (recall the notation of mapping cones from \[notation:total-complexes\]); to summarise, we have $$\label{eq:shuffling-sl2:imaegs-of-projectives}
\begin{aligned}
\mathbf L\Sh_s P(e) &= \{P(e)\to \degZero{P(s)}\} \simeq M(s) &
\mathbf R\Csh_s P(e) &= \{\degZero{P(s)}\to P(e)\} \\
\mathbf L\Sh_s P(s) &= P(s) &
\mathbf R\Csh P(s) &= P(s).
\end{aligned}$$ From the naturality diagram $$\label{eqn:adj-of-Ps}
\begin{tikzcd}[ampersand replacement=\&]
M(e) \rar["a"]\dar["\eta_{M(e)}"']\& P(s) \rar["b"]\dar["\Mtrx{1\\x}", "\eta_{P(s)}"'] \& P(e) \dar["\eta_{M(s)}"]\\
P(s) \rar["\Mtrx{1\\0}"] \& P(s)⊕P(s) \rar["\Mtrx{0 & 1}"]\rar \& 0
\end{tikzcd}$$ of $\eta\colon\id\Rightarrow\Theta_s$ we get that the morphisms $a\colon P(e)\into P(s)$ and $b\colon P(s)\to P(e)$ have images $$\label{eq:shuffling-sl2:images-of-generating-morphisms}
\Sh_s a\colon M(s)\into P(s)\quad\text{and}\quad\Sh_s b\colon P(s)\onto M(s).$$ Since $\Db(\CatO_0)$ is generated as a triangulated category by $P(e)$ and $P(s)$, this datum suffices to describe the behaviour of $\mathbf L\Sh_s$; we thus have shown:
The Artin braid group $B_1$ acts via $\Sh_s$ on $\Db(\CatO_0(\SL_2))$ by $$\begin{aligned}
B_1\isom \mathbf Z &\to \Aut(\Db(\CatO_0))\\
n &\mapsto
\underset{\strut}{
\begin{aligned}[t]
& P(s)\mapsto P(s),\\
& P(e)\mapsto
\begin{cases}
\bigl\{P(e)\into \smash[t]{\overbrace{P(s)\xto[ab]{} P(s)\to \dotsb\to \degZero{P(s)}}^{\text{$n$ times}}}\bigr\} & \text{if $n\geq 0$},\\
\bigl\{\smash{\underbrace{\degZero{P(s)}\xto{ab} P(s)\to \dotsb\to P(s)}_{\text{$-n$ times}}} \to P(e) \bigr\} & \text{if $n\leq 0$}
\end{cases}
\end{aligned}
}
\end{aligned}$$ with homological degree $0$ as indicated.
Spherical objects
-----------------
$\Db(\CatO_0)$ contains two spherical objects:
1. $P(s)$ is $0$-spherical since $\End_{\CatO}(P(s))\isom\mathbf C[x]/(x)^2$ with $x=ab$.
2. $L(e)$ is $2$-spherical; this can be seen from the projective resolution $L(e) \qis \{P(e)\to P(s)\to P(e)\}$.
A simple module never can be $0$- or $1$-spherical since simples do not extend themselves non-trivially.
Spherically twisting by P(s)
----------------------------
The non-trivial endomorphism $x\coloneqq ab$ of $P(s)$ factors through $P(e)$. Since $P(e)$ and $P(s)$ generate $D^\mathrm b(\mathcal O_0)$, this already implies that composition pairing from the Calabi-Yau-property (\[def:spherical-object:calabi-yau\]) is non-degenerate, so $P(s)$ is spherical. The images of projectives under the cotwisting functor are $$\begin{alignedat}{4}
T'_{P(s)}P(s) &= \{\Mtrx{1\\x}\colon \degZero{P(s)} → P(s)⊕P(s)\} &&\qis P(s)[-1] &&= \Sh_s P(s)[-1],\\
T'_{P(s)}P(e) &= \{a\colon \degZero{P(e)} → P(s)\} &&\qis M(s)[-1] &&= \Sh_s P(e)[-1].
\end{alignedat}$$ This proves half of the following:
\[thm:shuffling-is-twisting:sl2\] There is a natural isomorphism $𝐋\Sh_s[-1]\isom T'_{P(s)}$ of autoequivalences of $\Db(𝓞_0(\SL_2))$.
Let $A$ and $B$ be rings. Any right exact functor $F\colon \Mod{}[A] → \Mod{}[B]$ that preserves arbitrary direct sums is isomorphic to tensoring with the $A$-$B$-bimodule $FA$ [@Bass:K-theory Thm. II.2.3].
For abelian categories $𝓐$ and $𝓑$ with projective generators $P_𝓐$ and $P_𝓑$, Morita’s theorem allows us to identify $𝓐$ and $𝓑$ with $\Mod{}[\End_𝓐(P_𝓐)]$ and $\Mod{}[\End_𝓑(P_𝓑)]$ respectively. Then for any right exact functor $F\colon 𝓐 \to 𝓑$ commuting with arbitrary direct sums there is a natural isomorphism of functors $$F \isom -⊗_{\End_𝓐(P_𝓐)}\underbrace{\Hom_𝓑(P_𝓑, FP_𝓐)}_{\eqqcolon {M_{F}}}\colon 𝓐 \to 𝓑,$$ where ${M_{F}}$ becomes an $\End_𝓐(P_𝓐)$-$\End_𝓑(P_𝓑)$-bimodule by $a\ldot f\ldot b = Fa\circ f\circ b$ for $a\in \End_𝓐{P_𝓐}, b\in \End_𝓑(P_𝓑)$ and $f\in {M_{F}}$.
By the corollary there are natural isomorpisms $$\begin{aligned}
{2}
𝐋\Sh_s[-1] &\stackrel{\text{def}}{=} \{ \degZero{\id_𝓞} \Rightarrow 𝛩_s \} &&≅ -⊗_A \{ A \to {M_{𝛩_s}} \},\\*
T'_{P(s)} &\stackrel{\text{def}}{=} \{ \degZero{\id_𝓞} \Rightarrow \Xi'_{P(s)} \} &&≅ -⊗_A \{ A \to {M_{\Xi'_{P(s)}}} \},
\end{aligned}$$ such that it suffices to show ${M_{𝛩_s}} \isom {M_{\Xi'_{P(s)}}}$. Recall that $\Xi'_{P(s)} = \lin(\hom_𝓞(-, P(s)), P(s))$. By finite dimensionality, there is an isomorphism ${M_{\Xi'_{P(s)}}} ≅ P(s)^* ⊗_𝐂 P(s)$ of $A$-$A$-bimodules. Consider the canonical vector space basis $\{\epsilon_s, s\from e, s\from e\from s\}$ of $P(s)=\epsilon_sA$ and the dual basis of $P(s)^*$. Then a basis of ${M_{\Xi'_{P(s)}}}$ is given by all pairwise tensor products, on which the bimodule action is given schematically as follows: $$\label{eq:shuffling-twisting-isomorphic:sl2-tensor}
M_{\Xi'} \isom P(s)^* ⊗_𝐂 P(s)\colon \quad
\begin{tikzpicture}[mth, baseline=(B.base)]
\matrix (A) [
matrix of math nodes,
row sep=0mm,
column sep=-2mm,
text height=2ex, text depth=.5ex,
column 2/.style={anchor=base east},
column 4/.style={anchor=base west}
] {
~& (s←e←s)^* & & \epsilon_s &~\\
~& |[alias=B]| (s←e)^* & \quad⊗\quad & (s←e) &~\\
~& \epsilon_s^* & & (s←e←s). &~\\
};
\node[braced box={T}, fit=(A-1-2) (A-3-2)] {} ;
\node[braced box'={T'},fit=(A-1-4) (A-3-4)] {} ;
\draw
(A-1-1) edge[|->, out=195, in=165] node[swap]{$(e←[b]s)_*$} (A-2-1.160)
(A-2-1.200) edge[|->, out=195, in=165] node[swap]{$(s←[a]e)_*$} (A-3-1)
(A-1-5) edge[|->, out=-15, in=15] node{$∘(s←[a]e)$} (A-2-5.20)
(A-2-5.340) edge[|->, out=-15, in=15] node{$∘(s←[b]e)$} (A-3-5);
\end{tikzpicture}$$ To describe the $A$-$A$-bimodule action on ${M_{\Theta_s}} = \Hom_A(P, \Theta_s P)$ in terms of a vector space basis we introduce the following notation. Recall that $P=P(e)⊕P(s)$ and $\Theta_s P\isom P(s)^{⊕3}$. We enumerate the summands of $\Theta_s P = P(s)_1⊕P(s)_2⊕P(s)_3$. We then abbreviate, , the morphism $\Mtrx{0&0\\0&0\\0&x} \in \Hom_A(P, \Theta_s P)$, by $P(s)_3 \xleftarrow{x} P(s)$. For $\Theta_s$, the naturality diagram of $\eta\colon\id\Rightarrow\Theta_s$ shows that the images under $\Theta_s$ of the morphisms $a$ and $b$ generating $\End_\mathcal{C}(P)$ are $$\begin{tikzcd}[column sep={\the\algnRef}, nodes={inner xsep=0pt}, row sep=small]
P(e)\rar[phantom, "⊕"] & P(s)\ar[dl, "b"'] &[4em] P(s)_1 \rar[phantom, "⊕"] & P(s)_2 \rar[phantom, "⊕"] & P(s)_3 \ar[dll, "1"']\\
P(e)\rar[phantom, "⊕"] \ar[dr, "a"'] & P(s) \ar[phantom, "\xmapsto{\Theta_s}", r]& P(s)_1 \rar[phantom, "⊕"] \ar[dr, "1"'] & P(s)_2 \rar[phantom, "⊕"] & P(s)_3\\
P(e)\rar[phantom, "⊕"] & P(s) & P(s)_1 \rar[phantom, "⊕"] & P(s)_2 \rar[phantom, "⊕"] & P(s)_3\mathrlap{.} \\
\end{tikzcd}$$ A vector space basis of ${M_{𝛩_s}}$ is given by irreducible morphisms from one summand of $P$ to another in the schematic $$\label{eq:shuffling-twisting-isomorphic:sl2-hom}
{M_{𝛩_s}}\colon\quad
\begin{tikzpicture}[mth, baseline=(B.base)]
\matrix (A) [matrix of math nodes, row sep=0mm, column sep=2mm, text height=2ex, text depth=.5ex, column 2/.style={anchor=base east}, column 4/.style={anchor=base west}] {
{} &[-1em] P(s)_3← & & ←[1]P(s) &[-1em]~\\
{} & P(s)_1← &←& ← P(e) & |[alias=B]| ~\\
{} & P(s)_2← & & ←[x] P(s), &~\\
};
\node[braced box={T}, fit=(A-1-2) (A-3-2)] {} ;
\node[braced box'={T'},fit=(A-1-4) (A-3-4)] {} ;
\draw
(A-1-1) edge[|->, out=195, in=165] node[swap]{$\Theta_s(P(e)\xleftarrow{b} P(s))\circ $} (A-2-1.150)
(A-2-1.210) edge[|->, out=195, in=165] node[swap]{$\Theta_s(P(s)\xleftarrow{a} P(e))\circ$} (A-3-1)
(A-1-5) edge[|->, out=-15, in=15] node{$∘(P(s)←[a]P(e))$} (A-2-5.30)
(A-2-5.330) edge[|->, out=-15, in=15] node{$∘(P(e)←[b]P(s))$} (A-3-5);
\end{tikzpicture}$$ with the bimodule action as indicated. Comparison of and shows that the obvious isomorphism ${M_{\Theta_s}} \isom {M_{\Xi'_{P(s)}}}$ of vector spaces is an isomorphism of $A$-$A$-bimodules.
Spherically twisting by L(e)
----------------------------
Recall that $\mathbf{L}\Sh_s L(e) = L(e)[1]$. From \[rmk:twisting-E-by-itself\] it follows that if there is any isomorphism $T'_{L(e)} \isom \mathbf L\Sh_s[?]$, then the shift $?$ must be zero. Recall the projective resolution $L(e)\qis\{P(e)\to P(s)\to P(e)\}$. We obtain $$\begin{aligned}
\hom^\bullet(P(e), L(e))
&= \left\{
\left\langle
\begin{tikzcd}[baseline=(B.base)]
\&\& P(e) \ar[dll, "\id"'] \ar[d, phantom, "\strut" name=B]\\
P(e) \ar[r] \& P(s) \ar[r] \& P(e)
\end{tikzcd}
\right\rangle
\xrightarrow{\mathmakebox[\widthof{\scriptsize$\!\!\!\Mtrx{0\\1}\!\!\!$}]{1}}
\left\langle
\begin{tikzcd}[baseline=(B.base)]
\&\& P(e) \ar[dl, "a"'] \ar[d, phantom, "\strut" name=B]\\
P(e) \ar[r] \& P(s) \ar[r] \& P(e)
\end{tikzcd}
\right\rangle
\xrightarrow{\mathmakebox[\widthof{\scriptsize$\!\!\!\Mtrx{1 & 0}\!\!\!$}]{0}}
\left\langle
\begin{tikzcd}[baseline=(B.base)]
\&\& P(e) \ar[d, name=B, "\id"]\\
P(e) \ar[r] \& P(s) \ar[r] \& P(e)
\end{tikzcd}
\right\rangle
\right\}\notag\\*
&
\label{eq:twsting-sl2-by-L(e):Hom(P(e),L(e))}
\qis \{P(e)\to L(e)\},\\
\hom^\bullet(P(s), L(e))
&= \left\{
\left\langle
\begin{tikzcd}[baseline=(B.base)]
\&\& P(s) \ar[dll, "b"'] \ar[d, phantom, "\strut" name=B]\\
P(e) \ar[r] \& P(s) \ar[r] \& P(e)
\end{tikzcd}
\right\rangle
\xrightarrow{\!\!\!\Mtrx{0\\1}\!\!\!}
\left\langle
\begin{tikzcd}[baseline=(B.base)]
\&\& P(s) \ar[dl, "{\id, x}"'] \ar[d, phantom, "\strut" name=B]\\
P(e) \ar[r] \& P(s) \ar[r] \& P(e)
\end{tikzcd}
\right\rangle
\xrightarrow{\!\!\!\Mtrx{1 & 0}\!\!\!}
\left\langle
\begin{tikzcd}[baseline=(B.base)]
\&\& P(s) \ar[d, "b" name=B]\\
P(e) \ar[r] \& P(s) \ar[r] \& P(e)
\end{tikzcd}
\right\rangle
\right\}\notag\\*
\label{eq:twsting-sl2-by-L(e):Hom(P(s),L(e))}
&\qis 0,\end{aligned}$$ with the rightmost entry in degree zero. The angle brackets denote vector spaces generated by the indicated morphisms. From it follows $T'_{L(e)} P(s) = P(s) = T_{L(e)} P(s)$. For $P(e)$, the images under $T'_{L(e)}$ and $T_{L(e)}$ are the respective total complexes of the triple complexes $$\begin{aligned}
\notag
&\eqsp T'_{L(e)} P(e) &&\eqsp T_{L(e)} P(e)\\*
\notag
&=\bigl\{\bgroup\color{gray}\degZero{P(e)}\xto{\ev'}\egroup\lin\bigl(\hom^\bullet(P(e),L(e)), L(e)\bigr)\bigr\}
&&=\bigl\{\hom^\bullet\bigl(P(e),L(e)\bigr) \otimes L(e) \bgroup\color{gray}\xto{\ev}\degZero{P(e)}\egroup\bigr\}\\
\intertext{
with the gray $P(e)$ in degree $0$.
We write out the double complexes
$\lin(\hom^\bullet_{\CatO}(-,-), -)$ and $\hom^\bullet_{\CatO}(-,-)\otimes -$:
}
&=
\label{eq:cotwist-of-P(e)-wrt-L(e)}
\left\{
\begin{tikzcd}[row sep={.7cm,between origins}, column sep={1.2cm,between origins}, nodes={inner sep=1pt},ampersand replacement=\&]
P(e)_{\id} \ar[rr, "a"]\ar[dd,"0"']
\&\& P(s)_{\id} \ar[rr,"b"]
\&\& P(e)_{\id} \ar[dd,"0"]\\
\& |[color=gray]| P(e) \ar[gray, dddl, "\id"]\ar[gray, dr, "a"]\ar[gray, urrr, "\id" very near start, dashed]\\
P(e)_a \ar[rr, "a", cross line]\ar[dd,"\id"', dashed]
\&\& P(s)_a \ar[from=uu, cross line, "0"]\ar[rr,"b"]\ar[dd,"\id", dashed]
\&\& P(e)_a \ar[dd,"\id", dashed]\\\\
P(e)_{\id} \ar[rr, "a"']
\&\& P(s)_{\id} \ar[rr,"b"']
\&\& P(e)_{\id}
\end{tikzcd}
\right\}
&&=
\left\{
\begin{tikzcd}[row sep={.7cm,between origins}, column sep={1cm,between origins}, nodes={inner sep=1pt},ampersand replacement=\&]
P(e)_{\id} \ar[rr, "a"] \ar[dd,"\id"', dashed]
\&\& P(s)_{\id} \ar[rr,"b"]
\&\& P(e)_{\id} \ar[dd,"\id", dashed]\\\\
P(e)_b \ar[rr, "a"] \ar[dd,"0"']
\&\& P(s)_b \ar[from=uu, "\id", dashed] \ar[rr,"b"] \ar[dd,"0"']
\&\& P(e)_b \ar[dd,"0"]\\
\&\&\& |[color=gray]| P(e) \ar[gray, from=dlll, cross line, "\id"]\ar[gray, from=ul, "b"]\ar[gray, from=uuur, cross line, "\id"' near start, dashed]\\
P(e)_{\id} \ar[rr, "a"']
\&\& P(s)_{\id} \ar[rr,"b"']
\&\& P(e)_{\id}
\end{tikzcd}
\right\}\\
\intertext{
where the $P(-)$'s are indexed by basis elements from \eqref{eq:twsting-sl2-by-L(e):Hom(P(e),L(e))}
(\ie, by morphisms from $P(e)$).
The adjunction maps comprise the three gray morphisms.
Gauß elimination
(\ie, the removal of an identity together with the two identic summands at its ends)
along the dashed identities yields
}
\notag
&\qis \{P(e) \xto{a} \degZero{P(s)}\},
&&\qis \{\degZero{P(s)} \xto{b} P(e)\} \\
&\qis \mathbf L\Sh_s P(e),
&& \qis \mathbf{R}\Csh_s P(e),\end{aligned}$$ with $P(s)$ in degree zero. Comparing this to shows half of the following:
There are natural isomorphisms isomorphisms $T'_{L(e)} \isom \mathbf L\Sh_s$.
To show that both functors are isomorphic we have yet to show that the morphisms $\{\id_{P(e)}, \id_{P(s)}, a, b\}$ generating $\End_{\CatO}(P)$ have identical images under both functors. Recall the images $\Sh_s a\colon M(s)\into P(s)$ and $\Sh_s b\colon P(s)\onto M(s)$ from .
To compute $T'_{L(e)}b$ consider the triple complex with only the bottom three identities eliminated. A triple complex representing $T'_{L(e)} P(s)\simeq P(s)$ with partial cancellation is obtained similarly from . We thus obtain that $T'_{L(e)}b$ is the map $$\label{eq:twisting-by-L(e):action-on-b:triple-complexes}
\begin{tikzcd}[ampersand replacement=\&, row sep={.7cm,between origins}, column sep={2cm,between origins}]
\&[-5mm] \mathmakebox[\widthof{$P(e)_\id$}]{P(e)_b}
\ar[rr]
\ar[dd, equal, cross line]
\ar[brace', -, dd, start anchor=north west, end anchor=south west, xshift=-1ex, "T'_{L(e)} P(s) \qis{}" {name=TPs, anchor=east, xshift=-.8ex, font=\normalsize}]
\ar[dd, equal, cross line]
\&\&[-5mm] P(s)_b \ar[rr]\ar[dd, equal]
\&\&[-5mm] \mathmakebox[\widthof{$P(e)_\id$}]{P(e)_b} \ar[dd, equal]
\\
\&\& |[gray]| P(s)
\ar[urrr, gray]
\ar[dr, gray, equal]
\\
\& P(e)_\id \ar[rr]
\&\& P(s)_\id
\ar[from=uu, equal, cross line]
\ar[rr]
\&\& P(e)_\id
\ar[brace, -, from=uu, start anchor=north east, end anchor=south east]
\\
P(e)_\id
\ar[from=uuur, cross line, equal]
\&\& P(s)_\id
\ar[from=uuur, cross line, equal]
\ar[rr]
\&\& P(e)_\id
\ar[from=uuur, equal, cross line]\\
\phantom{P(e)_\id}
\ar[brace', -, xshift=-1ex, from=u, start anchor=north west, end anchor=south west, "T'_{L(e)} P(e) \qis{}" {name=TPe, anchor=east, xshift=-.8ex, font=\normalsize}]
\& |[gray]| P(e)
\ar[urrr, gray, equal]
\ar[from=uuur, cross line, gray, "b"']
\ar[from=ul, to=ur, cross line]
\&\&\& \phantom{P(e)_\id}
\ar[brace, -, from=u, start anchor=north east, end anchor=south east]
\ar[from=TPs, to=TPe, "T'_{L(e)}b"']
\end{tikzcd}$$ between the complexes representing $T'_{L(e)}P(s)$ and $T'_{L(e)}P(e)$. We have indicated the elements of $\hom^\bullet_{\CatO}(P(-), L(e))$ the summands of $T'_{L(e)} P(-)$ are indexed by. We pass to the total complexes of and choose quasi-isomorphic replacements $$\begin{tikzcd}[ampersand replacement=\&]
\&[-.6cm]
\& P(s) \ar[from=d, "\Mtrx{0\\0\\1}", xshift=-.5em] \ar[d, "\Mtrx{0\\-1\\1}", xshift=.5em] \ar[d, phantom, "\qis" rotate=90]
\\
T'_{L(e)}P(s) \ar[r, phantom, "\qis"] \ar[d, "T'_{L(e)}b"']\&
\smash{\Bigl\{}
P(e) \ar[d, "1"'] \ar[r, "\Mtrx{-1\\a\\0}"] \&
P(e)⊕P(s)⊕\textcolor{gray}{P(s)} \ar[r, "\Mtrx{a&1&\textcolor{gray}{1}\\0&b&\textcolor{gray}{b}}"] \ar[d, "\Mtrx{0&1&0\\0&0&\textcolor{gray}{b}}"] \&
P(s)⊕P(e) \ar[r, "\Mtrx{b & -1}"] \ar[d, "\Mtrx{0 & \textcolor{gray}{1}}"] \&
P(e)
\mathrlap{\Bigr\}}
\\
T'_{L(e)}P(e) \ar[r, phantom, "\qis"] \&
\smash{\Bigl\{}
P(e) \ar[r, "\Mtrx{a\\0}"] \&
P(s)⊕\textcolor{gray}{P(e)} \ar[r, "\Mtrx{b & 1}"'] \&
P(e)
\mathrlap{\smash{\Bigr\}}}
\\
\&
\& M(s) \ar[from=u, "\Mtrx{\can&0}", xshift=.5em] \ar[u, phantom, "\qis" rotate=90] \ar[u, "\Mtrx{\can\\-\can}", xshift=-.5em]
\end{tikzcd}$$ as indicated. This shows that $T'_{L(e)} b = \mathbf L\Sh_sb\colon P(s)\onto M(s)$ is the canonical quotient map indeed. A similar argument shows that $T'_{L(e)}a = \Sh_s a\colon M(s)\into P(s)$ is the canonical inclusion. Since all morphisms in $\Db(\CatO_0)$ are generated by $a$ and $b$, $T'_{L(e)}$ and $\mathbf L\Sh_s$ are naturally isomorphic as auto-equivalences of $\Db(\CatO_0)$.
𝐵ₙ-actions for 𝔰𝔩₃ and 𝔰𝔩ₙ {#sec:sln-case}
==========================
The Lie algebra $\SL_3$ has as its Weyl group the symmetric group $S_3=\{e, s, t, st, ts, w_0\}$. A quiver $Q_{\SL_3}$ and a homogeneous ideal $\mathfrak a_{\SL_3}$ of $\mathbf CQ_{\SL_3}$ such that $\CatO_0(\SL_3)\simeq\Mod{}[A_{\SL_3}]$ for the path algebra $A_{\SL_3}=Q_{\SL_3}/\mathfrak a_{\SL_3}$ is provided in [@stroppel:quivers; @Marko:quivers].
One sees that $Q_{\SL_2}$ is a full subquiver of $Q_{\SL_3}$ and $\mathfrak a_{\SL_3}\cap\mathbf CQ_{\SL_2}=\mathfrak a_{\SL_2}$. The thus induced inclusion $A(\SL_2)\into A(\SL_3)$ gives rise to an adjoint pair of functors $$\begin{aligned}
\Res_{\SL_3}^{\SL_2}\colon\CatO_0(\SL_3) &\xtofrom{\dashv} \CatO_0(\SL_2)\noloc \Ind_{\SL_2}^{\SL_3},\\
\begin{aligned}
P(e), P(t) &\mapsto P(e),\\
P(s), P(st), P(ts), P(w_0) &\mapsto P(s),\\
\end{aligned}
&
\phantom{{}\xtofrom{\dashv}{}}
\begin{aligned}
P(e) &\mapsfrom P(e),\\
P(s) &\mapsfrom P(s),
\end{aligned}\end{aligned}$$ which turns $\CatO_0(\SL_2)$ into a full subcategory of $\CatO_0(\SL_3)$. In particular, $\End_{\CatO_0(\SL_3)}(P(s)) \isom \mathbf{C}[x]/(x^2)$ and $P(s)$ is $0$-spherelike also in $\CatO_0(\SL_3)$.
The Calabi-Yau property from \[def:spherical-object:calabi-yau\] is not “local”, in the sense that an object can lose this property in a larger ambient category. For instance, there are non-trivial morphisms $P(s)\to P(t)$ and $P(t)\to P(s)$ in $\CatO_0(\SL_3)$ whose composition is zero, so $P(s)$ cannot be spherical. We shall present two possible remedies in this section.
Spherical subcategories
-----------------------
Consider a $k$-linear triangulated category $\mathcal T$.
An object $E\in\mathcal T$ is has a $\mathrm{S} E$ if the covariant functor $\Hom_\mathcal T(-, E)^*$ is representable by $\mathrm{S} E$ (here, the star indicates vector space dual). If a Serre dual can be chosen functorially and this functor is an auto-equivalence, $\mathcal T$ is said to have a $\mathrm S$.
The Calabi-Yau condition from \[def:spherical-object:calabi-yau\] requires that $E[d]$ be a Serre dual for $E$.
Assume that $E\in\mathcal T$ is a $d$-spherelike but not necessarily spherical object which has *some* Serre dual $\mathrm SE$. Since in particular $\End_\mathcal T(E)^*\isom\Hom_\mathcal T(E, \mathrm SE[-d])$, there is a morphism $x^*\colon E\to\mathrm SE[-d]$, which corresponds to the dual of the non-trivial endomorphism $x$ of $E$.
The of a spherelike object $E$ is $Q(E)≔\cone(x^*)$. Its ${}^\bot Q(E) ≔ \{X\in\mathcal T\colon\Hom_{\mathcal T}(X, Q(E))=0\}$ is a full triangulated subcategory of $\mathcal T$.
The *spherical subcategory* $\Sph(E)≔{}^\bot Q(E)$ of $E$ is the largest triangulated subcategory of $\Db(\mathcal T)$ in which $E$ is spherical.
For $\mathfrak g$ a semisimple complex Lie algebra and $\lambda$ a regular weight (for instance, $\lambda=0$), the auto-equivalence $\mathrm S\coloneqq\mathbf L\Sh_{w_0}^2$ is a Serre functor of $\Db(\CatO_\lambda)$ [@MS:Serre-functors Prop. 4.1].
The $0$-spherelike module $P(s)\in\Db(\CatO_0(\SL_3))$ has Serre dual $\mathrm SP(s)\isom P(s)^\vee$.
For this proof, we take the graded structure on $\CatO_0^{\mathbf Z}$ into account. Recall from \[def:graded-translation\] that $\Theta_s^2 ≅ \Theta_s⟨-1⟩⊕\Theta_s⟨1⟩$, and hence $\Sh_s\Theta_s\isom\Theta_s⟨-1⟩$. Recall that for $w<ws$, $\Sh_s$ maps standard factors $M(w)$ to $M(ws)$ and $M(ws)$ to $\begin{psmallmatrix}M(ws)\\M(w)/M(ws)\end{psmallmatrix}⟨-1⟩$. For $P(s)\in\Db(\CatO_0(\SL_3))$, one thus can compute $$\begin{aligned}
{2}
&\phantom{ {}\xmapsto{\Sh_s}{} }
P(s)
= \Theta_s M(s)
&
&\xmapsto{\Sh_s}
P(s)\langle -1\rangle =\mathmakebox[\algnRef][l]{\begin{psmallmatrix}M(s)\\M(e)\end{psmallmatrix}\langle -1\rangle}\\
&\xmapsto{\Sh_t}
\begin{psmallmatrix}M(ts)\\M(t)\end{psmallmatrix}\langle -1\rangle
&
&\xmapsto{\Sh_s}
\mathmakebox[\algnRef][l]{\begin{psmallmatrix}M(w_0)\\M(ts)\end{psmallmatrix}\langle -1\rangle}
= \Theta_t M(ts)\langle -1\rangle
\\
&\xmapsto{\Sh_t}
\begin{psmallmatrix}M(w_0)\\M(ts)\end{psmallmatrix}\langle -2\rangle
&
&\xmapsto{\Sh_s}
\begin{psmallmatrix}
& M(w_0)\\
M(ts) & M(st)/M(w_0)\\
M(t)/M(ts)
\end{psmallmatrix}⟨-3⟩
\\
&\xmapsto{\Sh_t}
\mathrlap{
\arraycolsep=3pt
\left(
\mkern-8mu
\begin{smallmatrix}
&&& \begin{smallmatrix} M(w_0)\\M(st)/M(w_0) \end{smallmatrix}
\\[-0.9em]
&\begin{smallmatrix}\phantom{()} \\ M(w_0) \end{smallmatrix}
& \left( \begin{smallmatrix} M(st)\\M(s)/(st) \end{smallmatrix} \middle/ \begin{smallmatrix} M(w_0)\\M(ts)/M(w_0) \end{smallmatrix} \right) \\[-1.225em]
\\[0.125em]
\left(
\begin{smallmatrix} M(t)\\M(e)/M(t) \end{smallmatrix} \middle/ \begin{smallmatrix} \phantom{()}\\ M(w_0) \end{smallmatrix}
\right)
\end{smallmatrix}
\mkern-3mu\right)
\langle -4\rangle
}
\\
&\mathmakebox[\widthof{${}\xmapsto{\Sh_t}{}$}][r]{{}={}}
\begin{psmallmatrix}
& L(w_0)\\\
& L(st)\quad L(ts)\\
L(w_0) & L(s)\quad L(t)\\
L(st) \quad L(ts) & L(e)\\
L(s)
\end{psmallmatrix}
\langle -4\rangle
&
&\mathmakebox[\widthof{${}\xmapsto{\Sh_t}{}$}][r]{{}={}}
P(s)^\vee.
\end{aligned}$$ The morphisms dual to the endomorphisms of $P(s)$ span $\End(P(s))^* \isom \Hom(P(s), \mathrm{S}P(s))$; these are $$\begin{tikzpicture}[baseline=(Ps-1-5.base),
font={\scriptsize},
every matrix/.append style={
column sep={1.4em,between origins},
anchor={base east},
row sep={1em,between origins},
},
every edge quotes/.append style={
execute at begin node={$\scriptstyle},
execute at end node={$}
}
]
\matrix(Ps)[matrix of math nodes,matrix anchor=Ps-1-5.base]{
&&&& L(s) &\\
& L(e) && L(st) && L(ts)\\
L(s) && |[gray]| L(t) &&|[gray]|L(w_0)\\
|[gray]| L(st) && |[gray]| L(ts)\\
& |[gray]| L(w_0)\\
};
\matrix(SPs)[right=7em of Ps-1-5.base, matrix of math nodes, matrix anchor=SPs-5-2.base]{
&&&& |[gray]|L(w_0) &\\
&&& |[gray]|L(st) && |[gray]|L(ts)\\
& |[gray]|L(w_0) && L(s) && |[gray]|L(t)\\
L(st) && L(ts) && L(e)\\
& L(s)\\
};
\node [braced box''={T}, fit=(Ps-1-5) (Ps-3-1) (Ps-2-6)] {};
\node [braced box'''={T'}, fit=(SPs-5-2) (SPs-4-1) (SPs-4-5)] {};
\draw[->] (Ps-1-5) to["\id^*" pos=.6] (SPs-5-2);
\draw[limit bb={1mm}{draw, cross line}, ->] (T) to[out=80, in=-100, looseness=2, "x^*" very near start] (T');
\end{tikzpicture}
,$$ where the grey composition factors belong to the kernel and cokernel of $x^*$.
The inclusion $\Db(\CatO_0(\SL_2))\into\Db(\CatO_0(\SL_3))$ factors through $\Sph(P(s))$.
Take the projective resolution $P(s)^\vee\simeq\bigl\{P(s)\to P(w_0)⟨-2⟩\to P(w_0)⟨-4⟩\bigr\}$; the asphericality $Q(P(s))≔\cone(x^*)$ of $P(s)$ then is the total complex $$Q \qis
\left\{
\begin{tikzcd}[nodes={inner sep={2pt}}]
&& P(s)⟨-2⟩\dar["x^*"]\\
P(s) \rar & P(w_0)⟨-2⟩ \rar & \degZero{P(w_0)⟨-4⟩}
\end{tikzcd}
\right\}$$ with the bottom right $P(w_0)$ in homological degree $0$. We claim that $$\Hom_{\Db(\CatO(\SL_3))}(P(w), Q)
\begin{cases}
= 0 & \text{if $w\in\{e, s\}$,}\\
\neq 0 & \text{if $w\in\{t, st, ts, w_0\}$.}
\end{cases}$$ Consider composition series for the modules involved in the above double complex. The $P(w_0)⟨-4⟩$ in degree $0$ has composition series $$P(w_0)⟨-4⟩ =
\begin{psmallmatrix}
& L(w_0)\\
& L(st)\quad L(ts)\\
\textcolor{gray}{L(s)} & \textcolor{gray}{L(w_0)}\quad L(w_0) & L(t)\\
\textcolor{gray}{L(st)}\quad \textcolor{gray}{L(ts)} & \textcolor{gray}{L(e)} & \textcolor{gray}{L(st)}\quad \textcolor{gray}{L(ts)}\\
\textcolor{gray}{L(w_0)} & \textcolor{gray}{L(s)}\quad \textcolor{gray}{L(t)} & \textcolor{gray}{L(w_0)}\\
& \textcolor{gray}{L(st)}\quad \textcolor{gray}{L(ts)}\\
& \textcolor{gray}{L(w_0)}
\end{psmallmatrix}⟨-4⟩,$$ where $\im(P(s)⟨-2⟩\xto{x^*} P(w_0)⟨-4⟩)+\im(P(w_0)⟨-2⟩\to P(w_0)⟨-4⟩)$ consists of the gray factors. S we see that any map $P(e), P(s)\to P(w_0)$ must factor through $P(s)$ and hence is null-homotopic. The black factors $L(w)$ generate images of morphisms $P(w)⟨-⟩\to⟨-4⟩$ which cannot factor through $P(s)⟨-2⟩$ or $P(w_0)⟨-2⟩$ and thus are not null-homotopic. We see that the triangulated subcategory $\Sph(P(s))$ of $\Db(\CatO_0(\SL_3))$ is generated by $P(s)$ and $P(e)$, which proves the claim.
We shall address another remedy for failure of the Calabi-Yau property of $P(s)$ in $\CatO_0(\SL_3)$.
Maximal parabolic subalgebras
-----------------------------
Consider the category $\CatO^\mathfrak p_0$ corresponding to the parabolic subgroup $W_\mathfrak p=⟨t⟩\isom S_2\times S_1$ of $W=S_3$ with minimal-length representatives $W^\mathfrak p=\{e, s, st\}$ of cosets in $W/W_\mathfrak p$. An algebra $A_\mathfrak p$ such that $\CatO^\mathfrak p_0\simeq\Mod{A_\mathfrak p}$ is given by the path algebra $A_\mathfrak p≔A/(\epsilon_t, e_{ts}, e_{w_0}) = \mathbf C\bigl[e \rightleftarrows s \rightleftarrows st\bigr]\Bigm/
\Bigl(
\begin{scriptaligned}
e\from s\from e &= 0, & e\from s\from st &= 0,\\[-0.5em]
st\from s\from e &= 0, & s\from e \from s &= s\from st\from s
\end{scriptaligned}
\Bigr)$, see \[sec:parabolic-subalgebras\]. The parabolic Verma modules and projectives have the following composition series: $$\begin{array}{ccc|cc@{{}={}}cc@{{}={}}c}
\toprule
M^𝔭(e) &
M^𝔭(s) &
M^𝔭(st) &
P^𝔭(e) &
\multicolumn{2}{c}{P^𝔭(s)} &
\multicolumn{2}{c}{P^𝔭(st)}\\
\midrule
\begin{smallmatrix}\vphantom{L(e)}\\L(e)\\L(s)\end{smallmatrix} &
\begin{smallmatrix}\vphantom{L(e)}\\L(s)\\L(st)\end{smallmatrix} &
\begin{smallmatrix}\vphantom{L(e)}\\\vphantom{L(e)}\\L(st)\end{smallmatrix} &
\begin{smallmatrix}\vphantom{L(e)}\\M(e)\end{smallmatrix} &
\begin{smallmatrix}M(s)\\M(e)\end{smallmatrix} &
\begin{smallmatrix}L(s)\\L(st)\quad L(e)\\L(s)\end{smallmatrix} &
\begin{smallmatrix}M(st)\\M(s)\end{smallmatrix} &
\begin{smallmatrix}L(st)\\L(s)\\L(st)\end{smallmatrix}\\
\bottomrule
\end{array}$$ Since $\Theta_s$ and $\mathbf L\Sh_s$ commute with $Z^\mathfrak p$, we immediately obtain the following images under translation and shuffling: $$\begin{array}{c|cr@{}>{{}}lcr@{}>{{}}l}
\toprule
M & \Theta_s M & \multicolumn{2}{c}{\mathbf L\Sh_s M} & \Theta_t M & \multicolumn{2}{c}{\mathbf L\Sh_t M}\\
\midrule
P^\mathfrak p(e) & P^\mathfrak p(s) & \{0 &\to M^\mathfrak p(s)\} & 0 & \{P^\mathfrak p(e) &\to 0\}\\
P^\mathfrak p(s) & P^\mathfrak p(s)⊕P^\mathfrak p(s) & \{0 &\to P^\mathfrak p(s)\} & P^\mathfrak p(st) & \{P^\mathfrak p(s) &\to P^\mathfrak p(st)\}\\
P^\mathfrak p(st) & P^\mathfrak p(s) & \{P^\mathfrak p(st) &\to P^\mathfrak p(s)\} & P^\mathfrak p(st)⊕P^\mathfrak p(st) & \{0 &\to P^\mathfrak p(st)\}\\
\bottomrule
\end{array}$$ All chain complexes have the right entry in degree $0$,
A module $M$ is $\Sh_w$-acyclic if and only the entry for $\mathbf L\Sh_w$ is concentrated in its $0$-th degree. These results thus are examples for \[caveat:parabolic-projectives-not-projective\]: albeit projective in the category $\CatO_0^\mathfrak p$, the objects $P^\mathfrak p(-)$ are not $\Sh_s$-acyclic and hence in particular not projective in $\CatO_0$.
The set $\{P^\mathfrak p(s), P^\mathfrak p(st)\}$ is an $\mathrm{A}_2$-collection of $0$-spherical objects in $\Db(\CatO^\mathfrak p_0)$.
From the above composition series one sees we get that $P^\mathfrak p(s)$ and $P^\mathfrak p(st)$ have endomorphism algebras isomorphic to $\mathbf{C}[x]/(x^2)$. In the following, all Hom-spaces are one-dimensional, and we see that that the composition pairings $$\begin{gathered}
\label{eq:composition-pairing:SL3:Ps-Pe}
\begin{multlined}[b][.8\columnwidth]
\circ: \Hom_\mathcal O(P^𝔭(e), P^𝔭(s)) \otimes \Hom_\mathcal O(P^𝔭(s), P^\mathfrak p(e))
{
\,
\tikz[remember picture, baseline=0]
\coordinate (start-arrow) at (0,0.65ex);
\\*[3mm]
\tikz[remember picture, overlay, baseline=0]
\draw[->]
coordinate (end-arrow) at (0,0.8ex)
coordinate (mid-arrow1) at ($(start-arrow)!0.5!(end-arrow)$)
coordinate (mid-arrow2) at ($(start-arrow)!0.25!(end-arrow)$)
coordinate (mid-arrow) at (mid-arrow1 |- mid-arrow2)
(start-arrow)
to[out=0, in=0, out looseness=0.5, in looseness=3.5] (mid-arrow)
to[in=180, out=180, in looseness=1, out looseness=3] (end-arrow);
\,
}\left\langle
\begin{tikzpicture}[x=.4cm,y=.4cm, baseline=(Fb.base),every node/.append style={font=\scriptsize}]
\node (Fa) at (0,1) {$L(s)$};
\node (Fb) at (1,0) {$L(e)$};
\draw[brace] ($(Fa)+(0.6,.7)$) to node[brace tip](Pa){} ($(Fb)+(1.1,0.1)$);
\node (Fc) at (-1,0) {$L(st)$};
\node (Fd) at (0,-1) {$L(s)$};
\node[font=\normalsize] at (0,-2.5) {$P^\mathfrak p(e)$};
\draw[brace'] (Fc.west |- Fd.south) to (Fd.south -| Fb.east);
\node[font=\normalsize] at (6,-2.5) {$P^\mathfrak p(s)$};
\node (Ta) at (6,0.5) {$L(s)$};
\node (Tb) at (6,-0.5) {$L(e)$};
\draw[brace] ($(Tb)-(1,.3)$) to node[brace tip](Pb){} ($(Ta)-(1,-.3)$);
\draw[brace'] (Ta.west |- Fd.south) to (Fd.south -| Ta.east);
\node[font=\normalsize] at (12,-2.5) {$P^\mathfrak p(e)$};
\node at (12,1) {$L(s)$};
\node (Sb) at (11,0) {$L(e)$};
\node (Sc) at (13,0) {$L(st)$};
\node (T') at (12,-1) {$L(s)$};
\draw[->, limit bb] (Pa.center) to[out=25,in=180] (Pb.center);
\draw[->, limit bb] (Ta) to[out=0,in=180] (T');
\draw[brace'] (Sb.west |- Fd.south) to (Fd.south -| Sc.east);
\end{tikzpicture}
\right\rangle
= \bigl\langle x_{P^\mathfrak p(s)}\bigr\rangle
\end{multlined}\\
\label{eq:composition-pairing:SL3:Ps-Pst}
\begin{multlined}[b][.8\columnwidth]
∘:\Hom(P^𝔭(st), P^𝔭(s)) \otimes \Hom(P^𝔭(s), P^\mathfrak p(st))
{
\,
\tikz[remember picture, baseline=0]
\coordinate (start-arrow) at (0,0.65ex);
\\*[3mm]
\tikz[remember picture, overlay, baseline=0]
\draw[->]
coordinate (end-arrow) at (0,0.8ex)
coordinate (mid-arrow1) at ($(start-arrow)!0.5!(end-arrow)$)
coordinate (mid-arrow2) at ($(start-arrow)!0.25!(end-arrow)$)
coordinate (mid-arrow) at (mid-arrow1 |- mid-arrow2)
(start-arrow)
to[out=0, in=0, out looseness=0.5, in looseness=3.5] (mid-arrow)
to[in=180, out=180, in looseness=1, out looseness=3] (end-arrow);
\,
}\left\langle
\begin{tikzpicture}[x=.4cm,y=.4cm, baseline=(Fb.base),every node/.append style={font=\scriptsize}]
\node[font=\normalsize] at (0,-2.5) {$P^\mathfrak p(s)$};
\node (Fa) at (0,1) {$L(s)$};
\node at (-1,0) {$L(e)$};
\node (Fb) at (1,0) {$L(st)$};
\node at (0,-1) {$L(s)$};
\draw[brace] ($(Fa)+(0.6,.7)$) to node[brace tip](Pa){} ($(Fb)+(1.1,.1)$);
\draw[brace'] (Fc.west |- Fd.south) to (Fd.south -| Fb.east);
\node[font=\normalsize] at (6,-2.5) {$P^\mathfrak p(st)$};
\node (Ta) at (6,1) {$L(st)$};
\node (Tb) at (6,0) {$L(s)$};
\node (Tc) at (6,-1) {$L(st)$};
\draw[brace] ($(Tc)-(1,.3)$) to node[brace tip](Pb){} ($(Tb)-(1,-.3)$);
\draw[brace] ($(Ta)+(1,.3)$) to node[brace tip](Pc){} ($(Tb)+(1,-.3)$);
\draw[brace'] (Ta.west |- Fd.south) to (Fd.south -| Ta.east);
\node[font=\normalsize] at (12,-2.5) {$P^\mathfrak p(s)$};
\node (Sb) at (12,1) {$L(s)$};
\node (Za) at (11,0) {$L(st)$};
\node (Sc) at (13,0) {$L(e)$};
\node (Zb) at (12,-1) {$L(s)$};
\draw[brace] ($(Zb)-(0.8,.2)$) to node[brace tip](Pd){} ($(Za)-(1.0,0)$);
\draw[->,limit bb] (Pa.center) to[out=45,in=180] (Pb.center);
\draw[->,limit bb] (Pc.center) to[out=0,in=235] (Pd.center);
\draw[brace'] (Sb.west |- Fd.south) to (Fd.south -| Sc.east);
\end{tikzpicture}
\right\rangle
= \bigl\langle x_{P^\mathfrak p(s)}\bigr\rangle
\end{multlined}\\[3mm]
\label{eq:composition-pairing:SL3:Pst-Pe}
\begin{multlined}[b][.8\columnwidth]
\circ: \Hom_\mathcal O(P^𝔭(s), P^𝔭(st)) \otimes \Hom_\mathcal O(P^𝔭(s), P^\mathfrak p(st))
{
\,
\tikz[remember picture, baseline=0]
\coordinate (start-arrow) at (0,0.65ex);
\\*[3mm]
\tikz[remember picture, overlay, baseline=0]
\draw[->]
coordinate (end-arrow) at (0,0.8ex)
coordinate (mid-arrow1) at ($(start-arrow)!0.5!(end-arrow)$)
coordinate (mid-arrow2) at ($(start-arrow)!0.25!(end-arrow)$)
coordinate (mid-arrow) at (mid-arrow1 |- mid-arrow2)
(start-arrow)
to[out=0, in=0, out looseness=0.5, in looseness=3.5] (mid-arrow)
to[in=180, out=180, in looseness=1, out looseness=3] (end-arrow);
\,
}\left\langle
\begin{tikzpicture}[x=.4cm,y=.4cm, baseline=(Fb.base),every node/.append style={font=\scriptsize, inner sep=2pt}]
\node[font=\normalsize] at (0,-2.5) {$P^\mathfrak p(st)$};
\node (Fa) at (0,1) {$L(st)$};
\node (Fb) at (0,0) {$L(s)$};
\node[braced box={Pa}, fit=(Fa) (Fb)]{};
\node at (0,-1) {$L(st)$};
\draw[brace'] (Fb.west |- Fd.south) to (Fd.south -| Fb.east);
\node[font=\normalsize] at (6,-2.5) {$P^\mathfrak p(s)$};
\node (Ma) at (6,1) {$L(s)$};
\node (Mb) at (5,0) {$L(st)$};
\node (Md) at (7,0) {$L(e)$};
\node (Mc) at (6,-1) {$L(s)$};
\draw[brace'] (Mb.-170) to node[brace tip'](Pb){} (Mc.south west);
\draw[brace] (Mb.170) to node[brace tip] (Ta){} (Ma.north west);
\draw[brace'] (Mb.west |- Fd.south) to (Fd.south -| Md.east);
\node[font=\normalsize] at (12,-2.5) {$P^\mathfrak p(st)$};
\node (Sb) at (12,1) {$L(st)$};
\node (Sc) at (12,0) {$L(s)$};
\node (Tb) at (12,-1) {$L(st)$};
\draw[->, limit bb] (Pa.center) to[out=0,in=-135] (Pb.center);
\draw[-, limit bb] (Ta) to[out=135, out looseness=1.5, in=135] (8,1.5);
\draw[->, limit bb] (8,1.5) to[out=-45, in=180] (Tb);
\draw[brace'] (Sb.west |- Fd.south) to (Fd.south -| Sc.east);
\end{tikzpicture}
\right\rangle
= \bigl\langle x_{P^\mathfrak p(st)}\bigr\rangle
\end{multlined}
\end{gathered}$$ are non-degenerate; hence $P^\mathfrak p(s)$ and $P^\mathfrak p(st)$ are $0$-spherical objects in $\Db(\CatO_0^\mathfrak p)$. Furthermore, we see from the composition series that the dimensions of the spaces $\Hom(P^\mathfrak p(-), P^\mathfrak p(-))$’s are as required; thus the set $\{P^\mathfrak p(s), P^\mathfrak p(st)\}$ is an $\mathrm A_2$-configuration indeed.
For $\mathfrak p$ as above, there are natural isomorphisms $T'_{P^\mathfrak p(s)} \isom \mathbf L\Sh_s[-1]$ and $T'_{P^\mathfrak p(st)} \isom \mathbf L\Sh_t[-1]$ of autoequivalences of $\CatO^\mathfrak p_0$.
The above shows that the respective images under $\Theta_s$ and $\Theta_t$ and $T'_{P^\mathfrak p(s)}$ and $T'_{P^\mathfrak p(st)}$ of the $P^\mathfrak p(-)$’s are isomorphic; hence $T'_{P^\mathfrak p(s)} M\qis\mathbf L\Sh_s[-1] M$ and $T'_{P^\mathfrak p(st)} M\qis\mathbf L\Sh_t[-1] M$ for all $M\in\Db(\CatO_0^\mathfrak p)$.
The proof that these isomorphisms of images form a natural isomorphism of functors is carried out analogously to \[thm:shuffling-is-twisting:sl2\]. The proof for $\Theta_t \isom T'_{P^(st)}$ is, m.m., the same as for $\Theta_s \isom T'_{P^(s)}$; we hence only show the latter. To that end, we show that $M_{\Xi'_{P^\mathfrak p(s)}} \isom M_{\Theta_s}$, where the notation is the same as in \[thm:shuffling-is-twisting:sl2\].
The $A_𝔭$-$A_𝔭$-bimodule $M_{\Xi'_{P^\mathfrak p(s)}} = P^\mathfrak p(s)^* ⊗_\mathbf C P^\mathfrak p(s)$ has a $\mathbf C$-basis given by all pairwise tensor products $$\label{eq:sl3-p:M-Xi}
\begin{tikzpicture}[mth, ampersand replacement=\&]
\matrix (A) [
matrix of math nodes,
row sep=0mm,
column sep=2mm,
text height=2ex,
text depth=.5ex,
column 1/.style={text width=\algnRef, align=right},
column 3/.style={text width=\algnRef-1em, align=left},
nodes={font=\normalsize}
] {
(s←e←s)^* \& \& e \\
(s←st)^* \&~~\& s←e \\
(s←e)^* \& \& s←st \\
e^* \& \& s←e←s \\
};
\node[braced box={T}, fit=(A-1-1) (A-4-1)] {} ;
\node[braced box'={T'},fit=(A-1-3) (A-4-3)] {} ;
\path[draw=none] (T) to node[anchor=base]{$\displaystyle ⊗$} (T');
\draw
(A-1-1.west) edge[|->, out=195, in=165] node[swap] {$(e←s)_*$} (A-2-1.west)
(A-2-1.base west) edge[|->, out=195, in=165] node[swap] {$(e←s)_*$} (A-4-1.west)
([xshift=-5em]A-1-1.west) edge[|->, out=195, in=165] node[swap] {$(st←s)_*$} ([xshift=-5em]A-3-1.west)
([xshift=-5em]A-3-1.base west) edge[|->, out=195, in=165] node[swap] {$(s←st)_*$} ([xshift=-5em]A-4-1.west)
(A-1-3.east) edge[|->, out=-15, in=15] node {$∘(s←e)$} (A-2-3.east)
(A-2-3.base east) edge[|->, out=-15, in=15] node {$∘(e←s)$} (A-4-3.east)
([xshift=4.5em]A-1-3.east) edge[|->, out=-15, in=15] node {$∘(s←st)$} ([xshift=4.5em]A-3-3.east)
([xshift=4.5em]A-3-3.base east) edge[|->, out=-15, in=15] node {$∘(st←s).$} ([xshift=4.5em]A-4-3.east);
\end{tikzpicture}$$ with the indicated $A_𝔭$-$A_𝔭$-bimodule action. The algebra $\End_{\CatO}(P^\mathfrak p)$ is generated by the following four elements; a diagram chase of morphisms through the relevant naturality diagrams shows that $\Theta_s$ acts on these by $$\begin{tikzcd}[column sep={\the\algnRef}, nodes={inner xsep=0pt}, row sep=small]
P^\mathfrak p = P(e)\rar[phantom, "⊕"]
& P(s)\ar[dl]\ar[dr, dashed]\rar[phantom, "⊕"]
& P(st) \ar[d, phantom, "" name=C]
&[4em] P(s)_1 \rar[phantom, "⊕"] \ar[d, phantom, "" name=C']
& P(s)_2 \rar[phantom, "⊕"]
& P(s)_3 \rar[phantom, "⊕"]\ar[dll, "1"']\ar[dr, dashed, "1"]
& P(s)_4 =\Theta_s P^\mathfrak p
\\
\phantom{P^\mathfrak p = {}} P(e)\rar[phantom, "⊕"] \ar[dr]
& P(s) \rar[phantom, "⊕"]
& P(st) \ar[dl, dashed] \ar[d, phantom, "" name=D]
& P(s)_1 \rar[phantom, "⊕"] \ar[dr, "1"'] \ar[d, phantom, "" name=D']
& P(s)_2 \rar[phantom, "⊕"]
& P(s)_3 \rar[phantom, "⊕"]
& P(s)_4 \ar[dll, "1"', dashed] \phantom{{}=\Theta_s P^\mathfrak p}
\\
\phantom{P^\mathfrak p = {}} P(e)\rar[phantom, "⊕"]
& P(s) \rar[phantom, "⊕"]
& P(st)
& P(s)_1 \rar[phantom, "⊕"]
& P(s)_2 \rar[phantom, "⊕"]
& P(s)_3 \rar[phantom, "⊕"]
& P(s)_4 \mathrlap{.}\phantom{{}=\Theta_s P^\mathfrak p} \ar[phantom, from=C, to=C', "\xmapsto{\Theta_s}"]\ar[phantom, from=D, to=D', "\xmapsto{\Theta_s}"]
\end{tikzcd}$$ A vector space basis of $M_{\Theta_s} = \Hom(P^\mathfrak p, \Theta_s P^\mathfrak p)$ is given by the morphisms between the summands of $P^\mathfrak p$ in the following schematic. Recall that the left $A^\mathfrak p$-action on $M_{\Theta_s}$ is given by $\phi\ldot m = \Theta_s(\phi)\circ m$ for $\phi\in A^{\mathfrak p}$ and $m\in M_{\Theta_s}$, so it acts on basis elements as follows:
we obtain that the $A^\mathfrak p$-$A^\mathfrak p$-bimodule action on its $\mathbf C$-basis is $$\label{eq:sl3-p:M-Theta}
\begin{tikzpicture}[mth, ampersand replacement=\&]
\matrix (A) [
matrix of math nodes,
row sep=0mm,
column sep=2mm,
text height=2ex,
text depth=.5ex,
] {
P^𝔭(s)_3← \& \& ←[1]P^𝔭(s)\\
P^𝔭(s)_1← \&~~\& ←P^𝔭(e)\\
P^𝔭(s)_4← \& \& ←P^𝔭(st) \\
P^𝔭(s)_2← \& \& ←[x]P^𝔭(s)\\
};
\path[draw=none]
node[braced box={T}, fit=(A-1-1) (A-4-1)] {}
node[braced box'={T'},fit=(A-1-3) (A-4-3)] {}
(T) to node[anchor=base]{$←$} (T');
\draw
(A-1-1.west) edge[|->, out=195, in=165] node[swap] {$𝛩_s(e←s)∘$} (A-2-1.west)
(A-2-1.base west) edge[|->, out=195, in=165] node[swap, pos=.45] {$𝛩_s(e←s)∘$} (A-4-1.west)
([xshift=-5em]A-1-1.west) edge[|->, out=195, in=165] node[swap] {$𝛩_s(st←s)∘$} ([xshift=-5em]A-3-1.west)
([xshift=-5em]A-3-1.base west) edge[|->, out=195, in=165] node[swap] {$𝛩_s(s←st)∘$} ([xshift=-5em]A-4-1.west)
(A-1-3.east) edge[|->, out=-15, in=15] node {$∘(s←e)$} (A-2-3.east)
(A-2-3.base east) edge[|->, out=-15, in=15] node {$∘(e←s)$} (A-4-3.east)
([xshift=4.5em]A-1-3.east) edge[|->, out=-15, in=15] node {$∘(s←st)$} ([xshift=4.5em]A-3-3.east)
([xshift=4.5em]A-3-3.base east) edge[|->, out=-15, in=15] node {$∘(st←s).$} ([xshift=4.5em]A-4-3.east);
\end{tikzpicture}$$ Comparing \[eq:sl3-p:M-Xi\] and \[eq:sl3-p:M-Theta\] shows that the isomorphism $M_{\Xi'_{P^\mathfrak p(s)}} \isom M_{\Theta_s}$ of vector spaces is an isomorphism of $A^\mathfrak p$-$A^\mathfrak p$-bimodules. It then follows that $T'_{P^\mathfrak p(s)} \simeq \{1 \to -\otimes M_{\Xi'_{P^\mathfrak p(s)}}\}$ and $\Sh_s = \{1 \to -\cong M_{\Theta_s}$ are naturally isomorphic functors.
We now transfer results for $\CatO^\mathfrak p_0$ from $\SL_3$ to $\SL_n$. Consider the parabolic subalgebra $\mathfrak p$ of $\SL_n$ corresponding to the subgroup $W_\mathfrak p=⟨s_2, \dotsc, s_{n-1}⟩=S_{n-1}\times S_{1}\leq S_n$ and the coset representatives $W^\mathfrak p=\{e, s_1, s_1s_2, \dotsc, s_1\dotsm s_{n-1}\}$; we abbreviate these representatives in $W^\mathfrak p$ by $\sigma_i≔ s_1\dotsm s_i$.
The category $\CatO^\mathfrak p_0(\SL_n)$ is equivalent to $\Mod{}[A^\mathfrak p(\SL_n)]$, where $A^\mathfrak p(\SL_n)$ is the path algebra quotient $A^\mathfrak p=C\bigl[e\rightleftarrows\sigma_1\rightleftarrows\dotsb\rightleftarrows\sigma_{n-1}\bigr]
\Bigm/
\Bigl(
\setlength\jot{-.3ex}
\begin{scriptaligned}
e\from\sigma_1\from e &= 0,\\
\sigma_i\from\sigma_{i+1}\from\sigma_i &= \sigma_i\from\sigma_{i-1}\from\sigma_i,\\
\sigma_i\from\sigma_{i\pm 1}\from\sigma_{i\pm 2} &= 0
\end{scriptaligned}
\Bigr)$ for $1 \leq i \leq n-2$.
We compute the composition series of Verma modules and indecomposable projectives in $\CatO^\mathfrak p_0$ using the generalised Kazhdan-Lusztig theorem [@Irving:filtered-category-OS Cor. 7.1.3; @CC:parabolic-KL Thm. 1.3]. In the setting of parabolic subgroups of the form $W_\mathfrak p=S_{k}\times S_{n-k}\leq S_n$ there is a handy graphical calculus for the computation of parabolic Kazhdan-Lusztig polynomials [@Brundan-Stroppel:Highest-weight-categories-I §5; @JS:Graphical-Kazhdan-Lusztig]. The thus obtained composition series are: $$\begin{array}{c|ccccc}
\toprule
& e & s_1 & s_1s_2 &\cdots & s_1\cdots s_{n-1}\\
\midrule
M^𝔭(-)
& \begin{smallmatrix}L(e)\\L(s_1)\end{smallmatrix}
& \begin{smallmatrix}L(\sigma_1)\\L(\sigma_2)\end{smallmatrix}
& \begin{smallmatrix}L(\sigma_2)\\L(\sigma_3)\end{smallmatrix}
& \cdots
& L(\sigma_{n-1})\\[1em]
P^𝔭(-)
& {}''
& \begin{smallmatrix}L(\sigma_1)\\L(e)\quad L(\sigma_2)\\L(\sigma_1)\end{smallmatrix}
& \begin{smallmatrix}L(\sigma_2)\\L(s_1)\quad L(\sigma_3)\\L(\sigma_2)\end{smallmatrix}
& \cdots
& \begin{smallmatrix}L(\sigma_{n-1})\\L(\sigma_{n-2})\\L(\sigma_{n-1})\end{smallmatrix}\\
\bottomrule
\end{array}$$ Since the arrows in $Q^\mathfrak p(\SL_n)$ are given by irreducible morphisms between $P^\mathfrak p(-)$’s, the relations of $Q^\mathfrak p(\SL_n)$ follow from these composition series.
Every Verma module $M^\mathfrak p(\sigma_i)$ fits uniquely into a short exact sequence $$M^\mathfrak p(\sigma_i)\into P^\mathfrak p(\sigma_{i+1})\onto M^\mathfrak p(\sigma_{i+1});$$ in particular, $\Theta_{s_i} M^\mathfrak p(\sigma_i)=P^\mathfrak p(\sigma_i)$ always is projective, which is not true in the non-parabolic category $\CatO_0$. We obtain the following list of images under translation and shuffling functors: $$ \let\Q P
\catcode`\P=\active
\def P{\mathmakebox[\widthof{$M$}][r]{\Q}}
\mathclap{
\begin{array}{l|lllllllll}
\toprule
M & \hfil\Theta_{s_1}\hfil M & \hfil \mathbf L\Sh_{s_1} M \hfil & \hfil\Theta_{s_2} M\hfil & \hfil\mathbf L\Sh_{s_2} M\hfil & \hfil\Theta_{s_3} M\hfil & \hfil\mathbf L\Sh_{s_3} M\hfill\mathllap{\cdots}\\
\midrule
P^\mathfrak p(e) & P^\mathfrak p(\sigma_1) & \hfill M^\mathfrak p(\sigma_1) \phantom{\}} & \hfil 0\hfil & \phantom{\{}P^\mathfrak p(e)[-1] & \hfil 0\hfil & \phantom{\{} P^\mathfrak p(e)[-1]\\
P^\mathfrak p(\sigma_1) & P^\mathfrak p(\sigma_1)^{⊕2} & \hfill P^\mathfrak p(\sigma_1) \phantom{\}} & P^\mathfrak p(\sigma_2) & \{P^\mathfrak p(\sigma_1) {\leaders\hbox{$\mskip -2mu\smash \relbar\mskip -2mu$}\hfill\mkern -17mu\rightarrow}P^\mathfrak p(\sigma_2)\} & \hfil 0\hfil & \phantom{\{} P^\mathfrak p(\sigma_1)[-1]\\
M^\mathfrak p(\sigma_1) & P^\mathfrak p(\sigma_1) & \{M^\mathfrak p(\sigma_1) \to P^\mathfrak p(\sigma_1)\} & P^\mathfrak p(\sigma_2) & \hfill M^\mathfrak p(\sigma_2) \phantom{\}} & \hfil 0\hfil & \phantom{\{} M^\mathfrak p(\sigma_1)[-1]\\
P^\mathfrak p(\sigma_2) & P^\mathfrak p(\sigma_1) & \{M^\mathfrak p(\sigma_2) \to P^\mathfrak p(\sigma_1)\} & P^\mathfrak p(\sigma_2)^{⊕2} & \hfill P^\mathfrak p(\sigma_2) \phantom{\}} & P^\mathfrak p(\sigma_3) & \{P^\mathfrak p(\sigma_2) \to P^\mathfrak p(\sigma_3)\}\\
M^\mathfrak p(\sigma_2) & \hfil 0\hfil & \phantom{\{} M^\mathfrak p(\sigma_2)[-1] & P^\mathfrak p(\sigma_2) & \{M^\mathfrak p(\sigma_2) \to P^\mathfrak p(\sigma_2)\} & P^\mathfrak p(\sigma_3) & \hfill M^\mathfrak p(\sigma_3) \phantom{\}}\\
P^\mathfrak p(\sigma_3) & \hfil 0\hfil & \phantom{\{} P^\mathfrak p(\sigma_3)[-1] & P^\mathfrak p(\sigma_2) & \{P^\mathfrak p(\sigma_3) {\leaders\hbox{$\mskip -2mu\smash \relbar\mskip -2mu$}\hfill\mkern -17mu\rightarrow}P^\mathfrak p(\sigma_2)\} & P^\mathfrak p(\sigma_3)^{⊕2} & \hfill P^\mathfrak p(\sigma_3) \phantom{\}}\\
M^\mathfrak p(\sigma_3) & \hfil 0\hfil & \phantom{\{} M^\mathfrak p(\sigma_3)[-1] & \hfil 0\hfil & \phantom{\{}M^\mathfrak p(\sigma_3)[-1] & P^\mathfrak p(\sigma_3) & \{M^\mathfrak p(\sigma_3) \to P^\mathfrak p(\sigma_3)\}\\
\hfil\vdots\hfil & \hfil\vdots\hfil & \hfil\vdots\hfil & \hfil\vdots\hfil & \hfil\vdots\hfil & \hfil\vdots\hfil & \hfill\ddots\\
\bottomrule
\end{array}
}$$
The set $\{P^\mathfrak p(\sigma_1),\dotsc,P^\mathfrak p(\sigma_{n-1})\}$ is an $\mathrm A_{n-2}$-configuration of $0$-spherical objects.
The composition series exhibit that $\Hom_{\CatO_0^\mathfrak p}(P^\mathfrak p(\sigma_i), P^\mathfrak p(\sigma_i)) \isom \mathbf C[x]/(x^2)$, where the nontrivial endomorphism $x$ is the degree $2$-map $$x\colon P^\mathfrak p(\sigma_i)\onto\operatorname{hd} P^\mathfrak p(\sigma_i)=\soc P^\mathfrak p(\sigma_i)\into P^\mathfrak p(\sigma_i),$$, and that $$\dim\Hom_{\CatO_0^\mathfrak p}(P^\mathfrak p(\sigma_j), P^\mathfrak p(\sigma_i)) =
\begin{smallcases}
2 & \text{if $i=j$},\\
1 & \text{if $|i-j|=1$}\\
0 & \text{otherwise.}
\end{smallcases}$$ It suffices to check non-degeneracy of the composition pairing for indecomposable projectives $P^\mathfrak p(\sigma_i\pm 1)$ that are connected by an arrow of $Q^\mathfrak p(\SL_n)$. In these two cases, the picture is analogous to :
$$\begin{gathered}
\circ\colon\Hom(P^\mathfrak p(\sigma_{i\pm 1}), P^\mathfrak p(\sigma_{i}))\otimes\Hom(P^\mathfrak p(\sigma_i), P^\mathfrak p(\sigma_{i\pm 1}))
{
\,
\tikz[remember picture, baseline=0]
\coordinate (start-arrow) at (0,0.65ex);
\\*[5mm]
\tikz[remember picture, overlay, baseline=0]
\draw[->]
coordinate (end-arrow) at (0,0.8ex)
coordinate (mid-arrow1) at ($(start-arrow)!0.5!(end-arrow)$)
coordinate (mid-arrow2) at ($(start-arrow)!0.25!(end-arrow)$)
coordinate (mid-arrow) at (mid-arrow1 |- mid-arrow2)
(start-arrow)
to[out=0, in=0, out looseness=0.5, in looseness=3.5] (mid-arrow)
to[in=180, out=180, in looseness=1, out looseness=3] (end-arrow);
\,
}
\Biggl\langle
\tikzset{x=.4cm,y=.4cm, baseline=(Fb.base),every node/.append style={font=\scriptsize, inner sep=2pt}}
\underbrace{
\begin{tikzpicture}[remember picture]
\node (Fa) at (0,1) {$L(\sigma_i)$};
\node at (-1.25,0) {$L(\sigma_{i+1})$};
\node (Fb) at (1.75,0) {$L(\sigma_{i-1})$};
\node (Fc) at (0,-1) {$L(\sigma_i)$};
\draw[brace] (Fa.50) to node[brace tip](Pa){} (Fb.10);
\end{tikzpicture}
}_{\textstyle P^\mathfrak p(\sigma_{i})}
\qquad
\underbrace{
\begin{tikzpicture}[remember picture]
\node (Ma) at (0,1) {$L(\sigma_{i+1})$};
\node (Mb) at (-1.25,0) {$L(\sigma_{i\pm 2})$};
\node at (1.25,0) {$L(\sigma_{i})$};
\node (Mc) at (0,-1) {$L(\sigma_{i+1})$};
\draw[brace'] (Mb.-170) to node[brace tip'](Pb){} (Mc.south west);
\draw[brace] (Mb.170) to node[brace tip] (Ta){} (Ma.north west);
\begin{scope}[overlay]
\draw[->] (Pa.center) to[out=70, out looseness=1.6, in=-135] (Pb.center);
\end{scope}
\end{tikzpicture}
}_{\textstyle P^\mathfrak p(\sigma_{i\pm 1})}
\qquad
\underbrace{
\begin{tikzpicture}[remember picture, trim left=(Gd.west)]
\node (Ga) at (0,1) {$L(\sigma_i)$};
\node (Gd) at (-1.75,0) {$L(\sigma_{i+1})$};
\node (Gb) at (1.25,0) {$L(\sigma_{i-1})$};
\node (Gc) at (0,-1) {$L(\sigma_i)$};
\draw[brace'] (Gc.-80)to node[brace tip'](Tb){} (Gb.-10) ;
\draw[-, limit bb] (Ta) to[out=135, in looseness=1.5, in=115] (-4, 0);
\draw[->, limit bb] (-4, 0) to[out=-65, out looseness=1.5, in=-70] (Tb);
\end{tikzpicture}
}_{\textstyle P^\mathfrak p(\sigma_{i})}
\Biggr\rangle
= \bigl\langle x_{P^\mathfrak p(\sigma_i)}\bigr\rangle
\end{gathered}$$
for $0<i<n-2$, and similar for $i=0$ and $n-1$. This shows that $\circ$ gives a non-degenerate pairing.
\[thm:main-result\] for the parabolic subalgebra $\mathfrak p\subseteq\SL_n$ corresponding to $W_\mathfrak p=S_{n-1}\times S_1< S_n$, the autoequivalences $\mathbf L\Sh_{s_i}[-1]$ and $T'_{P^\mathfrak p(\sigma_i)}$ of $\Db(\CatO^\mathfrak p_0)$ are naturally isomorphic.
Let $A^\mathfrak p_n≔A^\mathfrak p(\SL_n)$. Assume that the assertion holds for the respective subalgebra of $\SL_{n-1}$. One checks that the assignment $Q_{\SL_{n-1}} \to Q_{\SL_{n}}, \sigma_i \to \sigma_{i-1}$ of quivers gives rise to an isomorphism $A^\mathfrak p_n/(\epsilon_{\sigma_0}) \to A^\mathfrak p_{n-1}$ of path algebras. The thus defined maps $$\begin{aligned}
p\colon A^\mathfrak p_n &\onto A^\mathfrak p_n/(\epsilon_{\sigma_0}) \isom A^\mathfrak p_{n-1} &
i\colon A^\mathfrak p_{n-1} &\into A^\mathfrak p_n\\
\shortintertext{
induce fully faithful functors
}
p^* \colon \CatO^\mathfrak p_0(\SL_{n-1})&\to\CatO^\mathfrak p_0(\SL_{n}), &
i_*\colon\Proj{\CatO^\mathfrak p_0(\SL_{n-1})}&\to\Proj{\CatO^\mathfrak p_0(\SL_{n})},\\
P^ \mathfrak p(e)& \mapsto M^\mathfrak p(\sigma_1), &
P^\mathfrak p(\sigma_l)&\mapsto P^\mathfrak p(\sigma_l) \\
P^\mathfrak p(\sigma_k)&\mapsto P^\mathfrak p(\sigma_{k+1})
\end{aligned}$$ for $1\leq k< n-1$ and $0\leq l < n-1$. By induction, this shows that the assertion holds for the functors restricted to the triangulated subcategories $⟨P^\mathfrak p(\sigma_2), \dotsc, P^\mathfrak p(\sigma_{n-1})⟩$ (via $p^*$) and $⟨P^\mathfrak p(e), P^\mathfrak p(\sigma_1), P^\mathfrak p(\sigma_2)⟩$ (via $i_*$) of $\Db(\CatO^\mathfrak p_0(\SL_n))$. Hence, the assertion holds on $\CatO^\mathfrak p_0(\SL_n)$.
We know that the object $P^\mathfrak p(s)\in\CatO_0^\mathfrak p(\SL_3)$ is spherical, so one might ask if $\CatO^\mathfrak p_0(\SL_3)$ arises as the spherical subcategory $\Sph(P^\mathfrak p(s))$ of $P^\mathfrak p(s)\in\CatO_0(\SL_3)$. However, $P^\mathfrak p(s)$ is not spherelike in $\Db(\CatO(\SL_3))$, , we cannot assign a meaningful spherical subcategory to it.
Consider the projective resolution $P^\mathfrak p(s) \qis \{P(s)\to P(ts)\to P(s)\}$ in $\Db(\CatO(\SL_3))$. Using this resolution, we obtain the chain complex $$\begin{gathered}
\hom^\bullet_{\Db(\CatO_0)}(P^\mathfrak p(s), P^\mathfrak p(s))
\\
\qis\left\{
\left\langle
\begin{tikzcd}[baseline=(R.base), ampersand replacement=\&]
P(s) \rar\dar \& P(ts)\rar\dar \& P(s)\ar[d, "{\id, x}" name=R]\\
P(s) \rar \& P(ts)\rar \& P(s)
\end{tikzcd}
\right\rangle
\to
0
\to
\left\langle
\begin{tikzcd}[baseline=(R.base), ampersand replacement=\&]
P(s) \rar\ar[drr, "{\id}"] \& P(ts) \rar \& P(s) \dar[phantom, "\phantom{\id, x}" name=R]\\
P(s) \rar \& P(ts) \rar \& P(s)
\end{tikzcd}
\right\rangle
\right\},
\end{gathered}$$ , $\hom^\bullet_{\Db(\CatO_0)}(P^\mathfrak p(s), P^\mathfrak p(s))$ has total dimension $3$, so $P^\mathfrak p(s)$ is not spherelike. As a side remark, we notice that the inclusion $\Db(\CatO_0^\mathfrak p) \subset \Db(\CatO_0)$, given by mapping projectives to projectives, is not full.
The object $L(\sigma_{n-2})\in\Db(\CatO^\mathfrak p_0(\SL_n))$ can easily seen to be $2$-spherical object. However, its induced spherical twist functor does not yield an $\mathbf L\Sh_{(-)}$ for $n>2$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This article compiles the results of a Master’s thesis written with the advise Catharina Stroppel, whom the author wants to thank for her continuous support and the opportunity to dive into this topic.
[^1]: In case that $d=0$, we require that $\circ\colon\Hom^i_{\Db}(F, E) ⊗ \Hom^{d-i}_{\Db}(F, E) \to \Hom^d_{\Db}(E, E)⟨\id_E⟩ ≅ 𝐂$ be non/degenerate.
|
---
abstract: 'A recently developed linear algebraic method for the computation of perturbation expansion coefficients to large order is applied to the problem of a hydrogenic atom in a magnetic field. We take as the zeroth order approximation the $D \rightarrow \infty$ limit, where $D$ is the number of spatial dimensions. In this pseudoclassical limit, the wavefunction is localized at the minimum of an effective potential surface. A perturbation expansion, corresponding to harmonic oscillations about this minimum and higher order anharmonic correction terms, is then developed in inverse powers of $(D-1)$ about this limit, to 30th order. To demonstrate the implicit parallelism of this method, which is crucial if it is to be successfully applied to problems with many degrees of freedom, we describe and analyze a particular implementation on massively parallel Connection Machine systems (CM-2 and CM-5). After presenting performance results, we conclude with a discussion of the prospects for extending this method to larger systems.'
address:
- 'Department of Chemistry, Harvard University, 12 Oxford Street, Cambridge, MA 02138'
- 'Center for Computational Science, Boston University, 3 Cummington Street, Boston, MA 02215'
author:
- 'Timothy C. Germann and Dudley R. Herschbach'
- 'Bruce M. Boghosian'
title: |
[BU-CCS-941002]{}\
[chem-ph/9411003]{}\
[To Appear in [*Computers in Physics*]{} (1994)]{}
Dimensional Perturbation Theory on the Connection Machine
---
[H]{}
Introduction
============
The spatial dimension has long been treated as a variable parameter in analyzing critical phenomena and in other areas of physics.[@witten] However, only in the past ten years has this concept been extensively applied to atomic and molecular systems, particularly to develop dimensional scaling methods for electronic structure.[@dscal] The motivation for this unconventional approach is that the Schrödinger equation reduces to easily solvable forms in the limits $D \rightarrow
1$ and/or $D \rightarrow \infty$. When both limiting solutions are available, interpolation in $1/D$ may be used to approximate the physically meaningful $D=3$ result; this has yielded excellent results for correlation energies of two-electron atoms[@loeser] and for $H_2$ Hartree-Fock energies.[@tan] Alternatively, if the $D
\rightarrow \infty$ solutions are avaliable for both the problem of interest and a simpler model problem (e.g., Hartree-Fock) for which $D =
3$ results are easier to calculate, the latter may be used to “renormalize” some parameter (e.g., nuclear charge). Then the $D
\rightarrow \infty$ solution with the renormalized parameter may give a good approximation to the $D=3$ solution with the actual parameter value.[@renorm]
Our work deals with another widely applicable dimensional scaling method, a perturbation expansion in inverse powers of $D$ or a related function, about the solution for the $D \rightarrow \infty$ limit. That limit is pseudoclassical and readily evaluated, as it reduces to the simple problem of minimizing an effective potential function.[@dscal] For large but finite $D$, the first-order correction accounts for harmonic oscillations about this minimum, and higher-order terms provide anharmonic corrections. Dimensional perturbation theory has been applied quite successfully to the ground[@helium] and some excited states[@david93] two-electron atoms and to the hydrogen molecule-ion[@h2plus] using a “moment method” to solve the set of perturbation equations. However, this method is not easily extended to larger systems. It also requires a different program for each eigenstate, and does not directly provide an expansion for the wavefunction.
A recently developed linear algebraic method has overcome these shortcomings.[@dunn93] This is conceptually quite simple, so can easily be applied to systems with any number of degrees of freedom. It permits calculation of ground and excited energy levels using a single program and the wavefunction expansion coefficients are directly obtained in the course of computing the perturbation expansion for the energy. This method has thus far been applied to central force problems, including quasibound states for which the complex eigenenergy represents both the location and width of the resonance.[@tcg93]
The linear algebraic version of dimensional perturbation theory is also well suited to parallel computation. Here we demonstrate this for a prototype problem with two degrees of freedom, the hydrogen atom in a magnetic field. This system has received much attention; it exhibits chaotic behavior and poses difficulties that have challenged many theoretical techniques. Most theoretical approaches treat either the magnetic field or the Coulomb potential as a perturbation and therefore work best near either the low- or high-field limit, respectively. However, the leading terms of a perturbation expansion in inverse powers of $D$ include major portions of the nonseparable interactions in all field strengths. The efficacy of methods equivalent to the $1/D$ expansion has been demonstrated for the hydrogen atom in a magnetic field[@bender82] and for kindred problems with an electric field or crossed electric and magnetic fields,[@popov] although not in formulations suited to parallel computation. The present paper is devoted solely to implementing of the linear algebraic method on the Connection Machine, and to evaluating the performance of the computational algorithm as well as prospects for treating systems with more degrees of freedom. Numerical results for the ground and several excited states over a wide range of field strengths will be presented in a separate paper.[@next]
Theory
======
The Schrödinger equation for a nonrelativistic hydrogenic atom in a uniform magnetic field along the $z$ axis, in atomic units and cylindrical coordinates, is given by { - + + - } (,z) = E (,z). Here $m$ is the azimuthal quantum number and the magnetic field $B$ is measured in units $m_e^2 e^3 c/\hbar^3
\simeq 2.35 \times 10^9$ G.[@friedrich] We can eliminate the first derivative term with the substitution (,z) = \^[1/2]{} (,z), which gives { - ( + ) + + + - } (,z) = E (,z). \[eq:d3se\] The two good quantum numbers are $m$ and the $z$ parity $\pi_z$, although the observation of exponentially small level anticrossings suggests a “hidden” approximate symmetry.[@herrick82] The trivial (i.e. diagonal in each $m^\pi_z$ subspace) $mB/2$ term may be dropped, to be added at the end of the calculation.
Now we consider the more general case where $\rho$ is the radius of a $(D-1)$-dimensional hypersphere (with $D-2$ remaining angles), so that the total number of spatial dimensions is $D$ when we include the component $z$ parallel to the magnetic field. In a manner completely analogous to that described above, we may eliminate the first derivative in the radial part of the Laplacian, $$\frac{1}{\rho^{D-2}} \frac{\pd}{\pd \rho} \left( \rho^{D-2}
\frac{\pd}{\pd \rho} \right),$$ by introducing the substitution (,z) = \^[(D-2)/2]{} (,z), which leads to the same form as Eq. (\[eq:d3se\]), with $m^2 - 1/4$ replaced by $\Lambda(\Lambda+1)$ where $\Lambda = |m| +
\Half(D-4)$.[@herrick75] As before, we drop the $mB/2$ term. Since the resulting equation depends on $m$ and $D$ only through $\Lambda$, we see that interdimensional degeneracies[@herrick75] connect states $(|m|,D)$ with $(|m|-1,D+2)$. It proves convenient to define a scaling parameter as $\kappa \equiv D + 2|m| - 1$ and take $\Lambda =
\Half(\kappa-3)$. The entire $|m| = 0,1,2,\ldots$ spectrum for the real three-dimensional system can then be recovered simply by obtaining the solutions for $\kappa = 2,4,6,\ldots$, respectively.
On introducing dimensionally scaled variables defined by = \^2 , z = \^2 , = \^3 B, = \^2 E, \[eq:scal\] we obtain the dimension-dependent Schrödinger equation { - \^2 ( + ) + + - } (,) = (,), \[eq:dimschro\] where $\d = 1/\kappa$ is treated as a continuous perturbation parameter. The same form may be obtained simply by replacing $|m|$ in Eq. (\[eq:d3se\]) by $(\kappa-2)/2$; this is the procedure employed by Bender and coworkers.[@bender82]
We see that in the $\d \rightarrow 0$ limit, the problem reduces to finding the minimum of the effective potential V\_ (,) = + - . Clearly, the effective potential is minimized for $\tz = z_m = 0$, so we are left with the straightforward problem of minimizing a function of one variable to determine $\rho_m$ and $V_{\mbox{eff}} (\rho_m, z_m)$. In the zero-field limit, $\rho_m = (4Z)^{-1}$ and $V_{\mbox{eff}}
(\rho_m, z_m) = -2Z^2$ in the scaled energy units of Eq. (\[eq:scal\]); in unscaled units this becomes E = , the correct expression for the ground state energy in each azimuthal manifold. In the strong-field limit, $\rho_m \approx \tilde{B}^{-1/2}$ and $V_{\mbox{eff}} (\rho_m, z_m) \approx \tilde{B}/4$, or in unscaled units E B(|m|+1), which is simply the continuum threshold in each $m^{\pi_z}$ subspace. This appropriate behavior in both limits, which has been noted previously,[@bender82] is the motivation behind the definition of $\kappa$ which we have used.[@shifted]
For large but finite $\kappa$ the system will undergo harmonic oscillations about this minimum, so we introduce dimension-scaled displacement coordinates $x_1$ and $x_2$ through = \_m + \^[1/2]{}x\_1, = \^[1/2]{}x\_2. Substituting into Eq. (\[eq:dimschro\]) leads to { - ( + ) + ( \_1\^2 x\_1\^2 + \_2\^2 x\_2\^2 + v\_[0,0]{} ) + V\_ (\_m, z\_m) + . &&\
. \_[j=1]{}\^\^[j/2]{} } && (x\_1, x\_2) = (x\_1, x\_2), \[eq:normalse\] where \_1\^2 &=& - + ,\
\_2\^2 &=& ,\
v\_[0,0]{} &=& - ,\
v\_[1,-1]{} &=& 0.We next expand the wavefunction and energy in Eq. (\[eq:normalse\]) as &=& V\_ (\_m, z\_m) + \_[j=0]{}\^ \_[2j]{} \^j, \[eq:e\_exp\]\
(x\_1, x\_2) &=& \_[j=0]{}\^\_j(x\_1,x\_2) \^[j/2]{} and collect powers of $\d^{1/2}$ to obtain an infinite set of inhomogeneous differential equations, \_[j=0]{}\^p(\_j-\_j)\_[p-j]{}=0,p=0,1,2,…, \[eq:setofeq\] where
\[hams\] \_0&=&- ( + ) + \_1\^2 x\_1\^2 + \_2\^2 x\_2\^2 + v\_[0,0]{}\
\[ham0\] \_[j>0]{}&=& ( \_[l=0]{}\^[(j+2)/2]{} \^lv\_[j,j+2]{} x\_1\^[j+2-2l]{} x\_2\^[2l]{} ) + v\_[j,j]{} x\_1\^j + v\_[j,j-2]{} x\_1\^[j-2]{} \[hamj\]
with $\e_{2i+1}=0$. The case $p=0$ corresponds to a pair of independent harmonic oscillators, with the solution &&\_0=(\_1+)\_1+(\_2+)\_2+ v\_[0,0]{},\
&&\_0(x\_1,x\_2)=h\_[\_1]{}(\_1\^[1/2]{} x\_1)h\_[\_2]{}(\_2\^[1/2]{} x\_2), where $\nu_1$ and $\nu_2$ are quantum numbers for the normal modes corresponding to motion perpendicular to and parallel to the magnetic field, respectively, and the $h_i$ are the one-dimensional harmonic oscillator eigenfunctions. The correspondences between these large-dimension quantum numbers and the conventional labels for the low, intermediate, and high field cases will be detailed elsewhere.[@next]
Tensor method
=============
The higher-order terms in the energy and wavefunction expansions may be computed using the linear algebraic method.[@dunn93] The wavefunction expansion terms $\Phi_j(x_1,x_2)$ are expanded in terms of the harmonic oscillator eigenfunctions $h_i$, \_j(x\_1,x\_2) = \_[i\_1]{} \_[i\_2]{} \^[i\_1,i\_2]{}a\_j h\_[i\_1]{}(\_1\^[1/2]{} x\_1)h\_[i\_2]{}(\_2\^[1/2]{} x\_2), so that the representation $\a_j$ is a tensor of rank 2. The displacement coordinates $x_i$ are represented in this basis by the matrix \_i = (
[ccccc]{} 0 & & 0 & 0 &\
& 0 & & 0 &\
0 & & 0 & &\
0 & 0 & & 0 &\
& & & &
), \[eq:xrep\] and the Hamiltonian $\H_{j>0}$ is represented as a linear combination of direct (outer) products of these matrices, namely \_[j>0]{}= ( \_[l=0]{}\^[(j+2)/2]{} \^lv\_[j,j+2]{} \_1\^[j+2-2l]{} \_2\^[2l]{} ) + v\_[j,j]{} \_1\^j + v\_[j,j-2]{} \_1\^[j-2]{} .
Using these representations, the $j=0$ term of Eq. (\[eq:setofeq\]), $(\H_0 - \epsilon_0)\Phi_p$, becomes (\_0 - \_0 )\_p = ((\_1 \_1) (\_2 \_2)) \_p, where $\bK_i$ is a diagonal matrix with element $(j,j)$ equal to $j-\nu_i$, since the basis functions $h_{i_1}(\omega_1^{1/2}
x_1)h_{i_2}(\omega_2^{1/2} x_2)$ are eigenfunctions of $\H_0$. Let us pause at this point to be sure that there is no confusion regarding the notation. A “direct sum” is analogous to the direct product operation, i.e. ${\bf C} = {\bf A} \oplus {\bf B}$, where ${\bf A}$ and ${\bf B}$ are matrices, gives a rank 4 tensor with elements $C_{j_1 k_1
j_2 k_2} = A_{j_1 k_1} + B_{j_2 k_2}$. Then the multiplication ${\bf E}
= {\bf C} {\bf D}$, where ${\bf D}$ and ${\bf E}$ are rank 2 tensors, implies $E_{j_1 j_2} = C_{j_1 k_1 j_2 k_2} D_{k_1 k_2}$, where we are using the implied summation convention. If we define a rank 4 tensor $\bKt$ with elements \_[j\_1 k\_1 j\_2 k\_2]{} = {
[ll]{} 1, & j\_1 = k\_1 = \_1, j\_2 = k\_2 = \_2\
(\_1(j\_1 - \_1) + \_2 (j\_2 - \_2))\^[-1]{}, & j\_1 = k\_1 , j\_2 = k\_2 , j\_1 \_1 j\_2 \_2\
0, &
., \[eq:Ktilde\] then it may be verified that $\bKt((\omega_1 \bK_1) \oplus (\omega_2
\bK_2))$ is equal to the direct product $\bI \otimes \bI$ except for a zero in element $\nu_1 \nu_1 \nu_2 \nu_2$. However, due to the orthogonality condition \_0\^T \_p = \_[0,p]{}, \[eq:ortho\] where $\delta_{0,p}$ is the Kronecker delta, we have ((\_1 \_1) (\_2 \_2)) \_p = \_p, p > 0. Therefore if we multiply Eq. (\[eq:setofeq\]) on the left by $\bKt$, we obtain a recursive solution for $\a_p$, \_p = - \_[j=1]{}\^p (\_j - \_j ) \_[p-j]{}. \[eq:ap\] If we instead multiply Eq. (\[eq:setofeq\]) on the left by $\a_0^T$ and use the orthogonality condition of Eq. (\[eq:ortho\]), we obtain an expression for $\epsilon_p$ in terms of $\a_{j<p}$, \_p = \_[j=1]{}\^p \_0\^T \_j \_[p-j]{}. \[eq:ep\]
From a computational standpoint, the only operation with which we need to concern ourselves is the multiplication of a rank 4 tensor with a rank 2 tensor, $\bH_j \a_{p-j}$. According to Eq. (\[eq:Ktilde\]), the nonzero elements of $\bKt$ appear in Eq. (\[eq:ap\]) simply as scaling factors for each element of $\a_p$ after the summation has been completed.
Implementation
==============
Rewriting the recursion relations
---------------------------------
In order to solve Eqs. (\[eq:ap\]) and (\[eq:ep\]) recursively, we introduce the rank 2 tensor $\A_{jln}$ which is defined as \_[jln]{} (\_1\^[j-2l]{} \_2\^[2l]{}) \_n, \[eq:def\] so that the $\bH_j \a_{p-j}$ term which most concerns us may be expressed as a linear combination of these tensors, namely \_j \_[p-j]{} = ( \_[l=0]{}\^[(j+2)/2]{} \^lv\_[j,j+2]{} \_[j+2,l,p-j]{} ) + v\_[j,j]{} \_[j,0,p-j]{} + v\_[j,j-2]{} \_[j-2,0,p-j]{}. \[eq:H\_a\] In order for this change of notation to be beneficial, we need to find some simple recursion relation(s) relating the $\A_{jln}$ tensors for a given order to those tensors which have been used at lower orders. In addition, we hope that only a small subset of the tensors used for previous orders are needed, so that a managable number of these tensors (and the wavefunction tensors $\a_n$) need to be allocated in memory. Two recursion relations are immediately evident, namely
\[recurs\] \_[j+1,l,n]{} &=& (\_1 ) \_[jln]{},\[recur1\]\
\_[j+2,l+1,n]{} &=& (\_2\^2) \_[jln]{}. \[recur2\]
It remains to be seen how many terms may be computed using these relations, and conversely how many will need to be computed “from scratch.” We show in Fig. \[fig:matrices\] the tensors $\A_{jln}$ which appear in the first term of Eq. (\[eq:H\_a\]) for the first few orders $p$, and also indicate which recursion relations (if any) may be used to compute each tensor (with preference given to Eq. (\[recur1\]) if both are applicable). We see that Eq. (\[recur1\]) simply “shifts” the staircase-like pattern of tensors to the right, while Eq. (\[recur2\]) fills in all but three of the gaps left by this shift. Turning our attention to the second term of Eq. (\[eq:H\_a\]), we note that at order $p$ this term represents tensors which have appeared in the first term at order $p-2$, with the exception of $\A_{1,0,p-1}$ and $\A_{2,0,p-2}$. Similarly, the third term of Eq. (\[eq:H\_a\]) at order $p$ involves tensors which occured in the second term at order $p-2$, with the exception of $\A_{0,0,p-2}$, which is simply the wavefunction tensor $\a_{p-2}$ which is resident in memory. Therefore, at each order we have at most five tensors which cannot be computed from direct application of Eqs. (\[recurs\]).
In reality, the computation of these five tensors is no more difficult than the two standard recursion relations: \_[1,0,p-1]{} &=& (\_1 ) \_[p-1]{},\
\_[3,0,p-1]{} &=& (\_1\^2 ) \_[1,0,p-1]{},\
\_[3,1,p-1]{} &=& (\_2\^2) \_[1,0,p-1]{}, \[altrec\]\
\_[2,0,p-2]{} &=& (\_1\^2 ) \_[p-2]{},\
\_[4,2,p-2]{} &=& (\_2\^2) \[(\_2\^2)\_[p-2]{}\].We note that these relations, as well as the two standard recursion relations, only involve computation along one coordinate axis.
By properly ordering the computation steps, we may compute all of the tensors $\A_{jln}$ at order $p$ “in place” (i.e. overwriting all of the tensors from order $p-1$), thus minimizing the need for temporary storage. The second recursion relation uses tensors from order $p-2$, which must be temporarily stored during the order $p-1$ calculation. However, this relation is only used for $(p-2)/2$ tensors at order $p$, so the additional storage cost is minor. The algorithm may be summarized as follows (see also Fig. \[fig:algorithm\]): (1) Recursion Eq. (\[recur1\]) is applied to all tensors, working from right to left in Fig. \[fig:algorithm\] to avoid the need for temporary storage. Note that this step does [*not*]{} overwrite those tensors which we will need later for Eq. (\[recur2\]). (2) Recursion Eq. (\[recur2\]) is applied to tensors from order $p-2$ which are stored separately. By again working from right to left, we may replace the tensors in temporary storage as they are used with the corresponding tensors from order $p-1$, so that both the main set of tensors and the temporary set are updated as they are used. (3) Five original tensors (Eq. (\[altrec\])) are computed. (4) $\a_p$ is computed by taking a linear combination of tensors, and the contributions of those tensors which are necessary for $\a_{p+2}$ and $\a_{p+4}$ are now computed.
Implementing the recursion relations on the Connection Machine
--------------------------------------------------------------
We have implemented the computation of the energy eigenvalues, as described in the previous section, using a data-parallel algorithm on the CM-5 Connection Machine computer. The CM-5 computer consists of up to 16,384 processor nodes, each consisting of a Sun Sparc processor and four vector units for fast floating-point computation. It supports both the data-parallel and the message-passing styles of parallel computation.
The data-parallel capabilities of the CM-5 are supported in the form of high-level languages: C\* is a data-parallel extension of the C programming language; while CM Fortran is similar to the Fortran 90 draft standard, augmented by some features from High-Performance Fortran (HPF) such as the [FORALL]{} statement. For this work, we chose to use CM Fortran.
The basic parallel data type in CM Fortran is the usual Fortran array. When the CM Fortran compiler encounters a [DIMENSION]{} statement, it allocates memory to spread the array across the processors of the CM-5. If it is desired that certain axes of a multidimensional array be localized to a single processor, that can be arranged by a simple compiler directive, called the [LAYOUT]{} directive. Axes localized in this way are called [*serial axes*]{}, while those spread across the processor array are called [*news axes*]{}.
Operations on corresponding elements of arrays with identical layouts can then be performed by each processor in parallel. If there are more array elements than physical processors, the compiler arranges for each physical processor to contain a [*subgrid*]{} of multiple array elements, and array operations are then multiplexed over these elements as required. The compiler’s orchestration of this process is completely transparent to the CM Fortran user, who may then regard each array element as living in its own [*virtual processor*]{}.
In this picture, operations involving noncorresponding array elements require interprocessor communication. Data-parallel programming languages, such as CM Fortran, support many different varieties of this. For the problem at hand, however, only [*nearest-neighbor communication*]{} is used.
Nearest-neighbor communication is needed when, for example, you would like to take a linear combination of the array elements in (virtual) processors, $i-1$, $i$, and $i+1$. It is supported by a Fortran 90 (and CM Fortran) intrinsic function known as [CSHIFT]{}. If [A]{} is a CM Fortran array, and [i]{} and [j]{} are integers, then [CSHIFT(A,i,j)]{} is another such array, whose elements are shifted along axis [i]{} by an amount [j]{}. There is another variant of this function, known as [EOSHIFT]{}, that does basically the same thing, differing only in its treatment of the boundary elements of the array; whereas [CSHIFT]{} treats the boundaries as though the array were periodic, [EOSHIFT]{} has extra arguments for arrays of codimension one to be inserted at the boundary.
Our storage method uses the fact that the position matrices in the harmonic oscillator representation, given by Eq. (\[eq:xrep\]), are doubly banded. We store only the nonzero elements of the $i$th row in (virtual) processor $i$. Thus, one can think of the band below the diagonal as a one-dimensional array, [XL]{}, and that above the diagonal as another one-dimensional array, [XU]{}. Then, [XL(i)]{} and [XU(i)]{} contain the two nonzero elements in row [i]{}, and we can write $${\bf x}_{\alpha\beta} = {\tt XL}(\alpha)\delta_{\alpha-1,\beta} +
{\tt XU}(\alpha)\delta_{\alpha+1,\beta}.$$
The main operation needed for the implementation of the recursion relations, Eqs. (\[recurs\]), is the inner product of the position matrices with the matrices, $\A_{j,l,n}$. The latter are stored as dense three-dimensional arrays, with two [*news*]{} axes for the components of the matrices, and one [*serial*]{} axis representing the allowed combinations of $j$, $l$, and $n$. Thus, one can imagine these matrices as spread across the processor array, with corresponding components localized to the same (virtual) processor.
Consider one of these matrices; call it $\A_{\mu\nu}$ (where the Greek indices now label the [*tensor*]{} components and [*not*]{} the serial axis, $\{j,l,n\}$, which is being supressed for the moment). Then Eq. (\[recur1\]), which is the inner product of ${\bf x}$ with (the first component of) $\A$, is $$(\x_{\xi\mu} \otimes \bI)\A_{\mu\nu} =
{\tt XL}(\xi)\A_{\xi-1,\nu} +
{\tt XU}(\xi)\A_{\xi+1,\nu}.
\label{eq:ip}$$ Note that we can implement this in terms of two [CSHIFT]{} operations along the first axis of $\A$. Since this must be done for all values of $\mu$, we add this [*instance axis*]{} to the arrays, [XL]{} and [XU]{}.
The complete algorithm for implementing the inner product then begins by dimensioning the arrays as follows:
INTEGER m,n
DIMENSION XL(nmax,nmax), XU(nmax,nmax), AA(nmax,nmax)
The [XL]{} and [XU]{} arrays are then initialized as follows:
FORALL(m=1:nmax,n=1:nmax) XL(m,n) = SQRT(n-1)
FORALL(m=1:nmax,n=1:nmax) XU(m,n) = SQRT(n)
where the [FORALL]{} statements are parallel [DO]{}-loop constructions supported in High-Performance Fortran (and CM Fortran). For example, the first such statements cause each (virtual) processor to take its own index, decrement it by one, and take the square root. Alternatively, rather than allocate a separate array for [XU]{}, we can obtain it from [XL]{} by a simple [EOSHIFT]{} operation.
The inner product in Eq. (\[eq:ip\]) is then taken as follows:
XA = XL*EOSHIFT(AA,1,-1) + XU*EOSHIFT(AA,1,+1)
Using this method, the recursion relations, Eqs. (\[recurs\]) and (\[altrec\]), can be implemented, and the terms in Eq. (\[eq:H\_a\]) can be multiplied by the scalars, $v_{i,j}$, and summed in place. (Since the $\{j,l,n\}$ axis is serial, the arrays $\A_{j,l,n}$ are all aligned in the processor array.) Similarly, the sum over $j$ in Eq. (\[eq:ep\]) can be taken (without the premultiplication by ${\bf
a}_0$, which is independent of $j$.)
Finally, the result for $\epsilon_p$, given by Eq. (\[eq:ep\]), can be found quite easily. Since ${\bf a}_0$ is an eigenvector in our representation, we can effectively premultiply it by simply taking the first component of $\sum_j {\bf H}_j {\bf a}_{p-j}$. The statement
ep = XA(1,1)
reaches into the (virtual) processor containing the [(1,1)]{} component of the array [XA]{}, pulls out its value, and writes it into the scalar (stored on the control processor) quantity [ep]{}, from where it can be manipulated, output, etc.
Thus, the array extensions of Fortran 90 and High-Performance Fortran, as embodied in the CM Fortran language, give a convenient set of primitives for the efficient implementation of the dimensional perturbation theory algorithm.
Contraction of axis lengths
---------------------------
All of the wavefunction tensors $\a_n$ and the tensors $\A_{jln}$ used in their computation are allocated to the size of the final wavefunction tensor, $\a_{p_{max}}$, which permits calculation of energy expansion coefficients through $\epsilon_{p_{max}+1}$. However, we have not yet attempted to take advantage of the sparseness of these tensors. To illustrate the amount of room for improvement, Fig. \[fig:sparse\] indicates the nonzero elements of the first few wavefunction tensors for the case of a ground state ($\nu_1 = \nu_2 = 0$) calculation. Since we would lose much of the efficiency of the routines for the recursion relations described above if we attempted to treat these as general sparse tensors, we will instead look for smaller-scale improvements.
First, we see that since $\x_2$ only occurs in even powers, alternating columns are always zero and need not be stored. In addition to cutting the storage in half, this reduces the two next-nearest neighbor communications required for the $(\bI \otimes \x_2^2)$ multiplication to two nearest neighbor communications.
Looking along the other axis, we see that either even or odd rows, but not both in a given tensor, contain nonzero elements, so that the rows may be paired up as long as we maintain some sort of flag telling us whether the even or odd row of the pair is represented. This also carries an extra communication benefit in addition to the storage improvement; one of the two nearest-neighbor communications in the $(\x_1 \otimes \bI)$ multiplication is converted to an entirely local operation.
Performance
===========
We now turn to the mapping of these arrays to the individual processors on the Connection Machine, in the typical case where the array size exceeds the number of processors. The recursion relation of Eq. (\[recur1\]) may be applied independently to different columns since the communication is entirely within columns. Similarly, Eq. (\[recur2\]) may operate independently on separate rows. Since the former recursion relation is applied much more often than the latter, we would expect that the optimal array layout would be that in which each processor operates on a small number of long column segments, i.e. the first (row) axis is “more serial” while the second (column) axis is “more parallel.”
We can in fact obtain a quantitative measure of this balance. Suppose we have an $N \times N$ matrix to be allocated on a system of $M$ processors. If we assign $n$ columns (or segments of columns) to each processor, then the length of each of these column segments is $N^2/nM$, as shown in Fig. \[fig:layout\]. The first recursion relation may be implemented in CM Fortran as described in section IV. B, with the addition of an [IF]{} clause due to the contraction of the first axis, as described in the previous section. Using appropriately defined arrays [X0]{} and [XLU]{}, this may be accomplished as follows:
IF (odd rows nonzero) THEN
XA = X0*AA + XLU*EOSHIFT(AA,1,-1)
ELSE
XA = X0*AA + EOSHIFT(XLU*AA,1,+1)
ENDIF
Whichever branch of this clause is taken, there are three arithmetic operations on each array element, with a total cost of $3 N^2 t_A / M$, where $t_A$ represents the time for a single arithmetic operation. The [EOSHIFT]{} operation will move $n$ elements from any given processor into an adjacent processor, while the remaining $N^2/M - n$ elements require an on-processor move (except for the case $n = N/M$, where all elements are moved on-processor). Thus the [EOSHIFT]{} time is given by $$\begin{array}{ll}
n t_Q & \mbox{if $n = N/M$} \\
n t_Q + \left( \frac{N^2}{M} - n \right) t_M & \mbox{otherwise}
\end{array} ,$$ where $t_Q$ represents the queue waiting time for interprocessor communication and $t_M$ is the on-processor move time.
The second recursion relation can be effected by the CM Fortran statement
XA = X20*AA + EOSHIFT(X2LU*AA,2,-1) + X2LU*EOSHIFT(AA,2,+1)
where the definition of the constant arrays [X20]{} and [X2LU]{} is dependent on the contraction of the second axis, as described in the previous section. The arithmetic cost is $5 N^2M t_A / M$ and the cost of the two [EOSHIFT]{} operations is $$\begin{array}{ll}
\frac{2 N^2}{nM} t_Q & \mbox{if $n = N$} \\
\frac{2 N^2}{nM} t_Q + 2 \left( \frac{N^2}{M} - \frac{N^2}{nM}
\right) t_M & \mbox{otherwise}
\end{array} .$$
For a program which applies the first recursion relation $n_1$ times and the second relation $n_2$ times, the total time incurred by the two relations is $$t_{recur} = \frac{N^2}{M} [ (3 n_1 + 5 n_2) t_A +
(n_1 + 2 n_2) t_M ] + \left(n_1 n + \frac{2 n_2 N^2}{nM} \right)
(t_Q - t_M),$$ where we have neglected the two special cases for $n$ described above. The first term is independent of the specific layout, described by the parameter $n$. Since the only interprocessor data motion in the entire program arises from the recursion relations, the second term can be used to determine the optimal array layout. Since $t_Q > t_M$, we want to minimize the term $n_1 n + 2 n_2 N^2 / n M$. Setting its derivative with respect to $n$ equal to zero gives a simple expression for the optimal layout parameter $n_{opt}$, $$n_{opt} = \sqrt{\frac{2 n_2 N^2}{n_1 M}}.
\label{eq:nopt}$$ For example, a 30th order calculation using the implementation discussed here requires $N = 128$, $n_1 = 18441$, $n_2 = 925$. A 64-processor CM-5 contains 256 vector units (hence $M=256$). Thus, we would estimate $n_{opt} \approx 2.5$, in agreement with the empirical observation that $n=2$ results in faster run times than either $n=1$ or $n=4$ on a 64-processor CM-5 (see Table I). Special care must be paid to the special case $n=N/M$ when $M \leq N$, for a layout with the first axis entirely serial $(n=1)$ may be optimal despite the fact that Eq. (\[eq:nopt\]) gives a larger value for $n_{opt}$.
Table I presents timings and the corresponding floating point rates for 20th and 30th order calculations. The recursion relations involve substantial interprocessor communication; for the problem sizes considered here, we have found that approximately 40% of the CM execution time is spent performing arithmetic, the remainder being consumed by communication operations.
In spite of this, we were able to obtain a performance of nearly 20 Mflops per CM-5 processor node for the largest subgrid sizes considered here.[@note] Since the algorithm described here is fully scalable, we expect that if the subgrid size was kept fixed and the problem was ported to the largest CM-5’s existing today (1024 processor nodes), the total performance would be about 20 Gflops.
Keeping the subgrid size fixed, however, implies that the problem size grows with the hardware. Again referring to Table I, we see that if we move a fixed-size problem to larger machines, the per-processor performance degrades. This is because the correspondingly decreasing subgrid size means that a larger fraction of time is being spent on latencies (overhead costs).
Thus, massively parallel implementations of this algorithm will be useful if there is something to be gained by going to larger problem sizes. The most obvious way to do this is to increase the order of the perturbation series, and hence the sizes of the arrays involved, the number of recursion relations that must be used, and the accuracy of the results. Here, however, we run into a problem. Table II shows that at 30th order we have exhausted (or nearly exhausted) the significant digits of the IEEE standard double-precision floating-point numbers used in the calculation. Thus, to grow the problem in this direction, we would need to employ quadruple-precision arithmetic.
There are, however, other options for growing the problem size in a useful manner. Table III shows results for several different magnetic field strengths. Indeed, one of the more interesting things to study in chaotic systems of this sort is the continuous dependence of the energy levels on the magnetic field strength, or other system parameter. Variation of these parameters provides another dimension over which to parallelize, and thereby maintain large subgrid sizes.
Conclusions
===========
In Table II we give ground state expansion coefficients $\epsilon_{2j}$ for two field strengths, $B = 1$ and 1000, computed in double-precision arithmetic. Comparison with quadruple-precision calculations shows that the accumulation of roundoff error causes a linearly decreasing number of significant digits in the coefficients, most pronounced for those series with the slowest rate of growth in the coefficients.
In their work, Bender, Mlodinow, and Papanicolaou applied Shanks extrapolation to accelerate convergence.[@bender82] For the higher-order series we have computed, the large order divergent behavior due to singularities renders this method ineffective.[@b_and_o] Consequently, we employ Padé approximants to sum our series. Table III compares our $1/\kappa$ results for the ground state of the $m$ = 0 and $-1$ mainfolds with variational calculations[@rosner] and lower and upper bounds,[@handy] where available. As noted by Goodson and Watson, the order at which roundoff error leaves no significant digits in the expansion coefficient $\epsilon_{2j}$ may be determined by noting qualitative changes in the root and ratio tests.[@david93] We also give in Table III these maximum orders which are attainable using double-precision arithmetic. Results for excited states will be presented elsewhere.[@next]
For general problems with $\tau$ degrees of freedom, the $\a_n$ and $\A_n$ objects are tensors of rank $\tau$. It is clear that storing all of the tensor elements will rapidly become impractical; while a $128
\times 128$ array of double precision numbers consumes 131 kBytes of memory, a $128 \times 128 \times 128 \times 128$ object requires 2 GB. Even using a parallel I/O device such as the DataVault or Scalable Disk Array for temporary storage, it is evident that it will become necessary to incorporate the sparse nature of these data structures more explicitly and to reduce the communication (especially to external devices) at the expense of additional arithmetic wherever possible.
It has recently been shown[@dunn93] that the recursion relations may be reordered so that only the $\a_n$ tensors need to be stored. This approach has been used in the computation of expansion coefficients for the helium atom, a system with three degrees of freedom. Numerical estimates of the storage and computational costs associated with this approach indicate that large order calculations (20th order or higher) are at present restricted to at most six degrees of freedom, [*e.g.*]{} a three-electron atom.[@dunn93] However, a promising approach may be to combine low-order perturbation expansions with other methods. In particular, the recent development of expansions about the $D
\rightarrow 0$ limit[@bender92] may provide means to augment the $D
\rightarrow \infty$ results. Expansions about both of these limits can be enhanced by renormalization schemes.
[**Acknowledgements**]{}
T.C.G. gratefully acknowledges the award of a Computational Science Graduate Fellowship from the U.S. Department of Energy, and would like to thank Thinking Machines Corporation for their hospitality during the time this work was performed.
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[lrrrr]{} Architecture & $P$ & $n$ & CM time (sec) & MFLOPS/sec/node\
\
CM-5 (VU) & 32 & 2 & 0.896 & 5.012\
& 128 & 2 & 0.730 & 1.536\
\
CM-2 & 256 & 1 & 11.91 & 0.588\
& 512 & 1 & 7.11 & 0.492\
CM-5 (VU) & 4 & 1 & 22.78 & 19.668\
& 32 & 1 & 4.62 & 12.124\
& 64 & 1 & 3.81\
& 64 & 2 & 3.68 & 7.608\
& 64 & 4 & 3.75\
& 128 & 2 & 2.82 & 4.964\
[rdd]{} $j$ & &\
$-$1 & $-$1.577218587578393 & 1.910051627706109(3)\
0 & 0.63293278553615$_1$ & 3.61765949346725$_3$(2)\
1 & $-$0.3281655376303$_1$ & $-$1.61755795476809$_6$(3)\
2 & 0.189180754067$_1$ & 8.71028965022799$_1$(3)\
3 & $-$0.1202775104$_2$ & $-$5.8564694134795$_7$(4)\
4 & 0.10221091$_4$ & 4.518169171696$_0$(5)\
5 & $-$0.1685588$_6$ & $-$2.55253312427$_6$(6)\
6 & 0.46534$_7$ & $-$4.75598614937$_9$(7)\
7 & $-$1.486$_3$ & 3.36811883775$_9$(9)\
8 & 4.92$_6$ & $-$1.4285156162$_5$(11)\
9 & $-$1.6$_8$(1) & 5.3728029857$_7$(12)\
10 & 1.$_0$(2) & $-$1.876673877$_8$(14)\
11 & & 5.834807058$_2$(15)\
12 & & $-$1.29765909$_0$(17)\
13 & & $-$1.39749835$_0$(18)\
14 & & 4.7606391$_8$(20)\
15 & & $-$4.5262663$_0$(22)\
16 & & 3.365746$_4$(24)\
17 & & $-$2.179148$_6$(26)\
18 & & 1.21980$_8$(28)\
19 & & $-$5.17984$_0$(29)\
20 & & 3.7664$_0$(30)\
21 & & 2.6805$_1$(33)\
22 & & $-$4.395$_8$(35)\
23 & & 5.073$_1$(37)\
24 & & $-$4.92$_4$(39)\
25 & & 4.09$_7$(41)\
26 & & $-$2.6$_8$(43)\
27 & & 7.8$_6$(44)\
28 & & 1.$_5$(47)\
29 & & $-$4.$_0$(49)\
[rrdddcl]{} & & & &\
\
0.1 & 7 & 0.54752648 & 0.547527 & & &\
1 & 11 & 0.831169 & 0.831169 & & &\
2 & 12 & 1.022213 & 1.022214 & 1.0222138 &$<E_B<$& 1.0222142\
20 & 20 & 2.21539 & 2.215399 & 2.215325 &$<E_B<$& 2.215450\
200 & 24 & 4.7275& 4.727 & 4.710 &$<E_B<$& 4.740\
300 & 25 & 5.362 & 5.361 & 5.34 &$<E_B<$& 5.39\
1000 & 30 & 7.67 & 7.662 & 7.55 &$<E_B<$& 7.85\
&\
\
0.1 &10&0.20084567 & 0.2008456\
1 &16& 0.4565971 & 0.456597\
2 &19& 0.599613 & 0.599613\
20 &25&1.46551 & 1.4655\
200&33& 3.3473 & 3.3471\
300&35& 3.83485 & 3.8346\
1000&38& 5.640 & 5.63842\
[**Figures**]{}
Fig. 1. $\A_{jln}$ matrices arising from the first term in Eq.(\[eq:H\_a\]), for the first few orders $p$. Superscripts indicate which recursion relation, if any, of Eq.(\[recurs\]) may be used to compute that matrix, with preference given to Eq.(\[recur1\]).
Fig. 2. Algorithm for applying recursion relations, using order $p=8$ as an example. Displayed is the state of the matrices $\A_{jln}$ after each of the following steps: (a) status after order $p-1$; (b) apply relation Eq.(\[recur1\]); (c) apply relation Eq.(\[recur1\]); (d) compute three new elements. Open and filled circles denote matrices from order $p-1$ and $p$, respectively.
Fig. 3. Nonzero elements of the first few wavefunction expansion matrices $\a_p$ for the ground state case.
Fig. 4. Layout of an $N \times N$ matrix on $M$ processors.
$$\begin{array}{lrrrrrr}
p=1 \hspace{0.5in} &\A_{310} \\
&\A_{300} \vspace{0.4in} \\
p=2 && \A_{420} \\
&\A_{311} & ^a\A_{410} \\
&\A_{301} & ^a\A_{400} \vspace{0.4in} \\
p=3 && \A_{421} & ^a\A_{520} \\
&\A_{312} & ^a\A_{411} & ^a\A_{510} \\
&\A_{302} & ^a\A_{401} & ^a\A_{500} \vspace{0.4in} \\
p=4 && & & ^b\A_{630} \\
&& \A_{422} & ^a\A_{521} & ^a\A_{620} \\
&\A_{313} & ^a\A_{412} & ^a\A_{511} & ^a\A_{610} \\
&\A_{303} & ^a\A_{402} & ^a\A_{501} & ^a\A_{600} \vspace{0.4in} \\
p=5 && & & ^b\A_{631} & ^a\A_{730} \\
&& \A_{423} & ^a\A_{522} & ^a\A_{621} & ^a\A_{720} \\
&\A_{314} & ^a\A_{413} & ^a\A_{512} & ^a\A_{611} & ^a\A_{710} \\
&\A_{304} & ^a\A_{403} & ^a\A_{502} & ^a\A_{601} & ^a\A_{700} \\
\vspace{0.4in} \\
p=6 && & & & & ^b\A_{840} \\
&& & & ^b\A_{632} & ^a\A_{731} & ^a\A_{830} \\
&& \A_{424} & ^a\A_{523} & ^a\A_{622} & ^a\A_{721}
&^a\A_{820} \\
&\A_{315} & ^a\A_{414} & ^a\A_{513} & ^a\A_{612} & ^a\A_{711}
&^a\A_{810} \\
&\A_{305} & ^a\A_{404} & ^a\A_{503} & ^a\A_{602} & ^a\A_{701}
&^a\A_{800} \\
\end{array}$$
(280,500) (0,440)[a]{} (0,340)[b]{} (0,240)[c]{} (0,140)[d]{} (70,110)[(1,0)[190]{}]{} (70,210)[(1,0)[190]{}]{} (70,310)[(1,0)[190]{}]{} (70,410)[(1,0)[190]{}]{} (70,110)[(0,1)[80]{}]{} (70,210)[(0,1)[80]{}]{} (70,310)[(0,1)[80]{}]{} (70,410)[(0,1)[80]{}]{} (250,100)[$j$]{} (250,200)[$j$]{} (250,300)[$j$]{} (250,400)[$j$]{} (50,180)[$l$]{} (50,280)[$l$]{} (50,380)[$l$]{} (50,480)[$l$]{} (80,100)[3]{} (100,100)[4]{} (120,100)[5]{} (140,100)[6]{} (160,100)[7]{} (180,100)[8]{} (200,100)[9]{} (220,100)[10]{} (60,115)[0]{} (60,135)[2]{} (60,155)[4]{} (60,175)[6]{} (83,408)(20,0)[8]{}[(0,1)[4]{}]{} (68,419)(0,10)[7]{}[(1,0)[4]{}]{} (83,419)(20,0)[7]{} (83,429)(20,0)[7]{} (103,439)(20,0)[6]{} (143,449)(20,0)[4]{} (183,459)(20,0)[2]{}
(83,308)(20,0)[8]{}[(0,1)[4]{}]{} (68,319)(0,10)[7]{}[(1,0)[4]{}]{} (103,319)(20,0)[7]{} (103,329)(20,0)[7]{} (123,339)(20,0)[6]{} (163,349)(20,0)[4]{} (203,359)(20,0)[2]{} (83,319) (83,329) (103,339) (143,349) (183,359)
(83,208)(20,0)[8]{}[(0,1)[4]{}]{} (68,219)(0,10)[7]{}[(1,0)[4]{}]{} (103,219)(20,0)[7]{} (103,229)(20,0)[7]{} (123,239)(20,0)[6]{} (143,249)(20,0)[5]{} (183,259)(20,0)[3]{} (223,269) (83,219) (83,229) (103,239)
(83,108)(20,0)[8]{}[(0,1)[4]{}]{} (68,119)(0,10)[7]{}[(1,0)[4]{}]{} (83,119)(20,0)[8]{} (83,129)(20,0)[8]{} (103,139)(20,0)[7]{} (143,149)(20,0)[5]{} (183,159)(20,0)[3]{} (223,169)
[$\a_0$]{} $\left( \begin{array}{ccccccccc}
\X&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0&\cdots \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
&&&&\vdots&&&&\ddots
\end{array} \right)$
[$\a_1$]{} $\left( \begin{array}{ccccccccc}
0&0&0&0&0&0&0&0 \\
\X&0&\X&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
\X&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0&\cdots \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
&&&&\vdots&&&&\ddots
\end{array} \right)$
[$\a_2$]{} $\left( \begin{array}{ccccccccc}
0&0&\X&0&\X&0&0&0 \\
0&0&0&0&0&0&0&0 \\
\X&0&\X&0&\X&0&0&0 \\
0&0&0&0&0&0&0&0 \\
\X&0&\X&0&0&0&0&0 \\
0&0&0&0&0&0&0&0&\cdots \\
\X&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
&&&&\vdots&&&&\ddots
\end{array} \right)$
[$\a_3$]{} $\left( \begin{array}{ccccccccc}
0&0&0&0&0&0&0&0 \\
\X&0&\X&0&\X&0&\X&0 \\
0&0&0&0&0&0&0&0 \\
\X&0&\X&0&\X&0&\X&0 \\
0&0&0&0&0&0&0&0 \\
\X&0&\X&0&\X&0&0&0&\cdots \\
0&0&0&0&0&0&0&0 \\
\X&0&\X&0&0&0&0&0 \\
0&0&0&0&0&0&0&0 \\
\X&0&0&0&0&0&0&0 \\
&&&&\vdots&&&&\ddots
\end{array} \right)$
(300,300) (50,50)(0,100)[3]{}[(1,0)[200]{}]{} (50,50)(50,0)[5]{}[(0,1)[200]{}]{} (20,150)[$N$]{} (150,260)[$N$]{} (70,215)[$n$]{} (83,220)[(1,0)[17]{}]{} (67,220)[(-1,0)[17]{}]{} (121,190)[$\frac{N^2}{nM}$]{} (125,210)[(0,1)[40]{}]{} (125,180)[(0,-1)[30]{}]{}
|
---
abstract: 'We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at the initial time $t=0$. This expression is function of the time $t$ at which a measurement is performed to determine if the state has decayed and, if yes, in which one of the infinitely many possible final states. For large times the energy spread is, as expected, given by the decay width $\Gamma$ of the initial unstable state. However, if the measurement of the final state is performed at a time $t$ comparable to (or smaller than) the mean lifetime of the state $1/\Gamma$, then the uncertainty on the energy of the final state is much larger than the decay width $\Gamma$. Namely, for short times an uncertainty of the type $1/t$ dominates, while at large times the usual spread $\Gamma$ is recovered. Then, we turn to a generic two-body decay process and describe the energy uncertainty of each one of the two outgoing particles. We apply these formulas to the two-body decays of the neutral and charged pions and to the spontaneous emission process of an excited atom. As a last step, we study a case in which the non-exponential decay is realized ad show that for short times eventual asymmetric terms are enhanced in the spectrum.'
author:
- Francesco Giacosa
title: Energy uncertainty of the final state of a decay process
---
It is well known that an unstable quantum state, which we denote as $\left\vert S\right\rangle $, is not an energy eigenstate of the full Hamiltonian $H$ of the corresponding quantum system, but shows an energy spread of the order of its decay width $\Gamma$. One defines for such an unstable state a ‘spectral function’ $d_{S}(E)$, which represents the probability distribution of energy (thus implying the validity of the normalization $\int_{-\infty}^{+\infty}d_{S}(E)dE=1$). The survival probability amplitude, i.e. the amplitude that the unstable state $\left\vert
S\right\rangle $ prepared at the time $t=0$ has not yet decayed at the later instant $t,$ is given by the Fourier transform of the spectral function [@khalfin; @ghirardi],$$a(t)=\left\langle S\right\vert e^{-iHt}\left\vert S\right\rangle
=\int_{-\infty}^{+\infty}d_{S}(E)e^{-iEt}dE\text{ ;} \label{at}%$$ the survival probability $p(t)$ is given as the square of the amplitude, $p(t)=\left\vert a(t)\right\vert ^{2}$. (Note: natural units $c=\hslash=1$ are used.)
In the very well known, and in most cases very well accurate, Breit-Wigner (BW) limit [@ww; @breit], the spectral function takes the Lorentzian form $$d_{S}(E)\overset{\text{BW-limit}}{=}\frac{\Gamma}{2\pi}\frac{1}{(E-M)^{2}%
+\Gamma^{2}/4}, \label{bwdse}%$$ where $M$ is referred to as the ‘mass’ (or energy) of the unstable state $\left\vert S\right\rangle $. In this limit the exponential form $$a(t)\overset{\text{BW-limit}}{=}e^{-iMt-\Gamma t/2}\rightarrow p(t)=e^{-\Gamma
t}. \label{abw}%$$ is obtained by using Eq. (\[at\]). It is a fact that the BW form is not exact; the existence of an energy threshold, $d_{S}(E)=0$ for $E<E_{th}$, is necessary for the quantum system to be consistent; in turn, this property implies that $p(t)$ is not exponential for long times [@khalfin; @ghirardi], see also the discussions in Ref. [@mercouris] and refs. therein; for an indirect experimental proof of long-time deviations we refer to [@berillium] and for a direct one which makes use of decays of organic materials to [@rothe]. Deviations from the BW form for large values of the energy ($E\gg M$), which usually take place due to a form factor that makes the function $d_{S}(E)$ decreasing faster than $E^{-2}$ for large values of $E$, implies a non-exponential behavior of $p(t)$ at short times. These short-time deviations have been confirmed experimentally [@raizen1], which in turn have also led to the verification of the Zeno and Anti-Zeno effects [@raizen2]. (For theoretical details in Quantum Mechanics see Refs. [@ghirardi; @misra; @facchiprl; @duecan] and in the framework of Quantum Field Theory (QFT) Refs. [@duecan; @zenoqft; @pascazioqft]; for a general discussion of spectral functions and deviations from the BW limit in QFT see also Refs. [@salam; @achasov; @lupo; @lupo2; @e38].)
Quite recently, a vivid debate on the non-exponential weak decay of ions measured in Ref. [@gsianomaly] has taken place. This is an extremely interesting phenomenon because for the first time short-time deviations from the exponential decay have been seen in a microscopic and natural nuclear system. In Ref. [@gp] these deviations have been linked to a modification of the BW distribution due to interactions with the measuring apparatus. Other explanations, based on neutrino oscillations and energy level splitting of the initial state, have been put forward [@giunti]. At present, no consensus on the measured data exist.
Often the theoretical interest in the study of a decay has focussed on the determination of the survival probability $p(t)$ of the initial unstable state $\left\vert S\right\rangle $. In this work we turn our attention to the final states of the decay of the unstable state. To this end, we suppose to perform at the instant $t>0$ a measurement on the quantum system of the following type: we measure the probability that the quantum state has decayed and, if yes, in which one of its possible final states. In fact, an infinity of such finals states is present in each decay process; at a given instant $t$, the sum of all these probabilities must clearly be the decay probability $1-p(t).$ Still, the question of the value of each single probability of decay in a given final state (and not only the overall sum), is interesting and sheds light on the distribution of energy of the final state in a decay process.
In order to make our discussion quantitative, we need to fix an Hamiltonian. We use the quite general Lee Hamiltonian approach [@lee; @chiu], in which the unstable state $\left\vert S\right\rangle $ is coupled to an infinity set of final states $\left\vert k\right\rangle $: $$H=H_{0}+H_{1}\text{ ,}%$$ where the free Hamiltonian $H_{0}$ and the interacting Hamiltonian $H_{1}$ are given by $$H_{0}=M\left\vert S\right\rangle \left\langle S\right\vert +\int_{-\infty
}^{+\infty}dk\omega(k)\left\vert k\right\rangle \left\langle k\right\vert
\text{ , }H_{1}=\int_{-\infty}^{+\infty}\frac{dk}{\sqrt{2\pi}}gf(k)\left(
\left\vert k\right\rangle \left\langle S\right\vert +\left\vert S\right\rangle
\left\langle k\right\vert \right) \text{ .}%$$ The quantity $g$ is a coupling constant with the dimension of energy$^{1/2}$. The dimensionless function $f(k)$ specifies the mixing of the state $\left\vert S\right\rangle $ with the state $\left\vert k\right\rangle ;$ the energy $\omega(k)$ is the energy of the state $\left\vert k\right\rangle $ in the interaction free case. Moreover, the following normalizations hold: $\left\langle S|S\right\rangle =1,$ $\left\langle k|k^{\prime}\right\rangle
=\delta(k-k^{\prime})$, $\left\langle S|k\right\rangle =0,$ $1=\left\vert
S\right\rangle \left\langle S\right\vert +\int_{-\infty}^{+\infty}dk\left\vert
k\right\rangle \left\langle k\right\vert $. Note, we have taken for simplicity $k$ as a one-dimensional variable: if we think of a two-body decay of the state $\left\vert S\right\rangle $ in its rest frame, the ket $\left\vert
k\right\rangle $ describes a two-particle state, one of which is moving with momentum $k$ and the other one with momentum $-k.$ The generalization to a three-dimensional decay $\vec{k}$ is straightforward, but unnecessary for our purposes, see Ref. [@duecan] which we also refer to for further technical details.
At the initial time $t=0$ the state $\left\vert S\right\rangle $ is prepared. Then, the time evolution implies that at the instant $t$ the system is described by the state$$e^{-iHt}\left\vert S\right\rangle =a(t)\left\vert S\right\rangle
+\int_{-\infty}^{+\infty}\frac{dk}{\sqrt{2\pi}}a_{Sk}(t)\left\vert
k\right\rangle \text{ .} \label{te}%$$ A possible way to evaluate the previous expression makes use of the operator relation $e^{-iHt}=\frac{i}{2\pi}\int_{-\infty}^{+\infty}dEG(E)e^{-iEt}$ with $G(E)=\left[ E-H+i\varepsilon\right] ^{-1}$ [@facchiprl; @fpbook]. Then, the validity of Eq. (\[at\]) can be proven: $$a(t)=\frac{i}{2\pi}\int_{-\infty}^{+\infty}dEG_{S}(E)e^{-iEt}\text{ }%
=\int_{-\infty}^{+\infty}dEd_{S}(E)e^{-iEt}\text{ ,}%$$ where the propagator $G_{S}(E)$ of the unstable state $\left\vert
S\right\rangle $ reads $$G_{S}(E)=\left\langle S\left\vert G(E)\right\vert S\right\rangle =\frac
{1}{E-M+\Pi(E)+i\varepsilon}\text{ , }\Pi(E)=\int_{-\infty}^{+\infty}\frac
{dk}{2\pi}\frac{g^{2}f^{2}(k)}{E-\omega(k)+i\varepsilon}\text{ ,} \label{gse}%$$ and the spectral function emerges as the imaginary part of the propagator: $d_{S}(E)=\frac{1}{\pi}\left\vert \operatorname{Im}G_{S}(E)\right\vert .$ The quantity $\Pi(E)$ is referred to as the self-energy quantum contribution for the state $\left\vert S\right\rangle $.
The BW limit is obtained for $f(k)=1$ and $\omega(k)=k$, that implies $\Pi(E)=i\Gamma/2$, where the decay width $\Gamma$ reads: $\Gamma=g^{2}$. In this limit Eq. (\[abw\]) follows. For generic $f(k)$ and $\omega(k)$, the decay is not exponential, but is usually very well approximated by an exponential decay where the decay width is given by the Fermi golden-rule: $\Gamma=2\operatorname{Im}[\Pi(M)]=g^{2}f^{2}(k_{M})/\left\vert \omega
^{\prime}(k_{M})\right\vert $, where $\omega(k_{M})=M$. Note, the choice $\omega(k)=k$ means that the energy is not bounded from below. This is obviously unphysical and represents a mathematical trick. However, as long as the distribution function is peaked around $M$ and the low-energy threshold is far away from it, the error done by this approximation is (indeed very) small.
In this work we are interested in the probability to find the system described by a certain state $\left\vert k\right\rangle $ when performing a measurement at the instant $t.$ To this end, it is important to determine the coefficients $a_{Sk}(t)$ in Eq. (\[te\]). A direct evaluation delivers the following general result:$$a_{Sk}(t)=\left\langle k\right\vert e^{-iHt}\left\vert S\right\rangle
=i\frac{gf(k)}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\frac{dk}{2\pi}\frac
{G_{S}(E)}{E-\omega(k)+i\varepsilon}e^{-iEt}\text{ .} \label{askgen}%$$ It follows that the decay probability, i.e. the probability that the state has decayed at the time $t,$ is given by one minus the survival probability:$$\begin{aligned}
w(t) & =\int_{-\infty}^{+\infty}dk\left\vert a_{Sk}(t)\right\vert
^{2}=\text{ }\int_{-\infty}^{+\infty}dk\left\langle S\right\vert
e^{iHt}\left\vert k\right\rangle \left\langle k\right\vert e^{-iHt}\left\vert
S\right\rangle \nonumber\\
& =\left\langle S\right\vert e^{iHt}\left( 1-\left\vert S\right\rangle
\left\langle S\right\vert \right) e^{-iHt}\left\vert S\right\rangle
=1-p(t)\text{ } \label{w}%\end{aligned}$$ In addition to this, the previous expression also tells us that the probability density that a particular final state $\left\vert k\right\rangle $ is realized at the instant $t$ is given by the quantity $\left\vert
a_{Sk}(t)\right\vert ^{2}$. By a change of coordinate from $k$ to $\omega=\omega(k)$ we rewrite $w(t)$ as$$w(t)=\int_{-\infty}^{+\infty}d\omega\eta(t,\omega)\text{ ,} \label{etaint}%$$ where the quantity $\eta(t,\omega)d\omega$ is the probability that, by a measurement of the final state of the quantum state at the instant $t$, the quantum state has decayed and has an energy between $\omega$ and $\omega+d\omega$.
Let us in the following restrict to the BW limit, which, as mentioned above, is realized for $f(k)=1$ and $\omega(k)=k$. In this case a simple explicit calculation shows that the energy distribution $\eta(t,\omega)$ takes the form $$\eta(t,\omega)=\frac{\Gamma}{2\pi}\left\vert \frac{e^{-i\omega t}%
-e^{-i(M-i\Gamma/2)t}}{\omega-M+i\Gamma/2}\right\vert ^{2}. \label{eta}%$$ A direct evaluation of the integral (\[etaint\]) using the explicit BW form of Eq. (\[eta\]) shows the validity of the general Eq. (\[w\]) in the BW limit: $w(t)=\int_{-\infty}^{+\infty}d\omega\eta(t,\omega)=1-e^{-\Gamma t}$.
The validity of $\eta(t,\omega)$ in Eq. (\[eta\]) is general, as long as the exponential (or BW) limit is considered. It depends only on the two parameters $M$ and $\Gamma$ and not on the details of the decay process under study. In particular, Eq. (\[eta\]) does not depend on the fact that our variable $k$ is one dimensional; the same result would be obtained with a more realistic three-dimensional variable. The result also does not depend on the special form of the Lee-Hamiltonian, but would be obtained with each Hamiltonian, as long as the BW limit is taken. Quite remarkably, the result is applicable also in relativistic Quantum Field Theory because the nonrelativistic BW limit is typically a very good approximation also in this context, see details in Ref. [@duecan] in which the link between QFT and Lee models has been described. Clearly, when including deviations from the Breit-Wigner energy distribution, also $\eta(t,\omega)$ changes (see the end of the paper), but, as long as the exponential decay law well approximates the decay of an unstable state, Eq. (\[eta\]) represents a very good description of the probability density of the energy of the final state for a wide range of $\omega$.
\[ptb\]
[fig1.eps]{}
The energy distribution of Eq. (\[eta\]) was already implicitly present in the pioneering work of Ref. [@ww] where the natural broadening of spectral lines was studied, see also e.g. Refs. [@scully; @facchispont; @ford; @elattari; @kofman] and refs. therein. In particular, in the limit of large times, $t\rightarrow\infty$, one obtains: $$\eta(t\rightarrow\infty,\omega)=\frac{\Gamma}{2\pi}\left\vert \frac{1}%
{\omega-M+i\Gamma/2}\right\vert ^{2}=\frac{\Gamma}{2\pi}\frac{1}%
{(\omega-M)^{2}+\Gamma^{2}/4}\text{ .}%$$ Thus, in the long-time regime the energy distribution is the original energy distribution of the initial state (\[bwdse\]). The probability that the final state has an energy between $\omega$ and $\omega+d\omega$ is given by $\eta(t\rightarrow\infty,\omega)d\omega=d_{S}(\omega)d\omega$. This result has been shown in Ref. [@ford] and can be also easily derived with the general expressions written above; this is indeed a general outcome which is not restricted to the BW approximation.
In the present work we aim to discuss the distribution of energies of the final state also for early times and not only in the long-time limit. Namely, we study the form of $\eta(t,\omega)$ of Eq. (\[eta\]) as function of $\omega$ for different values of $t.$ The function has, for each value of $t$, a maximum for $\omega=M$. The value of the function at its maximum $$\eta(t,\omega=M)=\frac{2}{\pi\Gamma}\left( 1-e^{-\Gamma t}\right) ^{2}%$$ vanishes for $t\rightarrow0$ and increases for increasing $t$; this is in agreement with the fact that the overall area of $\eta(t,\omega)$ is a increasing function of $t.$ In Fig. 1 we plot the function $\eta(t,\omega)$ for $t=\tau$ and for $t=100\tau$ where $\tau=\Gamma^{-1}$ is the mean lifetime. It is visible that for $t=100\tau$ the limit $t\rightarrow\infty$ is well recovered. However, for $t=\tau$ the function $\eta(t=\tau,\omega)$ is sizably wider, implying a broadening of the energy uncertainty if a measurement of the final state (and its energy) is performed at such an early time.
In Fig. 2 we plot the function $\eta(t,\omega)/\eta(t,M)$ for $t=\tau/10,$ $\tau/2,\tau,3\tau,100\tau$. Being this quantity per construction normalized to 1 at the maximum $\omega=M$, the spread of the energy distribution of the final state is better visualized. For short times, $t\lesssim$ $3\tau$, the function $\eta(t,\omega)$ is spread over a wide range of energy, sizably larger than the natural expected energy spread $\Gamma.$ Moreover, a pattern of maxima and minima become visible for small values of $t$.
\[ptb\]
[fig2.eps]{}
In order to quantify the energy uncertainty of the final state, the variance is not usable because the integral $\int_{-\infty}^{\infty}\omega^{2}%
\eta(t,\omega)d\omega$ is divergent in the BW limit (more in general, it is dependent on the tails of the distribution). A well-defined quantity for our purposes is the width, at a given time $t$, of the distribution $\eta
(t,\omega)$ at mid height. We denote it as $\delta\omega=\delta\omega(t)$, mathematically expressed by the equation $$\frac{\eta(t,M)}{2}=\eta(t,M+\delta\omega/2)\text{ .} \label{deltaomega}%$$ The function $\delta\omega/\Gamma$ is plotted in Fig. 3 as function of $t$: for large $t$ the quantity $\delta\omega/\Gamma$ tends to $1,$ which is the ‘natural’ energy uncertainty of the initial state. Conversely, for short times $\delta\omega/\Gamma$ increases sizably. The following approximate form is valid for $t\lesssim3\tau$ $$\delta\omega\simeq\frac{2\cdot2.78}{t}\text{ .} \label{deltapprx}%$$ The numerical coefficient $2.78$ is the (nonzero) solution of the transcendental equation $y=\sqrt{2}\left\vert 1-e^{iy}\right\vert $, which follows from Eq. (\[deltaomega\]) in the short-time limit. Eq. (\[deltapprx\]) approximates the form of $\delta\omega$ better and better for decreasing time, as it is shown in Fig. 3. This fact is intuitively expected: for short times the energy uncertainty becomes dominant and this uncertainty is proportional to $1/t$. The full form of $\delta\omega$ interpolates between the $1/t$ behavior at short times and the constant limit $\Gamma$ for long times.
In Ref. [@elattari] a related phenomenon has been studied in the case of electron tunneling out of a quantum dot. The attention is focussed on the continuous monitoring of the unstable state: it is shown that an increase of the width of the emitted spectrum arises. This is similar to our result, although the physical situation has some important differences. Namely, in Ref. [@elattari] the continuous measurement (modelled by a term in the Hamiltonian) is acting on the unstable initial state ($\left\vert
S\right\rangle $ in our notation), while in our framework the (instantaneous) measurement is designed to detect the final decay product (one of the states $\left\vert k\right\rangle $). The broadening of our study is an intrinsic feature of the short-time evolution and is not associated to the measurement process, which is of the ideal type.
In Ref. [@kofman] a formalism which is similar to the one used in this work has been adopted to study atomic spontaneous emission. Different choices for the interaction of the unstable upper level to the ground state and to an emitted photon are extensively analyzed. We thus refer to this work for further important details and for the investigation of non-exponential decay law. However, in Ref. [@kofman] the emitted spectrum is investigated only in the limit $t\rightarrow\infty$ and not for earlier times. As we shall see later on, the spontaneous emission of an atom can be seen as a particular limit of our approach.
\[ptb\]
[fig3.eps]{}
A nice feature of the Lee Hamiltonian used in this work is that the formalism can be easily applied to the general case of a two-body decay, which we describe in the following: the final state $\left\vert k\right\rangle $ with energy $\omega$ describes two particles, with masses $m_{1}$ and $m_{2}$ respectively, flying back-to-back. The energies $\omega_{1}$ and $\omega_{2}$ of the first and the second particle read (in a general relativistic framework):$$\omega_{1}=\frac{\omega^{2}+m_{1}^{2}-m_{2}^{2}}{2\omega}\text{ , }\omega
_{2}=\frac{\omega^{2}-m_{1}^{2}+m_{2}^{2}}{2\omega}\text{ .}%$$ The constraint $\omega_{1}+\omega_{2}=\omega$ is fulfilled and, for each value of $\omega$, the energies $\omega_{1}$ and $\omega_{2}$ are uniquely determined. The opposite is not true, being $\omega=\omega_{1}\pm\sqrt
{\omega_{1}^{2}-m_{1}^{2}+m_{2}^{2}}$ and $\omega=\omega_{2}\pm\sqrt
{\omega_{2}^{2}+m_{1}^{2}-m_{2}^{2}}$. One can determine in a straightforward way also the general expression for the probability density distribution $\eta_{1}(t,\omega_{1})$ ($\eta_{2}(t,\omega_{2})$), which represents the probability that the first (second) particle has an energy between $\omega
_{1}+d\omega_{1}$ ($\omega_{2}+d\omega_{2}$) if a measurement at the instant $t$ on it is performed:$$\eta_{1}(t,\omega_{1})=\eta_{1}^{+}(t,\omega_{1})+\eta_{1}^{-}(t,\omega
_{1})\text{ , }\eta_{1}^{\pm}(t,\omega_{1})=\left\vert 1\pm\frac{\omega_{1}%
}{\sqrt{\omega_{1}^{2}-\Delta m^{2}}}\right\vert \eta\left( t,\omega_{1}%
\pm\sqrt{\omega_{1}^{2}-\Delta m^{2}}\right) \label{eta1}%$$$$\eta_{2}(t,\omega_{2})=\eta_{2}^{+}(t,\omega_{2})+\eta_{2}^{-}(t,\omega
_{2})\text{ , }\eta_{2}^{\pm}(t,\omega_{2})=\left\vert 1\pm\frac{\omega_{2}%
}{\sqrt{\omega_{2}^{2}+\Delta m^{2}}}\right\vert \eta\left( t,\omega_{2}%
\pm\sqrt{\omega_{2}^{2}+\Delta m^{2}}\right) \text{ ,} \label{eta2}%$$ where $\Delta m^{2}=m_{1}^{2}-m_{2}^{2}.$ Notice that, if we choose $\Delta
m^{2}>0$, the function $\eta_{1}(t,\omega_{1})$ is only defined for $\left\vert \omega_{1}\right\vert \geq\Delta m^{2}$. This limit is however irrelevant in practical cases because the relation $\omega_{1}>m_{1}$ holds (relativistically: $\omega_{1}=\sqrt{k^{2}+m_{1}^{2}}$ ). Eqs. (\[eta1\]) and (\[eta2\]) are of general validity and can be applied beyond the BW limit.
In most practical cases, when the peak of $d_{S}(E)$ is narrow and away from threshold, the relation between full energy $\omega$ and the energies of the two outgoing particles $\omega_{1}$ and $\omega_{2}$ can be simplified: $$\omega-M\simeq\frac{2M^{2}}{M^{2}-\Delta m^{2}}\left( \omega_{1}-\bar{\omega
}_{1}\right) =\frac{2M^{2}}{M^{2}+\Delta m^{2}}\left( \omega_{2}-\bar
{\omega}_{2}\right) \text{ , }%$$ where $$\bar{\omega}_{1}=\frac{M^{2}+\Delta m^{2}}{2M}\text{ , }\bar{\omega}_{2}%
=\frac{M^{2}-\Delta m^{2}}{2M}\text{ .} \label{o12b}%$$ Introducing the ‘decay widths’ $$\Gamma_{1}=\left( \frac{1}{2}-\frac{\Delta m^{2}}{2M^{2}}\right)
\Gamma\text{ , }\Gamma_{2}=\left( \frac{1}{2}+\frac{\Delta m^{2}}{2M^{2}%
}\right) \Gamma\text{ ,} \label{gamma12}%$$ the expression of the probability distributions $\eta_{1}(t,\omega_{1})$ and $\eta_{2}(t,\omega_{1})$ takes the simplified form$$\eta_{1}(t,\omega_{1})=\frac{2M^{2}}{M^{2}-\Delta m^{2}}\eta\left(
t,M+\frac{2M^{2}\left( \omega_{1}-\bar{\omega}_{1}\right) }{M^{2}-\Delta
m^{2}}\right) =\frac{\Gamma_{1}}{2\pi}\left\vert \frac{1-e^{i\frac
{2M^{2}(\omega_{1}-\bar{\omega}_{1}+i\Gamma_{1}/2)t}{M^{2}-\Delta m^{2}}}%
}{\omega_{1}-\bar{\omega}_{1}+i\Gamma_{1}/2}\right\vert ^{2}\text{ ,}%$$$$\eta_{2}(t,\omega_{2})=\frac{2M^{2}}{M^{2}+\Delta m^{2}}\eta\left(
t,M+\frac{2M^{2}\left( \omega_{2}-\bar{\omega}_{2}\right) }{M^{2}+\Delta
m^{2}}\right) =\frac{\Gamma_{2}}{2\pi}\left\vert \frac{1-e^{i\frac
{2M^{2}(\omega_{2}-\bar{\omega}_{2}+i\Gamma_{2}/2)t}{M^{2}+\Delta m^{2}}}%
}{\omega_{2}-\bar{\omega}_{2}+i\Gamma_{2}/2}\right\vert ^{2}\text{ ,}%$$ where in the r.h.s. of the latter equations the BW-limit has been again taken. The previous study of the function $\eta(t,\omega)$ can be easily repeated for $\eta_{1}(t,\omega_{1})$ and $\eta_{2}(t,\omega_{2}),$ which show the same qualitative properties described above, but with different values for the position and the width of the peak: namely, the distributions $\eta
_{1}(t,\omega_{1})$ and $\eta_{2}(t,\omega_{2})$ are peaked around the values $\bar{\omega}_{1}$ and $\bar{\omega}_{2}$ defined in Eq. (\[o12b\]) and have a time-dependent width at mid height given by the rescaled quantities$$\delta\omega_{1}=\frac{\Gamma_{1}}{\Gamma}\delta\omega\text{ , }\delta
\omega_{2}=\frac{\Gamma_{2}}{\Gamma}\delta\omega\text{ ,}%$$ where $\Gamma_{1}$ and $\Gamma_{2}$ have been defined in Eq. (\[gamma12\]) and $\delta\omega$ is the time-dependent function defined in Eq. (\[deltaomega\]) and plotted in Fig. 3. Then, for each particle an enhanced spread of the energy takes place for short times. For long times, being $\delta\omega\rightarrow\Gamma$, the limits $\delta\omega_{1}\rightarrow
\Gamma_{1}$ and $\delta\omega_{2}\rightarrow\Gamma_{2}$ are realized. The integrals over all the energies read: $$\int_{0}^{\infty}\eta_{1}(t,\omega_{1})d\omega_{1}=\int_{0}^{\infty}\eta
_{2}(t,\omega_{2})d\omega_{2}=1-p(t)=1-e^{-\Gamma t}\text{ ;}%$$ the overall probability to find the particle 1 (or 2) is, as it must, the overall decay probability.
We now turn to some examples from particle and atomic physics:
\(i) $\pi^{0}$ decay: the electromagnetic decay of the $\pi^{0}$ meson into two photons, $\pi^{0}\rightarrow\gamma\gamma,$ has a mean life time of $\tau
_{\pi^{0}}=(8.52\pm0.18)\cdot10^{-17}$ s [@PDG]. Here $m_{1}=m_{2}=0$ and the previous formulae simplify. Namely, each photon has an energy of $E_{\gamma}=\omega_{1}=\omega_{2}$, whose distribution is peaked for $\bar{\omega}_{1}=\bar{\omega}_{2}=M_{\pi^{0}}/2$, and, in virtue of the spreading described above, the energy uncertainty per photon is given by $\delta E_{\gamma}=\delta\omega/2$ with $\delta\omega$ given in Eq. (\[deltaomega\]). Thus, $\delta E_{\gamma}$ is larger than $\Gamma_{\pi^{0}%
}/2$ (where $\Gamma_{\pi^{0}}=1/\tau_{\pi^{0}}$) if a measurement at a time of $t\lesssim3\tau_{\pi^{0}}$ is performed. This is however difficult because of the very short times involved.
\(ii) $\pi^{+}$ decay: the decay $\pi^{+}\rightarrow\mu^{+}\nu_{\mu}$ is a weak two-body decay process with a lifetime $\tau_{\pi^{+}}=(2.6033\pm
0.0005)\cdot10^{-8}$ s. Plugging in the nominal masses (neglecting neutrino mixing and masses) implies that $\delta E_{\mu}=\delta\omega_{1}%
\simeq0.213\delta\omega,$ thus the energy spread of the muon energy is smaller than $\delta\omega/2$ because of the sizable muon mass. On the other hand, for the energy uncertainty of the neutrino one has $\delta E_{\nu}=\delta
\omega_{2}\simeq0.787\delta\omega,$ thus larger than $\delta\omega/2$. While the neutrino can hardly be detected, a measurement of the muon at such an early time could be feasible.
\(iii) Atomic spontaneous emission: an atom in which an electron is in an excited state decays to its ground state by emitting a photon. This is indeed the original framework in which the decay was studied [@ww; @breit] (see also e.g. Refs. [@scully; @facchispont; @ford; @kofman] and refs. therein) and can be seen as a particular limit of our formalism. Let $\Delta E$ be the energy difference of the two energy levels: the initial ‘mother’ state is the excited atom, with a central mass $M=m_{M}=m_{D}+\Delta E$, and the final state consists of two particles, the ‘daughter’ state (ground-state stable atom, with mass $m_{D}$) and a photon. Due to the fact that $\Delta E\ll
m_{D},$ it follows that $\delta E_{D}=\delta\omega_{1}\simeq0$ and $\delta
E_{\gamma}=\delta\omega_{2}\simeq\delta\omega$ (valid to a very good level of approximation). In this case the uncertainty on the energy of the photon is the whole energy uncertainty of the quantum system. Thus, the distribution energy of the photon is given by $\eta_{2}(t,\omega_{2})=\eta\left(
t,\omega-m_{D}\right) $, which is centered on $\Delta E$ and has for large times a spread of $\delta E_{\gamma}=\delta\omega=\Gamma$, where $\Gamma$ is the decay width of the spontaneous emission process: this energy uncertainty is nothing else than the well-known natural broadening of the spectral line due to the uncertainty principle. For decreasing $t$ the quantity $\delta
E_{\gamma}$ shows the broadening expressed in Fig. 3. Thus, if it were possible to perform a measurement at a time scale of the order of $\Gamma
^{-1}$ (typically in the range of $10^{-9}$ s, but dependent on the considered atom and energy levels), then the broadening of $\delta\omega$ could be eventually visible. (Indeed, another known effect of spectral line broadening is the so-called impact pressure broadening, in which other particles interrupts the decay process. A question is if such pressure broadening can be related to the effect described in this work: to which extent can the other particles, and thus the environment, make a measure of the unstable excited atom? This surely interesting topic is left as an outlook).
\[ptb\]
[fig4.eps]{}
As a last step we go beyond the exponential limit by studying a case in which the function $f(k)$ takes the form$$f^{2}(k)=\left( 1+\alpha k\right) \theta(E_{0}^{2}-(E-M)^{2}). \label{fk}%$$ For simplicity, we still keep the relation $\omega(k)=k$ valid. The step function in Eq. (\[fk\]) implies that the state $\left\vert S\right\rangle $ couples to a continuum of states $\left\vert k\right\rangle $ limited to a band of energy $(M-E_{0},M+E_{0})$ [@gp]. In virtue of the introduced thresholds the survival probability $p(t)$ is not an exponential for short and long times. Moreover, the parameter $\alpha,$ which has the dimension $[E^{-1}]$, introduces an asymmetry in the coupling to this band. As a consequence, the spectral function $d_{S}(x)$ is not symmetric around the peak located at $M$. The self-energy contribution takes the form (after reabsorbing an inessential constant) $\Pi(E)=\frac{g^{2}}{2\pi}(1+\alpha E)\log\left[
\frac{E-M-E_{0}}{E-M+E_{0}}\right] $. Note, in the limit $\alpha=0$ and $E_{0}\rightarrow\infty$ we recover the self-energy in the exponential case: $\Pi(E)=ig^{2}/2$.
The expression of the decay width as given by the Fermi golden rule reads now $\Gamma=g^{2}(1+\alpha M)$ and the corresponding mean life time is $\tau=1/\Gamma$. In this case, being the decay not exponential, $\Gamma$ is only an approximate quantity. This is clearly visible in Fig. 4, upper panel, where the survival probability $p(t)$ is plotted for a suitable numerical choice ($g=0.95\sqrt{\Gamma},$ $M-E_{0}=2.52\Gamma,$ $\alpha=0.0396\tau$) and compared to the exponential decay $e^{-\Gamma t}.$ The typical quadratic behavior of $p(t)$ is realized; then, after some oscillations, the exponential limit is reached (deviations for large times take place as well, but are not relevant here).
For the purposes of our work, we study the function $\eta(t,\omega)$ for this system for different times. Its analytic expression can be derived by the previously presented general formulae and reads$$\eta(t,\omega)=\frac{\operatorname{Im}\Pi(\omega)}{\pi}\left\vert
\int_{-\infty}^{\infty}\mathrm{dE}d_{S}(E)\frac{e^{-i\omega t}-e^{-iEt}%
}{\omega-E}\right\vert ^{2}\text{.}%$$ (This expression is formally valid for each choice of $f(k),$ as long as $\omega(k)=k$. In the case of Eq. (\[fk\]): $\operatorname{Im}\Pi
(\omega)=g^{2}\left( 1+\alpha\omega\right) /2$ for $\omega\in$ $(M-E_{0},M+E_{0})$ .) The function $\eta(t,\omega)/\eta(t,M)$ is plotted in Fig. 4, lower panel, for different values of time. For large times, $t=100\tau,$ the equality $\eta(t\rightarrow\infty,\omega)=d_{S}(\omega)$ is realized (thick solid line). It is then visible that the function $d_{S}(\omega)$ has the usual form and that the asymmetry due to the parameter $\alpha$ introduced in Eq. (\[fk\]) is hardly noticeable. However, when going to smaller times, the expected broadening takes place, thus showing the generality of this effect. On top of that, an interesting phenomenon emerges: the asymmetry gets enhanced. For $t=0.40\tau$ (thin solid line) the function $\eta(t=0.40\tau,\omega)$ this is evident: even the maximum does not take place at $M$ but is shifted to a higher value.
One important remark about the model of Eq. (\[fk\]) is necessary: two additional discrete energy levels emerge for each value of the coupling constant $g$ (for small $g,$ one emergent stable state has an energy just below $M-E_{0}$ and the other just above $M+E_{0}$.) As a consequence, the survival probability $p(t)$ shows in general for large times oscillations which involve these additional discrete levels, see also Ref. [@kofman] for the description of this phenomenon. However, for the numerical values used in Fig. 4 the presence of these two additional stable states is negligible: for very large $t$ the survival amplitude reads $a(t)\simeq Z_{1}e^{-iE_{1}%
t}+Z_{2}e^{-iE_{2}t},$ where $Z_{1}=0.92\cdot10^{-6}$ and $Z_{2}%
=1.4\cdot10^{-5}$ are very small numbers, meaning that the oscillations are extremely suppressed (and, moreover, have practically no influence on the short-time behavior of the system); the energies $E_{1}=M-E_{0}-1.3\cdot
10^{-7}\Gamma$ and $E_{2}=M+E_{0}+2.5\cdot10^{-6}\Gamma$ are the emergent discrete levels (very close the the energy thresholds of Eq. (\[fk\])). In conclusion, for the illustrative purposes of our analysis the use of the simple model of Eq. (\[fk\]) is acceptable, although the subtlety of the emerging stable states should be kept in mind when this model is studied; we also refer to Ref. [@wolkanowski], where this issue has been analyzed both analytically and numerically in great detail in dependence of the coupling constant $g$.
The emergence of discrete energy levels for each value of $g$ is indeed a peculiarity of Eq. (\[fk\]) due to the sharp boundaries. A more realistic form of $f^{2}(k)$ is given by $$f^{2}(k)=(1+\alpha k)\frac{\sqrt{k-(M-E_{0})}}{k^{2}+\Lambda^{2}}%
\theta(k-(M-E_{0}))\text{ ,} \label{fknew}%$$ in which a phase space factor $\sqrt{k-(M_{0}-E_{0})}$ renders the function continuous close to the left energy threshold $(M-E_{0})$ and a smooth cutoff behavior for large $k$ replaces the right threshold. For this form, no additional stable state emerges as long as $g$ does not exceed a critical value. (Quite interestingly, the emergence of an additional stable state when the coupling constant exceeds a certain value takes place also in relativistic quantum field theory, see Ref. [@e38].) A numerical study of the system with Eq. (\[fknew\]) shows the same qualitative features as Fig. 4. Then, we are led to think that the described properties (broadening and distortion for small times) are rather general, and hold also for more complicated choices of the function $f(k)$ and $\omega(k)$.
In conclusions, we have studied the energy uncertainty of the final state of a decay process, finding that it can be much larger than the natural decay width if a measurement of the final state and its energy is performed at time comparable to (or smaller than) the lifetime of the unstable state, see Figs. 1, 2, and 3. Further outlooks are listed in the following. (i) Systematic study for a class of models with short- and long-time deviations from the exponential decay law. The simple case studied here leading to Fig. 4 shows that this subject is potentially very interesting. (ii) Study of the energy spread of the final state of decay processes in which the initial unstable state can decay, along the line of Ref. [@duecan], in more than one decay channel. (iii) In this work the measurement has been still considered of the ideal type; a description of the measurement in a more realistic way, along the line of Ref. [@shimizu], is an important work for the future. In this framework one has to make particular attention to the details of the type of measurement performed.
**Acknowledgments:** the author thanks G. Pagliara and T. Wolkanowski for useful discussions and the Foundation of the Polytechnical Society of Frankfurt for support through an Educator fellowship.
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|
---
author:
- 'A. Wyttenbach, D. Ehrenreich, C. Lovis, S. Udry, F. Pepe'
date: 'Received 2015-01-23; accepted 2015-03-14'
title: |
Spectrally resolved detection of sodium in the atmosphere of\
HD189733b with the HARPS spectrograph[^1]
---
[Atmospheric properties of exoplanets can be constrained with transit spectroscopy. At low spectral resolution, this technique is limited by the presence of clouds. The signature of atomic sodium (), known to be present above the clouds, is a powerful probe of the upper atmosphere, where it can be best detected and characterized at high spectral resolution.]{} [Our goal is to obtain a high-resolution transit spectrum of HD 189733b in the region around the resonance doublet of at 589 nm, to characterize the absorption signature previously detected from space at low resolution.]{} [We analyze archival transit data of HD 189733b obtained with the HARPS spectrograph ($\mathcal{R}=115\,000$) at ESO 3.6-meter telescope. We perform differential spectroscopy to retrieve the transit spectrum and light curve of the planet, implementing corrections for telluric contamination and planetary orbital motion. We compare our results to synthetic transit spectra calculated from isothermal models of the planetary atmosphere.]{} [We spectrally resolve the D doublet and measure line contrasts of $0.64\pm0.07\%$ (D2) and $0.40\pm0.07\%$ (D1) and FWHMs of $0.52\pm0.08~\AA$. This corresponds to a detection at the 10-$\sigma$ level of excess of absorption of $0.32\pm0.03\%$ in a passband of $2\times0.75\ \AA$ centered on each line. We derive temperatures of $2\,600\pm600$ K and $3270\pm330$ K at altitudes of $9\,800\pm2\,800$ and $12\,700\pm2\,600$ km in the D1 and D2 line cores, respectively. We measure a temperature gradient of $\sim0.2$ K km$^{-1}$ in the region where the sodium absorption dominates over the haze-absorption, from comparison with theoretical models. We also detect a blueshift of $0.16\pm0.04\ \AA$ (4 $\sigma$) in the line positions. This blueshift may be due to winds blowing at $8\pm2$ [km s$^{-1}$]{} in the upper layers of the atmosphere.]{} [We demonstrate the relevance of studying exoplanet atmospheres with high-resolution spectrographs mounted on 4-meter-class telescopes. Our results pave the way towards in-depth characterization of physical conditions in the atmospheres of many exoplanetary systems with future spectrographs such as ESPRESSO on the VLT or HiReS and METIS on the E-ELT.]{}
Introduction
============
After two decades and over a thousand exoplanet detections, we have entered an era of characterization of these remote worlds. More and more exoplanets have precise mass and radius measurements thanks to simultaneous radial-velocity and transit observations, providing us with estimates of mean densities and constraints on possible bulk compositions. Meanwhile, detections and studies of exoplanet atmospheres represent the only direct observable window we have on the physical and chemical properties of these planets. Characterization of exoplanets and comparative planetology are now progressing well by virtue of cutting-edge instrumentation. Among these observations, spectra of exoplanets are among the best products for in-depth comprehension of atmospheric and surface conditions.
Shortly after the first indirect exoplanet discoveries, early observational campaigns encouraged by theoretical work [@Marley1999; @Seager2000; @Brown2001], attempted to detect reflected light from the dayside [@Charbonneau1999] or absorption through limb transmission [@Moutou2001]. Observations of HD209458, the unique transiting system known at that time, with the Space Telescope Imaging Spectrograph (STIS) on board the 2.4-m Hubble Space Telescope (HST), allowed @Charbonneau2002 to detect atmospheric sodium () at 589 nm. This measurement was performed with a spectral resolution of $\mathcal{R}\equiv\lambda/\Delta\lambda\sim5\,500$ (or $\sim55$ [km s$^{-1}$]{}). Extra absorption of 0.023$\pm0.006\%$ and 0.013$\pm0.004\%$ have been measured during transits over spectral bins of 12 Å and 38 Å, respectively. The doublet lines D1 ($\lambda589.5924$ nm) and D2 ($\lambda588.9951$ nm) are not resolved in these bins.
In addition to important detections of other atomic and molecular signatures [*e.g.* @Vidal2003; @Deming2013], the sodium signature in HD209458b has remained one of the most robust examples of atmospheric characterization [@Madhusudhan2014; @Heng2014; @Pepe2014]. It has been confirmed by independent analysis of the same data set [@Sing2008a; @Sing2008b]. In the early days, ground-based observations of transiting systems with high-resolution spectrographs mounted on 8-meter-class telescope were also quickly tested in the perspective of confirming detections and resolving transmission spectra, unfortunately without any positive results [*e.g.* @Narita2005]. Indeed, even with high-enough signal, no good observational strategy or data analysis were found to overcome terrestrial atmospheric variation and telluric lines contamination that transmission spectra undergo. Later, careful data (re-)analysis made by @Redfield2008 and @Snellen2008 enabled to beat down systematics due to Earth atmosphere and led to the first ground-based detection of exoplanet atmospheric signature. Indeed the doublet was detected for the first time in HD189733b with the High Resolution Spectrograph (HRS; $\mathcal{R}\sim60\,000$) mounted on the 9-m Hobby-Eberly Telescope and confirmed in HD209458b using the High Dispersion Spectrograph (HDS; $\mathcal{R}\sim45\,000$) on the 8-m Subaru telescope. Furthermore, the analysis of the space-based detection of the sodium signature of these two planets by @Sing2008b, @Vidal2011, and @Huitson2012, demonstrated the potential to resolve the transmitted star light by the planet atmosphere. Despite these encouraging works, the presence of scattering hazes and clouds [@Lecavelier2008; @Lecavelier2008b; @Pont2013] in several exoplanets prevented the detection of major chemical constituents at low-to mid-resolution even from space [*e.g.* @Kreidberg2014; @Sing2015]. The whole potential of high-resolution spectroscopy with $R\sim100\ 000$ applied to transiting and non-transiting system was shown when $\mathrm{H_2O}$ and CO molecular bands were detected on account of cross-correlation on hundreds of resolved individual transitions [@Snellen2010; @Snellen2014; @Brogi2012; @Birkby2013]. This state-of-the-art method brought back attention to the importance of transit spectroscopy observations from the ground.
Transit spectroscopy with ground-based telescopes is often used to gauge the Rossiter-McLaughlin effect during an exoplanet transit. This is accomplished with high-precision velocimeters. As data are acquired during transit, we can think about measurements of exoplanet atmosphere properties with the same data. However, observations with simultaneous-calibrated or self-calibrated velocimeter are not equally valuable because of the use of an iodine cell. The absorption of the light by the iodine cell act as a contaminant. The iodine lines prevent measurements of several spectral features especially around the lines. Thus the use of stabilized spectrographs with simultaneous calibration makes possible to measure the Rossiter-McLaughlin effect and the transmission spectrum of an exoplanet at the same time.
In this paper, we will describe new results obtained with the HARPS spectrograph by analyzing existing high-quality Rossiter-McLaughlin observations of HD189733b. For the first time, features in an exoplanet atmosphere are resolved with a 4-m-class telescope. In Sect. \[Sec\_Obs\], we will present the observations, the data reduction, the telluric correction and the method to measure exo-atmospheric signals. In Sect. \[Sec\_Results\], we will describe the results obtained on the measure of the D absorption excess in HD189733b. In Sect. \[Sec\_Discuss\], we will discuss the implication of our detection, eventually we will compare our results with theory.
Observations, data reduction and methods {#Sec_Obs}
========================================
HARPS observations
------------------
HD189733 [@Bouchy2005] was observed with the HARPS echelle spectrograph (High-Accuracy Radial-velocity Planet Searcher) on the ESO 3.6 m telescope, La Silla, Chile. Data were retrieved from the ESO archive from programs 072.C-0488(E), 079.C-0828(A) (PI:Mayor) and 079.C-0127(A) (PI: Lecavelier des Etangs). The data sets are described in Table \[table1\] (nights 1,3,4 were employed by @Triaud2009 to measure the Rossiter-McLaughlin effect). In total, four sequences of data cover at least partially the transit of the planet. Among those four sequences, two were obtained using low cadence (600 s exposures), the two others using high cadence (300 s exposures). One low-cadence sequence was affected by bad weather and missed the second half of the transit. For this reason, we decided not to take this sequence (2006-07-29) into account. Our analysis is based on the remaining three data sequences totalizing 99 spectra and 3 planetary transits. Based on the transit ephemeris of @Agol2010 (see Table \[table2\]), we identified 46 spectra fully obtained “in-transit” (i.e. fully between the first and fourth contacts). The 53 remaining spectra constitute our “out-of-transit” sample.
[lccccccccc]{}
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& Date & $\#$ Spectra$^1$ & Exp. Time \[s\] & Airmass & Seeing & SNR$^2$ & SNR$^3$ & Program & Analysis\
$\mathrm{Night\ 1}$ & 2006-09-07 & 20 (9/11) & 900 to 600 & 1.6-2.1 & 0.7-1.0 & $\sim170$ & $\sim32$ & 072.C-0488(E) & Yes
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\
$\mathrm{Night\ 2}$ & 2007-07-19 & 39 (18/21) & 300 & 2.4-1.6 & 0.6-0.8 & $\sim110$ & $\sim20$ & 079.C-0828(A) & Yes\
$\mathrm{Night\ 3}$ & 2007-08-28 & 40 (19/21) & 300 & 2.2-1.6 & 0.7-2.0 & $\sim100$ & $\sim18$ & 079.C-0127(A) & Yes\
$\mathrm{Night\ 4}$ & 2006-07-29 & 12 (5/7) & 600 & 1.8-1.6 & 0.9-1.2 & $\sim140$ & $\sim27$ & 072.C-0488(E) & No\
\[table1\]
Data reduction
--------------
The HARPS observations were reduced with the version 3.5 of the HARPS Data Reduction Software. Spectra are extracted order by order and flat-fielded with the daily calibration set. Each spectral order is blaze-corrected and wavelength-calibrated. All spectral orders from a given two-dimensional echelle spectrum are merged and resampled with a 0.01 $\AA$ wavelength step into a one-dimensional spectrum. Flux is conserved during this step. A reduced HARPS spectrum covers the region between 380 nm and 690 nm with a spectral resolution of $\mathcal{R}\sim115\,000$ or 2.7 [km s$^{-1}$]{}. All spectra are referred to the Solar System barycenter rest frame and wavelengths are given in the air.
As we will see in Sect. \[subSec\_TransSpec\], the next step is to co-add the in-transit spectra on the one hand, and the out-of-transit spectra on the other hand to build the master-in and master-out spectra. The ratio of the master-in and the master-out yields the transmission spectrum. Then, any changes in the stellar line shape or position during the observations can create spurious signals in the transmission spectra, which can become false-positive of atmospheric signals (i.e., with similar amplitude). This is the case for changes in the line spread function (LSF) of the instrument. This is where the spectrograph design is important: HARPS is a fiber-fed spectrograph stabilized in temperature and pressure, ensuring that changes in the point spread function (PSF) between two consecutive observations lead to negligible changes in the LSF at time scales of one night. Nonetheless, change in the line positions are due to the radial-velocity variation of the star during the transit of the planet ($\sim 50$ m s$^{-1}$). This must be removed before computing the transmission spectra. Actually, omitting this step for spectral shifts of 0.01 Å or larger creates artificial signals. We simply correct from this effect by shifting the spectra to the null stellar radial velocity (i.e. the mid-transit stellar radial velocity) according to a Keplerian model of the targeted planetary system (see the model parameters in Table \[table2\]). Another source of change in the stellar line shape is the Rossiter-McLaughlin effect. As the transit occurs, the planet occults different parts of the stellar disk which have different intrinsic line shapes and shifts. False-positive features will then appear in the transmission spectrum even if the stellar spectra are well aligned. A way to overcome this difficulty is to register spectra uniformly during the transit, so that this effect will be averaged out during a full transit.
[lcclc]{}
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Parameters & Symbol & Values & Units &References
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\
Transit epoch (BJD) & $T_0$ & $2454279.436714\pm0.000015$ & days & @Agol2010\
Orbital Period & $P$ & $2.21857567\pm0.00000015$ & days & @Agol2010\
Planet/star area ratio & $(R_p/R_s)^2$ & $0.02391\pm0.00007$& & @Torres2008\
Transit duration & $t_T$ & $0.07527\pm0.00037$ & days & @Triaud2009\
Impact parameter & $b$ & $0.6631\pm0.0023$ & $R_{\star}$ & @Agol2010\
Orbital semi-major axis & $a$ & $0.0312\pm0.00037$ & AU & @Triaud2009\
Orbital inclination & $i$ & $85.710\pm0.024$ & degrees & @Agol2010\
Orbital eccentricity & $e$ & 0 & & fixed\
Longitude of periastron & $\omega$ & 90 & degrees & fixed\
Stellar velocity semi-amplitude & $K_1$ & $200.56\pm0.88$ & $km\ s^{-1}$ & @Boisse2009\
Systemic velocity & $\gamma$ & $-2.2765\pm0.0017$ & $km\ s^{-1}$ & @Boisse2009\
Stellar mass & $M_{\star}$ & $0.823\pm0.029$ & $M_{\odot}$ & @Triaud2009\
Stellar Radius & $R_{\star}$ & $0.756\pm0.018$& $R_{\odot}$ & @Torres2008\
Planet Mass & $M_P$ & $1.138\pm0.025$ & $M_J$ & @Triaud2009\
Planet Radius & $R_P$ & $1.138\pm0.027$ & $R_J$ & @Torres2008\
\[table2\]
Telluric correction {#TellCorr}
-------------------
High-resolution spectra recorded from the ground bear the imprint of Earth’s atmosphere. In the visible domain covered with HARPS, water vapor and molecular oxygen are the main contributors to this time-variable telluric contamination. Indeed, variation in the transmission of Earth’s atmosphere during a night depends on the airmass and on water column variations in the air. In addition to the contaminating water lines around the doublet, we expect the observed spectra to also possess a signature of the telluric sodium [@Vidal2010]. Even if the telluric sodium undergoes seasonal variations and possibly does not follow the water absorption levels, we make the assumption that, within a night, this telluric sodium absorption behave and can be corrected as other telluric features [@Snellen2008; @Zhou2012]. Furthermore, visual inspection of our telluric spectra did not reveal any obvious telluric sodium features.
Telluric contamination produce ubiquitous features in transmission spectra. However, the telluric features can be removed by subtracting high-quality telluric spectra obtained on the same nights. As our observed spectra are referred to the Solar System barycentric rest frame, the telluric lines are shifted during the three different nights by -0.26 Å, 0.09 Å and -0.19 Å compared to their rest frame. Methods described in @Vidal2010 and @Astudillo-Defru2013 permit us to build high-quality telluric spectra night by night using the HARPS observations themselves. Both methods consider that the variation of telluric lines follows linearly the airmass variation. This is a consequence of the usual hypothesis of a radiative transfer in a plane parallel atmosphere. It implies that apart from different constructions of telluric spectrum these methods yields equivalent telluric spectra. Similarly, we build for each night a reference telluric spectrum $T(\lambda)$ such as $$T(\lambda) = T^1(\lambda) \equiv e^{Nk_{\lambda}}$$ corresponding to a zenithal atmospheric transmission (airmass $\sec z = 1$). The zenithal optical depth $Nk_{\lambda}$ is derived at each wavelength $\lambda$ by linear regression on the logarithm of the normalized flux $\log F(\lambda)$ as a function of $\sec z$ [@Astudillo-Defru2013]. The computed telluric spectra (see Fig. \[TellCorrFig\]) are consistent with the atmospheric transmission obtained by @Hinkle2003. Note that the telluric spectra are built with the out-of-transit sample only. This is important in order not to over-correct for the possible exoplanet atmosphere we are looking for. However, for night 2, we had to take into account the in-transit sample because the base line was insufficient to compute a high-quality telluric spectrum (during this night, there was no observation before transit).
Taking advantage of the scaling relation between the telluric line strength and the airmass, we can correct spectra for telluric contamination. Corrections are made by rescaling all spectra as if they had been observed at the same airmass. To obtain this spectrum $F_{ref}$ at airmass $\sec z_{ref}$, we simply divided the observed spectra $O_{\sec z}(\lambda)$ taken at airmass $\sec z$ by the telluric reference spectrum to the power $\sec z-\sec z_{ref}$: $$F_{ref}(\lambda) = O_{\sec z}(\lambda)/T(\lambda)^{\sec z-\sec z_{ref}}\ .$$ We fix the reference airmass to the average airmass of the in-transit spectra. This ensures both a telluric correction and minimal flux changes in the reduced spectra. If the quality and the stability of the observation nights are good enough which typically depends on a constant water atmospheric content, this method is sufficient to correct down to the statistical noise level. However, the quality of the nights are generally not sufficient as second-order variations of telluric lines are present and telluric residual can be still visible in transmission spectra (see Fig. \[TellCorrFig\] and Sect. \[subSec\_TransSpec\]). Indeed these observation were not planned to observe exoplanetary atmospheres but Rossiter-McLaughlin effects. This is less demanding in terms of sky quality and requires shorter time baseline. Later in our analysis, we will perform a second telluric correction to remove possible telluric residuals.
Transmission spectrum {#subSec_TransSpec}
---------------------
Transit spectroscopy is a differential technique requiring the acquisition of spectra during and outside transit events. Ground-based observations out-of-transit ($O_{out}$) contain the star light absorbed by the terrestrial atmosphere. Observations during transit ($O_{in}$) additionally contain the exoplanet atmosphere transmission diluted into the stellar flux. Stacking telluric-corrected spectra in-transit ($F_{in}$) and out-of-transit ($F_{out}$) allows to obtain master-in ($\mathcal{F}_{in}$) and master-out spectra ($\mathcal{F}_{out}$), respectively. Classical methods obtain transmission spectra ($\mathfrak{R}^{'}=\mathfrak{R}-1=\mathcal{F}_{in}/\mathcal{F}_{out}-1$) of the exoplanet atmosphere by dividing night by night the master spectrum in-transit by the master spectrum out-of-transit [@Redfield2008]. We can then analyze exoplanet atmospheres features along wavelengths of interest. Such methods do not consider changes in radial velocity of the *planet*.
Here, we present a modified approach with respect to @Redfield2008 to correct for this effect. As the planet transits its star, its radial velocity changes typically from $-15$ [km s$^{-1}$]{} to $+15$ [km s$^{-1}$]{}. Therefore, for the sodium the planetary atmospheric absorption lines move from the blue-shifted to the red-shifted part of the stellar lines. In the optical, near the sodium doublet this shift is $\lesssim0.5\ \AA$ during the full transit. We correct this effect in our analysis by shifting the planetary signal to the null radial velocity in the *planet* rest frame (i.e. the radial velocity of the planet in the middle of the transit). Instead of taking the ratio of master spectra, we divide each spectrum in-transit by the master-out, then shift the residual with the computed planetary radial velocity and sum up all the individual transmission spectra. We normalized this sum to unity by a linear fit to the continuum outside the regions of interests. This allows us to compute the transmission spectra as $$\mathfrak{R}^{'}(\lambda) = \frac{\mathcal{F}_{in}}{\mathcal{F}_{out}} -1 = \sum\limits_{in} \frac{F_{in}(\lambda)}{\sum\limits_{out} F_{out}(\lambda)}\Big|_{Planet\,RV\,shift} - 1\ .$$ With this correction, we expect to see the exoplanetary signal at wavelengths corresponding to the systemic velocity rest frame (and not in the solar system rest frame).
A transmission spectrum (with or without radial-velocity correction) corrected from the airmass effect (Sect. \[TellCorr\]) can still present some telluric line residuals. This is due to water column variations in the air above the telescope. Then, we applied a second telluric correction to correct from telluric features in the transmission spectra. Because both the transmission spectra and the telluric spectra contain telluric features at corresponding wavelengths, these two spectra are correlated. To eliminate these correlations and the residuals of telluric lines in the transmission spectrum, a linear fit is made between all corresponding fluxes in wavelength of the transmission spectrum and the telluric reference spectrum. The transmission spectrum is then divided by the fit solution. As only telluric residuals are correlated with the telluric spectrum, doing this iteratively eliminates telluric residuals in the transmission spectra without affecting the other part of the spectra [@Snellen2008; @Snellen2010; @Khalafinejad2013]. Practically, one iteration is sufficient to tame atmospheric pollution down to the photon-noise level.
![Example of the effect of the Earth atmosphere on exoplanetary transmission spectra. This shows the importance of a precise telluric correction. Upper panel: Calculated telluric spectrum $T(\lambda)$ (top), transmission spectrum without any correction (middle), and transmission spectrum with the telluric correction (bottom) for night 2. The telluric lines are dominating all features in the uncorrected transmission spectrum. Lower panel: Transmission spectra with telluric corrections already performed for night 3. Once we apply the telluric correction described in Sect. \[TellCorr\] (airmass effect) we get a transmission spectra with weak telluric lines residuals (top). Since residuals of telluric lines are still visible, we apply a second telluric correction (correction of features due to water column variation, see Sect. \[subSec\_TransSpec\]). This affords us to get a transmission spectrum corrected from the effect of Earth atmosphere (bottom). Note the different vertical scales. Some evident telluric lines are emphasized with vertical lines. The telluric lines are shifted from one night to the other because the spectra are referred to the Solar System barycenter rest frame and were not observed at the same barycentric Earth radial velocity.[]{data-label="TellCorrFig"}](HD189733_Na1_telluric_correction){width="47.00000%"}
Binned atmospheric absorption depth {#subSec_BinTS}
-----------------------------------
In order to compare our results with previous detections of sodium in HD189733b [@Redfield2008; @Jensen2011; @Huitson2012] or in other exoplanets [@Charbonneau2002; @Snellen2008; @Sing2008a; @Langland2009; @Wood2011; @Sing2012; @Zhou2012; @Murgas2014; @Nikolov2014; @Burton2015], we calculate the relative absorption depth across various bins in wavelength. Taking into account the systemic velocity, we average the flux around the central passbands ($C$) centered on both lines of the sodium doublet. Here and later in our analysis, all the averages are weighted by the estimated errors on the reduced spectra, which are taken to be random photon noise obeying Poisson statistics. We compare the integrated flux in the transmission spectrum $\mathfrak{R}^{'}(C)$ to bins of similar band widths taken in the transmission spectrum continuum. @Snellen2008 chose adjacent control passbands on the blue ($B$) and red ($R$) side of the central passband. As at high-resolution exoplanet atmospheric lines are likely to be resolved, we prefer to choose absolute reference passbands outside the sodium doublet, but still on the two side (B and R) of the transition [see *e.g.* @Charbonneau2002]. Relative depths due to exo-atmospheric absorption are then obtained by the difference of fluxes between the central and the reference passband, $$\delta(\Delta\lambda) = \overline{\mathfrak{R}^{'}(C)} - \frac{\overline{\mathfrak{R}^{'}(B)} + \overline{\mathfrak{R}^{'}(R)}}{2}\ .$$ As small passbands encompass only one line, we average the absorption depth of the two D lines. When the central passband includes the two lines, we adjusted it on the center of the doublet.
Transmission light curve {#light_curve_method}
------------------------
This method consists in directly deriving the relative transmission light curve as a function of time [@Charbonneau2002; @Snellen2008]. Thus, the absorption excess due to the exoplanet atmospheric limb can be seen as a relative flux decrease during the transit. It differs from the method presented in Sect. \[subSec\_TransSpec\]. The spectrophotometry is performed on individual spectra for a given spectral bin. We obtain a time-dependent information but the wavelength dependence is lost.
For each telluric-corrected stellar spectrum ($F$), we derive the relative flux at the D lines by comparing fluxes inside passbands in the center, red and blue part of each lines (see Sect. \[subSec\_BinTS\] and \[Sec\_SodiumD\] for the description of the passbands): $$\mathcal{F}_{rel}(t,\Delta\lambda)= \frac{2\times\overline{F(C)}}{\overline{F(B)} + \overline{F(R)}}\ .$$ We then normalized the relative time serie to unity. For the smaller passbands, we average, spectrum by spectrum, the relative fluxes of the two D lines.
Because we always compare together same parts of each spectrum, the relative flux should be constant with time except if absorption by the planetary atmosphere is present during the transit. With this method, we cannot apply any planetary radial-velocity correction. Taking into account the radial-velocity effect would change the central passband, thus it would change the relative flux (not because there is an absorption, but because we do not have a constant central passband). However this is potentially an issue only for the smallest passbands for which we can expect a loss of absorption.
Another type of telluric correction is applied to the relative flux sequence. As the telluric absorption changes with airmass, it is expected to see correlations between relative fluxes and the airmass, especially when the passbands contain telluric lines. The first correction applied on individual stellar spectra (Sect. \[TellCorr\]) should have already mitigated this effect. However, some light curve still show correlations with airmass. To remove any residual airmass effect, we model the flux variation as a linear function of the airmass and remove the linear trend. If possible, we only consider out-of-transit data to perform our fit.
@Astudillo-Defru2013 discussed whether or not fitting a transit model to the data can change the measurement of the absorption depth. They showed that the absorption depth is well measured by simple average differences. Indeed, differential stellar limb-darkening do not affect significantly measurement in such narrow passbands. The transmitted flux or relative depth is then given by the difference of the average of the relative flux in and out-of-transit: $$\delta(\Delta\lambda)= \frac{\quad\overline{\mathcal{F}_{rel}(t_{In}})\quad}{\overline{\mathcal{F}_{rel}(t_{Out})}} - 1\ .$$
Integrating the transit light curve over the in-transit duration makes it possible to compare the result with the binned atmospheric absorption depths described in Sect. \[subSec\_BinTS\]. Both approaches give similar results, because of the normalization processes and the use of photon noise weighted averages in the two respective methods.
Systematic effects {#Stat_methods}
------------------
To measure the uncertainty on the relative absorption depth $\delta(\Delta\lambda)$, we propagated the errors of the reduced spectra through our analysis. We estimated the errors on the reduced spectra as random photon noise obeying Poisson statistics. Systematic effects, on the other hand, are likely to contribute to the total noise budget. We have used different statistical methods, such as empirical Monte-Carlo [@Redfield2008] or bootstrapping, to estimate the impact of correlated noise. The basic principle of these methods is to randomize data, to artificially create new set of observations, and to feed them to our data analysis.
Firstly, we followed the empirical Monte-Carlo technic presented in @Redfield2008. The idea is to see whether or not the measurement is an artifact of the data or if it is really created by the transit and hence due to the planetary companion. We select a sub-sample of spectra of all the spectra available in a night. The choice of this sub-sample is then fixed (we call it a scenario). Next, this new sample is randomly divided in two parts to create an “in-transit“ and an “out-of-transit” simulated data sets. These two data sets always contain the same number of spectra.
As in @Redfield2008, we explored three scenarios per night. In the first scenario (“out-out”), the spectra are only selected among the nominal out-of-transit spectra. The amount of spectra in the simulated in-transit data compared to the number of spectra in the simulated out-of-transit data are in the same proportion as in the nominal observations. As we choose randomly only among the out-of-transit spectra, we expect to find the distribution of computed relative absorption depth centered at zero. Similarly, an “in-in” scenario is created, but the random in-transit and out-of-transit data are chosen only in the spectrum observed during the transit. As the planetary signal should be present inside every in-transit observation, the final “in-in” distribution is also expected to be centered at zero. Finally an “in-out” scenario is executed. All the out-of-transit observation form a fixed master-out (actually the same than the nominal master-out). We choose randomly a sub-sample of in-transit spectra. Here, we did not fix the number of spectra of the in-transit sub-sample. This later one is composed by half to the totality of the number of nominal in-transit spectra.
How to interpret the computed distribution? @Redfield2008 considered the standard deviation of the “out-out” distribution as the global error on the transmitted signal. Nonetheless, @Astudillo-Defru2013 considered instead the standard deviation of the “in-out” distribution as the error on the transmitted signal. Here, we follow @Redfield2008 and take the “out-out” scenario to infer the error on our measured absorption depth, because it is independent of any planetary signal. Nevertheless, the errors are overestimated with the standard deviation of the “out-out” scenario, because in each iteration only a fraction of the data is used. Thus, to get correct errors on our absorption depth, we divide the standard deviation of the “out-out” scenario by the square root of the ratio of the total number of spectra to the number of spectra out-of-transit.
Secondly, we calculate false alarm probabilities. We randomize spectra and recreate transmission spectra and light curves (i.e. by randomizing the time sequence), and quantify how many randomized data sets yield a signal as significant as the real measurement. This is done night by night and by considering all the original spectra.
{width="97.00000%"}
Results and analysis {#Sec_Results}
====================
D detection {#Sec_SodiumD}
------------
As we focus on the two transitions of the sodium doublet (D2 at $\lambda$5889.951 $\&$ D1 at $\lambda$5895.924 $\AA$), we restrict the wavelength range for our analysis to the domain from 5870 to 5916 $\AA$. Taking into account the systemic velocity of -2.2765 [km s$^{-1}$]{}, these lines are, in the solar system barycentric rest frame, at $\lambda$5889.906 and $\lambda$5895.879 $\AA$, respectively.
To compute absorption depths, we define different central passbands (C) composed by two sub-bands centered on each sodium transition lines. Total band width of 6=2$\times3$ $\AA$, 3=2$\times1.5$ $\AA$, 1.5=2$\times0.75$ $\AA$, 0.75=2$\times0.375$ $\AA$ and 0.375=2$\times0.188$ $\AA$ were chosen (the 6=2$\times3$ $\AA$ is actually 2$\times2.98~\AA$). We add to these bands a larger passband of 12 $\AA$ to perform the comparison of our results to those of @Redfield2008 [@Jensen2011; @Huitson2012]. As the 12 $\AA$ band encompasses the two sodium lines, we adjusted it on the center of the doublet. For every bands, we choose a unique reference passband corresponding to the blue (B) and red (R) adjacent passbands of the central passband of 12 $\AA$ (i.e. $B=5874.89-5886.89\ \AA$, $R=5898.89-5910.89\ \AA$).
Transmission spectrum analysis
------------------------------
{width="47.00000%"} {width="47.00000%"}
The total signal-to-noise ratio (SNR) per extracted pixel of the “in” and “out” master spectra ($\mathcal{F}_{in}$ and $\mathcal{F}_{out}$) ranges from 400 to 550 for all nights, in the continuum around 589 nm. A pixel on the HARPS detector represents $\sim$ 0.8 [km s$^{-1}$]{} (or 0.016 $\AA$ at 589 nm). The total SNR for each master is about 850 when co-adding the nights. Thus, a final SNR on the transmission spectra ($\mathfrak{R}^{'}$) of about 1200 per extracted pixel is reached. For the transmission spectra, the SNR in the core of the sodium lines decreases to $\sim$ 260. It is therefore important to co-add the fluxes over tens to hundreds pixels to reach a sufficient SNR for atmospheric detections. We compute different transmission spectra, notably with and without planetary radial-velocity correction. The standard deviation of the transmission spectra was about 2200 ppm for each night in the continuum. When adding the nights together the standard deviation goes down to 1450 ppm. As we can see in Fig. \[TSpectrum\], the two exoplanetary sodium lines peak out from the continuum. Without planetary radial-velocity correction, the atmospheric absorption is best detected in our 1.5=2$\times0.75~\AA$ passband at a level of $0.309\pm0.034\%$ (9 $\sigma$). Our detection in the 12 $\AA$ passband is measured at a level of $0.060\pm0.008\%$ (7.5 $\sigma$). Here and in the following, we define absorption as a negative relative depth. With planetary radial-velocity correction the signal in these same 1.5=2$\times0.75$ and 12 $\AA$ passbands are $0.320\pm0.031\%$ (10.3 $\sigma$) and $0.056\pm0.007\%$ (8 $\sigma$) respectively (see Table \[table3\]). As the radial-velocity correction consists of a wavelength shift smaller than 0.25 $\AA$, it is normal that the integrated signal on these passbands are consistent with each other. Furthermore, our measurements are in agreement with the detection by @Huitson2012 from space ($0.051\pm0.006\%$ for the 12 $\AA$ passbands) and by @Redfield2008 and @Jensen2011 from ground ($0.067\pm0.020$ and $0.053\pm0.017$ for the 12 $\AA$ passband, respectively).
These absorption signals can be interpreted as equivalent relative altitudes. Altitudes can be inferred by considering the transmission at the limb. Indeed, the atmospheric absorption (which depends on wavelength) is due to an optically thick layer (presence of an absorber) at a certain height above the measured planetary radius (in broad band light curve). The height of this layer is generally a few times bigger than the atmospheric scale height of the planet ($H=kT/\mu g$), because of the variation with wavelength of the cross section of the absorber [@Lecavelier2008]. The absorption depths measured in the 1.5=2$\times0.75~\AA$ passband correspond to equivalent altitude of $5100\pm500$ km assuming unresolved features. This is about 27 atmospheric scale height of HD189733b ($H=190$ km for 1140 K, the equilibrium temperature of HD189733b, assuming a mean molecular weight of $\mu=2.3$ and an albedo of 0.2).
In order to understand the impact of our radial-velocity correction it is then worth to look at the smallest passband. At first glance, integrations over 0.75=2$\times0.375$ $\AA$ and 0.375=2$\times0.188$ $\AA$ passbands give us smaller signals than the signals we acquire without radial-velocity correction. This is not expected, since the radial-velocity correction should add up all the absorptions signals and thus strengthen the signal. A closer look to the transmission spectrum with a Gaussian fitting to each line shows that the exoplanetary sodium lines are blueshifted by about $0.16\ \AA$ in the planet rest frame. Integrating the lines including this shift (with the smallest passbands blueshifted) enables the full signal to be recovered.
What can produce the blueshift of the exoplanetary sodium lines we measure and is it significant? A global error on the planetary radial-velocity shift is likely to be lower than 3 [km s$^{-1}$]{} (0.06 $\AA$) considering the HARPS precision on the stellar radial velocity propagated to the planet. Our $0.16\ \AA$ value is by far smaller than the $\sim0.75\ \AA$ measured by @Redfield2008. Since this specific blueshift would have been easily measured with our data, we do not confirm the value reported by these authors. Nevertheless, our detected $0.16\pm0.04\ \AA$ shift is significant and most likely real, since it is seen even on the transmission spectra without radial velocity correction. A possible physical explanation is that a net blueshift is imprinted by winds within the exoplanet atmosphere [see @Snellen2010 for a first detection of winds in exoplanets]. Theoretical models [*e.g.* @Kempton2014; @Showman2013] indeed predict a net blueshift due to both planetary rotation and winds in the terminator of the atmosphere (the part we are probing with transmission spectroscopy). The blueshift we measured corresponds to a wind speed of $8\pm2$ km/s. Models typically predict a 3 km/s blueshift. However, these winds are estimated for pressures down to $\sim10^{-5}-10^{-6}$ bar. As we will see later, our sodium detection allows us to probe pressures of $\sim10^{-7}-10^{-9}$ bar ($0.1-0.001\mu$bar) at altitude of 10000 km. We can therefore expect stronger winds than 3 km/s in the higher atmosphere. This calls for additional theoretical works on winds in the upper atmospheres of exoplanets, and new data to confirm the blueshift value.
{width="47.00000%"} {width="47.00000%"}
[lcccccc]{}
------------------------------------------------------------------------
$\Delta\lambda~[\AA]$ & 0.375=2$\times0.188$&0.75=2$\times0.375$&1.5=2$\times0.75$&3=2$\times1.5$&6=2$\times3$&12\
$\#$ pixel &$\sim2\times12$&$\sim2\times25$&$\sim2\times50$&$\sim2\times98$&$\sim2\times194$&$\sim780$\
$\mathrm{Night\ 1}$ &0.764$\pm$0.095&0.532$\pm$0.064&0.378$\pm$0.046&0.202$\pm$0.025&0.115$\pm$0.014&0.077$\pm$0.010
------------------------------------------------------------------------
\
$\mathrm{Night\ 2}$ &0.495$\pm$0.112&0.431$\pm$0.077&0.198$\pm$0.054&0.028$\pm$0.029&0.015$\pm$0.017&0.018$\pm$0.011\
$\mathrm{Night\ 3}$ &0.560$\pm$0.104&0.500$\pm$0.071&0.415$\pm$0.050&0.221$\pm$0.027&0.113$\pm$0.015&0.080$\pm$0.010\
$\mathrm{All\ Nights}$ &0.571$\pm$0.065&0.472$\pm$0.044&0.320$\pm$0.031&0.141$\pm$0.017&0.075$\pm$0.010&0.056$\pm$0.007\
\[table3\]
[lcccccc]{}
------------------------------------------------------------------------
$\Delta\lambda~[\AA]$ & 0.375=2$\times0.188$&0.75=2$\times0.375$&1.5=2$\times0.75$&3=2$\times1.5$&6=2$\times3$&12\
$\#$ pixel &$\sim2\times12$&$\sim2\times25$&$\sim2\times50$&$\sim2\times98$&$\sim2\times194$&$\sim780$\
$\mathrm{Night\ 1}$ &0.606$\pm$0.179&0.468$\pm$0.095&0.339$\pm$0.053&0.207$\pm$0.031&0.127$\pm$0.018&0.064$\pm$0.013
------------------------------------------------------------------------
\
$\mathrm{Night\ 2}$ &0.244$\pm$0.199&0.165$\pm$0.106&0.143$\pm$0.059&0.040$\pm$0.034&0.027$\pm$0.021&0.026$\pm$0.014\
$\mathrm{Night\ 3}$ &0.521$\pm$0.224&0.661$\pm$0.119&0.457$\pm$0.066&0.274$\pm$0.039&0.133$\pm$0.023&0.091$\pm$0.016\
$\mathrm{All\ Nights}$ &0.478$\pm$0.113&0.451$\pm$0.060&0.325$\pm$0.033&0.184$\pm$0.019&0.102$\pm$0.012&0.062$\pm$0.008\
\[table4\]
Transmission light curve analysis
---------------------------------
We compute the transmission light curve of HD189733b for every observation nights following Sect. \[light\_curve\_method\] and for every passbands described in Sect. \[Sec\_SodiumD\]. For our analysis, the transit ephemeris are fixed. They are given by the parameters described in Table \[table2\]. For the 12 $\AA$ passband, we estimate errors on each individual spectrum of 250 ppm to 800 ppm. The propagation of these errors to the time series provides us uncertainties of about 100–150 ppm on the values of the baseline and on the absorption depth. This allows us to determine absorption depths due to sodium in the exoplanet atmosphere during each night at a level between 1.9 and 5.7 $\sigma$ (see Table \[table4\]). Note that when we subtract the best-fit absorption depth, the standard deviation of the residuals is about 550 ppm for each night. When co-adding data of the three nights together, we obtain an absorption signal of $0.062\pm0.008\%$ (7.8 $\sigma$) for the 12 $\AA$ passband (Fig. \[TSlightcurve\]). Our best measurement is achieved for the passband of $1.5=2\times0.75\ \AA$ and delivers a value of $0.325\pm0.033\%$ (9.8 $\sigma$). This is in perfect agreement with our transmission spectrum method and shows both the complementarity and the compatibility of the two methods. Compatibility because the measured absorption level are within a 1 $\sigma$ error bars [see @Astudillo-Defru2013] and complementarity because the second method carries line information and demonstrates that the additional absorption is indeed observed during the transit (see also Sect. \[Stat1\]). As expected, a small decrease on the measured absorption signal is observed for the smallest passband (0.375=2$\times0.188$ $\AA$), which width is commensurable with the wavelength shift imprinted by the radial velocity of the planet. This effect thus results in a flux loss, hence to a weaker signal.
Systematic effects {#Stat1}
------------------
Possible systematic errors can be present in the data. To estimate such effects, we perform an empirical Monte-Carlo (EMC) analysis for each of our observed transits following Sect. \[Stat\_methods\]. This analysis makes us confident that our measured signals is indeed due to the exoplanetary transit and not spurious (*e.g.* due to other astrophysical effects as stellar rotation, observational conditions, or instrumental effects). The EMC is performed for both the transmission spectrum and the light curve methods using 3000 iterations for each scenario described in Sect. \[Stat\_methods\] (“in-in”,“out-out”,“in-out”) and for each night. The standard deviation of the distributions does not increase significantly when increasing the number of iterations.
The results are shown in Fig. \[StatFig\] and the errors are summarized with all the different methods in Fig. \[TScompar\]. We can see that all the “in-in” and “out-out” distribution are centered on zero showing that the measured absorption signal comes indeed from the transit and has no other sources of explanation. The “in-out” distributions are well centered at the value determined by the transmission spectrum and light curve methods. The errors given by the EMC are larger than the nominal errors (yielding from the propagation of the photon noise), but only by a factor of $\lesssim 2$. This shows that while systematic errors are present in our data they do not undermine our detection. The empirical Monte-Carlo simulation described above strengthens the transit origin of the detected signal.
We also computed false alarm probabilities for each night for both the transmission spectrum and the light curve methods. We performed 10000 iterations and found false-alarm probabilities to our sodium detection of $\sim 0.5\%$, $\sim 9\%$ and $\lesssim0.01\%$ for night 1, 2, and 3 respectively (see Fig. \[StatFig2\]). Adding all the nights yields a false-alarm probabilities $\lesssim0.01\%$.
Discussion {#Sec_Discuss}
==========
Summary of the D detection
--------------------------
A summary of our results is shown in Fig. \[TScompar\]. All the different methods to measure the transit depth $\delta(\Delta\lambda)$ in a given wavelength bin $\Delta\lambda$ are shown for each night individually and also for the combined data set, for the $1.5=2\times0.75$ $\AA$ and 12 $\AA$ passbands.
A number of aspects must be pointed out: First, the correction of the telluric lines is mandatory. In fact, water lines around the sodium D doublet imprint spurious systematics in any possible direction. It depends if the transit is observed at low or high airmass compared to the out-of-transit data and at which value of the barycentric Earth radial velocity (BERV). Indeed, due the BERV, the telluric lines will not always fall at the same place in the stellar spectra. For a fixed choice of passbands, a given telluric line may not be always included. Then, the ratio with the reference passbands, which contain other telluric lines, can be affected in any direction. For example, during night 3, the signal measured without telluric correction is, by chance, the same signal measured after telluric correction, as the positions of telluric lines in the transmission spectrum do not impact the sodium signal extraction. It should be noted that the correction for night 2 is less robust than the one of the two other nights, because of the lack of a baseline before the transit. This implies that we probably over-corrected the data taken during this night, hence the measured value is under estimated. Secondly, the absorption depths measured with transmission-spectra and light-curve methods are consistent with each other within the same night. Thirdly, systematic effects are different from night to night. Nights 2 and 3 were probably more “stable” than night 1. Systematics in night 1 are probably also bigger due to the low-cadence of the observations. Even if similar signal-to-noise ratios are reached in the three different nights, it seems that it is better to have a high-cadence observational strategy (provided all the spectra have a SNR above a certain threshold). We also note that the measured values are different for different nights. At this level of precision, and assuming we are efficiently correcting for systematics, we cannot exclude intrinsic variability related to the stellar or planetary properties. Our results favor a variation from night to night within 1–2 $\sigma$. We did not take into account the potential effect of a differential limb darkening or of the Rossiter-McLaughlin effect. However, the impact of differential limb-darkening should be an order of magnitude lower than the achieved precision [@Charbonneau2002; @Redfield2008]. On the other hand, the Rossiter-McLaughlin effect can mimic an exoplanetary atmospheric absorption and imprint features at similar level in a transmission spectrum. Nonetheless it should average out during a whole transit if the exposures are equally distributed over the transit [@Dravins2014]. This is the case for our three analyzed transit events. Finally, we would like to point out that the precision obtained from the coaddition of all the nights is comparable with HST/STIS precision (with the same number of observed transits) or with 10-m sizes telescopes. This highlights the potential of ground-based high resolution observations of exoplanetary atmospheres from 4-m telescopes.
[lcccc]{}
------------------------------------------------------------------------
$\mathrm{Model\ Atmospheres}$ & Wavelength Ranges \[$\AA$\] & $\#$ D’s FWHM & Corresponding Altitude \[km\] & Fitted Temperature \[K\]\
$\mathrm{Model\ 0}$ & 5870.00–5882.22 & –& 0 & 1140\
& 5903.24–5916.00 & & &\
$\mathrm{Model\ 1}$ & 5882.22–5889.22 & 2–14 & 1500$\pm$1500 & 1630$\pm$70\
$\mathrm{(}$“$\mathrm{Wing'shoulders}$”$\mathrm{)}$ & 5890.26–5895.20 & & &\
& 5896.24–5903.24 & & &\
$\mathrm{Model\ 2a\ (D1\ core})$ & 5895.20–5895.46 & 1–2 & 2700$\pm$800 & 1700$\pm$320\
& 5895.98–5896.24 & & &\
$\mathrm{Model\ 2b\ (D2\ core})$ & 5889.22–5889.48 & 1–2 & 3800$\pm$900 & 2170$\pm$320\
& 5890.00–5900.26 & & &\
$\mathrm{Model\ 3a\ (D1\ core})$ & 5895.46–5895.67 & 0.2–1 & 5100$\pm$3100 & 2220$\pm$340\
& 5895.77–5895.98 & & &\
$\mathrm{Model\ 3b\ (D2\ core})$ & 5889.48–5889.69 & 0.2–1 & 7900$\pm$5500 & 3220$\pm$270\
& 5889.79–5890.00 & & &\
$\mathrm{Model\ 4a\ (D1\ core})$ & 5895.67–5895.77 & $\leqslant0.2$ & 9800$\pm$2800 & 2600$\pm$600\
$\mathrm{Model\ 4b\ (D2\ core})$ & 5889.69–5889.79 & $\leqslant0.2$ & 12700$\pm$2600 & 3270$\pm$330\
\[table5\]
As mentioned before, we obtain the same precision on our detection ($0.056\pm0.007\%$) as on the HST/STIS measurements of @Huitson2012 ($0.051\pm0.006\%$) on a relatively wide passband (here the “12 $\AA$”). We can take advantage of the HARPS high spectral resolution ($\mathcal{R}\sim115\,000$), compared to HST/STIS ($\mathcal{R}\sim5\,500$) to characterize the narrow line cores of the doublet and, as a consequence, exploring higher regions in the atmosphere of HD189733b.
The presence of haze in the HD 189733b atmosphere has been previously deduced from low-resolution spectra covering much broader spectral regions than the one considered here [*e.g.* in @Lecavelier2008; @Pont2013]. Rayleigh scattering by hazes dominates the transmission spectrum in the optical, only allowing the narrow core of the sodium lines to be observed at high spectral resolution. This is indeed what we observe in our data.
As described in Sect. \[Sec\_SodiumD\], we compute different absorption depths $\delta(\Delta\lambda)$ for different passbands (summary in Table \[table3\] and \[table4\]). The absorption depth increases for narrow cores passbands. This shows that most of the signal we measure originates in the narrow line cores ($\leq1\AA$), even if a small portion of the absorption comes from a broader component. Actually, a Gaussian fitted to each lines yields a full width at half maximum (FWHM) of $0.52\pm0.08~\AA$ (see Fig. \[TSpectrum\]). A comparison to the resolution element of 0.05 $\AA$ (2.7 [km s$^{-1}$]{}) shows that the lines are resolved by a factor of $\sim10$ (see Fig.\[ResLine\]). The same fit allows us to measure absorption depths in the core of the two D lines of $0.64\pm0.07\%$ and $0.40\pm0.07\%$ for the D2 and D1 lines, respectively.
The D sodium doublet as a probe to the upper atmosphere
-------------------------------------------------------
We compare our measurements with transmission spectroscopy models. We use the $\eta$ model described in @Ehrenreich2006 [@Ehrenreich2012A; @Ehrenreich2014]. The model computes the opacity along the line of sight grazing the atmospheric limb of planet and integrate over the whole limb. The planetary atmosphere is modeled as a perfect gas in hydrostatic equilibrium composed by 93$\%$ of molecular hydrogen and, 7$\%$ of helium, and solar abundance of atomic sodium (volume mixing ratio of $10^{-6}$). Several model atmospheres are simulated with isothermal temperature profiles for temperature ranging from 1000 K to 3600 K by step of 100 K. To increase the resolution in temperature, we interpolate linearly between the models. The sodium line profiles are modeled with Voigt functions resulting from the convolution of a Doppler thermal profile (dominating the line cores) and a Lorentzian profile accounting for the natural and collisional broadenings in the line wings. Collisional or pressure broadening is completely dominating the line shapes far (over scales of $\sim100$ nm) from the cores [@Iro2005]. The half-width at half-maximum (HWHM) of the pressure broadened Lorentzian follows the prescriptions of @Burrows2000 and @Iro2005, which are valid over the wavelength range we are studying here. The modeled transmission spectra, calculated at a resolution of $0.01\AA$, are convolved with the average HARPS Line Spread Function (LSF), which is well represented by a Gaussian with width of $0.05\AA$. In contrast with HST/STIS, the comparison between the models and our data is eased by the fact that HARPS resolves individual lines arising in the planet atmosphere by a factor $\sim10$ (see above).
Before comparing our transmission spectrum to models, we first fitted to our data a simple haze model. Knowing that the Rayleigh slope in our wavelength range analysis is negligible, hazes can be well represented by an absorption cut-off at a given level. This level must be compatible with our measured continuum, which corresponds to the wavelength ranges of model 0 in Table \[table5\]. We want also to determine if the sodium wings absorption dominates the haze absorption in the transmission spectrum. The “wings’ shoulders” region is described by the wavelength ranges of model 1 in Table \[table5\]. Therefore, the maximum altitudes of the cloud deck which is compatible with the continuum level is given by a constant fit to the wavelength regions used to fit model atmospheres 0 and 1 to the data. This constant level gives a Bayesian information criterion (BIC) of 1597 in the wings region (region 1).
To compare our observed transmission spectrum to models, our approach is to fit the different regions of the measured transmission spectrum around the sodium doublet with model atmospheres with different isothermal profiles. We divided the transmission spectrum in separated wavelength ranges around each line of the sodium doublet, each corresponding to altitude slices. The altitude slices are given by the maximum between the altitude errors on the data and the difference in altitudes in the given ranges. These ranges are listed in Table \[table5\]. We fit these different parts of the spectrum with different isothermal models, probing separately the line cores (models 4), the region encompassed within $1\times$FWHM (excluding the line cores; models 3), the region between $1\times$ and $2\times$ the FWHM (models 2), the line wings (excluding the previous regions; model 1), and the continuum (model 0). All the fits are obtained with a $\chi^2$ minimization over the grids of atmospheric models obtained by varying the temperature and a general offset in relative flux. In the continuum wavelength range we cannot see a significant sodium absorption due to absorbing wings. Then, model 0 was chosen with a fixed temperature of 1140 K (the equilibrium temperature of HD189733b). The fit of this model to the continuum allows us to determine the offset in planetary radius between the data and the models. We fix this offset for the subsequent model adjustments. We fit models 1, 2, 3, and 4 to their respective spectral regions, simply considering the model temperature as a free parameter. The fit of model 1 gives us a temperature of 1630 K for a BIC of 1542. The comparison with the BIC of 1597 obtained with a constant haze level is very strong evidence ($\Delta \mathrm{BIC}\geq 20$) in favor of a scenario where the wings absorption dominates the haze absorption. Thus, part of our measured sodium absorption is due to the wings and correspond to lower altitudes and temperatures. The three other models (2, 3 and 4) investigate different parts of the line cores inside 1 $\AA$ (2 FWHM, see Table \[table5\]). The best fit models are shown in Fig. \[Fit\]. For each range of altitudes, we therefore derive one temperature. The resulting temperature profile as a function of altitude is shown in Fig. \[Tprofile\]. The temperature linearly increases with altitude with a gradient of $\sim0.2$ K km$^{-1}$ ($0.2\pm0.1$ K km$^{-1}$). Note that our measured temperatures are underestimated due to the use of isothermal models, which give, for a same upper temperature, more extended atmospheres than models with positive temperature gradient.
Our high-resolution measurement of sodium in HD189733b, allows us to probe a new atmospheric region (between $\sim1.1-1.2$ planetary radius, corresponding to pressures of $10^{-7}-10^{-9}$ bar), above the result previously obtained at lower resolution [@Huitson2012]. The present study suggests that a part of the sodium absorption takes place up to the (lower) thermosphere, where heating by the stellar X/EUV photons occurs [@Lammer2003; @Yelle2004; @Vidal2011; @Koskinen2013a]. Meanwhile our interpretation relies on hydrostatic models, which could be questionable at very high altitudes. Further theoretical analysis of these results is developed in a separate paper [@Heng2015 accepted in ApJ].
Conclusions {#Sec_Conclu}
===========
We have presented an analysis of all the available transit observations of HD189733b with HARPS on ESO 3.6 m telescope. We carefully corrected for the change in radial velocity of the planet and for telluric contamination, fully exploiting the high spectral resolution and the stability of HARPS. We detect excess of absorption due to D lines in our transmission spectrum. We measure line contrasts of $0.64\pm0.07\%$ (D2) and $0.40\pm0.07\%$ (D1) and FWHMs of $0.52\pm0.08~\AA$. Sodium is clearly detected over several different passbands and the signatures are robust against different statistical tests. Moreover, these detections are consistent and comparable in terms of precision with those obtained from space-borne or 10 m class ground-based facilities.
High-resolution permits us to measure a blueshift in the line positions of $0.16\pm0.04\ \AA$. We interpret it as wind in the upper layer of the atmosphere of a velocity of $8\pm2$ km/s. Since we resolved the sodium lines, we used it as a probe of the lower pressure part of the atmosphere. We measured a non-isothermal temperature profile inside our range of probed altitudes, where the sodium absorption is peaking out above the haze present in the atmosphere. We found that the temperature increases towards the thermosphere. A temperature gradient of $\sim0.2$ K km$^{-1}$ is measured. The existence of a positive temperature gradient does not depend on assumptions about the haze layer. In fact, the gradient can be measured from the resolved line cores and in particular from the difference in absorption levels between the D1 and the D2 lines. This shows that beyond the detections of atomic species, transit spectroscopy allows characterization of physical conditions present in atmospheres.
These transit spectroscopy data were already used by @Triaud2009 and by @Collier2010 to study the Rossiter-McLaughlin effect. Furthermore, with these same observations, @DiGloria2015 [to be submitted] detected a slope in the planet-star radius ratios that @Pont2008 [@Pont2013] interpreted as Rayleigh scattering. These studies demonstrate the relevance of studying exoplanet atmospheres with high-resolution spectrographs mounted on 4-meter-class telescopes. This is especially important considering that several such facilities will be built in the coming years to prepare for the follow-up of exoplanet candidates from incoming space missions such as K2 (on-going), TESS, CHEOPS, and PLATO.
In the near future, the HARPS-like instrument ESPRESSO, will be mounted on the Very-Large Telescope (VLT) incoherent focus. ESPRESSO could be used to probe exoplanetary atmospheres in the optical. This will bring a more complete census of the chemical composition and study of aeronomic properties of a large variety of exoplanets, including those with larger bulk densities or with fainter host stars. On the infrared side, several spectrographs will be built soon such as SPIROU, CARMENES, IRD and CRIRES+. Based on recent results [*e.g.* @Birkby2013; @Snellen2014] a strong development of atmospheric detections and characterization in the infrared is foreseeable. Eventually, high-resolution spectra of exoplanets with broad spectral coverage will allow inventory of the atmospheric chemical composition, the exploration of aeronomic properties such as pressure-temperature profile and lead to a better understanding of the formation and evolution of exoplanets. This is also a motivation for building high-resolution spectrograph on the European Extremely-Large-Telescope *e.g.* HiReS and METIS [@Udry2014; @Brandl2014; @Snellen2015]. In the meantime, extensive studies have to be done with existing facilities and data especially in the optical domain [*e.g.* @Hoeijmakers2014] to help observers make the best use of future facilities. The present study is a significant step in this direction. It could be extended to the search of other species or as a systematic search in other hot-jupiter atmospheres.
This work has been carried out within the frame of the National Centre for Competence in Research ‘PlanetS’ supported by the Swiss National Science Foundation (SNSF). The authors acknowledge the financial support of the SNSF. We thank M. Mayor, A.H.M.J. Triaud and A. Lecavelier for obtaining the data. We thank K. Heng, S. Khalafinejad, E. Di Gloria, V. Bourrier and R. Allart for discussion and insight. We gratefully acknowledge our referee, I.A.G. Snellen, for valuable comments that improved our manuscript.
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[^1]: using observations with the Harps spectrograph from the ESO 3.6m installed at La Silla, Chile, under the allocated programmes 072.C-0488(E), 079.C-0828(A) and 079.C-0127(A).
|
---
abstract: 'It is becoming clear that the atmospheres of the young, self-luminous extrasolar giant planets imaged to date are dusty. Planets with dusty atmospheres may exhibit detectable amounts of linear polarization in the near-infrared, as has been observed from some field L dwarfs. The asymmetry required in the thermal radiation field to produce polarization may arise either from the rotation-induced oblateness or from surface inhomogeneities, such as partial cloudiness. While it is not possible at present to predict the extent to which atmospheric dynamics on a given planet may produce surface inhomogeneities substantial enough to produce net non-zero disk integrated polarization, the contribution of rotation-induced oblateness can be estimated. Using a self-consistent, spatially homogeneous atmospheric model and a multiple scattering polarization formalism for this class of exoplanets, we show that polarization on the order of 1% may arise due to the rotation-induced oblateness of the planets. The degree of polarization for cloudy planets should peak at the same wavelengths at which the planets are brightest in the near-infrared. The observed polarization may be even higher if surface inhomogeneities exist and play a significant role. Polarized radiation from self-luminous gas giant exoplanets, if detected, provides an additional tool to characterize these young planets and a new method to constrain their surface gravity and masses.'
author:
- |
Mark S. Marley$^{1}$[^1] and Sujan Sengupta$^{2}$[^2]\
$^{1}$NASA Ames Research Center, MS-245-3, Moffett Field, CA 94035, U.S.A.\
$^{2}$Indian Institute of Astrophysics, Koramangala 2nd Block, Bangalore 560 034, India\
date: 'Submitted 2011 March 10.'
title: Probing the Physical Properties of Directly Imaged Gas Giant Exoplanets Through Polarization
---
\[firstpage\]
polarization – scattering – planets and satellites: atmospheres – stars:atmosphere.
Introduction
============
Several young, self-luminous gas giant planets have been detected by direct imaging [@Cha04; @Mar08; @Mar10; @Lag10; @Laf10] around nearby stars. These objects are now being characterized by photometry and even spectroscopy [@Bow10; @Pat10; @Cur11; @Bar11] in an attempt to characterize their atmospheres and constrain the planetary masses. In the next few years many more such planets are almost certainly to be detected by ground-based adaptive optics coronagraphs, such as the P1640 coronagraph on Palomar, the Gemini Planet Imager, and SPHERE on the VLT [@Bei10].
The characterization of the mass of a given directly imaged planet can be problematical, since such planets typically lie at large star-planet separations (tens of AU and greater) and are thus not amenable to detection by radial-velocity methods. Instead masses must be estimated either by comparison of photometry and spectroscopy to planetary evolutionary and atmospheric models or by their gravitational influence on other planets or disk [e.g. @Kal05; @Fab10]. Model comparisons as a method for constraining mass can be ambiguous, however. Evolution models which predict luminosity as a function of age have yet to be fully tested in this mass range for young planets and at very young ages ($< 100$ Myr) the model luminosity can depend on the unknown initial conditions [@Mar07; @For08]. The masses of the planets around HR 8799 estimated by cooling models are apparently inconsistent with standard model spectra [@Bow10; @Bar11; @Cur11] and can lead to rapid orbital instabilities if circular, face-on orbits are assumed [@Fab10]. Finally the mass of the planetary mass companion to the brown dwarf 2M1207 b [@Cha04] inferred from fitting of spectral models to observed near-infrared colors is discrepant with the mass inferred from the companion’s luminosity and the age of the primary [@Moh07].
Discrepancies such as these may arise because young exoplanets exist in a gravity-effective temperature ($g,T_{\rm eff}$) regime in which both the evolutionary and atmospheric models have yet to be validated. Fits of photometry and spectroscopy to predictions of atmosphere models depend upon the veracity of the models themselves, which—in the $T_{\rm eff}$ range of interest—in turn sensitively depend upon model cloud profiles, which are as yet uncertain. Extensive experience with fitting models to brown dwarf spectra and photometry [@Cus08; @Ste09] reveals that while effective temperature can be fairly tightly constrained, gravity determinations are usually less precise, uncertain in some cases by almost an order of magnitude in $g$. While there are low gravity spectral indicators recognized from surveys of young objects [@cruz09; @kirk06] these have yet to be calibrated by studies of binary objects which allow independent measures of mass. Ideally for a single object with a given radius $R$, evolution model luminosity, which (for a known parallax) constrains $R^2T_{\rm eff}^4$ would be fully consistent with $(g, T_{\rm eff})$ constraints from atmosphere model fitting. But, as noted above, this is often not in fact the case, as the derived luminosity, mass, and radii of the companion to 2M1207 b as well as the HR 8799 planets are not fully internally self-consistent with standard evolution models.
Given the likely future ubiquity of direct detections of young, hot Jupiters and the clear need for additional independent methods to constrain planet properties, we have explored the utility of polarization as an additional method for characterizing self-luminous planets.
Polarization of close-in giant exoplanets whose hot atmosphere favours the presence of silicate condensates, is discussed by [@Sea00] and by [@sengupta]. While these authors considered the polarization of the combined light from an unresolved system of star and planet, [@Sta04] presented the polarization of the reflected light of a resolved, directly-imaged Jupiter-like exoplanet. Since the polarized light of a close-in exoplanet is combined with the unpolarized continuum flux of the star which cannot be resolved, the amount of observable polarization in such case is extrmeley low – of the order of magnitude of planet-to-star flux ratio. Polarization measurements of directly-imaged exoplanets in reflected light is also challenging. The removal of scattered light from the primary star must be precise in both polarization channels so that the planet’s intrinsic polarization (which is a differential measurement) can be accurately determined. In any case no extrasolar planet has yet been imaged in scattered light, an accomplishment that will likely require a space-based coronagraph [e.g., @Boc11]. Measuring polarization of thermally emitted radiation—as we propose here—is also difficult but does not require a planet to be close to the star (where the starlight suppression is most difficult) so that it is bright in reflected light. Furthermore extrasolar planets have already been imaged which raises the possibility of polarization observations.
It is clear from comparisons of model spectra to data that most of the exoplanets directly imaged to date have dusty atmospheres [@Mar08; @Bow10; @Laf10; @Bar11; @Cur11; @Ske11]. Clear atmospheres lacking dust grains can be polarized, but only at blue optical wavelengths where gaseous Rayleigh scattering is important [@Sen09]. Since even the hottest young exoplanets will not emit significantly in the blue, grain scattering must be present for there to be measurable polarization in the near-infrared where warm giant planets are bright [@Sen10]. There are two temperature ranges within which we expect a gas giant exoplanet to possess significant atmospheric condensates. The first is L-dwarf like planets (roughly $1000 <
T_{\rm eff} < 2400\,\rm K$) where iron and silicate grains condense in the observable atmosphere. The lower end of this range in the planetary mass regime is as yet uncertain. The second temperature range occurs in cool planets with atmospheric water clouds ($T_{\rm eff} < 400\,\rm K$). There have yet been no confirmed detections of such planets. Here we will focus on the first category since such objects are brighter, more easily detectable, and the comparison to the field dwarfs is possible.
Although survey sizes are fairly small, linear polarization of field L dwarfs has been detected. [@Men02] and [@Oso05] both report that a fraction of L dwarfs, particularly the later, dustier spectral types, are intrinsically polarized. [@Sen10] find that the observed polarization can plausibly arise from emission of cloudy, oblate dwarfs, although to produce the required oblateness (20% or more) the dwarfs must have fairly low gravity for a field dwarf ($g\sim 300\,\rm m\,s^{-2}$) and rapid rotation. The required rotation periods are brisk, as little as 2 hours or less, but are compatible with observed rotational velocities in at least some cases (see [@Sen10] for a discussion). [@Sen10] further find that the near-infrared polarization is greatest at $T_{\rm eff} \sim 1600\,\rm K$ where their model condensate clouds are both optically thick and still prominent enough in the photosphere to maximally affect the polarization.
Surface inhomogeneities can also give rise to a net polarization [@Men04] and experience from the solar system confirms that irregularly spaced clouds are to be expected. Both Jupiter’s and Saturn’s thermal emission in the five-micron spectral window is strongly modulated by horizontally inhomogeneous cloud cover and it would not be surprising to find similar morphology in the atmospheres of exoplanets. In the presence of surface inhomogeneity, the asymmetry that produces the net non-zero disk-integrated polarization would increase and hence a combination of oblate photosphere and surface inhomogeneity can give rise to detectable levels of polarization.
Exoplanets are even better candidates than L dwarfs to have an oblate shape and be polarized. With a lower mass and roughly the same radius as a brown dwarf (and thus a lower gravity), a rapidly rotating planet can be significantly oblate and consequently produce a polarization signal even without surface inhomogeneities. Here we explore the conditions under which the thermal emission from a warm, young exoplanet may be polarized and consider the scientific value of measuring exoplanet polarization. We first look at the issue of oblateness, then present a set of cloudy model atmosphere calculations relevant to planets amenable to direct detection and discuss under which conditions their thermal emission may be polarized. Finally we discuss our findings and explore how the characterization of an extrasolar planet may be enhanced by polarization observations.
Young Giant Exoplanets
======================
Evolution
---------
When giant planets are young they are thermally expanded and boast larger radii and smaller gravity. [@For08] have computed evolution models for gas giant planets with masses ranging from 1 to $10\,\rm M_J$ for ages exceeding $10^6\,\rm yr$ and we can use their results to predict the oblateness of thermally expanded young Jupiters with various rotation rates. Those authors modeled two types of evolution models. The first variety, termed ‘hot starts’, was most traditional and assumed the planets formed from hot, distended envelopes of gas which rapidly collapsed. This calculation is comparable to that of most other workers in the field. They also presented calculations for planets fomed by the core accretion planet formation process (see [@Lis07]) which (depending on details of the assumed boundary condition for the accretion shock) produces planets that are initially much smaller and cooler than in the ‘hot start’ scenario.
For the calculations here we choose to use the ‘hot start’ evolutionary calculation. We do this for several reasons. First, these models provide a reasonable upper limit to the radius at young ages and thus bound the problem. Second, at the large orbital separations that will, at least initially, be probed by ground based adaptive optics coronagraphic imaging, the core accretion mechanism may be inefficient at forming planets. Thus the gaseous collapse scenario may be more relevant choice. Finally the three planets observed around HR 8799 are all much brighter than predicted by the [@For08] cold-start, but not the hot-start, cooling tracks.
Figure 1 presents model evolution tracks for non-irradiated giant exoplanets from [@For08]. On this figure planets age from the right to the left as effective temperature falls, the planets contract, and their surface gravity, $g$, increases. The dashed lines denote isochrones. This figure guides our selection of atmosphere models to evaluate for polarization studies. Groundbased coronagraphic searches for planets are expected to foucus on stars younger than about 200 Myr [e.g., @McB11]. From the figure we see that at ages of 10 to 200 Myr we expect exoplanets with masses falling between 1 and $10\,\rm M_J$ to have $g$ roughly in the range of 15 to $200\,\rm m\,s^{-2}$.
Shape
-----
Both Jupiter and Saturn are oblate. The fractional difference, $f=1-R_p/R_e$, between their equatorial and polar radii, known as oblateness, are 0.065 and 0.11 respectively. The extent to which their equators bulge outwards depends on their surface gravity, $g$, and rotation rate, $\Omega$, as well as their internal distribution of mass. The Darwin-Radau relationship [@Bar03] connects these quantities of interest: $$\begin{aligned}
\label{obl1}
f=\frac{\Omega^2R_e}{g}\left[\frac{5}{2}\left(1-\frac{3K}{2}\right)^2+
\frac{2}{5}\right]^{-1}\end{aligned}$$ Here $K=I/(MR_e^2)$, $I$ is the moment of inertia of the spherical configuration, and $M$ and $R_e$ are the mass and equatorial radii.
The relationship for the oblateness $f$ of a stable polytropic gas configuration under hydrostatic equilibrium is also derived by [@cha33] and can be written as $$\begin{aligned}
\label{obl2}
f=\frac{2}{3}C\frac{\Omega^2R_e}{g}\end{aligned}$$ where $C$ is a constant whose value depends on the polytropic index.
The above two relationships provide the same value of oblateness for any polytropic configuration. Equating Eq. (1) and Eq. (2), we obtain $$\begin{aligned}
C=\frac{3}{2}\left[\frac{5}{2}\left(1-\frac{3K}{2}\right)^2+ \frac{2}{5}\right].\end{aligned}$$ Substituting the value of $K$ for a polytrope of index $n$ gives the value of the corresponding $C$. For example, $K=0.4, 0.261, 0.205, 0.155, 0.0754 $ for $n=0,1,1.5,2,3$ respectively. The corresponding values of $C$ derived by [@cha33](p. 553, Table 1). are 1.875, 1,1399, 0.9669, 0.8612, 0.7716 respectively.
The interiors of gas giant planets can be well approximated as $n=1$ polytropes. For the observed mass, equitorial radii, and rotation rates of Jupiter and Saturn, expression (2) predicts, with $n=1$, an oblateness of 0.064 and 0.11, in excellent agreement with the observed values. Figure 2 presents the oblateness computed employing Eq. (2) as applied to 1 and $10\,\rm M_J$ planets at three different ages, 10, 100, and 1,000 Myr using the [@For08] hot-start cooling tracks. Also shown is the oblateness (0.44) at which a uniformly rotating $n =1.0 $ polytrope becomes unstable [@Jam64]. Clearly for rotation rates comparable to those seen among solar system planets we can expect a substantial degree ($f>0.10$) of rotational flattening. As gas giants age and contract the same rotation rate produces much less oblate planets. However for young, Jupiter mass planets rotation rates of 7 to 10 hours can easily produce $f\sim 0.2$ even for planets as old as 100 Myr. More rapid rotation rates may produce even greater degrees of flattening. L dwarfs, with much higher surface gravity, must have even more rapid rotation rates to exhibit even modest flattening [@Sen10].
Polarization of Young Exoplanets
================================
To explore the degree of polarization expected for various planet masses and ages we considered a selection of one-dimensional, plane-parallel, hydrostatic, non-gray, radiative-convective equilibrium atmosphere models with sixty vertical layers [@Ack01; @Mar02; @Freed08] for specified effective temperatures, $800< T_{\rm eff} < 1200 \, \rm K$ and surface gravities $g = 30$ and $100\,\rm m \,sec^{-2}$. We focus on this apparently limited parameter range since all gas giant exoplanets with masses below $10\,\rm M_J$ will have cooled below 1200 K by an age of 30 Myr (see Figure 1 and also [@For08]). The median age for nearby ($< 75\,\rm pc$) young stars that are likely targets for planet imaging surveys is $50\,\rm Myr$ [@McB11]. For our study we choose a lower limit of 800 K, well below the $T_{\rm eff}$ at which Sengupta & Marley (2010) predicted maximal polarization for field L dwarfs. At such temperatures dust clouds, if present globally across the disk, will lie at high optical depth and we expect produce a smaller polarization signal than the warmer objects.
Indeed 800 K is well below the field dwarf L to T transition temperature of 1200 to 1400 K ([@Ste09] and references therein) by which point most signs of clouds have departed. However there exists growing evidence that there is a gravity dependence to the effective temperature at which clouds are lost from the atmosphere and certainly the planets such as those orbiting HR 8799 are still dusty at effective temperatures near 1000 K [@Bow10]. Observation of a polarization signal in a cooler exoplanet would provide powerful evidence for atmospheric dust.
Some of the more massive young exoplanets ($M>8\,\rm M_J$) may have gravities in excess of our $100\,\rm m
\,sec^{-2}$ upper limit, but as we show below little oblateness-induced polarization is expected at high gravity in this $T_{\rm eff}$ range (see also @Sen10). For example a surface gravity of $g=100\,\rm m\,sec^{-2}$ and $T_{\rm
eff}= 1000\,\rm K$ approximately describes an $8\,\rm M_J$ planet at an age of 100 Myr while values of $30\,\rm m\,sec^{-2}$ and 800 K are expected for a $2\,\rm M_J$ planet at an age of 60 Myr. We choose these values and a few others to illustrate the parameter space and the sensitivity of the results to variations in gravity and effective temperature.
Each model includes atmospheric silicate and iron clouds computed with sedimentation efficiency [@Ack01] $f_{\rm sed}=2$. Preliminary studies by our group suggest that even dustier models with $f_{\rm sed}\sim 1$ might be necessary to reproduce the HR 8799 planets. However our previous work [@Sen10] has demonstrated that while $f_{\rm sed}= 1$ atmospheres do show greater polarization than $f_{\rm sed} = 2$, the difference is slight when integrated over the disk. Other cloud modeling approaches are reviewed by [@Hel08]. Some of these alternative cloud modeling formulations, such as those employed by Helling and collaborators [e.g., @Hel06; @Hel08b], predict a greater abundance of small particles high in the atmosphere than the Ackerman & Marley approach. Such a haze of small particles could potentially produce a larger polarization signal than we derive here. Polarization measurements may thus help provide insight into the veracity of various approaches.
As in [@Sen10] we employ the gas and dust opacity, the temperature-pressure profile and the dust scattering asymmetry function averaged over each atmospheric pressure level derived by the atmosphere code in a multiple scattering polarization code that solves the radiative transfer equations in vector form to calculate the two Stokes parameter $I$ and $Q$ in a locally plane-parallel medium [@Sen09]. For each model layer we fit a Henyey-Greenstein phase function to the particle scattering phase curve predicted by a Mie scattering calculation. A combined Henyey-Greenstein-Rayleigh phase matrix [@Liu06] is then used to calculate the angular distribution of the photons before and after scattering. In the near-infrared the contribution of Rayleigh scattering by the gas to the overall scattering is negligible and the scattering is treated in the Henyey-Greenstein limit with the particle phase function computed from Mie theory. Specifically the off diagonal terms of the scattering phase matrix are described by [@Whi79] and are very similar to the pure Rayleigh case. For the diagonal elements the Henyey-Greenstein elements are used. In the limit of the scattering asymmetry parameter approaching zero the matrix converges to the Rayleigh scattering limit. Finally, the angle dependent $I$ and $Q$ are integrated over the rotation-induced oblate disk of the object by using a spherical harmonic expansion method and the degree of polarization is taken as the ratio of the disk integrated polarized flux ($F_Q$) to the disk integrated total flux ($F_I$). The detailed formalisms as well as the numerical methods are provided in [@Sen09].
Figures 3 and 4 illustrate typical input properties of the models employed here. Figure 3 shows a model temperature-pressure profile along with iron and silicate condensate grain sizes as computed by our cloud model. Figure 4 shows the mean layer single scattering albedo, $\overline{\omega_0}$, and scattering asymmetry parameter, $\overline{\cos \theta}$, as a function of wavelength near the peak opacity of the cloud. Within strong molecular bands the single scattering albedo approaches zero since gas absorption dominates over the cloud opacity. Below the cloud base and far above the cloud both the albedo and asymmetry parameters are essentially zero in the near infrared as gaseous Rayleigh scattering makes little contribution to the opacity at those wavelengths. Note that for the computed particle sizes the cloud is strongly forward scattering at wavelengths where cloud opacity dominates molecular absorption in agreement with a recent study by [@Dek11].
Results and Discussions
=======================
Figure 5 presents the computed thermal emission and polarization spectra of an approximately 10 Myr old 2 Jupiter mass planet assuming rotation periods of 5 and 6 hrs. The striking dependence of polarization on the rotation rate arises from the sensitivity of oblateness to rotation period as seen in Figure 2. Generally speaking the degree of polarization is highest at those wavelengths of low gaseous opacity where the cloud is visible while at other wavelengths, inside of atomic and molecular absorption bands, flux emerges from higher in the atmosphere and is less influenced by cloud scattering. While the degree of polarization peaks at the shortest wavelengths shown (from the influence of gaseous Rayleigh scattering), there is very little flux at optical wavelengths. However in the near-infrared, where windows in the molecular opacity allow flux to emerge from within the clouds, the computed degree of polarization approaches 1%. In these spectral regions the planets will be bright, the contrast with the primary star favorable, and thus the polarization may be more easily detectable at this level. Beyond about $2.2\,\rm \mu m$ thermal emission emerges from above the cloud tops and thus there is no signature of the scattering and the net polarization is near zero. This pattern of polarization is diagnostic of atmospheric clouds and is easily distinguished from other sources of polarization, for example a circumplanetary disk.
Figures 6 and 7 show warmer model cases for the same gravity with similar behavior. These cases would apply to quite young planets at an age of less than ten million years, but illustrate that the degree of polarization does not dramatically increase at higher effective temperatures. Figure 8 shows the variation with gravity. With a fixed rotation period of five hours, models with $g$ of 56 and $100\,\rm m\,s^{-2}$ show very little polarization at any wavelength. These models would correspond to approximately 4 to 6 Jupiter mass planets at ages greater than 10 million years, perhaps typical of the planet types that may be directly imaged. The sensitivity of polarization to gravity seen in this figure illustrates the promise of polarization, in the right circumstances, to provide a new constraint on exoplanet mass.
Figure 9 generalizes these trends, showing the predicted polarization in $I$ and $J$ bands as a function of the rotational period $P_{rot}$. For a fixed surface gravity and viewing angle, $i$, the degree of polarization does not vary substantially within the range of $\rm T_{eff}$ between 800 and 1200 K. The polarization profiles in both bands increase with decreasing rotation period and the polarization is generally greater in $J$ than in $I$ band. As is the case for brown dwarfs [@Sen10], for a given rotation period the polarization decreases with lower $i$.
All of the cases shown in Figure 9 have an oblateness less than 0.44, the stability limit for an $n=1$ polytrope. For $g=30\,\rm m\,s^{-2}$ the stability limit is reached at a rotation period of about 4 hours, slightly less than the lower limit shown on the figure. Such short rotation periods may in fact be a natural consequence of giant planet formation in a circumstellar binary as the angular momentum of accreting gas naturally produces rapid rotation rates [@War10].
We conclude that a self-luminous gas giant planet–even with a homogeneous cloud distribution–will exhibit notable polarization (greater than a few tenths of percent) in the near infrared if the planet is (1) cloudy, (2) significantly oblate, and (3) viewed at a favorable geometry. An oblate shape is the easiest to obtain at low masses and modest rotation rates or higher masses and more rapid rotation rates. Higher effective temperatures, which would produce more dust higher in the atmosphere and more polarization [@Sen10], are generally excluded by the evolution for ages greater than a few million years. More massive planets, which take longer to cool, have higher gravity and thus a smaller oblateness and less polarization (Figures 2 & 9) for a given rotation rate. Given these considerations we believe the cases we have presented here are among the more favorable for homogenous cloud cover. While we have not considered every combination of parameters, the models presented here along with perturbations of those models we have also studied lead us to conclude that uniformly cloudy planets will not present polarization greater than a few percent and polarization is most likely to be found for young, low mass, rapidly rotating planets.
However inhomogeneous cloud cover, which we have not modeled, may also affect the polarization spectra. Indeed an inhomogeneous distribution of atmospheric dust (e.g., Jupiter-like banding) would not be unexpected. Such banding may provide further asymmetry [@Men04] and hence increase (or even decrease) the net non-zero polarization. A non-uniform cloud distribution may be the mechanism that underlies the L to T-type transition among brown dwarfs [@Ack01; @Bur02; @Marl10] and variability has been detected in some transition dwarfs [@Art09; @Rad10]. Cloud patchiness is also observed in images of thermal emission from Jupiter and Saturn taken in the M-band (five-micron) spectral region [e.g. @Wes69; @Wes74; @Ort96; @Bai05], so patchiness may indeed be common. Polarization arising from patchy clouds still requires the presence of some clouds of course, thus any polarization detection provides information on the presence of condensates and–by extension–constrains the atmospheric temperature.
CONCLUSIONS
===========
The next decade is expected to witness the discovery of a great many self-luminous extrasolar giant planets [@Bei10]. The masses, atmospheric composition and thermal structure of these planets will be characterized by photometry and spectroscopy. For some systems, other constraints, such as dynamical interactions with dust disks or potential instabilities arising from mutual gravitational interactions [e.g., @Fab10] may also contribute. Here we demonstrate that measurable linear polarization in $I$ or $J$ bands reveals the presence of atmospheric condensates, thereby placing limits on atmospheric composition and temperature. Polarization of thermal emission from a homogeneously cloudy planet is most favored for young, low mass, and rapidly rotating planets. A diagnostic characteristic of cloud-induced polarization is that the polarization peaks in the same spectral bandpasses as the flux from the planet because photons are emerging from within the cloud itself as opposed to higher in the atmosphere (Figures 5 through 8).
Assuming that our atmospheric and condensate cloud models are reasonably accurate, we conclude that any measured polarization greater than about 1% likely can be attributed to inhomogeneities in the global cloud deck. While we have not considered every possible model case, we find that our most favorable plausible cases do not produce notably greater polarization. Other cloud models [e.g., @Hel06; @Hel08b] which incorporate more small particles high in the atmosphere may well produce a different result, thus polarization may help to distinguish such cases. For a fixed rottion period, the oblateness and thus polarization increases with decreasing surface gravity. In such situations polarization may provide a new constraint on gravity and mass. However for gravity in excess of about 50 $\rm m\,s^{-2}$ and for $T_{\rm eff} < 1200\,\rm K$ (corresponding to planet masses greater than about $4\,\rm M_J$) we do not expect detectable amounts of polarization. Warmer and higher gravity field L dwarfs, can show measurable polarization since such cloud decks are higher in the atmosphere. For directly imaged exoplanets, however, we do not expect to encounter such high effective temperatures. For exoplanets with plausible $T_{\rm eff} < 1200\,\rm K$, Figure 9 shows that even if the rotation period is as rapid as 4.5 hrs. and the viewing angle is $90^o$ at which the polarization is maximum, the percentage degree of polarization in thermal emission is not more that a few times of $10^{-2}$.
The aim of our study was to better understand the information conveyed by polarization about the properties of extrasolar giant planets directly imaged in their thermal emission. We have found that in some cases polarization can provide additional constraints on planet mass, atmospheric structure and cloudiness. Combined with other constraints, polarization adds to our understanding, although there remain ambiguities. A study of the polarization signature of partly cloudy planets would yield further insight into the value of polarization measurements for constraining extrasolar giant planet properties.
Acknowledgements
================
We thank the anonymous referee for helpful comments that improved the manuscript. MSM recognizes the NASA Planetary Atmospheres Program for support of this work.
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![Evolution through time in $T_{\rm eff}$ – $g$ space of non-irradiated, metallicity $[M/H]=0.0$ giant planets of various masses. Solid lines are evolution tracks at various fixed masses ranging from 1 to $10\,\rm M_J$. Dashed lines are isochrones for various fixed ages since an arbitrary ‘hot start’ initial model [@For08]. \[fig1\]](f1)
![Rotationally induced oblateness as a function of rotational period for 1 and $10\,\rm M_J$ planets with three different ages (10 - solid, 100 - dashed, and 1000 Myr - dotted). Horizontal line is the stability limit for $n=1$ polytropes assuming solid body rotation. More rapidly rotating planets would form a triaxial ellipsoidal shape and eventually bifurcate. \[fig2\]](f2)
![Atmospheric temperature ($T$, lower scale) and cloud particle size, $r_{\rm eff}$, (upper scale) as a function of pressure, $P$, for one adopted model atmosphere with $T_{\rm eff} = 1000\,\rm K$, $g=30\,\rm m\,s^{-2}$, and $f_{\rm sed}=2$. From Figure 2 this corresponds to approximately a $2.5\,\rm M_J$ planet at an age of a few million years. Particle sizes are shown for iron and silicate (forsterite) grains, the two most significant contributors to cloud opacity. Shown is the “effective radius” ($r_{\rm eff}$) of the particles, a mean size precisely defined in Ackerman & Marley (2001). \[fig3\]](f3)
![Scattering properties as a function of wavelength, $\lambda$, of a model layer near the 1 bar pressure level for the model atmosphere shown in Figure 3. Shown are the layer single scattering albedo, $\overline{\omega_0}$, solid, and the layer asymmetry parameter, $\overline{\cos \theta}$, dashed. In strong molecular bands gaseous absorption dominates over scattering, thus lowering the mean layer albedo. \[fig4\]](f4)
![The emergent flux (A) and the disk-integrated degree of linear polarization $P(\%) $ (B) of non-irradiated exoplanets at different wavelengths at viewing angle $i=90^\circ$ (equatorial view). In (B), the top solid line represents the polarization profile for a rotational period $P_{\rm rot}= 5\, \rm hr$ while the bottom solid line represents that for 6 hr. Note that while the polarization can be high at blue wavelengths there is very little flux there. \[fig5\]](f5)
![Same as figure 5 but with $T_{\rm eff}=1000K$. This is the result for the model characterized in Figures 3 and 4. \[fig6\]](f6)
![Same as figure 5 but with $T_{\rm eff}=1200K$. \[fig7\]](f7)
![The emergent flux (A) and the disk-integrated degree of linear polarization (B) of non-irradiated exoplanets for a fixed $T_{\rm eff}=1000$K but for different surface gravities. In (B), the solid lines from top to bottom represent the polarization profiles for surface gravity $g=30$, 56 and $100\,\rm m\,s^{-2}$ respectively. The difference in the emergent flux (A) for a fixed effective temperature but surface gravity varying over this range is not noticeable on this scale. \[fig8\]](f8)
![ Scattering polarization profiles of non-irradiated exoplanets with different rotational periods. The solid lines represent the percentage degree of linear polarization in J-band while the broken lines represent that in I-band. A variety of model cases are shown, all assume $f_{\rm sed}=2$. Cases are shown for viewing angle $i=90^\circ$ (equator view) and $i=60$ and $45^\circ$. \[fig9\]](f9)
[^1]: E- mail: Mark.S.Marley@NASA.gov
[^2]: E-mail: sujan@iiap.res.in
|
---
abstract: 'We perform an analysis of the combined effects of geometry and a magnetic field for the case of a charged particle on a helicoid. The effective quantum potentials for a charged spinless particle confined on a helicoid for two simple magnetic field configurations are derived. These potentials depend nontrivially on the surface curvature and the external magnetic field. We find that the *qualitative* behavior of the effective potential can be altered by changing the strength of the applied magnetic field. The application of a magnetic field results in effective potentials that are either repulsive or attractive, depending on the magnitude of the magnetic field and the angular momentum of the particle. Finally, for the case of effective potentials that have a minimum, we also obtain approximate expressions for the energy levels valid when the particle is near a minimum, and these are found to be similar in form to the energy levels of a particle in a harmonic oscillator potential.'
author:
- Miguel Antonio Sulangi
- 'Quirino M. Sugon, Jr.'
date: 'November 24, 2012'
nocite: '[@*]'
title: The effect of the geometric potential and an external magnetic field on a charged particle on a helicoid
---
\[sec:level1\]Introduction
==========================
Recently, there has been an emergence of methods for experimentally producing nanostructures with novel geometries. A number of studies have demonstrated qualitatively unusual behavior of charge carriers in curved nanostructures in strong external electric and magnetic fields. Physical phenomena that emerge from a combination of strong electric and magnetic fields and surface curvature include Aharonov-Bohm oscillations [@aB99], formation of Landau levels [@hA92; @jK92], and the quantum Hall effect [@eP07]. Also, graphene systems such as fullerines—which are curved nanostructures—exhibit unusual phenomena induced by strong magnetic fields such as changes in the energy band gap [@uC04; @bL07] and in the magnetoresistance [@nS07; @aV07].
The helicoid is one surface that has been the subject of intense mathematical study. In particular, the effects of surface curvature on the wavefunction of a particle constrained on a helicoid have already been studied by Atanasov *et al.* [@vA09]. However, the problem of a charged particle constrained on the surface of a helicoid in external magnetic fields has not yet been studied. The motivation for doing this is to uncover interesting properties and methods of controlling charged particles on helicoid-shaped nanostructures using magnetic fields.
In this paper, we consider the *additional* effect of an external magnetic field on a charged *spinless* particle on a helicoid. We derive exact expressions for the effective potentials for a charged particle confined on a helicoid for two simple circumferential magnetic field configurations. From these expressions we look at the effect of increasing or decreasing the magnetic field strength on the behavior of the effective potentials; in particular we pay especially close attention to whether the potentials become *attractive* or *repulsive* upon increasing or decreasing the magnetic field strength. We also obtain approximate expressions for the energy levels of the particle for certain values of the angular momentum. What we find is that changing the magnetic field strength alters the behavior of the effective potentials and consequently that of the particle; it is possible to make the potential attractive or repulsive simply by changing the field strength. This points to a way of controlling the average behavior of an ensemble of charged particles on helicoid-shaped nanostructures in the non-interacting limit, a result that will possibly be of much use in areas such as microelectronics and nanochemistry.
However, we consider only a very limited subset of all possible magnetic field configurations. This is because only a small number of field configurations lend themselves to a purely analytic approach. For the majority of configurations the resulting equations of motion are non-separable and require numerical methods to solve. Our focus is on the effective potentials, which govern the quantum dynamics of the charged particles on the helicoid; we will not concern ourselves with the extremely difficult task of deriving exact solutions of the Schrodinger equation.
The effective potential from geometry and magnetic fields
=========================================================
The Schrodinger equation describing the *surface* wavefunction $\psi_s$ of a charged spinless particle with mass $m$ and charge $Q$ on a two-dimensional curved surface in a magnetic field is given by [@gF08]
$$i\hbar\partial_t\psi_s = \frac{1}{2m}\left[-\frac{\hbar^2}{\sqrt{g}}\partial_a(\sqrt{g}g^{ab}\partial_b\psi_s ) + \frac{iQ\hbar}{\sqrt{g}}\partial_a(\sqrt{g}g^{ab}A_b)\psi_s\right] + \frac{1}{2m}\left[2iQ\hbar g^{ab}A_a\partial_b\psi_s + Q^2g^{ab}A_aA_b\psi_s \right] + V_s\psi_s. \label{E:TDSEheli}$$
Note that Einstein summation notation has been used. In Eq. (\[E:TDSEheli\]) $\hbar$ is Planck’s constant, $\partial_i$ is the partial derivative operator taken with respect to the variable $i$, and $A_j$ denotes the component of the magnetic vector potential $\mathbf{A}$ in the direction of $j$. We allow both $i$ and $j$ are to be equal to $\rho$ and $z$. If we consider only a magnetic field that is constant in time, then there are no time-dependent terms in the full equation of motion of the wavefunction, and we may therefore work with the time-independent version of Eq. (\[E:TDSEheli\]):
$$E\chi_s = \frac{1}{2m}\left[-\frac{\hbar^2}{\sqrt{g}}\partial_a(\sqrt{g}g^{ab}\partial_b\chi_s ) + \frac{iQ\hbar}{\sqrt{g}}\partial_a(\sqrt{g}g^{ab}A_b)\chi_s\right] + \frac{1}{2m}\left[2iQ\hbar g^{ab}A_a\partial_b\chi_s + Q^2g^{ab}A_aA_b\chi_s \right] + V_s\chi_s \label{E:TISEheli}$$
where $$\label{E:separatedheli}
\psi_s = \chi_s e^{-iEt/\hbar}$$ and $E$ is the energy of the particle. Now we fill in the blanks and put in the expressions for the metric components that enter into the surface equation of motion. These will be specific to the helicoid.
A helicoid can be parametrized by the following set of equations [@aG93]: $$\begin{aligned}
x &=& \rho\cos(\omega z), \label{E:xparaheli} \nonumber \\
y &=& \rho\sin(\omega z), \label{E:yparaheli} \nonumber \\
z &=& z \label{E:zparaheli}\end{aligned}$$ In these equations $\omega=2\pi S$, where $S$ is the number of complete twists (i.e., $2\pi$-turns) per unit length of the helicoid and $\rho$ is the radial distance from the $z$-axis. For convenience, we use the coordinates $(z, \rho)$ to characterize a point on the helicoid. The infinitesimal line element on the helicoid is given by $$\label{E:lineelementheli}
ds^2 = d\rho^2 + (1 + \omega^2\rho^2)dz^2.$$ For ease we let $$a(\rho) = \sqrt{1 + \omega^2\rho^2}.$$ The metric components are thus given by $$\begin{aligned}
g_{\rho \rho} &=& 1, \nonumber \\
g_{zz} &=& 1 + \omega^2\rho^2 = a^2, \nonumber \\
g_{z \rho} &=& 0, \nonumber \\
g_{\rho z} &=& 0, \label{E:metriczheli}\end{aligned}$$ and the square root of the determinant of the metric is given by $$\label{E:metricdeterminantheli}
\sqrt{g} = \sqrt{1 + \omega^2\rho^2} = a.$$
This gives rise to the following expressions for the principal curvatures: $$\begin{aligned}
\kappa_1 &=& \frac{\omega}{1+\omega^2\rho^2}, \nonumber \\
\kappa_2 &=& -\frac{\omega}{1+\omega^2\rho^2}. \label{E:curvatureheli}\end{aligned}$$ As a result, the mean curvature $M$ vanishes: $$\label{E:meancurvatureheli}
M = \frac{1}{2}(\kappa_1 + \kappa_2) = 0.$$ This vanishing of $M$ is the reason for calling the helicoid a minimal surface. The Gaussian curvature $K$, on the other hand, is nonvanishing: $$\label{E:gaussiancurvatureheli}
K = \kappa_1\kappa_2 = -\frac{\omega^2}{(1+\omega^2\rho^2)^2}.$$ Therefore the curvature-induced potential $V_s$ is given by $$\label{E:curvpotentialheli}
V_s = -\frac{\hbar^2}{2m}\left(M^2 - K \right) = -\frac{\hbar^2}{2m}\frac{\omega^2}{(1+\omega^2\rho^2)^2}.$$
We now turn our attention to the Hamiltonian $H$ of the particle. To clean up our analysis we separate the Hamiltonian into two parts: $H_{em}$, which contains the terms describing the interaction with the magnetic field, and $H_{curv}$, which contains the non-electromagnetic terms—which in our case include the kinetic energy terms and the curvature-induced potential. The non-electromagnetic part of $H$ is given by
$$\label{E:hamiltoniancurv}
H_{curv}\chi_s = \frac{1}{2m}\left[\frac{-\hbar^2}{a} \left( \partial_z( \frac{1}{a}\partial_z\chi_s ) + \partial_{\rho}(a\partial_{\rho}\chi_s ) \right) \right] + V_s\chi_s.$$
Since the wavefunction has to be normalized with respect to the infinitesimal area $d\rho dz$—and not with respect to $ad\rho dz$, as Eq. (\[E:metriczheli\]) would suggest—we make the substitution $\chi_s \to \frac{1}{\sqrt{a}}\chi_s$ in Eq. (\[E:hamiltoniancurv\]), which would only affect terms involving derivatives with respect to $\rho$. After lengthy algebra, we arrive at this expression for $H_{curv}$:
$$\label{E:hamiltoniancurvnorm}
H_{curv}\chi_s = -\frac{\hbar^2}{2m} \left[\frac{\partial^2\chi_s}{\partial\rho^2} + \frac{1}{1+\omega^2\rho^2} \frac{\partial^2\chi_s}{\partial z^2} + \frac{\omega^2}{2(1+\omega^2\rho^2)^2} \left(1 + \frac{\omega^2\rho^2}{2} \right)\chi_s \right].$$
Note that this expression is simply the *full* Hamiltonian for an uncharged particle on a helicoid, as derived in the study by Atanasov *et al.* [@vA09].
$H_{em}$ meanwhile is given by
$$H_{em}\chi_s = \frac{iQ\hbar}{2m} \left[ \frac{1}{1+\omega^2\rho^2} \frac{\partial A_z}{\partial z} + \frac{\partial A_{\rho}}{\partial \rho} + \frac{\omega^2\rho}{1+\omega^2\rho^2}A_{\rho} \right]\chi_s + \frac{2iQ\hbar}{2m}\left[\frac{A_z}{1+\omega^2\rho^2}\frac{\partial\chi_s}{\partial z} + A_{\rho}\frac{\partial\chi_s}{\partial\rho} \right] + \frac{Q^2}{2m}\left( \frac{A_z^2}{1+\omega^2\rho^2} + A_{\rho}^2 \right)\chi_s. \label{E:hamiltonianem}$$
If we work with magnetic vector potentials in which $A_{\rho} = 0$, then $H_{em}$ *remains* the same even after the wavefunction $\chi_s$ has been adjusted with the $1/\sqrt{a}$ factor, since the term involving the derivative of the wavefunction with respect to $\rho$ vanishes.
We now have the full Hamiltonian, and consequently the time-independent Schrodinger equation $E\chi_s = H\chi_s$, for a charged spinless particle confined on a helicoid. The Schrodinger equation we have derived is a partial differential equation in two variables and is extremely difficult to solve in general. However, for some field configurations, separation of variables can be done, and we are left with a simpler ordinary differential equation. We now consider specific field configurations.
Behavior of the effective potential
===================================
Constant circumferential magnetic field
---------------------------------------
A magnetic field that is circumferential and of constant magnitude everywhere can be described by the vector equation $$\mathbf{B} = B\mathbf{\hat{\phi}},$$ where $B$ is a constant. This is not a very realistic field configuration, as it follows from Maxwell’s equations that no *physically realizable* current density $\mathbf{j}$ will produce this magnetic field; such a current density would have to take on an infinite value at the center of the wire. Nevertheless, this field configuration is useful for two reasons: 1) It provides the simplest test case for the formalism that we have developed for the charged particle on a helicoid, and the calculation is instructive and reveals a lot about the interplay between the geometric potential and the magnetic field; and 2) it is a reasonable approximation when one considers physically realizable circumferential fields that are sufficiently far from sources, in which case the magnetic field does $\emph{not}$ change much over the relevant length scale. An example of this would be the magnetic field produced by a current-carrying wire (which we will consider in the next section); when we consider portions of the helicoid sufficiently far from the current, the field does not change appreciably with changes in the distance, and therefore in this regime it can be reasonably approximated by a constant circumferential field.
### The Schrodinger equation and the effective potentials
We choose the most convenient vector potential $\mathbf{A}$ that gives $\mathbf{B} = B\mathbf{\hat{\phi}}$. The components of $\mathbf{A}$ are given by $$\label{E:avectorpotentialheli}
A_z = -B\rho, \quad A_{\rho} = 0.$$ Substituting this into Eq. (\[E:hamiltonianem\]) and combining with Eq. (\[E:hamiltoniancurvnorm\]), we have the following expression for the full Schrodinger equation:
$$E\chi_s = -\frac{\hbar}{2m}\left(\frac{\partial^2\chi_s}{\partial\rho^2} + \frac{1}{1+\omega^2\rho^2} \frac{\partial^2\chi_s}{\partial z^2}\right) - \frac{\hbar^2}{2m}\left[\frac{\omega^2}{2(1+\omega^2\rho^2)^2} \left(1 + \frac{\omega^2\rho^2}{2}\right) \right] \chi_s - \frac{2iQ\hbar B\rho}{2m(1+\omega^2\rho^2)} \frac{\partial\chi_s}{\partial z} + \frac{Q^2B^2\rho^2}{2m(1+\omega^2\rho^2)} \chi_s. \label{E:hamiltonsub}$$
The absence of $z$-dependent terms in Eq. (\[E:hamiltonsub\]) gives us license to separate $\chi_s$ in the following manner [@LL77]: $$\label{E:splitwaveheli}
\chi_s = e^{ikz}\gamma(\rho).$$ The wavevector $k$ is related to the $z$-component of the momentum $p_z$ by the relation $k = p_z/\hbar$. Since $\phi = \omega z$ for a helicoid, it can be shown that $p_z = \omega L_{\phi}$, where $L_{\phi}$ is the angular momentum about the $z$-axis. From the periodicity of the system, we require that $e^{ikz}$ be periodic with period $1/S$; this implies that $k = 2\pi S l = \omega l$, where $l \in \mathbb{Z}$.
Putting Eq. (\[E:splitwaveheli\]) into Eq. (\[E:hamiltonsub\]), we have
$$E\gamma = -\frac{\hbar}{2m}\frac{d^2\gamma}{d\rho^2} - \frac{\hbar}{2m}\left(\frac{\omega^2}{2(1+\omega^2\rho^2)}\right)\left(1 + \frac{\omega^2\rho^2}{2} \right)\gamma + \frac{\hbar^2 l^2\omega^2}{2m(1+\omega^2\rho^2)}\gamma + \frac{2Q\hbar B\rho l\omega}{2m(1+\omega^2\rho^2)}\gamma + \frac{Q^2 B^2 \rho^2}{2m(1+\omega^2\rho^2)}\gamma. \label{E:eqmotionheli}$$
After simplifying, $V$ can be shown to be given by the following expression:
$$V(\rho) = \left(-\frac{\hbar^2\omega^2}{8m} \right)\left\{\frac{1}{(1+\omega^2\rho^2)^2} + \frac{1}{1+\omega^2\rho^2}\left[1 - 4l^2 + \frac{4Q^2B^2}{\hbar^2\omega^4} \right]\right\} + \left(-\frac{\hbar^2\omega^2}{8m} \right)\left\{\frac{\rho}{1+\omega^2\rho^2}\left(-\frac{8QBl}{\hbar\omega} \right) -\frac{4Q^2B^2}{\hbar^2\omega^4} \right\}. \label{E:potentialheli}$$
To simplify the form of Eq. (\[E:potentialheli\]), we introduce the following parameters: $$\begin{aligned}
\tau &=& \frac{8QB}{\hbar}, \\
\quad \xi &=& \frac{\hbar^2}{8m}.\end{aligned}$$ We add that $\tau$ is a convenient proxy for the applied magnetic field strength. Noting that $\tau^2 = 64Q^2B^2/\hbar^2$, Eq. (\[E:potentialheli\]) becomes, in terms of $\tau$ and $\xi$,
$$\label{E:potentialhelisimple}
V(\rho) = -\xi\omega^2\left\{\frac{1}{(1+\omega^2\rho^2)^2} + \frac{1}{1+\omega^2\rho^2}\left[1 - 4l^2 + \frac{\tau^2}{16\omega^4} \right] - \frac{\rho}{1+\omega^2\rho^2}\left(\frac{\tau l}{\omega} \right) - \frac{\tau^2}{16\omega^4} \right\}.$$
Clearly the behavior of $V$ depends on the applied magnetic field strength and on the angular momentum mode of the particle. We note that when $\tau = 0$, we regain the effective potential of a particle on a helicoid $\emph{without}$ an external applied field, as found in a previous study [@vA09]. We plot the potentials for a particle on a helicoid for both positive and negative $\mathbf{B}$ in Figure 5.2 and Figure 5.3, respectively.
\[F:graph\_heli1\]
\[F:graph\_heli2\]
### Behavior of the effective potential
Let us consider the case $l = 0$. The effective potentials are identical for both positive and negative magnetic field strengths. For the zero-field case, the potential is attractive towards the origin—in fact, there is a shallow potential well near the origin. When the magnetic field strength is increased or decreased, the potential well near the origin becomes deeper, such that for extremely strong fields, the particle is practically confined at the origin.
The case $l = 1$ is more interesting. When the magnetic field is zero, the effective potential is repulsive from a certain distance from the origin onwards. If we have $QB > 0$ (that is, either $Q>0$ and $\mathbf{B}$ is in the direction of $\hat{\phi}$, *or*, equivalently, $Q<0$ and $\mathbf{B}$ is in the opposite direction) *and* the field magnitude $B$ is increased, the potential remains repulsive. The effect thus is that a charged particle is much less likely to be found near the origin. However, different behavior occurs when we have $QB<0$ and then increase the magnitude of $B$. For a sufficiently strong magnetic field with this orientation, the effective potential becomes attractive near the origin, and potential wells form. These mean that charged particles are much more likely to be found near these areas, where the effective potential is a minimum. As the magnetic field is increased in magnitude, the potential wells become deeper; for very strong magnetic fields, the particle becomes localized near (but not $\emph{at}$) the origin. Also, the distance at which the potential is a minimum becomes less when the magnetic field strength is increased. When $l = 2$, the results are for the most part qualitatively similar to that when $l = 1$—only this time, when $QB>0$, the potential is repulsive for nearly all $\rho$; and when $QB<0$, the potential minima are shifted some distance outward.
For a large ensemble of charge carriers in the non-interacting limit distributed over a helicoid, the $QB<0$ scenario would cause most of these charge carriers to be located along a narrow strip near the radius $\rho_{min}$ where the potential is a minimum. This can be realized by placing, say, negatively charged carriers such as electrons on a helicoid and applying a magnetic field $\mathbf{B}$ in the $-\hat{\phi}$ direction. If instead we have $QB>0$—which can be realized by reversing the direction of the magnetic field in the previous example—most of these charge carriers would be located far from the origin, near the outer edge of the helicoid-shaped strip, thanks to the repulsive nature of the effective potential. A possible application of this result would be the use of a magnetic field to control the flow of electric current through a nanostructure; by adjusting the applied magnetic field, one can either make the current stay on a strip near the origin, or make it localized primarily near the outer edge of the helicoid.
### Energy levels
In this subsection we perform a rudimentary calculation of the approximate energy levels of the charged particle on a helicoid in a constant circumferential magnetic field. We do this to see the dependence of the energies on the applied magnetic field strength, and this description should be applicable in general for potentials with a minimum. We restrict our analysis to the case $l = 0$, since for higher values of the angular momentum solving for the minima of the effective potentials entails solving a quartic equation. In those cases it is far more worthwhile to attack the problem using numerical methods instead.
For $l = 0$, Eq. (\[E:potentialhelisimple\]) becomes
$$\label{E:potential_0}
V(\rho) = -\xi\omega^2\left\{\frac{1}{(1+\omega^2\rho^2)^2} + \frac{1}{1+\omega^2\rho^2}\left[1 + \frac{\tau^2}{16\omega^4} \right] - \frac{\tau^2}{16\omega^4} \right\}.$$
We expand Eq. (\[E:potential\_0\]) as a Taylor series about its minimum. Evidently it has a minimum at $\rho = 0$, as can be seen in Figs. (5.2) and (5.3). We consider only small deviations from $\rho = 0$ so that we can ignore third- and higher-order terms in the Taylor expansion. The effective potential near $\rho = 0$ is thus given by
$$\label{E:potentialapprox}
V(\rho) \approx -2\xi\omega^2 + \xi\omega^4\left(3 + \frac{\tau^2}{16\omega^4}\right)\rho^2.$$
Replacing the exact potential with the approximate one, we have the following expression for the Schrodinger equation:
$$\label{E:schrodingersimple}
-\frac{\hbar^2}{2m}\frac{d^2\gamma}{d\rho^2} + \xi\omega^4\left(3 + \frac{\tau^2}{16\omega^4}\rho^2\right)\gamma = \left(E + 2\xi\omega^2\right)\gamma.$$
Eq. (\[E:schrodingersimple\]) is similar in form to the Schrodinger equation for a particle in a harmonic oscillator potential $\frac{1}{2} m\omega^2_0\rho^2$ (with characteristic frequency $\omega_0$), provided that $$\label{characteristicfrequency}
\omega_0 = \sqrt{\frac{2\xi\omega^4}{m}\left(3 + \frac{\tau^2}{16\omega^4}\right)}.$$ The energy levels for this system are thus similar to that of a particle in a harmonic oscillator potential. We have $$\label{energylevels}
E_n = \hbar\sqrt{\frac{2\xi\omega^4}{m}\left(3+\frac{\tau^2}{16\omega^4}\right)}\left(n + \frac{1}{2}\right) - 2\xi\omega^2,$$ or, expressing this in terms of the original variables $Q$ and $B$, $$\label{origenergylevels}
E_n = \frac{\hbar^2\omega^2}{2m}\sqrt{3 + \frac{4Q^2B^2}{\hbar^2\omega^4}}\left(n + \frac{1}{2}\right) - \frac{\hbar^2\omega^2}{4m}.$$ The term $-\hbar^2\omega^2/4m$ in the energies is due to the constant term $-\tau^2/16\omega^4$ in the effective potential given in Eq. (\[E:potentialhelisimple\]), which adds a time-dependent phase factor in the wavefunction but has no effect on the expectation values of any dynamical variable.
Eq. (\[energylevels\]) gives the approximate energy levels for a charged particle with angular momentum corresponding to $l = 0$. Clearly, increasing $\mathbf{B}$ increases the energy $E_n$ for all $n$; for large $B$ the energy scales as $E \sim B$.
Magnetic field from a current-carrying wire
-------------------------------------------
Perhaps the most physically relevant and experimentally realizable field configuration is that of a magnetic field produced by a current-carrying wire. We place the wire in such a way that it is in the middle of the helicoid; that is, the wire coincides with the $z$-axis. If a current $I$ in the $z$-direction flows through the wire, then it will create a circumferential magnetic field, given by $$\label{E:magneticfieldwire}
\mathbf{B} = \frac{\mu_0 I}{2\pi\rho}\hat{\phi}.$$ For convenience we let $$\delta = \frac{\mu_0}{2\pi}I.$$ The components of the vector potential $\mathbf{A}$ that gives rise to this magnetic field configuration are given by $$\begin{aligned}
A_z &=& -\delta\ln\rho, \nonumber \\
A_{\rho} &=& 0. \label{E:vectorpotentialwire}\end{aligned}$$ Repeating the same steps done in the previous section, we have the following Schrodinger equation for the surface wavefunction $\chi_s$:
$$E\chi_s = -\frac{\hbar}{2m}\left(\frac{\partial^2\chi_s}{\partial\rho^2} + \frac{1}{1+\omega^2\rho^2} \frac{\partial^2\chi_s}{\partial z^2}\right) - \frac{\hbar^2}{2m}\left[\frac{\omega^2}{2(1+\omega^2\rho^2)^2} \left(1 + \frac{\omega^2\rho^2}{2}\right) \right] \chi_s - \frac{2iQ\hbar\delta\ln\rho}{2m(1+\omega^2\rho^2)} \frac{\partial\chi_s}{\partial z} + \frac{Q^2\delta^2(\ln\rho)^2}{2m(1+\omega^2\rho^2)} \chi_s. \label{E:schrodingereqwire}$$
As before, we exploit the periodicity of the system and the absence of $z$-dependent terms in Eq. (\[E:schrodingereqwire\]). We introduce the ansatz $\chi_s = e^{ikz}\gamma(\rho)$ and substitute this into the Schrodinger equation. After a lot of algebra, we have
$$\label{E:schrodingereqfinal}
-\frac{\hbar}{2m}\frac{d^2\gamma}{d\rho^2} - \frac{\hbar^2\omega^2}{8m}\left[\frac{1}{(1+\omega^2\rho^2)^2} + \frac{1 - 4l^2}{1+\omega^2\rho^2}\right]\gamma + \frac{2Q\delta\hbar\omega l \ln(\rho) }{m(1+\omega^2\rho^2)}\gamma + \frac{Q^2\delta^2(\ln\rho)^2}{2m(1+\omega^2\rho^2)}\gamma = E\gamma.$$
All the terms in the left hand side except the first represent the effective potential $V$ of the system. If we introduce the dimensionless variables $$\begin{aligned}
\zeta = 8Q\delta/\hbar, \nonumber \\
\lambda ' = \hbar^2/8m, \end{aligned}$$ it can be shown that $V$ can be expressed in the simpler form $$\label{E:potentialwire}
V = -\lambda ' \omega^2\left[\frac{1}{(1+\omega^2\rho^2)^2} + \frac{1 - 4l^2 - \frac{\zeta l \ln\rho}{\omega} - \frac{\zeta^2(\ln\rho)^2}{16\omega^2} }{1+\omega^2\rho^2} \right].$$ In Figure 5.4 we plot the effective potential of the particle for a variety of angular momentum modes and magnitudes of the source current (as represented by the variable $\delta$).
\[F:graph\_3\]
When $l = 0$, the zero-field potential is attractive throughout, and is a minimum at the origin. Let us consider the case where $\zeta > 0$ (that is, either $I>0$ and $Q>0$, or $I<0$ and $Q<0$. We would then see that radius at which the effective potential is a minimum is shifted outward. For a wide range of current magnitudes, it is found that potential wells are formed near the origin. When the magnitude of the source current is increased, the radius at which the potential is a minimum is moved farther away from the center of the helicoid. For a large ensemble of charge carriers, the application of a magnetic field from a line current source would mean that the charge carriers will be clustered along a strip where the effective potential is a minimum.
The effective potentials for the higher angular momentum modes $l = 1$ and $l = 2$ display interesting behavior as well. The effective potential when there is no magnetic field is repulsive throughout the helicoid—which means that the particle will be more likely to be located on the outer rim of the helicoid. The addition of a magnetic field from a current source however causes the potential to be attractive near the origin and repulsive far from it. In these higher angular momentum modes the potential wells are much deeper and narrower than in the $l = 0$ case. The distances at which the effective potentials are minima for these angular momentum modes are also much smaller than in the $l = 0$ case. These properties are especially apparent in the $l = 2$ case, where the potential wells near the origin are much deeper and narrower than those corresponding to the $l = 0$ and $l = 1$ modes.
This implies that for higher angular momentum modes, the particle is effectively localized along a thin strip of the helicoid corresponding to where the potential is a minimum. If we consider a large number of charge carriers, this would mean that we will see most of the charged particles to be clustered around a narrow strip of the helicoid. For higher angular momentum modes the width of this strip is much smaller than in the $l = 0$ mode, and the strip will be found much closer to the center of the helicoid.
This system can be realized experimentally by creating a helical nanostructure, putting charge carriers on the nanostructure, and placing a tiny current-carrying wire through the center of the helicoid. Since the resulting potential wells are deep, the application and modulation of a magnetic field provides a method for the control of current in this helicoid-shaped nanostructure.
Conclusions
===========
We have considered the quantum mechanics of a charged particle on a helicoid in an external magnetic field. We derive the equations of motion for the wavefunction of a charged particle confined for two simple circumferential magnetic field configurations, and from these obtain expressions for the effective potentials that govern the behavior of the particle. We plot these potentials and examine their behavior for different values of angular momentum and the applied magnetic field strength. We were also able to derive approximate expressions for its energy levels for certain values of the angular momentum, to see the dependence of the energies on the applied magnetic field strength.
These effective potentials are dependent on both the surface curvature and the strength of the applied magnetic field. By changing the strength of the magnetic field, we observe that it is possible to change the qualitative behavior of the potential, which can range from repulsive to attractive for varying magnetic field strength and direction. Meanwhile, the energy levels of the charged particle on a helicoid are found to be similar in form to that of a particle in a harmonic oscillator potential, for potentials which have a minimum. For the case of a large ensemble of charge carriers on a helicoid-shaped nanostructure in the non-interacting limit, the application of an applied magnetic field would lead, for a range of magnetic field strengths, to the clustering of charge carriers on a strip near areas where the effective potential is a minimum. By increasing or decreasing the magnetic field strength, we find that it is possible to control where the strip is located and how thick this strip is.
There are many avenues for further research in this topic. A lot of electric and magnetic field configurations which do not yield separable equations of motion require a closer examination, ideally with numerical methods. Also, the effect of the curvature on the nonrelativistic quantum dynamics of charged particles has not yet been studied for many surfaces. More complex effects arising from the combination of the geometric potential and the electromagnetic field may result.
M.A.S. is very grateful to Germelino Abito for his support and advice in the initial stages of this project, and to Raphael Guerrero and Job Nable for very helpful discussions.
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|
---
abstract: 'I review recent developments in the direct and indirect detection of dark matter and new candidates beyond the WIMP paradigm.'
author:
- 'Jonathan L. Feng'
title: 'Dark Matter Phenomenology[^1]'
---
[ address=[Department of Physics and Astronomy, University of California, Irvine, California 92697, USA]{} ]{}
WIMP Dark Matter
================
In recent years, the amount of dark matter in the Universe has become precisely known, but its particle identity remains a mystery. Current observational constraints require that the bulk of dark matter be non-baryonic, cold or warm, and stable or long-lived. These constraints are easy to satisfy, and viable candidates have been proposed with masses and interaction strengths that span many, many orders of magnitude.
At the same time, there are strong reasons to focus on candidates with masses around the weak scale ${m_{{{\rm weak}}}}\sim 100~{{{\rm GeV}}}$. Despite significant progress since this scale was first identified in the work of Fermi in the 1930’s, the origin of ${m_{{{\rm weak}}}}$ is still unknown, and every attempt to understand it so far introduces new particles at this scale.
Furthermore, the relic density of such particles naturally reproduces the required dark matter density. If a new (heavy) particle $X$ is initially in thermal equilibrium, it can be shown that its relic density today is $$\Omega_X \propto \frac{1}{\langle \sigma_{{{\rm ann}}} v \rangle}
\sim \frac{m_X^2}{g_X^4} \ ,
\label{omega}$$ where $\langle \sigma_{{{\rm ann}}} v \rangle$ is the thermally-averaged annihilation cross section, and we have parametrized it in terms of a mass scale $m_X$ and coupling constant $g_X$ that characterize $XX$ annihilation. Including all relevant numerical factors, for weakly-interacting massive particles (WIMPs) with $m_X \sim {m_{{{\rm weak}}}}$ and $g_X \sim {g_{{{\rm weak}}}}\simeq 0.65$, the resulting relic density is $\Omega_X \sim 0.1$, near the required value for dark matter $\Omega_{{{\rm DM}}} \simeq 0.23$. This remarkable coincidence is the “WIMP miracle.” It implies that particle physics theories designed to explain the origin of the weak scale often naturally contain particles with the right relic density to be dark matter.
For WIMPs $X$ to have the right relic density, they must annihilate through $XXff$ interactions, where, in the simplest cases, $f$ denotes any of the known standard model particles. This implies that dark matter can be detected through $Xf \to Xf$ scattering (direct detection), or through its annihilations $XX \to f \bar{f}$ (indirect detection). Both strategies are currently vigorously pursued, and some recent highlights are briefly reviewed in the next two sections.
Direct Detection
================
Dark matter scattering is either spin-independent or spin-dependent [@Goodman:1984dc]. Current bounds and supersymmetric predictions for spin-independent scattering off nucleons are shown in [Fig. \[fig:direct\]]{}. As can be seen, current bounds do not test the bulk of supersymmetric parameter space. The experiments are improving rapidly, however, and in the coming year, sensitivities to cross sections of ${\sigma_{{{\rm SI}}}}\sim 10^{-45}~{{{\rm cm}}}^2$ are possible.
![Current bounds [@Gaitskell] on spin-independent WIMP-nucleon cross sections ${\sigma_{{{\rm SI}}}}$ from XENON10 [@Angle:2007uj] (solid red) and CDMS [@Ahmed:2008eu] (dashed blue), along with predictions from the general minimal supersymmetric standard model (MSSM) [@Baltz:2002ei] (shaded) and minimal supergravity [@Baltz:2004aw] (outlined). \[fig:direct\]](direct){height=".3\textheight"}
How significant will this progress be? As evident in [Fig. \[fig:direct\]]{}, the full range of supersymmetric predictions will not be probed for the foreseeable future. However, many well-known supersymmetric theories predict ${\sigma_{{{\rm SI}}}}\sim 10^{-44}~{{{\rm cm}}}^2$. Supersymmetric theories suffer from flavor and CP problems: the introduction of squarks and sleptons with generic flavor mixing and weak scale masses induces contributions to $K-\bar{K}$ mixing, $\mu \to e \gamma$, the electric dipole moments of the neutron and electron, and a host of other flavor- or CP-violating observables that badly violate known constraints. One generic solution to this problem is to assume heavy squarks and sleptons, so that they decouple and do not affect low-energy observables.
In general, the dominant contributions to neutralino annihilation are $\chi \chi \to q \bar{q}, l \bar{l}$ through $t$-channel squarks and sleptons, and $\chi \chi \to W^+ W^-, Z Z$ through $t$-channel charginos and neutralinos. In these decoupling theories, the first diagrams are ineffective, and so annihilation takes place through the second class of diagrams. Essentially, two parameters enter these diagrams: the neutralino’s mass and its Higgsino content. To keep the relic density constant, larger $\chi$ masses are compensated by larger Higgsino components. In these models, then, the supersymmetry parameter space is greatly reduced, with ${\sigma_{{{\rm SI}}}}$ essentially determined by the $\chi$ mass. More detailed study shows that ${\sigma_{{{\rm SI}}}}$ is in fact fairly constant, with values near $10^{-44}~{{{\rm cm}}}^2$, irrespective of mass.
In the next year or so, then, direct detection will probe many well-known supersymmetric models with widely varying motivations, from focus point models [@Feng:2000gh] to split supersymmetry [@Pierce:2004mk]. So far, direct detection experiments have trimmed a few fingernails off the body of supersymmetry parameter space, but if nothing is seen in the coming few years, it is arms and legs that will have been lopped off.
In addition to the limits described above, the DAMA experiment continues to find a signal in annual modulation [@Drukier:1986tm] with period and maximum at the expected values [@Bernabei:2008yi]. From a theorist’s viewpoint, the DAMA/LIBRA result has been puzzling because the signal, if interpreted as spin-independent elastic scattering, seemingly implied dark matter masses and scattering cross sections that have been excluded by other experiments. Inelastic scattering has been put forward as one solution [@TuckerSmith:2001hy]. More recently, astrophysics [@Gondolo:2005hh] and channeling [@Drobyshevski:2007zj; @Bernabei:2007hw], a condensed matter effect that effectively lowers the threshold for crystalline detectors, have been proposed as possible remedies to allow elastic scattering to explain DAMA without violating other constraints. If these indications are correct, the favored parameters are $m_X \sim
5~{{{\rm GeV}}}$ and ${\sigma_{{{\rm SI}}}}\sim 10^{-39}~{{{\rm cm}}}^2$. This mass is lower than typically expected, but even massless neutralinos are allowed if one relaxes the constraint of gaugino mass unification [@Dreiner:2009ic]. The cross section is, however, very large; it may be achieved in corners of MSSM parameter space [@Bottino:2007qg], but is more easily explained in completely different frameworks, such as those discussed below.
Indirect Detection
==================
Indirect searches look for particles produced when dark matter particles decay or pair annihilate. In contrast to direct detection, there have been many reported anomalies in indirect detection, which have been interpreted as possible evidence for dark matter.
![The positron fraction measurements and predictions of pulsars with various parameters [@Grasso:2009ma] (left) and the total $e^+ +e^-$ flux measured by ATIC, Fermi, and other experiments [@Abdo:2009zk] (right). \[fig:fermi\]](pos_ratio_pulsars "fig:"){height=".25\textheight"} ![The positron fraction measurements and predictions of pulsars with various parameters [@Grasso:2009ma] (left) and the total $e^+ +e^-$ flux measured by ATIC, Fermi, and other experiments [@Abdo:2009zk] (right). \[fig:fermi\]](fermi "fig:"){height=".25\textheight"}
Recently, the PAMELA and ATIC Collaborations have measured the positron fraction $e^+/(e^+ + e^-)$ [@Adriani:2008zr] and the total $e^+ + e^-$ flux [@:2008zzr], respectively, at energies in the range of 10 GeV to 1 TeV. These measurements revealed excesses above the expected cosmic ray background from GALPROP [@Strong:2009xj], which have been interpreted as dark matter signals. The PAMELA results are, however, consistent with astrophysical sources, such as the predicted fluxes from pulsars, as derived both long before and after [@Hooper:2008kg; @Yuksel:2008rf; @Profumo:2008ms; @Malyshev:2009tw; @Grasso:2009ma] the release of the PAMELA data. Results from a recent study scanning over pulsar characteristics are given in [Fig. \[fig:fermi\]]{}. The ATIC results have now been supplemented by high statistics data from the Fermi LAT Collaboration [@Abdo:2009zk], which sees no evidence for a bump (see [Fig. \[fig:fermi\]]{}). Additional data on both cosmic rays and gamma rays will provide further insights ([[*e.g.*]{}]{}, recent gamma ray data already further disfavor some dark matter proposals [@Profumo:2009uf]) and are eagerly anticipated from these experiments and many others.
Hidden Dark Matter
==================
The DAMA and other anomalies are not easy to explain with canonical WIMPs. This has motivated new candidates. All solid evidence for dark matter is gravitational, and there is also strong evidence against dark matter having strong or electromagnetic interactions. A logical alternative to the WIMP paradigm, then, is hidden dark matter, that is, dark matter that has no standard model gauge interactions. Hidden dark matter has been explored for decades [@Kobsarev:1966]. By considering this possibility, though, one seemingly loses (1) a connection to central problems in particle physics, such as the problem of electroweak symmetry breaking, (2) the WIMP miracle, and (3) the non-gravitational signals discussed above, which are most likely required if we are to identify dark matter.
In fact, however, hidden dark matter may have all three of the virtues listed above. Consider, for example, supersymmetric theories with gauge-mediated supersymmetry breaking (GMSB) [@Dine:1994vc; @Dine:1995ag]. These models preserve the many virtues of supersymmetry, while elegantly solving the flavor and CP problems mentioned above. Although minimal GMSB models contain only a supersymmetry-breaking sector and the MSSM, in models that arise from string theory, hidden sectors are ubiquitous. As a concrete example, we consider GMSB with an additional hidden sector, as depicted in [Fig. \[fig:sectors\]]{}.
In these models, supersymmetry is broken when a chiral field in the supersymmetry breaking sector gets a vacuum expectation value $\langle
S \rangle = M + \theta^2 F$. The resulting superpartner masses in the MSSM and hidden sectors are $m \sim \frac{g^2}{16 \pi^2} \frac{F}{M}$ and $m_X \sim \frac{g_X^2}{16 \pi^2} \frac{F}{M}$, and so $$\frac{m_X}{g_X^2} \sim \frac{m}{g^2} \sim
\frac{F}{16 \pi^2 M} \ ;
\label{mxgx}$$ that is, $m_X/g_X^2$ is determined solely by the supersymmetry-breaking sector. As this is exactly the combination of parameters that determines the thermal relic density of [Eq. (\[omega\])]{}, the hidden sector automatically includes a dark matter candidate that has the desired thermal relic density, irrespective of its mass. This is the “WIMPless miracle” — in these models, a hidden sector particle naturally has the desired thermal relic density, but it has neither a weak-scale mass nor weak force interactions [@Feng:2008ya; @Feng:2008mu].
The WIMPless framework opens up many new possibilities for dark matter signals without sacrificing the main motivations for WIMPs. For example, if there is an unbroken U(1) gauge symmetry in the hidden sector, these dark matter particles self-interact through Rutherford scattering, with a wide range of observable implications [@Ackerman:2008gi; @Feng:2009mn].
![Direct detection cross sections for spin-independent $X$-nucleon scattering as a function of dark matter mass $m_X$. The black lines are Super-K limits [@Desai:2004pq] and projected sensitivities. The magenta shaded region is DAMA-favored given channeling and no streams [@Petriello:2008jj], and the medium green shaded region is DAMA-favored at 3$\sigma$ given streams but no channeling [@Gondolo:2005hh]. The light yellow shaded region is excluded by the direct detection experiments indicated. The predictions of WIMPless models (with connector mass $m_Y = 400~{{{\rm GeV}}}$ and $0.3 < \lambda_b < 1$) lie in the blue shaded region. From Ref. [@Feng:2008qn]. \[fig:superkdirect\]](superkdirect_no_chi){height=".3\textheight"}
Alternatively, if there are connector particles, WIMPless dark matter can explain the DAMA/LIBRA signal [@Feng:2008dz]. Suppose the dark matter particle $X$ couples to standard model fermions $f$ via exchange of a connector particle $Y$ through interactions ${\cal L} =
\lambda_f X \bar{Y}_L f_L + \lambda_f X \bar{Y}_R f_R$. The Yukawa couplings $\lambda_f$ are model-dependent. If the dark matter couples mainly to 3rd generation quarks, the dominant nuclear coupling of WIMPless dark matter is to gluons via a loop of $b$-quarks. The predictions of this WIMPless dark matter model are given in [Fig. \[fig:superkdirect\]]{}. In this WIMPless scenario, the small mass required by DAMA is as natural as any other because of the WIMPless miracle. The large cross section is also easily achieved, because the heavy $Y$ propagator flips chirality without Yukawa suppression, in contrast to the case of neutralinos. This explanation is testable through searches for exotic 4th generation quarks at the Tevatron and LHC.
I thank my collaborators for their contributions to the work discussed here, and Marvin Marshak and the organizers of CIPANP 2009 for a beautifully organized conference. This work was supported in part by NSF grant PHY–0653656.
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|
---
abstract: 'Starting with the well-known NFW dark matter halo distribution, we construct a simple polytropic model for the intracluster medium which is in good agreement with high resolution numerical hydrodynamical simulations, apply this model to a very large scale concordance dark matter simulation, and compare the resulting global properties with recent observations of X-ray clusters, including the mass-temperature and luminosity-temperature relations. We make allowances for a non-negligible surface pressure, removal of low entropy (short cooling time) gas, energy injection due to feedback, and for a relativistic (non-thermal) pressure component. A polytropic index $n=5$ ($\Gamma=1.2$) provides a good approximation to the internal gas structure of massive clusters (except in the very central regions where cooling becomes important), and allows one to recover the observed $M_{500}-T$, $L_x-T$ and $T/n_e^{2/3}\propto T^{0.65}$ relations. Using these concepts and generalizing this method so that it can be applied to fully three-dimensional N-body simulations, one can predict the global X-ray and SZE trends for any specified cosmological model. We find a good fit to observations when assuming that twelve percent of the initial baryonic mass condenses into stars, the fraction of rest mass of this condensed component transferred back to the remaining gas (feedback) is $3.9\times10^{-5}$, and the fraction of total pressure from a nonthermal component is near ten percent.'
author:
- 'Jeremiah P. Ostriker and Paul Bode'
- Arif Babul
title: 'A Simple and Accurate Model for Intra-Cluster Gas '
---
Introduction {#sec:intro}
============
Gas in clusters of galaxies can be observed to large cosmological distances by a variety of techniques, from X-rays (Bremsstrahlung) to radio (S-Z effect). But to utilize these observations it is necessary to have a model for the state of the gas in clusters that is (a) motivated by sound physical reasons, (b) able to fit the observational data and (c) is simple enough (i.e., [*not*]{} a numerical simulation!) to be applied broadly. In this paper we attempt to present such a model for the gaseous component in clusters of galaxies in order to provide predictions for global properties such as temperature and X-ray luminosity.
For the dark matter (DM), the widely utilized @NFW, or NFW, model satisfies all of the above criteria. Although we now know that it is not universal in two senses— large variance in the central concentrations [@AFKK99; @Jing00; @BKSSKKPD01; @KKBP01; @FKM04; @TKGK04] and trends in properties with time and halo mass [@WBPKD02; @Ricotti03; @ZJMB03; @WOB05; @SMS05]— it remains an extremely useful first basis for analyzing and summarizing the properties of dark matter distributions. We know, however that the gas in clusters does [*not*]{} follow the dark matter profile. The well established central density profiles for dark matter halos are roughly power laws: density $\rho$ depends on radius $r$ as $\rho \propto r^{-\alpha}$, with $\alpha$ typically 1.0 (the NFW value) but ranging from 0.5 to 1.5, depending on circumstances. But the gas profiles show a definite core [$\alpha \rightarrow 0$; for a recent review see @Voit04] and, as we shall see, one would overestimate the X-ray luminosity by a large factor if one were to use the NFW or a steeper profile.
The construction of a satisfactory model will be guided by a few observed properties. First, the gas is essentially a trace component (approximately 1/7 of total mass) and resides, in close to hydrostatic equilibrium, in a potential which is well represented by NFW [or its variants– cf. @Zhao96]. Furthermore, we know that there are efficient means of redistributing energy/entropy within the cluster gas via, for example, turbulence [@KN03b] induced by merger shocks [@BN98] and galaxy wakes [@SAP99; @Sakelliou00]. Additionally, other processes such as conduction [@KN03a; @DJSBR04], cosmic ray transport, and magneto-sonic wave transport [@Cen05] may also be operating. Also, appropriate boundary conditions are required, since in both simple analytic models [eg. @Bert85] and detailed numerical simulations [eg. @BN98; @Frenk99] the hydrostatic portion of the cluster gas is terminated at an outer shock where the pressure is balanced by the momentum flux of the infalling gas.
A number of steps have already been made toward constructing such a model. @MSS98 derived an analytic expression for a gas distribution in hydrostatic equilibrium with an NFW potential, assuming isothermality. A more general expression for a polytropic equation of state was derived by @SSM98; a similar functional form has been compared in detail with hydrodynamic simulations by @AYMG03.
In the simplest of such models the source of the gas heating is gravitational, i.e. the gas energy comes from the same collapse and virialization processes which determine the dark matter profile; thus the energy per unit mass in the gas should be approximately the same as in the dark matter. This leads to an expected self-similar scalings of mass $M$ and luminosity $L$ with temperature $T$ of $M\propto T^{3/2}$ and $L\propto T^2$ [@Kaiser86]. However, this expectation is in contradiction with the observed relation, which is steeper [@EdgeStew91; @Markev98]. @Kaiser91 proposed that non-gravitational energy injection could lead to the observed relations. This possibility has been explored in the type of analytic model described here using an NFW profile for the dark matter [@SSM98; @WFN00; @SKSS04; @LCM05; @ALS05; @SMGS05], and with other profiles [@BBP99; @BBLP02]. The breaking of self-similarity can also be cast as the modification of the initial gas entropy by thermal and nonthermal processes, as explored in NFW-like potentials by @TozNor01, @KomSel01, @VBBB02, and @DosSD02; on the other hand @RoyNath03 argue that the entropy imparted to the gas from gravitational processes alone is larger than previously thought. Another impact on the gas energy comes from the fact that approximately one tenth of the baryons in a typical cluster are now in stellar form. So one must allow for both removal of the mass of this gas and of the associated energy (or entropy) of this gas [@VB01; @TozNor01; @VBBB02; @ScanOh04; @BV05]. Since the removed gas had short cooling times, low entropy, and low total energy, the mean energy per unit mass of the remaining gas is higher than before star and galaxy formation.
An issue not dealt with in these studies is non-thermal sources of pressure. Turbulence may provide in excess of 10% of the total pressure in Coma [@SFMBB04]; similar amounts of turbulent support have been seen in simulations [@NorBry99; @FKNG04]. Clusters should also contain a population of relativistic particles arising from shocks, as recently reviewed by @Miniati04, @Sarazin04, and @Bykov05. Magnetic fields may also be dynamically important [@CarTay02; @EVP05; @Bykov05].
The basic goal of this paper is to start with a population of dark matter halos from an N-body simulation, for which the DM density profiles can easily be measured, and deduce the global properties of the hot baryonic component in a physically well-motivated manner. The ideal method should be as simple as possible while including all the relevant components: hydrostatic equilibrium inside a dark matter halo potential; gas energy per unit mass similar to that of the dark matter, but modified by removal of low entropy gas and by feedback; appropriate outer boundary conditions; and pressure support from a non-thermal component. Some other processes necessary for detailed models will not be included because they are not in general required for obtaining global properties; though the results obtained here may need to be modified for those clusters having distinctly cooler cores [@AllFab98]. Finally, we will drop the limitation of a spherical NFW model and generalize to any case for which the dark matter potential is known. While several of the papers quoted above have allowed for some of these effects, none has included all in a fashion that can be adapted to an arbitrary gravitational potential.
The next section reviews properties of the NFW model; §\[sec:igasep\] derives the properties of an initially parallel gaseous component and §\[sec:poly\] derives these properties after the gas rearranges itself; §\[sec:constr\] presents the resulting profiles and compares global properties with observed clusters. All of these sections assume spherical symmetry. In §\[sec:3d\] we generalize the polytropic model in order to remove geometrical constraints, concluding our discussion in §\[sec:conclusion\].
The NFW Profile {#sec:nfw}
===============
@NFW have proposed a universal profile for dark matter halos, which we will first review here to establish nomenclature. Formally, the NFW profile extends to infinity and has logarithmically diverging mass; we will instead truncate the profile at the virial radius. In this section we first establish the properties of a distribution of matter with a truncated NFW profile. Assume that the density $\rho$ depends on radius $r$ as $$\label{eqn:rho}
\rho(r) = \left\{ \begin{array}{ll}
\frac{\rho_1r_1^3}{r(r+r_1)^2} & r\leq r_v \hspace{1cm} ,\\
0 & r>r_v
\hspace{1cm} .
\end{array} \right.$$ The virial radius $r_v$ is the radius within which the mean density is 200 times the critical density $\rho_c$. The parameters $C \equiv r_v/r_1$ and $\rho_1$ define the model. It follows that the mass $M$ is distributed inside $r_v$ as:
$$\begin{aligned}
\label{eqn:mass}
M(r)&=& 4 \pi\ \rho_1 r_1^3\ g(x) \hspace{1cm}, \\
g(x)&\equiv& \ln(1+x)-\frac{x}{(1+x)} \hspace{1cm},\end{aligned}$$
where $x\equiv r/r_1$. Thus $M_{tot} \equiv M(r_v) = 4\pi \ \rho_1 r_1^3\ g(C)$ is the total mass. The rotation curve, $V_c^2(r)=GM(r)/r=4 \pi G \rho_1 r_1^2 g(x)/x$ provides a useful label for the mass distribution; it has a maximum at $x_{c,max}\approx 2.163$, so the maximum circular velocity is
$$\begin{aligned}
\label{eqn:circvmax}
V_{c,max}^2&=&4 \pi G \rho_1 r_1^2 G_{max} \hspace{1cm}, \\
G_{max}&\equiv& g(x_{c,max})/x_{c,max}\approx 0.2162
\hspace{1cm} .\end{aligned}$$
Note that a given $M_{tot}$ and $V_{c,max}$ can be used to completely define the matter distribution (as an alternative to $C$ and $\rho_1$).
The gravitational potential from this mass distribution is $$\label{eqn:phi}
\Phi(x) = -\frac{V_{c,max}^2 }{G_{max}}f(x)
\hspace{1cm} ,$$ where $$f(x) = \left\{ \begin{array}{ll}
\frac{\ln (1+x)}{x} -\frac{1}{1+C} & x\leq C \\
\left[ \frac{\ln (1+C)}{C} -\frac{1}{1+C} \right]\frac{C}{x} & x>C \\
\end{array} \hspace{1cm} \right.$$ The total gravitational energy is then given by
$$\label{eqn:W}
W_0 = \frac{1}{2}\int_0^{M_{tot}}{\Phi(r)dM}
= - G_{max}^{-1}V_{c,max}^2 M_{tot} H(C) \hspace{1cm} ,$$
where $$H(C)\equiv \left[- \frac{\ln (1+C)}{(1+C)}+
\frac{C(1+C/2)}{(1+C)^2} \right] \frac{1}{g(C)} \hspace{1cm} .$$
Assuming velocities are isotropic for the bulk of the matter distribution, the velocity dispersion of the dark matter in 1-D, $\sigma^2(r)$, obeys the equation $\frac{d}{dr}(\rho \sigma^2)=-\rho\frac{d}{dr}\Phi$, which has the solution
$$\label{eqn:vdisp}
\sigma^2= \frac{V_{c,max}^2 }{G_{max}} S_C(x)x(1+x)^2 \hspace{1cm} ,$$
where $$\begin{aligned}
\label{eqn:sofx}
S_C(x)= S_C(C) - \int_x^C{\frac{x' - (1+x')\ln (1+x')}{x'^3(1+x')^3}}dx'
\hspace{1cm} ,\end{aligned}$$
with $S_C(C)$ a positive constant, the value of which will be determined below. The corresponding pressure is simply $P=\rho \sigma^2 = 4 \pi G \rho_1^2 r_1^2 S_C(x)$, so the average pressure is
$$\label{eqn:pdef}
\overline{P} = \frac{1}{\frac{4}{3}\pi r_v^3} \int_0^{r_v}{ P d {\bf r}}
= {12 \pi G \rho_1^2 r_1^2 C^{-3}}\int_0^C{S_C(x)x^2 dx} \hspace{1cm} ,$$
and the surface pressure is $$\label{eqn:psurf}
P_s = 4 \pi G \rho_1^2 r_1^2 S_C(C) \hspace{1cm} .$$
A boundary surface pressure is required because of the jump in density and pressure at $r=r_v$, and would be provided in any realistic physical model by the momentum flux from infalling matter at the boundary. One way to quantify the pressure term represented by $S_C(C)$ is to estimate the momentum per unit area transported in by infalling matter. If the rate of mass accretion is $\dot{M}_{tot}$ and the accreted mass (starting at rest from the turnaround radius) is moving at freefall velocity $v^2_{ff}=V^2_c(r_v)$, then $ P_s = \dot{M}_{tot} v_{ff} / (4 \pi r_v^2)$. In the self-similar solution of @Bert85 $M_{tot} \propto t^{2/3}$. This should be approximately correct, so we will take $\dot{M}_{tot} = 2qM_{tot}/(3t)$, with deviations allowed for with use of the correction factor $q$. Clusters of galaxies are by definition in overdense regions, so the surroundings will always appear to be close to the critical density $\rho_c$, hence an appropriate time is $t = (6\pi G\rho_c)^{-1/2}$; thus we will adopt $$\label{eqn:surfpres}
P_s = \frac{\sqrt{2}}{3}qV^2_c(r_v) \rho_s
\left( \frac{\rho_c}{\rho_s} \frac{\bar{\rho}}{\rho_s} \right)^{1/2}
\hspace{1cm} ,$$ where $\rho_s$ is the density at $r_v$, and $\bar{\rho}$ is the mean density inside $r_v$. Combining this with Eqn. (\[eqn:psurf\]), it follows that $$\label{eqn:scofc}
S_C(C) = \left( \frac{2}{3}q^2 \frac{g^3(C)}{C^5} \frac{\rho_c}{\rho_1}
\right)^{1/2}
=\sqrt{2}q\frac{g^2(C)}{C^4}\left(\frac{\bar{\rho}}{\rho_c}\right)^{-1/2}
=q\frac{g^2(C)}{10C^4}
\hspace{1cm} .$$
Alternatively, one may take the expression for the radial velocity dispersion in an NFW halo derived by @LM01 and evaluate it at the virial radius $x=C$. @LM01 solved the Jeans equation for an NFW density profile assuming a constant velocity anisotropy; for the simplest case of isotropic orbits, their Eqn. (14) yields $$\label{eqn:lmscofc}
\begin{array}{c} S_C(C) =
\frac{\pi^2}{2} - \frac{{\rm ln}C}{2} - \frac{1}{2C} -
\frac{1}{2(1+C)^2} - \frac{3}{1+C} +
\left( \frac{1}{2} + \frac{1}{2C^2} - \frac{2}{C} - \frac{1}{1+C} \right)
{\rm ln}(1+C) + \\
\frac{3}{2}{\rm ln}^2(1+C) + 3{\rm Li}_2(-C)
\end{array}$$ with the dilogarithm Li$_2(x)=\int_x^0{\rm ln} (1-y)d{\rm ln} y$. For the low concentration halos considered here, this gives pressures similar to Eqn. (\[eqn:scofc\]) with $q\approx 4$. We will use Eqn. (\[eqn:lmscofc\]) in the following, but have found simply using $q=4$ gives very similar results. An intriguing feature of the NFW model is that the coarse-grained phase space density $\rho(x)/\sigma^3(x)$ is a power law in radius, following $x^{-15/8}$ [@TayNav01; @WABBD04]. Using Eqn. (\[eqn:lmscofc\]) to set the surface pressure, the phase space density found by combining Eqns. (\[eqn:rho\]) and (\[eqn:vdisp\]) does display in this power law behavior, with the appropriate slope. If we instead use (for example) $q=1$, then the same power law still holds inside $0.5r_v$; only nearer the surface is there a significant deviation. This insensitivity to the exact choice of surface pressure helps explain why the approximation $q\approx 4$ works well.
Initial Gas Energy and Pressure {#sec:igasep}
===============================
The goal of this paper is to populate with gas, in a physically plausible fashion, the potential well created by a dark matter halo with known mass, radius, concentration, and maximum circular velocity. Let the ratio of initial gas mass to total mass equal the cosmic average, i.e. the total gas mass $M_{g,i} = \Omega_b M_{tot}/\Omega_m$. We begin by assuming that initially the two components have a parallel distribution— which is what would be expected in the absence of energy transport mechanisms. Thus the gas pressure is by hypothesis $P_g=(M_{g,i}/M_{tot})P$.
Now we apply the Virial Theorem to the whole, with allowance for the non-negligible surface pressure: $W_0 +2T_0-4\pi r_v^3 P_s = 0$. Since the kinetic energy $T_0$ is related to the mean pressure (in the absence of significant bulk motions) by $2T_0=4\pi r_v^3 \overline{P}$, we have, using Eqn. (\[eqn:pdef\]), $$W_0 = -4\pi r_v^3 \left(\overline{P}-P_s \right)
= -3\langle \sigma^2 \rangle M_{tot}(1-\delta_s)
\hspace{1cm} ,$$ where $\delta_s\equiv P_s/\bar{P}$ is the surface pressure in terms of the mean. When combined with Eqn. (\[eqn:W\]), this becomes $$\label{eqn:sigma}
3 \langle \sigma^2 \rangle (1 - \delta_s) = G_{max}^{-1} V_{c,max}^2 H(C)
\hspace{1cm} .$$ Note this is taken from the definition of $\langle \sigma^2 \rangle
\equiv \int{\sigma^2 dM}/M_{tot}
= (\frac{4}{3}\pi r_v^3 \overline{P})/M_{tot}$. Rewriting Eqn. (\[eqn:sigma\]) using Eqns. (\[eqn:W\]) and (\[eqn:vdisp\]) gives the dimensionless form of the Virial Theorem: $$\label{eqn:sk}
3(1-\delta_s)\int_0^C{S_C(x)x^2 dx} = H(C) g(C)
\hspace{1cm} .$$ It follows that $$\label{eqn:delta}
\delta_s = \frac{S_C(C)}{S_C(C) + C^{-3}H(C)g(C)}
\hspace{1cm} .$$ Given $S_C(C)$, $\delta_s$ is known, and Eqn. (\[eqn:sigma\]) can be rewritten as $\langle \sigma^2 \rangle = K(C) V_{c,max}^2$, where $$\begin{aligned}
\label{eqn:sigc}
K(C)=\frac{1}{3}H(C)\frac{G_{max}^{-1}}{(1-\delta_s)} \hspace{1cm} ,\end{aligned}$$ and the potential energy from Eqn. (\[eqn:W\]) is now $$\label{eqn:WofC}
W_0 = -3 K(C) (1-\delta_s) M_{tot} V_{c,max}^2
\hspace{1cm} .$$ Thus the kinetic and potential energies can be specified, once the NFW parameters for a dark matter halo have been specified.
Earlier we assumed a monatomic gas component, initially distributed in the same manner as the dark matter. What is the total energy of the gas? Treating it as a tracer of negligible mass, that is, assuming the gravitational potential is totally determined by the dark matter, $E_g=\onehalf M_{g,i} \langle 3 \sigma ^2 \rangle + 2\left(
{M_{g,i}}/{M_{tot}}\right)W_0$. Combining this with Eqn. (\[eqn:WofC\]) gives $$\label{eqn:egasjnos}
E_g=-\frac{3}{2}M_{g,i}V_{c,max}^2 K(C) (3-4\delta_s)
\hspace{1cm} .$$ Also, the gas surface pressure from Eqns. (\[eqn:psurf\]) and (\[eqn:circvmax\]) is $$\label{eqn:psurfC}
P_{s,gas} = \frac{M_{g,i}V_{c,max}^2}{4\pi r_1^3}
\frac{S_C(C)}{G_{max}g(C)}
\hspace{1cm} .$$ Thus both the gas energy and surface pressure are now known in terms of the dark matter halo parameters.
Allowance for Stellar Mass Dropout {#sec:dropout}
----------------------------------
Star formation will change the amount of energy in the remaining gas, because that portion of the the gas which collapses and is removed will have a lower entropy and a shorter cooling time than is typical. In our idealized halo, this is the gas which would end up in the central region, so we will estimate the change in energy by removing a fraction of the core; this removed fraction (corresponding to the mass in stars) has the shortest cooling time and lowest entropy. @FHP98 estimated that for clusters the stellar mass inside the virial radius is roughly 0.19$h^{0.5}$ times the gas mass, and @SanPon03 find a median stellar to gas ratio of 0.21; but lower values have been found by @BPBK01 and @LMS03. We will adopt a stellar to gas mass ratio of 0.12 independent of cluster mass, in reasonable agreement with the results of the latter two papers, assuming a LCDM model with $h$=0.7. For simplicity we will assume enough gas is turned into stars for this ratio to hold generally. In other words, the ratio of the mass in stars to the final gas mass is $f_s=0.12$, with $M_g=M_{g,i}/(1+f_s)$ remaining in the gaseous state. This will be done by removing all gas inside a radius $x_s$, found by solving $g(x_s)=g(C)f_s/(1+f_s)$. The initial energy in this remaining gas is found by integrating the previous expressions for the energy over only the mass $M_g$ outside of $x_s$: $$\label{eqn:egasj}
E_g=-\frac{3}{2}(1+f_s)M_g V_{c,max}^2 \left[ K(C) (3-4\delta_s) +
K_s(x_s)/G_{max} \right]$$ with $$K_s(x_s) = \frac{1}{g(C)} \left[ \int_0^{x_s}S_C(x)x^2 dx -
\frac{2}{3}\int_0^{x_s}\frac{f(x)x dx}{(1+x)^2} \right]$$ which, while unfortunately not as simple as before, can still be determined once the dark matter halo parameters are given. By removing this gas we are both lowering the gas mass and, more critically, increasing the energy per unit mass of the remaining gas [@VB01].
Other Changes to the Gas Energy {#sec:otherc}
-------------------------------
The gas energy will change due to the the work done by any increase or decrease of the gas volume. To calculate this latter term, we will assume that the surface pressure does not change with radius, i.e. it is always given by Eqn. (\[eqn:psurfC\]). Let $C_f$ be the final outer radius of the gas distribution in units of $r_1$; then $$\label{eqn:delep}
\Delta E_P=-\frac{4\pi}{3}r^3_1\left( C^3_f-C^3 \right)P_{s,gas}(C)
=-\frac{1}{3}(1+f_s)M_gV_{c,max}^2\left( C^3_f-C^3 \right)
\frac{S_C(C)}{G_{max}g(C)}
\hspace{1cm} .$$
In addition, we can expect there to be energy input to the gas from feedback processes. These are primarily of two kinds: wind and supernova shock energy deposited in the hot gas, and heating due to output from accreting massive black holes in the centers of the massive galaxies [@ScanOh04]. Dynamical friction on galaxies moving through the gas at somewhat trans-sonic speeds may also be a source of energy input [@Ost99; @ElZKK04; @FKNG04]. To a good first approximation all of these effects are proportional to the gas mass (there is no evidence that the efficiency with which gas is transferred to stars varies strongly and systematically with $V^2_{c,max}$ for moderately rich clusters). Let $\epsilon$ be a measure of the efficiency with which gas is heated by the condensed component; the energy input can then be written as $$\label{eqn:delef}
\Delta E_f = \epsilon f_s M_g c^2
\hspace{1cm} .$$ The value of $\epsilon$ from energy output of supernovae can be estimated as the product of the fraction of mass turned into stars ($f_s$=0.12), the number of supernovae per solar mass expected from a Salpeter initial mass function ($0.007M_\odot^{-1}$), and the energy input per supernova ($10^{51}$ erg); this gives $\epsilon=2.8\times 10^{-6}$ or $\epsilon f_s=3\times 10^{-7}$. While assuming perfect efficiency in transferring this energy into the gas is unrealistic, we find that by itself this amount of energy has little impact on the gas profile, consistent with previous results [@BBP99; @VS99; @BBLBCF01; @SKSS04]. The energy input from AGN is more substantial. For a galaxy hosting a supermassive black hole, the ratio of black hole to stellar mass will be $\approx$ 0.0013 [@KG00; @MerFer01], so the ratio of black hole to total gas mass will be roughly $1.6\times 10^{-4}$. Observational constraints give the efficiency with which energy is released from accreting black holes to be $\approx 0.10$ [@YuT02]; adopting a conversion efficiency to mechanical energy of 30% [@InSa01] leads to an efficiency $\epsilon f_s=4.7\times 10^{-6}$ or $\epsilon=3.9\times 10^{-5}$. We will adopt this last number in the rest of the paper. This is equivalent to an energy input of 2 keV per particle or 4 keV per baryon. Another way of looking at this is to divide $\epsilon f_s M_gc^2$ by a Hubble time to compute a typical luminosity; for a halo with a total mass of a few times $10^{14}M_\odot$, this is of order $10^{44}$erg/s. In other words, the energy input from black holes is the same magnitude as the observed energy radiated in X-rays.
Polytropic Rearrangement {#sec:poly}
========================
Now assume that the gas rearranges itself (changing its density profile) through unspecified processes into a polytropic distribution with polytropic index $n$ or adiabatic index $\Gamma=1+n^{-1}$. The commonly addressed cases are ($\Gamma,n$) = ($1,\infty$) for isothermality, and (5/3,3/2) or (4/3,3) for non-relativistic or relativistic isentropic fluids respectively. We know that there is turbulence [@KN03a] induced by merger shocks [@BN98] and galaxy wakes [@SAP99; @Sakelliou00]. Other processes driving the rearrangement could include, for example, conduction [@KN03a; @DJSBR04] or wave transport of energy [via gravity, Alfven, or magnetosonic waves, e.g. @Cen05]. This rearrangement means that the outer gas radius could be larger or smaller than $r_v$.
For central pressure $P_0$ and density $\rho_0$, a polytropic distribution requires that the gas pressure $P'$ and density $\rho'$ after rearrangement are related by $$\label{eqn:polytrop}
P' = P'_0\left( \rho'/\rho'_0 \right)^{(1+1/n)}
= P'_0\left( \rho'/\rho'_0 \right)^\Gamma
\hspace{1cm} ,$$ and the central isothermal gas sound speed is defined as $V^2_{s0}={P'_0}/{\rho'_0}$. Note that $\Gamma$ is not in fact the actual ratio of specific heats; we require only that the gas has arranged itself in polytropic fashion, as in Eqn. (\[eqn:polytrop\]). Fig$.$\[fig:gbrvrho\], kindly provided by Greg Bryan, shows results from a high resolution adiabatic AMR simulation of a massive ($\sim 10^{15}h^{-1}M_\odot$) cluster. The pressure–density relation in this calculation closely fits that of a $\Gamma =1.15$ polytrope, and lends support for the proposal that turbulent mixing (the only one of the processes listed above that was included in this computation) can lead to a fairly tight polytropic relation. SPH simulations by @LBKQHW00 resulted in a pressure-density relation well described by a polytropic equation of state with $\Gamma \approx 1.2$; similar results are reported by @AYMG03 and @BMSDDMTTT04. This result holds for both adiabatic and radiative simulations [but see @KTJP04], and agrees well with the effective $\Gamma$ derived from observed clusters by @FRB01. @SMGS05 find that $\Gamma=1.2$ offers the best consistency with the assumption that the specific energy of the hot gas equals that of the dark matter. Interestingly, the purely adiabatic, spherical, and self-similar collapse solution of @Bert85 was also polytropic with $\Gamma \approx 1.17$.
An additional contribution to the pressure may come from a relativistic component; such a component could be created for example at shock fronts, converting part of the gas energy into cosmic rays. While the relativistic portion of the gas will contain a negligible fraction of the mass, it may contribute significantly to the total gas energy and pressure. We allow for the fact that, in addition to the gas pressure $P'$ there may be a nonthermal component having pressure $P_{rel} = \delta_{rel}P'$ with total pressure $$\begin{aligned}
P_{tot}& = & P' + P_{rel} = P' (1+\delta_{rel}) .\end{aligned}$$ It is not obvious if $\delta_{rel}$ is maximum in the center, where there may be injection of a relativistic fluid by an AGN, or in the outer parts of the cluster, where there may be injection of relativistic particle energy in boundary shocks. Thus, for simplicity we will take $\delta_{rel}$ = constant.
Given these relations, the equation of equilibrium for a spherically symmetric distribution, $dP_{tot}/dr=-\rho d\Phi/dr$, becomes $$\begin{aligned}
(1+\delta_{rel})V^2_{s0}\frac{\rho'_0}{\rho'}
\frac{d}{dr}\left( \frac{\rho'}{\rho'_0} \right)^{(1+1/n)}
= -\frac{GM_{tot}(r)}{r^2} \hspace{1cm} .\end{aligned}$$ Thus $$\begin{aligned}
\left( \frac{\rho'}{\rho'_0} \right)^{1/n} - 1
= -\frac{ \left[\Phi(r)-\Phi(0)\right] }{ V^2_{s0}(1+n)(1+\delta_{rel}) }
= -\frac{ \beta j(x) }{(1+n)(1+\delta_{rel})}
\hspace{1cm} .\end{aligned}$$ This last is from Eqn. (\[eqn:phi\]) with
$$\label{eqn:beta}
\beta \equiv \frac{4 \pi G \rho_1 r_1^2}{V^2_{s0}}=\frac{V_{c,max}^2}{V^2_{s0}}G_{max}^{-1}$$
and $$j(x)\equiv \left\{ \begin{array}{ll}
1 - \frac{\ln(1+x)}{x} & x\leq C \\
1-\frac{1}{1+C}-\left[\ln(1+C)-\frac{C}{1+C}\right]x^{-1} & x>C
\hspace{1cm} .
\end{array} \right.$$
Thus,
$$\label{eqn:grhovr}
\rho'(r)=\rho'_0 \left[ 1-\frac{\beta j(x)}{(1+n)(1+\delta_{rel})}
\right]^n = \rho'_0\theta^n
\hspace{1cm} ,$$
where $$\theta = 1-\frac{\beta j(x)}{(1+n)(1+\delta_{rel})}$$
is the familiar polytropic variable defined by @Chandra. Eqn. (\[eqn:grhovr\]) was first derived by @MSS98 for the isothermal case, and @WFN00 more generally. The final gas radius can be smaller or larger than the initial value. Denoting this radius in units of $r_1$ as $C_f$, the gas mass can be written $M_g=4\pi r_1^3 \rho'_0 L$, where $$L = L(n,\beta,C,C_f)\equiv \int_0^{C_f} {\theta^n x^2 dx}
\hspace{1cm} .$$ The thermal component will contribute a factor of $\frac{3}{2}\int P'd^3x$ to the kinetic energy, and the relativistic component $3\int \delta_{rel} P' d^3x$, so the rearranged total gas energy is $E_g'=\frac{3}{2}(1+2\delta_{rel})\int_0^{M_g}{V_s^2 dM}
+\int_0^{M_g}{\Phi_{tot}dM}$. Defining two more integrals
$$\begin{aligned}
\label{eqn:i2int}
I_2&=& I_2(n,\beta,C,C_f)\equiv \int_0^{C_f}{f(x)\theta^n x^2 dx} \\
I_3&=& I_3(n,\beta,C,C_f)\equiv \int_0^{C_f}{\theta^{1+n} x^2 dx} \end{aligned}$$
we now have $$\label{eqn:egeqn}
E_g'=-M_{g}V^2_{s0} \left[-\frac{3}{2}(1+2\delta_{rel})\frac{I_3}{L} +
\beta \frac{I_2}{L}
\right]$$ as the final energy.
Constraints on the final temperature {#sec:constr}
====================================
Suppose we have a dark matter halo for which the relevant properties— $M_{tot}, r_v, C, V_{c,max}$, etc.— are known. From the previous section, the final distribution of the gas can be determined as a function of the two unknowns $\beta$ and $C_f$; thus to specify the final gas temperature and density distribution it remains only to constrain these two parameters. The first constraint is from conservation of energy: the final gas energy will equal the initial energy plus changes to due star formation, expansion or contraction, and feedback; i.e. $E_g+\Delta E_p+\Delta E_f =E_g'$. Combining this with Eqns. (\[eqn:egasj\]), (\[eqn:delep\]), (\[eqn:delef\]), and (\[eqn:egeqn\]) yields $$\label{eqn:eqen}
\begin{array}{cc}
\frac{3}{2}(1+f_s) \left[
G_{max}K(C)\left(3-4\delta_s\right) + K_s(x_s) \right]
- G_{max}\epsilon f_s\frac{c^2}{V_{c,max}^2} \\
+ \frac{1}{3}(1+f_s)\frac{S_C(C)}{g(C)}\left( C^3_f-C^3 \right) =
\frac{I_2}{L}
-\frac{3}{2}(1+2\delta_{rel})\frac{I_3}{\beta L}
\end{array}$$ (keeping in mind that $L$, $I_2$, and $I_3$ are functions of $\beta$ and $C_f$). A second constraint comes from the fact that the surface pressure of the gas must match the exterior pressure, which we have fixed at $P_{s,gas}(C)$. This gives $$\label{eqn:eqpres}
(1+f_s)\frac{S_C(C)}{g(C)}\beta L = (1+\delta_{rel}) \left[ 1-
\frac{\beta j(C_f)}{(1+n)(1+\delta_{rel})} \right]^{1+n}
\hspace{1cm} .$$
Thus, given $r_v$, $C$, and $V_{c,max}$ for the dark matter halo, and appropriate choices for $\Gamma$, $f_s$, $\epsilon$, and $\delta_{rel}$, Eqns. (\[eqn:eqen\]) and (\[eqn:eqpres\]) can be solved for $\beta$ and $C_f$, and the central temperature $$kT_0 = \frac{\mu m_p}{G_{max}\beta}V^2_{c,max}$$ is known, as is the density parameter $\rho'_0$, so the gas distribution is fully specified. The expected X-ray luminosity $L_X$ can then be calculated. Following @BBP99, we will include both Bremsstrahlung and recombination, which becomes important for temperatures below 4 keV, by using the cooling function $$\Lambda(T) = 2.1\times 10^{-27} T^{1/2}
\left[ 1 + (1.3\times 10^6/T)^{3/2} \right]
\rm{ cm^3 erg\, s^{-1} . }$$
Simulated Halo Catalogue {#sec:halocat}
------------------------
The plausibility of this procedure can be evaluated by trying it out on a population of many dark matter halos and comparing the results to observed clusters. The halos we use here come from an N-body simulation designed to be in concordance with observational constraints. The simulation is of a periodic cube 1500$h^{-1}$Mpc on a side containing N$=1260^3=2\times 10^9$ particles. The cosmology was chosen to be a standard LCDM power law model with the following parameters: baryon density $\Omega_b=0.047$; Cold Dark Matter density $\Omega_{\rm CDM}=0.223$ (hence total matter density $\Omega_m=0.27$); cosmological constant $\Omega_\Lambda=0.73$ (thus spatially flat); Hubble constant given by $h=H_0/(100{\rm km~s^{-1} Mpc^{-1}})=0.70$ (hence $\Omega_bh^2=0.02303$); primordial scalar spectral index $n_s=0.96$; and linear matter power spectrum amplitude $\sigma_8=0.84$. These values are consistent within one standard deviation to those derived either from WMAP data or from WMAP combined with smaller angular scale CMB experiments and galaxy data [@SpergWMAP]. The initial conditions were generated using the publicly available code GRAFIC2 [@Bert01] to compute initial particle velocities and displacements from a regular grid. Since the memory required to hold a $1260^3$ grid is 8 gigabytes, it was necessary to modify the single level portion of this program by adding message-passing commands in order to distribute the mesh among several processors.
The simulation was carried out with the TPM (Tree-Particle-Mesh) code [@BO03], using 420 processors on the Terascale Computing System at the Pittsburgh Supercomputing Center; it took not quite five days of actual running time. The box size and particle number determine the particle mass of $1.26\times 10^{11}h^{-1} M_\odot$. The cubic spline softening length was set to $17h^{-1}$kpc. A standard friends-of-friends (FOF) halo finding routine was run on the redshift $z=0$ box, using a linking length $b=0.2$ times the mean interparticle separation [@LC94]; this yielded 575,125 halos with both a FOF mass above $2\times 10^{13}h^{-1} M_\odot$ and a virial mass above $1.75\times 10^{13}h^{-1} M_\odot$. The PM mesh used in TPM contained 1260$^3$ cells, and at redshift zero all PM cells with an overdensity above 39 were being followed at full resolution, so these objects had the full force resolution of TPM. For the range of parameters used here, clusters with $kT>2$keV contained more than 200 particles within $r_v$.
For each halo, the position of the most bound particle is taken to be the cluster center. Then $M_{tot}$ and $r_v$ are measured, as are $V_{c,max}$ and the radius of maximum circular velocity $r_{c,max}$; this latter gives the concentration $C=2.163r_v/r_{c,max}$. This defines the equivalent NFW model halo, i.e. that NFW model closest to the computed dark matter halo. With this information, the procedure outlined above can be carried out on each halo to compute the gas density and temperature.
Resulting Profiles {#sec:resprof}
------------------
Given this set of halos, it remains to specify $\Gamma$, $f_s$, $\epsilon$, and $\delta_{rel}$. Let us first consider the appropriate $\Gamma$. Fig$.$\[fig:tprof\] shows the projected temperature profile for different values of $\Gamma$; this profile was computed by integrating the emission-weighted temperature along the line of sight. To normalize the curves, the mean temperature $\langle T\rangle$ was calculated by evenly weighting all radii inside $r_{v}/2$; this was done to correspond with the method of @DeGM02, who measured the mean profile for clusters with and without cooling flows— shown in the Figure as filled and open circles, respectively. Examination of Fig$.$\[fig:tprof\] shows that $\Gamma$=1.2–1.4, corresponding to polytropes with index $n$=2.5–5, provides adequate fits to the outer parts of the clusters, within which resides most of the gaseous mass [see also the discussion in @SMGS05]. @AYMG03 have shown that a $\Gamma=1.18$ model is a good fit to the average temperature profile measured by @MFSV98; this latter measurement has been confirmed by @DeGM02, @PJKT05, and @VMMJFvS05. As discussed above (§\[sec:poly\]), $\Gamma=1.2$ is also a good fit to hydrodynamical simulations [@LBKQHW00; @LNNBBM02; @AYMG03; @BMSDDMTTT04; @KTJP04]. The lack of an isothermal core will not lead to a serious overestimation of luminosity or emission-weighted temperature because the volume of this central region is small. This was tested by taking the $\Gamma$=1.2 profile and imposing an isothermal core, matching the density and pressure at $0.2r_v$; the resulting changes in emission-weighted temperature and luminosity were generally less than 10%. However, neglecting cooling will reduce the scatter in the $M_{500}-T$ and $L_x-T$ relations [@MBBPH04].
The effect of the polytropic rearrangement on the radial profile of the gas can be seen in Fig$.$\[fig:profs\]. The example halo used here has physical parameters $M_{tot}=4\times 10^{14}h^{-1} M_\odot$, $r_v=1.2 h^{-1}$Mpc, $C=4$, and $V_{c,max}=1200$km/s. The top two panels show the temperature (relative to $T_{ew}$, the mean emission-weighted temperature inside a radius $R_{500}$ containing an density of $500\rho_c$) and density (relative to $200\rho_c$) as a function of radius. It is instructive to compare to the original NFW distribution, shown as a dot-dashed line; in this case the central temperature goes to zero, as the density profile has a cusp. The polytropic rearrangement (taking $\Gamma=1.2$ and $f_s=\epsilon=\delta_{rel}=0$, shown as a dotted line) increases the central temperature while decreasing the density, removing the cusp (with a correspondingly dramatic lowering of the X-ray luminosity, as we shall see). This is seen more clearly in the third panel, which shows the ratio of gas to dark matter mass interior to a given radius, in terms of the cosmic average: inside the dark matter core radius, the gas fraction declines sharply. These temperature and density profiles result in the “entropy” profile shown in the final panel of Fig$.$\[fig:profs\], taking the definition of entropy to be $T\rho^{-2/3}$. The polytropic profile has a slope close to $r^{1.1}$ near the virial radius, and is shallower nearer the cluster core; this behavior has in fact been observed in a wide range of clusters [@PSF03; @PA05; @PJKT05]. This behavior has been derived before in analytic models assuming the gas is shock heated [@TozNor01], and is also seen in hydrodynamic simulations [@LBKQHW00; @BMSDDMTTT04; @KTJP04].
This change in profile has a strong impact on other observable cluster properties, as is shown in Fig$.$\[fig:rels\]. In these plots the temperature is taken to be the mean emission-weighted $T$ inside $R_{500}$, $T_{ew}$. In clusters with more complicated structure this measure may not coincide well with the spectroscopically measured $T$, as pointed out by @MRMT04, who provide an alternative measure. However, for the simplified models here, the difference between emission weighting and the @MRMT04 spectroscopic-like measure is only a few percent at most. The lines show the median value as a function of $T_{ew}$, found using the dark halo catalogue described in §\[sec:halocat\]. The first impact of the polytropic rearrangement is to increase the observed temperatures. This is clearly demonstrated in the left-hand panel, which gives the mass-temperature relation. The points are from @ReipBohr02, as adjusted by @MBBPH04; here the mass is $M_{500}$, the mass inside a sphere containing mean density 500$\rho_c$. The polytropic model distribution (dotted line) resembles that assuming an NFW profile, only shifted to higher $T$; note assuming an NFW gas profile leads to significant disagreement with the observed relation, while switching to a polytropic model provides much superior agreement. This is also true in the right-hand panel, which shows the bolometric X-ray luminosity as a function of $T$; the data points are the subset of the ASCA cluster catalogue [@Horner01] described in @MBBPH04. The polytropic model, without the central cusp, yields a lower luminosity than the NFW profile. The slopes of both the $L_x-T$ and $M_{500}-T$ relations retain the same self-similar values, however.
The next physical input is the fraction of gas which collapses into stars. As discussed above, this is roughly one eighth the the gas mass inside the virial radius, or $f_s=0.12$. Since the stars in clusters are old, this fraction will hold for all moderately low redshifts. As shown as short-dashed lines in Fig$.$\[fig:profs\], assuming $f_s=0.12$ for a typical cluster (keeping $\Gamma=1.2$) increases the temperature slightly and reduces the gas density. Since the temperature change is not large, this has little effect on the $M_{500}-T$ relation. However, gas removal for star formation, which increases the mean energy per particle for the remaining gas, leads to lower densities and so has a significant impact on the $L_x-T$ relation. For the most massive (hottest) clusters, the predicted luminosity is in fact close to that observed; however, the self-similar slope of the relation is still preserved, so for less massive clusters $L_x$ is overestimated.
The next required physical input is the amount of energy from feedback coming from supernovae and active galactic nuclei, discussed in §\[sec:otherc\]. The results of including feedback of $\epsilon = 3.9\times 10^{-5}$ in the $\Gamma=1.2, f_s=0.12$ model are shown as long-dashed lines in Figs$.$\[fig:profs\] and \[fig:rels\]. As one would expect, the radial profile has a higher temperature and lower density. However, the effect of feedback differs from those considered previously, because the resulting relations are no longer self-similar. For massive clusters with $V_{c,max}>1000$km/s or $kT>10$keV, feedback is of little importance because the added energy is small compared to the gravitational energy, but for smaller masses it can have a significant impact. One can see a steeper slope in the $M_{500}-T$ relation, but the most significant effect is on the luminosity, which in shape now more closely resembles the observed distribution.
The remaining physical effect left to include is nonthermal pressure. We will take $\delta_{rel}=0.1$; the nonthermal sources of pressure may in fact contribute a few tens of percent of the total [@Miniati04]. The results can be seen by comparing the solid ($\delta_{rel}=0.1$) and long-dashed ($\delta_{rel}=0$) lines in Figs$.$\[fig:profs\] and \[fig:rels\]. With this additional support, less kinetic energy is required at a given pressure. Thus, while the density profile is little changed, the resulting gas distribution is somewhat cooler, and the emission weighted temperature is lower at a fixed $M_{500}$ or $L_x$.
The departure from self-similar scaling is shown further in Fig$.$\[fig:mdprofs\], which displays the radial profiles of temperature, gas density, gas fraction, and entropy for clusters of mass $M_{tot} = 10^{15}, 5\times10^{14}, 2.5\times10^{14}, 1.25\times10^{14}$, and $6.25\times10^{13} h^{-1}M_\odot$. Star formation and feedback were included with $f_s=0.12$ and $\epsilon=3.9\times 10^{-5}$, but not a relativistic component. For the least massive cluster we took $C=4$, $V_{c,max}=700$km/s, and $r_v=650h^{-1}$kpc, scaling for the others as $C\propto M^{-0.13}$, $V_{c,max}\propto M^{1/3}$, and $r_v\propto M^{1/3}$; this is in reasonable agreement with our N-body cluster catalog. With these parameters, the emission weighted temperatures inside $R_{500}$ are $kT_{ew}$=10.1, 6.7, 4.5, 3.1, and 2.2 keV, respectively. For decreasing mass, the temperature and density profiles become increasingly shallow, leading to a faster decrease in X-ray luminosity. The ejection of gas following feedback energy injection leads to a gas fraction (relative to the universal value) less than unity at the virial radius; with the full halo catalog and these parameters we find for halos in the range $1-2\times10^{14}h^{-1}M_\odot$, the gas fraction at the virial radius is $0.72\pm 0.09$ (one standard deviation). This result is in agreement with simulations including both heating and cooling: @MTKP02 and @KNV05 find that for halos with $M \approx 10^{14}h^{-1}M_\odot$ the hot gas fraction inside a radius enclosing overdensity $\approx 100\rho_c$ is in the range 0.6–0.7, while @EBMMTDDSTT04 find slightly higher values of 0.7–0.8. Observational estimates give similar values with a higher scatter [@Evrard97; @MME99; @SanPon03]. Both these observational and the theoretical studies suggest that more massive clusters have higher hot gas fractions, behavior which is reproduced here.
The bottom panel in Fig$.$\[fig:mdprofs\] displays the entropy $T/n_e^{2/3}$, with $n_e$ the electron density in units of cm$^{-3}$ (and $T$ in keV), which can be compared directly with the observations listed above. The entropy profiles have a slope close to the observed value of $r^{1.1}$ at the virial radius, but this slope quickly becomes shallower for smaller radii. However, we have not taken into account cooling, which would be more important near the center, steepening the inner entropy profile [@MBBPH04]. It has been observed that the entropy in clusters scales as $T^{0.65}$ [@PSF03; @PA05; @PJKT05], rather than linearly in temperature as one would expect for self-similar scaling. Both of these scalings are shown in Fig$.$\[fig:mdents\]; the upper panel gives $T/n_e^{2/3}/T_{ew}$ and the lower panel $T/n_e^{2/3}/T_{ew}^{0.65}$. It is clear from this Figure that the observed $T^{0.65}$ scaling is followed quite closely. However, at radii near $r_v$ the observational picture is unclear; with a sample of 14 nearby clusters, @Neumann05 found that the outer regions followed self-similar scaling and may be affected by the accretion of cooler material. Also, for poorer systems than considered in this paper ($kT<2$keV), we find that this scaling breaks down; such groups may have different entropy profiles than are seen in richer systems [@MFBGH05].
Generalization to an Arbitrary Dark Matter Potential {#sec:3d}
====================================================
The virtue of the method presented in this paper is not just in the equations presented in the preceding sections, whose purpose was to establish physical and mathematical principles and assess the plausibility of the results, but also in its ability to efficiently model complex asymmetrical systems containing substantial substructure. In this section we relax the previous assumption of spherical symmetry and apply the same method to more complex DM potentials. Suppose that a cluster potential is known from an accurate DM integration; the cluster will likely be aspherical and contain significant substructure. Then, if we are satisfied by the ability of an equilibrium polytrope to model the gas in a cluster of galaxies, the integration of the equation of equilibrium $\vec{\nabla}P_{tot}=-\rho\vec{\nabla}\phi$ gives $$(1+\delta_{rel})\frac{\Gamma}{\Gamma-1}\frac{P(\vec{r})}{\rho(\vec{r})} =
-\phi(\vec{r}) +
(1+\delta_{rel})\frac{\Gamma}{\Gamma-1}\frac{P_0}{\rho_0} + \phi_0
\hspace{1cm} .$$ The last two terms comprise a constant of integration; here $\phi_0$ is the potential minimum located at position $\vec{r}=\vec{r_0}$, and the pressure and density at this point are designated by $P_0$ and $\rho_0$. Then, making the definition $$\label{eqn:3dtheta}
\theta(\vec{r}) \equiv 1 + \frac{\Gamma-1}{(1+\delta_{rel})\Gamma}
\frac{\rho_0}{P_0}\left(\phi_0-\phi(\vec{r})\right)
\hspace{1cm} ,$$ the pressure and density are simply
$$\begin{aligned}
\label{eqn:p0rho0}
P(\vec{r}) = P_0 \theta(\vec{r})^\frac{\Gamma}{\Gamma-1} \\
\rho(\vec{r}) = \rho_0 \theta(\vec{r})^\frac{1}{\Gamma-1}\end{aligned}$$
where $\theta(\vec{r})$ is essentially the same polytropic variable defined by @Chandra. Thus for an equilibrium polytropic gas residing in a known potential $\phi(\vec{r})$, the determination of the structure is reduced to the determination of the two numbers $P_0$ and $\rho_0$. Adopting the approach taken in the previous sections, these constants can be determined by satisfying two equations of constraint on the final energy and the surface pressure.
We carried out this procedure on the same N-body simulation used previously in the following manner. A set of particles identified as a cluster is placed in a nonperiodic 3-D grid. The grid cell size $l$ is set to four times the N-body particle spline softening length, as scales smaller than this can be affected by numerical resolution issues; increasing or decreasing $l$ by a factor of two had little impact on the results. The dark matter density in each cell $k$, $\rho_k$, is found from the particle positions using cloud-in-cell (CIC), and the gravitational potential $\phi_k$ on the mesh is calculated from the density using a nonperiodic FFT [@HE81]. The position of the cell with the minimum potential $MIN(\phi_k)=\phi_0$ is taken to be the center of the cluster, $\vec{r_0}$. The cluster velocity is estimated as the mean velocity of the 125 particles closest to $\vec{r_0}$; this mean is subtracted from all the particle velocities. Then the DM kinetic energy per unit volume $t_k=\rho v^2$ is found in each cell: as with the mass, the kinetic energy of each particle is distributed among 8 cells using CIC. The virial radius $r_v$ and DM mass $M_{tot}$ are found from the density distribution. The $N_{cl}$ cells inside $r_v$ are identified, as are the $N_{b}$ cells in a buffer region of width $r_{buf}$ surrounding the cluster with centers in the range $r_v<r<r_v+r_{buf}$. The buffer width was set at 9 cells, or $r_{buf}=153h^{-1}$kpc for the simulation used here. The gas surface pressure is taken to be the mean value (assuming velocities are isotropic) inside this buffer region: $$\label{eqn:3dpsurf}
P_s = N_{b}^{-1}\sum\limits_{k=1}^{N_{b}}
\frac{1}{3} \frac{\Omega_b}{\Omega_m} t_k$$ where the sum is over all cells in the buffer region. Assuming gas traces the DM, the gas mass inside $r_v$ is originally $M_{g,i}=\frac{\Omega_b}{\Omega_m}M_{tot}$. As before, the portion of this gas which is turned into stars is $f_sM_{g,i}/(1+f_s)$. To decide which portion of the gas becomes stars, cells are ranked by binding energy $\rho_k\phi_k+\onehalf t_k$; starting with the most bound cell, the initial $\rho_k$ and $t_k$ are set to zero for each cell in turn until the gas mass removed, $\frac{\Omega_b}{\Omega_m}\rho_k l^3$, totals to $f_sM_{g,i}/(1+f_s)$. The original gas mass inside $r_v$ is then $$\label{eqn:3dmass}
M_g = \frac{M_{g,i}}{1+f_s} =
\sum\limits_{k=1}^{N_{cl}} \frac{\Omega_b}{\Omega_m}\rho_k l^3$$ the sum being over all cells inside $r_v$, and the initial gas energy is $$\label{eqn:3dinite}
E_g = \sum\limits_{k=1}^{N_{cl}} \frac{\Omega_b}{\Omega_m}
\left\{ \phi_k\rho_k+\onehalf t_k \right\} l^3
\hspace{1cm} .$$ As before, this energy can be supplemented by feedback energy $\Delta E_f = \epsilon f_s M_g c^2$.
As in the 1-D case, this gas is assumed to rearrange itself into a polytropic distribution with $\Gamma=1.2$. It only remains to specify $P_0$ and $\rho_0$, which are fixed by the final energy and surface pressure. For a given initial choice of ($P_0,\rho_0$), the final gas density and pressure can be found after calculating $\theta_k$ for each cell from Eqn. (\[eqn:3dtheta\]). As before, the initial energy may be changed by the inflow or outflow of gas. The final radius of the gas, $r_f$, is found by moving outwards from the cluster center until mass $M_g$ is enclosed, i.e. $$\label{eqn:3dfm}
M_g=\sum\limits_{k=1}^{N_{f}} \rho_0 \theta_k^\frac{1}{\Gamma-1}l^3
\hspace{1cm} ,$$ where the sum is over the $N_{f}$ cells inside $r_f$. Similarly to Eqn. (\[eqn:delep\]), we will assume that the surface pressure does not change with radius, so the change in energy due to expansion or contraction is proportional to the change in volume. This means $\Delta E_P = (4\pi/3)(r_v^3-r_f^3)P_s$, with $P_s$ given by Eqn. (\[eqn:3dpsurf\]). Now we have all the information required for the first constraint on ($P_0,\rho_0$), namely the conservation of energy, where the final gas energy is $$\label{eqn:3dfe}
E_f=\sum\limits_{k=1}^{N_{cl}} \left\{
\rho_0 \theta_k^\frac{1}{\Gamma-1} \phi_0
+ \frac{3}{2}P_0\theta_k^\frac{\Gamma}{\Gamma-1} \right\}l^3
= E_g + \Delta E_f + \Delta E_P
\hspace{1cm} .$$ The second constraint is the mean pressure in the $N_{b,f}$ buffer cells between $r_f$ and $r_f+r_{buf}$; this is assumed to match the original value: $$\label{eqn:3dfp}
N_{b,f}^{-1} \sum\limits_{k=1}^{N_{b,f}} P_0\theta_k^\frac{\Gamma}{\Gamma-1}
= P_s
\hspace{1cm} .$$ Thus after an initial estimate for ($P_0,\rho_0$), it is now possible to iterate to a solution satisfying Eqns. (\[eqn:3dfe\]) and (\[eqn:3dfp\]). This solution provides the full three dimensional pressure and density of the gas with allowance for substructure, triaxiality, etc.
An example of the resulting gas distribution for one halo taken from the catalog is given in Fig$.$\[fig:3dexmpl\]. The simulation particle positions are shown in the upper left-hand panel; there are several large substructures in the process of merging with the main objects. The volume shown is a cube 6.4$h^{-1}$Mpc on a side. The upper right-hand panel shows the projected gas surface density obtained by the method just described. With the gas density and temperature, maps can be made for the X-ray emission and SZE, as shown in the lower panels. The scale is linear, with black a factor of 100 below white. The results of this procedure employed on the entire set of halos used before are displayed in Fig$.$\[fig:3drels\]. A stellar fraction $f_s=0.12$ and feedback $\epsilon=3.9\times 10^{-5}$ were included. For each plot, the median value is shown as a solid line and the shaded region encloses 68% of the halos at that temperature. Also shown as dashed lines are the results of the method of §\[sec:constr\] based on the NFW profile; the scatter seen using this latter method is somewhat smaller than that shown for the full 3-D method, as the latter realistically includes substructure and triaxiality. We are neglecting cooling, which would increase the scatter further, and tend to increase the luminosity at a given temperature [@MBBPH04]. Use of the NFW approximation appears to have little effect on either the $M_{500}-T$ or the $L_x-T$ relation, relative to using the full particle distribution. Examination of individual clusters shows that the spherically averaged gas profiles resulting from the N-body potentials are slightly shallower in both temperature and density, although the amount of gas inside $R_{500}$ is the same for both methods; the larger clumping factor when using the true density, with triaxiality and substructure, increases the luminosity enough to compensate for this.
Conclusion {#sec:conclusion}
==========
The NFW model has provided a useful description of the distribution of matter inside collisionless DM halos, such as those hosting X-ray clusters. This success inspires hope that a similarly concise description can be found for the hot gaseous component. Given a population of dark matter halos from an N-body simulation (or from some semi-analytic model such as extended Press-Schechter), can we deduce the global properties of the baryonic component inside each halo? In this paper we have worked towards providing a prescription which is simple enough to apply broadly while remaining physically well motivated.
In the model presented here, the gas is assumed to initially have energy per unit mass equivalent to that of the dark matter; this energy can be modified by removal of low entropy gas (to form stars), addition of feedback energy expected from supernovae and accreting black holes, and mechanical work done as the gas expands or contracts. The gas is assumed to redistribute itself into a polytropic distribution in hydrostatic equilibrium with the DM potential; given the constraints on the total energy and the surface pressure at the virial radius, the gas distribution is entirely specified.
We applied two variations of this method to a catalogue of cluster-sized dark matter halos drawn from a large cosmological N-body simulation. In the first variant, the mass, virial radius, and concentration of each halo was measured, and the mass profile was assumed to follow a spherically symmetric NFW profile. To determine the gas distribution in this case means solving Eqns. (\[eqn:eqen\]) and (\[eqn:eqpres\]). We then allowed for complex, nonspherical profiles and substructure by using the full set of particle positions and velocities in each N-body halo to determine the potential and kinetic energy. These two methods give similar results, but assuming a spherical NFW profile gives slightly lower temperatures on average and gives significantly less scatter in the mass-temperature and X-ray luminosity-temperature than is observed.
Simply assuming the gas follows the dark matter leads to too low temperatures and too high central densities, since the DM profile has a cusp. The polytropic rearrangement increases the central temperature while decreasing the density, removing the cusp. Removing low entropy gas for star formation further increases temperatures and reduces density. However, neither of these processes changes the self-similar nature of the model. Including energy from feedback does change this, because in massive clusters the energy input will be small compared to the total binding energy, while for smaller masses it can have more of an impact. We also implemented a simple approximation for including nonthermal pressure support; including a relativistic component in this way leads to somewhat lower temperatures and slightly higher densities.
Essentially two dimensionless numbers are required to prescribe the state of the gas in a given DM potential: the fraction of gas mass transformed to a condensed (primarily stellar) form (determined by observations to have the value $f_s\approx0.12$); and the feedback from the condensed component, for which a plausible estimate for the energy output from supernovae and black holes that is trapped in the cluster gas is $\epsilon f_s \approx 0.12\cdot 3.9\times 10^{-5} = 4.7\times10^{-6}$.
The utility of fully understanding the properties of the intergalactic medium in clusters can be seen in Fig$.$\[fig:nofm\], which shows the cumulative temperature function; the data points are from @IRBTK02. The lines (the line types are the same as in Fig$.$\[fig:profs\] and Fig$.$\[fig:rels\]) demonstrate how, as different processes are included, the resulting temperature function can change quite dramatically. (Note that the curves are the number density at $z=0$, whereas at the highest $T$ objects are sufficiently rare such that the data points actually reflect the cluster density at $z>0$, which is lower). There is an apparent conflict in that no single model seems to satisfy all the observations simultaneously but this may not be a serious problem, and may in fact reflect the strength of the models. In order to reproduce the observed $M_{500}-T$ and $L_x-T$ relations requires a high $T$ and low density, in other words a significant amount of star formation and feedback. But this seems to predict too high a number density of clusters for a given T, as is seen in the plot of $n(>kT)$. However, rather than a problem with the gas model, this may simply be due to our choice of cosmological parameters $\Omega_m$ and $\sigma_8$ when generating the cluster catalogue. The mass function from our N-body simulation is also too high when compared to that observed by SDSS [see Fig$.$2 of @YBB05]. Thus an accurate mass-temperature relation would also lead to an overestimate in the temperature function, so the failure to reproduce the observed $n(>kT)$ relation reflects a failure of the cosmological model; lowering $\sigma_8$ and/or $\Omega_m$ would alleviate this problem without significantly altering the predicted $M_{500}-T$ and $L_x-T$ relations. A 10% reduction in $\sigma_8$ would reduce the number of clusters with $kT>4$ by roughly a factor of two. This points out the usefulness of the cluster number density as a probe of cosmological parameters, but also the necessity of including all the relevant physics accurately.
A variety of telescopic surveys in many wavelength bands will soon greatly multiply the number of galaxy clusters catalogued, particularly at higher redshifts. Unlocking the power of these new observational datasets as cosmological probes will require sophisticated theoretical predictions. In the future we plan to apply the methods developed here to explore the properties of clusters at higher redshifts and make detailed predictions for X-ray and SZE surveys in many different cosmological models.
Many thanks to Greg Bryan for kindly supplying Fig$.$\[fig:gbrvrho\], and to Ian McCarthy for helpful discussions and for making available the observational data from @MBBPH04, as well as the anonymous referee for a careful and helpful reading of the manuscript. Computations were performed on the National Science Foundation Terascale Computing System at the Pittsburgh Supercomputing Center, with support from NCSA under NSF Cooperative Agreement ASC97-40300, PACI Subaward 766. Computational facilities at Princeton were provided by NSF grant AST-0216105. Research support for AB comes from the Natural Sciences and Engineering Research Council (Canada) through the Discovery grant program. AB would also like to acknowledge support from the Leverhulme Trust (UK) in the form of the Leverhulme Visiting Professorship at the University of Oxford.
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abstract: 'The full parameter space of Hermitian quark mass matrices with four texture zeros is explored by using current experimental data. We find that all ten free parameters of the four-zero quark mass matrices can well be constrained. In particular, only one of the two phase parameters plays an important role in CP violation. The structural features of this specific pattern of quark mass matrices are also discussed in detail.'
address: |
Institute of High Energy Physics, Chinese Academy of Sciences,\
P.O. Box 918 (4), Beijing 100039, China\
([*Electronic address: xingzz@mail.ihep.ac.cn*]{})
author:
- '[**Zhi-zhong Xing**]{} and [**He Zhang**]{}'
title: |
Complete Parameter Space of Quark Mass Matrices\
with Four Texture Zeros
---
The texture of quark mass matrices, which can significantly impact on the pattern of quark flavor mixing, is completely unknown in the standard electroweak model. A theory more fundamental than the standard model is expected to allow us to determine the concrete structure of quark mass matrices, from which six quark masses, three flavor mixing angles and one CP-violating phase can fully be calculated. Attempts in this direction (e.g., those starting from supersymmetric grand unification theories and from superstring theories) are encouraging but have not proved to be very successful. Phenomenologically, a very common approach is to devise simple textures of quark mass matrices that can predict some self-consistent and experimentally-favored relations between quark masses and flavor mixing parameters [@Review]. The flavor symmetries hidden in such textures might finally provide us with useful hints about the underlying dynamics responsible for the generation of quark masses and the origin of CP violation.
Without loss of generality, the quark mass matrices $M_{\rm u}$ (up-type) and $M_{\rm d}$ (down-type) can always be taken to be Hermitian in the standard model or its extensions which have no flavor-changing right-handed currents [@F79]. Physics is invariant under a common unitary transformation of Hermitian $M_{\rm u}$ and $M_{\rm d}$ (i.e., $M_{\rm u,d} \rightarrow S M_{\rm u,d} S^\dagger$ with $S$ being an arbitrary unitary matrix). This freedom allows a further arrangement of the structures of quark mass matrices, such that $$M_{\rm u} \; = \; \left ( \matrix{
{\bf 0} & C_{\rm u} & {\bf 0} \cr
C^*_{\rm u} & \tilde{B}_{\rm u} & B_{\rm u} \cr
{\bf 0} & B^*_{\rm u} & A_{\rm u} \cr} \right ) \;
% (1)$$ and $$M_{\rm d} \; = \; \left ( \matrix{
D_{\rm d} & C_{\rm d} & {\bf 0} \cr
C^*_{\rm d} & \tilde{B}_{\rm d} & B_{\rm d} \cr
{\bf 0} & B^*_{\rm d} & A_{\rm d} \cr} \right ) \;
% (2)$$ or $$M'_{\rm d} \; = \; \left ( \matrix{
{\bf 0} & C_{\rm d} & D'_{\rm d} \cr
C^*_{\rm d} & \tilde{B}_{\rm d} & B_{\rm d} \cr
{D'}^*_{\rm d} & B^*_{\rm d} & A_{\rm d} \cr} \right ) \;
% (3)$$ hold [@FX99]. We see that $M_{\rm u}$ has two texture zeros and $M_{\rm d}$ or $M'_{\rm d}$ has one texture zero [@Note]. Because the texture zeros of quark mass matrices in Eqs. (1)–(3) result from some proper transformations of the flavor basis under which the gauge currents keep diagonal and real, there is no loss of any physical content for quark masses and flavor mixing. But it is impossible to further obtain $D_{\rm d} = 0$ for $M_{\rm d}$ or $D'_{\rm d} =0$ for $M'_{\rm d}$ via a new physics-irrelevant transformation of the flavor basis [@FX99]. In other words, $D_{\rm d} = D'_{\rm d} = 0$ can only be a physical assumption. This assumption leads to the well-known four-zero texture of Hermitian quark mass matrices, which has the up-down parallelism and respects the chiral evolution of quark masses [@F87].
Although a lot of interest has been paid to the four-zero texture of Hermitian quark mass matrices [@4zero; @FX03], a complete analysis of its parameter space has been lacking. This unsatisfactory situation is partly due to the fact that many authors prefer to make instructive analytical approximations in analyzing the consequences of $M_{\rm u}$ and $M_{\rm d}$ on flavor mixing and CP violation. Some non-trivial parts of the whole parameter space of $M_{\rm u,d}$ were unfortunately missed or ignored in such analytical approximations for a long time, as pointed out in Ref. [@FX03].
The purpose of this short note is to make use of current experimental data to explore the complete parameter space of the four-zero quark mass matrices $M_{\rm u}$ and $M_{\rm d}$. We find that the ten free parameters of $M_{\rm u}$ and $M_{\rm d}$ can well be constrained. In particular, only one of the two phase parameters plays an important role in CP violation. We shall also discuss the structural features of $M_{\rm u}$ and $M_{\rm d}$ in detail.
Let us concentrate on the four-zero texture of Hermitian quark mass matrices given in Eqs. (1) and (2) with $D_{\rm d}=0$. The observed hierarchy of quark masses ($m_u \ll m_c \ll m_t$ and $m_d \ll m_s \ll m_b$) implies that $|A_{\rm q}| > |\tilde{B}_{\rm q}|, |B_{\rm q}| > |C_{\rm q}|$ (for q = u or d) should in general hold [@F87]. Note that $M_{\rm q}$ can be decomposed into $M_{\rm q} = P^\dagger_{\rm q} \overline{M}_{\rm q} P_{\rm q}$, where $$\overline{M}_{\rm q} \; = \; \left ( \matrix{
{\bf 0} & |C_{\rm q}| & {\bf 0} \cr
|C_{\rm q}| & \tilde{B}_{\rm q} & |B_{\rm q}| \cr
{\bf 0} & |B_{\rm q}| & A_{\rm q} \cr} \right ) \;
% (4)$$ and $P_{\rm q} = {\rm Diag} \{1, \exp(i\phi^{~}_{C_{\rm q}}),
\exp(i\phi^{~}_{B_{\rm q}} + i\phi^{~}_{C_{\rm q}}) \}$ with $\phi^{~}_{B_{\rm q}} \equiv \arg (B_{\rm q})$ and $\phi^{~}_{C_{\rm q}} \equiv \arg (C_{\rm q})$. For simplicity, we neglect the subscript “q” in the following, whenever it is unnecessary to distinguish between the up and down quark sectors. The real symmetric mass matrix $\overline{M}$ can be diagonalized by use of the orthogonal transformation $$O^T \overline{M} O \; = \; \left ( \matrix{
\lambda_1 & 0 & 0 \cr
0 & \lambda_2 & 0 \cr
0 & 0 & \lambda_3 \cr} \right ) \; ,
% (5)$$ where $\lambda_i$ (for $i=1,2,3$) are quark mass eigenvalues. Without loss of generality, we take $\lambda_3 >0$ and $A >0$. Then ${\rm Det} (\overline{M}) = -A|C|^2 < 0$ implies that $\lambda_1 \lambda_2 < 0$ is required. It is easy to find that $\tilde{B}$, $|B|$ and $|C|$ can be expressed in terms of $\lambda_i$ and $A$ as $$\begin{aligned}
\tilde{B} & = & \lambda_1 + \lambda_2 + \lambda_3 - A \; ,
\nonumber \\
|B| & = & \sqrt{\frac{(A -\lambda_1) (A -\lambda_2)
(\lambda_3 -A)}{A}} \;\; ,
\nonumber \\
|C| & = & \sqrt{\frac{-\lambda_1 \lambda_2 \lambda_3}{A}} \;\; .
% (6)\end{aligned}$$ The exact expression of $O$ turns out to be [@FX03] $$O \; =\; \left ( \matrix{ \cr
\displaystyle
\sqrt{\frac{\lambda_2 \lambda_3 (A-\lambda_1)}{A (\lambda_2 - \lambda_1)
(\lambda_3 - \lambda_1)}}
& \displaystyle
\eta \sqrt{\frac{\lambda_1 \lambda_3 (\lambda_2 -A)}
{A (\lambda_2 - \lambda_1) (\lambda_3 - \lambda_2)}}
& \displaystyle
\sqrt{\frac{\lambda_1 \lambda_2 (A -\lambda_3)}
{A (\lambda_3 - \lambda_1) (\lambda_3 - \lambda_2)}} \cr\cr\cr
\displaystyle
- \eta \sqrt{\frac{\lambda_1 (\lambda_1 -A)}{(\lambda_2 - \lambda_1)
(\lambda_3 - \lambda_1)}}
& \displaystyle
\sqrt{\frac{\lambda_2 (A-\lambda_2)}
{(\lambda_2 - \lambda_1) (\lambda_3 - \lambda_2)}}
& \displaystyle
\sqrt{\frac{\lambda_3 (\lambda_3 -A)}
{(\lambda_3 - \lambda_1) (\lambda_3 - \lambda_2)}} \cr\cr\cr
\displaystyle
\eta \sqrt{\frac{\lambda_1 (A -\lambda_2) (A -\lambda_3)}
{A (\lambda_2 - \lambda_1) (\lambda_3 - \lambda_1)}}
& \displaystyle
- \sqrt{\frac{\lambda_2 (A-\lambda_1) (\lambda_3 -A)}
{A (\lambda_2 - \lambda_1) (\lambda_3 - \lambda_2)}}
& \displaystyle
\sqrt{\frac{\lambda_3 (A -\lambda_1) (A -\lambda_2)}
{A (\lambda_3 - \lambda_1) (\lambda_3 - \lambda_2)}} \cr\cr}
\right ) \; ,
% (7)$$ in which $\eta \equiv \lambda_2/m_2 = +1$ or $-1$ corresponding to the possibility $(\lambda_1, \lambda_2) = (-m_1, +m_2)$ or $(+m_1, -m_2)$. The Cabibbo-Kobayashi-Maskawa (CKM) flavor mixing matrix [@CKM], which measures the non-trivial mismatch between diagonalizations of $M_{\rm u}$ and $M_{\rm d}$, is given by $V \equiv O^T_{\rm u} (P_{\rm u} P^\dagger_{\rm d}) O_{\rm d}$. Explicitly, we have $$V_{i\alpha} \; =\; O^{\rm u}_{1i} O^{\rm d}_{1\alpha} +
O^{\rm u}_{2i} O^{\rm d}_{2\alpha} e^{i\phi_1} +
O^{\rm u}_{3i} O^{\rm d}_{3\alpha} e^{i(\phi_1 + \phi_2)} \; ,
% (8)$$ where the subscripts $i$ and $\alpha$ run respectively over $(u,c,t)$ and $(d,s,b)$, and two phases are defined as $\phi_1 \equiv \phi^{~}_{C_{\rm u}} - \phi^{~}_{C_{\rm d}}$ and $\phi_2 \equiv \phi^{~}_{B_{\rm u}} - \phi^{~}_{B_{\rm d}}$.
It is well known that nine elements of the CKM matrix $V$ have six orthogonal relations, corresponding to six triangles in the complex plane [@Review]. Among them, the unitarity triangle defined by $V^*_{ub}V_{ud} + V^*_{cb}V_{cd} + V^*_{tb}V_{td} =0$ is of particular interest for the study of CP violation at $B$-meson factories [@PDG]. Three inner angles of this triangle are commonly denoted as $$\begin{aligned}
\alpha & = & \arg \left ( - \frac{V^*_{tb}V_{td}}{V^*_{ub}V_{ud}}
\right ) \;\; , \nonumber \\
\beta & = & \arg \left ( -\frac{V^*_{cb}V_{cd}}{V^*_{tb}V_{td}}
\right ) \;\; , \nonumber \\
\gamma & = & \arg \left ( -\frac{V^*_{ub}V_{ud}}{V^*_{cb}V_{cd}}
\right ) \;\; .
% (9) \end{aligned}$$ So far the angle $\beta$ has unambiguously been measured from the CP-violating asymmetry in $B^0_d$ vs $\bar{B}^0_d\rightarrow J/\psi K_{\rm S}$ decays [@Browder]. The angles $\alpha$ and $\gamma$ are expected to be detected at KEK and SLAC $B$-meson factories in the near future. Given the four-zero texture of quark mass matrices, these three angles depend on the CP-violating phases $\phi_1$ and $\phi_2$. We shall examine the explicit dependence of $(\alpha, \beta, \gamma)$ on $(\phi_1, \phi_2)$ in the following.
Now we explore the whole parameter space of $M_{\rm u}$ and $M_{\rm d}$ with the help of current experimental data. There are totally ten free parameters associated with $M_{\rm u,d}$: $A_{\rm u,d}$, $|B_{\rm u,d}|$, $\tilde{B}_{\rm u,d}$, $|C_{\rm u,d}|$ and $\phi_{1,2}$. In comparison, there are also ten observables which can be derived from the four-zero texture of quark mass matrices: six quark masses ($m_u, m_c, m_t$ and $m_d, m_s, m_b$) and four independent parameters of quark flavor mixing (typically, $|V_{us}|$, $|V_{cb}|$, $|V_{ub}/V_{cb}|$ and $\sin 2\beta$). Thus there is no problem to determine the complete parameter space of $M_{\rm u,d}$.
\(1) The first step of our numerical calculations is to find out the allowed ranges of $A_{\rm u}/m_t$, $A_{\rm d}/m_b$, $\phi_1$ and $\phi_2$ by using Eqs. (6)-(9). For this purpose, we adopt the following reasonable and generous values of quark mass ratios at the electroweak scale $\mu = M_Z$ [@Review; @PDG]: $$\begin{aligned}
\frac{m_c}{m_u} & = & 270 - 350 \; , ~~~~~
\frac{m_s}{m_d} \; = \; 17 - 25 \; ;
\nonumber \\
\frac{m_t}{m_c} & = & 260 - 320 \; , ~~~~~
\frac{m_b}{m_s} \; = \; 35 - 45 \; .
% (10)\end{aligned}$$ The predictions of $M_{\rm u}$ and $M_{\rm d}$ for the CKM matrix elements are required to agree with current experimental data [@Browder; @Buras]: $$\begin{aligned}
|V_{us}| & = & 0.2240 \pm 0.0036 \; , ~~~~~~~~~
\left | \frac{V_{ub}}{V_{cb}} \right | \; =\; 0.086 \pm 0.008 \; ,
\nonumber \\
|V_{cb}| & = & (41.5 \pm 0.8) \times 10^{-3} \; , ~~~~~
\sin 2\beta \; =\; 0.736 \pm 0.049 \; .
% (11)\end{aligned}$$ Note that the results of $V$ may involve a four-fold ambiguity arising from four possible values of $(\eta_{\rm u}, \eta_{\rm d})$ in $O_{\rm u}$ and $O_{\rm d}$, as one can see from Eq. (7). To be specific, we first choose $\eta_{\rm u} = \eta_{\rm d} =+1$ in our numerical analysis and then discuss the other three possibilities.
The numerical results for $A_{\rm u}/m_t$ vs $A_{\rm d}/m_b$ and $\phi_1$ vs $\phi_2$ are illustrated in Fig. 1. We see that the most favorable values of these four quantities are $A_{\rm u}/m_t \sim 0.94$, $A_{\rm d}/m_b \sim 0.94$, $\phi_1 \sim 0.5\pi$ and $\phi_2 \sim 1.96\pi$. Fig. 1(a) confirms that the (3,3) elements (i.e., $A_{\rm u} \sim m_t$ and $A_{\rm d} \sim m_b$) are the dominant matrix elements in $M_{\rm u}$ and $M_{\rm d}$. The strong constraint on $\phi_1$ comes from the experimental data on $|V_{us}|$ and $\sin 2\beta$; while the tight restriction on $\phi_2$ results from current data on $|V_{cb}|$. Because of $\sin \phi_1 \gg |\sin \phi_2|$ as shown in Fig. 1(b), the strength of CP violation in the CKM matrix is mainly governed by $\phi_1$. In many analytical approximations, $\phi_1 \approx 0.5\pi$ and $\phi_2 =0$ have typically been taken [@4zero; @FX03].
We find that both $A_{\rm u}/m_t$ and $A_{\rm d}/m_b$ are insensitive to the signs of $\eta_{\rm u}$ and $\eta_{\rm d}$. In other words, the allowed ranges of $A_{\rm u}/m_t$ and $A_{\rm d}/m_b$ are essentially the same in $(\eta_{\rm u}, \eta_{\rm d}) = (\pm 1, \pm1)$ and $(\pm 1, \mp1)$ cases. While $\phi_1$ is sensitive to the signs of $\eta_{\rm u}$ and $\eta_{\rm d}$, $\phi_2$ is not. To be explicit, we have $$\begin{aligned}
(\eta_{\rm u}, \eta_{\rm d}) & = & (+1, +1): ~~~~~
\phi_1 \sim 0.5\pi \; , ~~~ \phi_2 \lesssim 2\pi \; ,
\nonumber \\
(\eta_{\rm u}, \eta_{\rm d}) & = & (+1, -1): ~~~~~
\phi_1 \sim 1.5\pi \; , ~~~ \phi_2 \gtrsim 0 \; ,
\nonumber \\
(\eta_{\rm u}, \eta_{\rm d}) & = & (-1, +1): ~~~~~
\phi_1 \sim 1.5\pi \; , ~~~ \phi_2 \lesssim 2\pi \; ,
\nonumber \\
(\eta_{\rm u}, \eta_{\rm d}) & = & (-1, -1): ~~~~~
\phi_1 \sim 0.5\pi \; , ~~~ \phi_2 \gtrsim 0 \; .
% (12)\end{aligned}$$ The dependence of $\phi_1$ on $\eta_{\rm u}$ and $\eta_{\rm d}$ can easily be understood. Indeed, $\tan\beta \propto \eta_{\rm u}\eta_{\rm d} \sin\phi_1$ holds in the leading-order analytical approximation with $|\sin\phi_2| \ll 1$. Thus the positiveness of $\tan\beta$ requires that $\sin\phi_1$ and $\eta_{\rm u}\eta_{\rm d}$ have the same sign.
\(2) The second step of our numerical analysis is to determine the relative magnitudes of four non-zero matrix elements of $M_{\rm u,d}$ by using Eq. (6) and the results for $A_{\rm u}/m_t$ and $A_{\rm d}/m_b$. The numerical results for $|B_{\rm u}|/A_{\rm u}$ vs $|B_{\rm d}|/A_{\rm d}$, $\tilde{B}_{\rm u}/|B_{\rm u}|$ vs $\tilde{B}_{\rm d}/|B_{\rm d}|$ and $|C_{\rm u}|/\tilde{B}_{\rm u}$ vs $|C_{\rm d}|/\tilde{B}_{\rm d}$ are shown in Fig. 2. One can see that the most favorable values of these six quantities are $|B_{\rm u}|/A_{\rm u} \sim 0.25$, $\tilde{B}_{\rm u}/|B_{\rm u}| \sim 0.3$, $|C_{\rm u}|/\tilde{B}_{\rm u} \sim 0.003$ and $|B_{\rm d}|/A_{\rm d} \sim 0.25$, $\tilde{B}_{\rm d}/|B_{\rm d}| \sim 0.4$, $|C_{\rm d}|/\tilde{B}_{\rm d} \sim 0.06$. A remarkable feature of our typical results is that $A_{\rm q}$, $|B_{\rm q}|$ and $\tilde{B}_{\rm q}$ (for q = u or d) roughly satisfy a geometric relation: $|B_{\rm q}|^2 \sim A_{\rm q} \tilde{B}_{\rm q}$ [@FX03]. In addition, $|B_{\rm u}| \gg m_c$ and $|B_{\rm d}| \gg m_s$ hold. While a very strong hierarchy exists between (1,2) and (2,2) elements of $M_{\rm u,d}$, there is only a weak hierarchy among (2,2), (2,3) and (3,3) elements of $M_{\rm u,d}$. Such a structural property of quark mass matrices must be taken into account in model building.
To be more explicit, let us illustrate the texture of $\overline{M}_{\rm u,d}$ by choosing $m_c/m_u = 320$, $m_t/m_c = 290$, $m_s/m_d =21$ and $m_b/m_s = 40$. We obtain $$\begin{aligned}
\overline{M}_{\rm u} & \approx & A_{\rm u} \left ( \matrix{
{\bf 0} & 0.0002 & {\bf 0} \cr
0.0002 & 0.067 & 0.24 \cr
{\bf 0} & 0.24 & {\bf 1} \cr} \right ) \sim \;
A_{\rm u} \left ( \matrix{
{\bf 0} & \varepsilon^6 & {\bf 0} \cr
\varepsilon^6 & \varepsilon^2 & \varepsilon \cr
{\bf 0} & \varepsilon & {\bf 1} \cr} \right ) \; ,
\nonumber \\
\overline{M}_{\rm d} & \approx & A_{\rm d} \left ( \matrix{
{\bf 0} & 0.0059 & {\bf 0} \cr
0.0059 & 0.089 & 0.24 \cr
{\bf 0} & 0.24 & {\bf 1} \cr} \right ) \sim \;
A_{\rm d} \left ( \matrix{
{\bf 0} & \varepsilon^4 & {\bf 0} \cr
\varepsilon^4 & \varepsilon^2 & \varepsilon \cr
{\bf 0} & \varepsilon & {\bf 1} \cr} \right ) \; ,
% (13)\end{aligned}$$ where $\varepsilon \approx 0.24$ in this special case [@Note2]. Such a four-zero pattern of quark mass matrices depends on a small expansion parameter and is quite suggestive for model building. For example, one may speculate that $\overline{M}_{\rm u}$ and $\overline{M}_{\rm d}$ in Eq. (13) could naturally result from a string-inspired model of quark mass generation [@Ibanez] or from a horizontal U(1) family symmetry and its perturbative breaking [@Flavor].
Note that the numerical results in Fig. 2 and Eq. (13) have been obtained by taking $\eta_{\rm u} = \eta_{\rm d} =+1$. A careful analysis shows that $\tilde{B}_{\rm q}$ (for q = u or d) is sensitive to the sign of $\eta_{\rm q}$, but $|B_{\rm q}|$ and $|C_{\rm q}|$ are not. In view of Eq. (6), we find that the sign of $\eta = \lambda_2/m_2$ may significantly affect the size of $\tilde{B}$ if its $\lambda_2$ and $\lambda_3 -A$ terms are comparable in magnitude. In contrast, the dependence of $|B|$ on $\eta$ is negligible due to $A \gg m_2$; and $|C|$ is completely independent of the sign of $\eta$.
\(3) The final step of our numerical analysis is to examine the outputs of three CP-violating angles $(\alpha, \beta, \gamma)$ and the ratio $|V_{ub}/V_{cb}|$ constrained by the four-zero texture of quark mass matrices. We plot the result for $|V_{ub}/V_{cb}|$ vs $\sin 2\beta$ in Fig. 3(a) and that for $\alpha$ vs $\gamma$ in Fig. 3(b). The correlation between $\alpha$ and $\gamma$ is quite obvious, as a result of $\alpha + \beta + \gamma = \pi$. Typically, $\alpha \sim 0.5\pi$ holds. The possibility $\alpha \approx \gamma$, implying that the unitarity triangle is approximately an isoceless triangle [@FH], is also allowed by current data and quark mass matrices with four texture zeros. Note that the size of $\sin 2\beta$ increases with $|V_{ub}/V_{cb}|$. This feature can easily be understood: in the unitarity triangle with three sides rescaled by $|V_{cb}|$, the inner angle $\beta$ corresponds to the side proportional to $|V_{ub}/V_{cb}|$.
Finally we mention that the outputs of $|V_{ub}/V_{cb}|$ and $(\alpha, \beta, \gamma)$ are completely insensitive to the sign ambiguity of $\eta_{\rm u}$ and $\eta_{\rm d}$.
In summary, we have analyzed the complete parameter space of Hermitian quark mass matrices with four texure zeros by using current experimental data. It is clear that the four-zero pattern of quark mass matrices can survive current experimental tests and its parameter space gets well constrained. We find that only one of the two phase parameters plays a crucial role in CP violation. The (2,2), (2,3) and (3,3) elements of the up- or down-type quark mass matrix have a relatively weak hierarchy, although their magnitudes are considerably larger than the magnitude of the (1,2) element. Such a structural feature of the four-zero quark mass matrices might serve as a useful starting point of view for model building.
We remark that the phenomenological consequences of quark mass matrices depend both on the number of their texture zeros and on the hierarchy of their non-vanishing entries. The former are in general not preserved to all orders or at any energy scales in the unspecified interactions which generate quark masses and flavor mixing [@X03]. But an experimentally-favored texture of quark mass matrices at low energy scales (such as the one under discussion) is possible to shed some light on the underlying flavor symmetry and its breaking mechanism responsible for fermion mass generation and CP violation at high energy scales.
This work was supported in part by National Natural Science Foundation of China.
: While our paper was being completed, we received a preprint by Zhou [@Zhou], in which all possible four-zero textures of quark mass matrices are classified and computed. The analyses, results and discussions in these two papers have little overlap.
[99]{}
For a recent review with extensive references, see: H. Fritzsch and Z.Z. Xing, Prog. Part. Nucl. Phys. [**45**]{}, 1 (2000).
H. Fritzsch, Nucl. Phys. B [**155**]{}, 189 (1979).
H. Fritzsch and Z.Z. Xing, Nucl. Phys. B [**556**]{}, 49 (1999); G.C. Branco, D. Emmanuel-Costa, and R. Gonz$\rm\acute{a}$lez Felipe, Phys. Lett. B [**477**]{}, 147 (2000).
Because $M_{\rm u}$ and $M_{\rm d}$ (or $M'_{\rm d}$) are Hermitian, a pair of off-diagonal texture zeros in each mass matrix have been counted as one zero.
H. Fritzsch, Phys. Lett. B [**184**]{}, 391 (1987); H. Fritzsch and Z.Z. Xing, Phys. Lett. B [**413**]{}, 396 (1997); Phys. Rev. D [**57**]{}, 594 (1998).
See, e.g., D. Du and Z.Z. Xing, Phys. Rev. D [**48**]{}, 2349 (1993); L.J. Hall and A. Rasin, Phys. Lett. B [**315**]{}, 164 (1993); H. Fritzsch and D. Holtmansp$\rm\ddot{o}$tter, Phys. Lett. B [**338**]{}, 290 (1994); H. Fritzsch and Z.Z. Xing, Phys. Lett. B [**353**]{}, 114 (1995); P.S. Gill and M. Gupta, J. Phys. G [**21**]{}, 1 (1995); Phys. Rev. D [**56**]{}, 3143 (1997); H. Lehmann, C. Newton, and T.T. Wu, Phys. Lett. B [**384**]{}, 249 (1996); Z.Z. Xing, J. Phys. G [**23**]{}, 1563 (1997); K. Kang and S.K. Kang, Phys. Rev. D [**56**]{}, 1511 (1997); T. Kobayashi and Z.Z. Xing, Mod. Phys. Lett. A [**12**]{}, 561 (1997); Int. J. Mod. Phys. A [**13**]{}, 2201 (1998); J.L. Chkareuli and C.D. Froggatt, Phys. Lett. B [**450**]{}, 158 (1999); Nucl. Phys. B [**626**]{}, 307 (2002); A. Mondrag$\rm\acute{o}$n and E. Rodriguez-J$\rm\acute{a}$uregui, Phys. Rev. D [**59**]{}, 093009 (1999); H. Nishiura, K. Matsuda, and T. Fukuyama, Phys. Rev. D [**60**]{}, 013006 (1999); G.C. Branco, D. Emmanuel-Costa, and R. Gonz$\rm\acute{a}$lez Felipe, in Ref. [@FX99]; S.H. Chiu, T.K. Kuo, and G.H. Wu, Phys. Rev. D [**62**]{}, 053014 (2000); H. Fritzsch and Z.Z. Xing, Phys. Rev. D [**61**]{}, 073016 (2000); Phys. Lett. B [**506**]{}, 109 (2001); R. Rosenfeld and J.L. Rosner, Phys. Lett. B [**516**]{}, 408 (2001); R.G. Roberts, A. Romanino, G.G. Ross, and L. Velasco-Sevilla, Nucl. Phys. B [**615**]{}, 358 (2001).
H. Fritzsch and Z.Z. Xing, Phys. Lett. B [**555**]{}, 63 (2003).
N. Cabibbo, Phys. Rev. Lett. [**10**]{}, 531 (1963); M. Kobayashi and T. Maskawa, Prog. Theor. Phys. [**49**]{}, 652 (1973).
Particle Data Group, K. Hagiwara [*et al.*]{}, Phys. Rev. D [**66**]{}, 010001 (2002).
T. Browder, talk given at the 21st International Symposium on Lepton and Photon Interactions at High Energies (LP 03), August 2003, Batavia, Illinois, USA.
A.J. Buras, hep-ph/0307203; and references therein.
A different expansion of $\overline{M}_{\rm u,d}$ has been given in Ref. [@FX03] in terms of two different small parameters.
D. Cremades, L.E. Ib$\rm\acute{a}\tilde{n}$ez, and F. Marchesano, hep-ph/0212064; and references therein.
C.D. Froggatt and H.B. Nielsen, Nucl. Phys. B [**147**]{}, 277 (1979); L.E. Ib$\rm\acute{a}\tilde{n}$ez and G.G. Ross, Phys. Lett. B [**332**]{}, 100 (1994).
I am grateful to H. Fritzsch and C. Hamzaoui for pointing out this possibility to me.
Z.Z. Xing, hep-ph/0307359; and references therein.
Y.F. Zhou, hep-ph/0309076.
|
---
abstract: |
We present deep, near-infrared images of the circumbinary disk surrounding the pre–main-sequence binary star, A, obtained with NICMOS aboard the Hubble Space Telescope. The spatially resolved proto-planetary disk scatters $\sim$1.5% of the stellar flux, with a near-to-far side flux ratio of $\sim$1.4, independent of wavelength, and colors that are comparable to the central source ($\Delta\left(M_{F110W}-M_{F160W}\right) = 0.10 \pm
0.03 $, $\Delta\left(M_{F160W}-M_{F205W}\right) = -0.04 \pm 0.06 $); all of these properties are significantly different from the earlier ground-based observations. New Monte Carlo scattering simulations of the disk emphasize that the general properties of the disk, such as disk flux, near side to far side flux ratio and integrated colors, can be approximately reproduced using ISM-like dust grains, without the presence of either circumstellar disks or large dust grains, as had previously been suggested. A single parameter phase function is fitted to the observed azimuthal variation in disk flux, providing a lower limit on the median grain size of $a > 0.23$ . Our analysis, in comparison to previous simulations, shows that the major limitation to the study of grain growth in T Tauri disk systems through scattered light lies in the uncertain ISM dust grain properties. Without explicit determination of the scattering properties it is not possible to differentiate between geometric, scattering and evolutionary effects. Finally, we use the 9 year baseline of astrometric measurements of the binary to solve the complete orbit, assuming that the binary is coplanar with the circumbinary ring. We find that the estimated 1$\sigma$ range on disk inner edge to semi-major axis ratio, $3.2 < R_{in}/a < 6.7$, is larger than that estimated by previous SPH simulations of binary-disk interactions.
author:
- 'C. McCabe, G. Duchêne & A.M. Ghez'
title: NICMOS images of the GG Tau Circumbinary Disk
---
Introduction
============
GG Tau A is one of the few T Tauri systems in which a disk has been spatially resolved at both millimeter and near-infrared wavelengths and is therefore an ideal system for detailed studies of proto-planetary disk geometry and composition. As the second brightest millimeter object in the survey of Beckwith et al. (1990), GG Tau was an early target for millimeter wave interferometry studies (Simon & Guilloteau 1992; Kawabe et al. 1993; Dutrey, Guilloteau & Simon 1994; Guilloteau, Dutrey & Simon 1999, hereafter G99). These observations showed the emission to be from a massive ($0.13 M_{\odot}$) disk surrounding GG Tau A, the closest pair of stars in the GG Tau quadruple stellar system[^1], with the bulk of the emission arising from a distinct ring structure extending from 180 to 260 AU (assuming a distance of 139 pc; Bertout, Robichon & Arenou, 1999). The circumbinary disk was subsequently detected via near-infrared scattered light in multi-wavelength, ground-based adaptive optics (AO) images (Roddier et al. 1996, hereafter known as R96) and in space-based Hubble Space Telescope (HST) 1 polarimetric images (Silber et al. 2000) and WFPC2 optical images (Krist, Stapelfeldt & Watson 2002).
Analysis of the ground-based near-infrared images resulted in disk colors that appear to be redder than the central stars (R96). Such a red color excess could either indicate that the circumbinary disk has a substantial population of large dust grains, or that circumstellar disks, which are coplanar with the circumbinary disk, redden the stellar light before it scatters off the circumbinary disk. Comparing the variation of disk magnitude with color around the disk, Roddier et al. find a trend for fainter regions of the disk to be redder, leading them to suggest that circumstellar disks are the cause of the observed red color excess. The existence of circumstellar disks around the individual components in GG Tau A is supported by observations of near-infrared excesses and strong hydrogen emission lines in each component (White et al. 1999). No information, however, is currently available regarding their orientation and in fact many disks in binary systems show evidence of non-coplanarity (e.g., Stapelfeldt et al. 1998; Monin, Ménard & Duchêne, 1998; Jensen et al. 2000; Wood et al. 2001). To test the circumstellar disk hypothesis, Wood, Crosas & Ghez (1999) ran Monte Carlo scattering simulations of the GG Tau circumbinary ring with and without circumstellar disks being present. Using the Kim, Martin & Hendry (1994) interstellar medium (ISM) grain properties and the disk geometry derived by Dutrey et al. (1994), they find that the circumstellar disks are required to match the disk to star flux ratios and near side to far side flux ratios observed in the ground-based AO images. While these simulations roughly reproduce the estimated quantities due to the large observational uncertainties, they predict blue rather than red disk color excesses. In order to make further progress in understanding the role of circumstellar disks and the possibility of large particles, much higher signal to noise measurements of the circumbinary disk are required at multiple wavelengths.
In this paper we present new high angular resolution, multi-wavelength observations of the GG Tau system taken with NICMOS aboard HST. These deep, space-based observations offer a more stable point spread function (PSF) than that obtainable with ground-based AO systems, allowing a more accurate determination of the disk properties. The observations are outlined in §\[obs\] and the data reduction and PSF subtraction method used are explained in §\[reduc\]. Newly derived stellar and circumbinary disk properties are presented in §\[res\] and are discussed in sections §\[discus\] and §\[implications\] in light of additional Monte Carlo simulations.
Observations {#obs}
============
GG Tau A ($\alpha = 04^{h} 32^{m} 30^{s}.3$, $\delta =
+17^{\circ} 31^{'} 41^{''}, J2000$) was observed using the near-infrared camera, NICMOS, aboard HST on 1997 October 10. Deep images were obtained with the F110W ($\lambda_{o}=1.03 \micron$, $\Delta\lambda=0.55 \micron$), F160W ($\lambda_{o}=1.55 \micron$, $\Delta\lambda=0.40 \micron$), and F205W ($\lambda_{o}=1.90 \micron$, $\Delta\lambda=0.60 \micron$) filters[^2]. In the F110W and F160W filters, GG Tau was imaged 9 times on NIC1, the NICMOS camera with the smallest field of view, 11$\arcsec \times$ 11$\arcsec$, and pixel scale of 0.$\arcsec$043 per pixel (Thompson 1995). Each image was offset by 2.$\arcsec$2 in a spiral dither pattern in order to reduce residual flat field uncertainties and the dither size was set to a non-integral number in order to reduce intra-pixel sensitivity variations, while also keeping the circumbinary disk in the field of view. Each individual dither image was integrated for 128 seconds in the multiaccum mode. The F205W filter data were taken using NIC2, which has a 19.$\arcsec$2 $\times$ 19.$\arcsec$2 field of view and a pixel scale of 0.$\arcsec$075 per pixel. Due to the significant thermal background at this wavelength a dither-chop observing pattern was employed, whereby each image is composed of one 128 second integration on GG Tau followed by a 32$\arcsec$ chop and a 128 second integration on a blank piece of sky. Nine images were obtained in this manner.
Because the aim of our research was to detect faint, diffuse emission close to bright sources, a good characterization of the point spread function was required. To this end, images of a calibrator star, NTTS 042417+1744, were obtained in the same manner as GG Tau in each of the above mentioned filters. This weak-lined T Tauri star has a number of properties which were used to select it as a calibrator: no infrared excess indicative of circumstellar material (Walter et al. 1988), known to be single from previous speckle interferometry observations (Ghez et al. 1993), only half a magnitude brighter than GG Tau Aa in the J band (1.25 ), and a spectral type of K1, which is similar to the stellar components of the binary star (K7 and M0.5 for GG Tau Aa and Ab respectively; White et al. 1999).
Data Analysis {#reduc}
=============
Image Processing
----------------
Initial data reduction was carried out by STScI, providing a quick analysis of the data quality. The images were recalibrated with the latest calibration reference files using the STScI’s pipeline routine CALNICA (Bushouse, Skinner & MacKenty, 1996), which performs standard image reduction tasks: removal of bias and dark current levels, interpolation over known bad pixels, flatfielding, and removal of all cosmic ray hits. Once the data were calibrated, each individual image was sub-pixelated by a factor of 2.
In order to obtain estimates of the properties of the central binary (§\[orbit\]) and the circumbinary disk (§\[cbdisk\]), a model of the central binary star was generated through PSF fitting and then subtracted from the individual images. Both the empirical PSF (observations of NTTS 042417+1744) and TinyTim model PSFs (Krist, 1993) were used. The PSF-fitting was carried out for each of the nine GG Tau images by minimizing the chi-squared between the PSF model and the data in the regions dominated by the binary star (the central area of $\sim 0.67\
arcsec^{2}$) and, in the case of the empirical PSF, the diffraction spikes (an additional $\sim 1.65\ arcsec^{2}$ comprising 4 rectangular regions covering the diffraction spikes beyond 2distance from the central binary). For the empirical PSF, all nine PSF estimates were evaluated and the one that produced the best fit, i.e., the smallest residuals, was selected for each GG Tau image. Although the TinyTim routine allowed us to match the colors of the two GG Tau A stellar components as well as the focus position of the observations (see §3.2), the empirical PSF models consistently produced smaller residuals than those created using TinyTim, and thus the final values for the central binary and the circumbinary disk were derived using the empirical PSF model subtracted images. The difference in the quality of the fits between the empirical PSF and the TinyTim PSF can be understood in terms of (a) small scale PSF variations due to the telescopes “breathing”, the change in the focus of the telescope on timescales of an orbit due to a varying thermal load, and (b) the presence of scattered light off the telescope optics (Krist and Hook, 1999). Neither of these effects are incorporated into the TinyTim PSF models used, and both are measured with the empirical PSF. The nine model subtracted images were combined together to form an average subtracted image at each wavelength, which are shown in Figure \[mosaic\]. Azimuthal averages of the model subtracted images are plotted in the first row of Figure \[profile\].
Bias and Uncertainty Estimates
------------------------------
### Measurement Uncertainties
A map of the measurement uncertainties was created by taking the standard deviation of the mean from all nine PSF subtracted images of GG Tau. Azimuthal averages of these statistical uncertainties are shown in the second row of Figure 2; while these values provide a good estimate of the uncertainties associated with the measurement and subtraction process, they do not take into account the presence of any systematic effects.
### Systematic Biases and Associated Uncertainties {#bias}
Two systematic effects that alter the structure of the PSF have been identified. The first, which affects all three band passes, arises because the observed PSF, NTTS 042417+1744, is slightly bluer than either component of GG Tau. The second is an inadvertent focus offset between the object and PSF in the F205W data. These effects create a systematic bias and an additional source of uncertainty, both of which were modeled and accounted for.
In order to investigate the consequences of the small color differences between the object and PSF, we generated polychromatic models of GG Tau Aa, Ab and NTTS 042417+1744 using TinyTim, matching the spectral shape of these objects using previous photometry from Ghez et al. (1997) and Walter et al. (1988). The PSF fitting and subtraction process was run on the TinyTim model images in order to produce a model of the bias term (i.e., the subtraction residuals caused by this color mismatch); azimuthal averages of this color bias are plotted in the first row of Figure \[profile\]. To obtain an estimate of the uncertainty associated with this measurement of the color bias, the method was repeated using a PSF that varied in spectral type by the assumed uncertainty in the calibrators spectral type (one subclass). The difference in residuals between using a PSF of spectral type K0 and K1 is our estimate of the statistical uncertainty on the color bias term (plotted in the second row of Figure \[profile\]); the uncertainty on the color bias is significantly smaller than the measurement uncertainties.
While this method was sufficient for the F110W and F160W data, the F205W data required an additional systematic effect to be accounted for. Observations of GG Tau on NIC2 were run in parallel mode with polarization observations (which are not presented here) on NIC1. Due to the unexpected thermal problems, each NICMOS camera had a different optimal focus position and therefore parallel observations were taken at non-optimal focus positions. Since these observations were scheduled prior to the recognition of this problem, the polarization data were arbitrarily regarded as the prime observation. Therefore both the polarization and F205W observations were made with the NIC1 focus setting, resulting in slightly out of focus F205W observations. Unfortunately, the F205W data on the PSF were taken in a separate orbit with no parallel observations and therefore at the optimal NIC2 focus setting. This mismatch in focus between the object and calibrator in the F205W data set was quantified using the same TinyTim modeling used to determine the color bias in the F110W and F160W data, but incorporating both the color difference and mean focus offset. As shown in the third column of Figure \[profile\], the focus bias is the source of a considerably large systematic error. To estimate the systematic uncertainty associated with the measurement of this bias, a polychromatic TinyTim model of GG Tau was created with the focus set at the [*maximum*]{} focus position of the data set, and then subtracted using a K0 type PSF set at the [*mean*]{} focus position of the PSF data set. As focus and color mismatches can cause similar problems, this model set-up maximizes the difference between the PSF for the object and calibrator data sets. In contrast with the F110W and F160W data sets, the uncertainty associated with the bias term in the F205W data dominates the measurement uncertainties.
### Combining Statistical and Systematic Effects
All of the results presented in §\[res\] are obtained from analysis of images that have had estimates of the systematic bias subtracted out and uncertainties that have been combined in quadrature, as described in equations \[eq:final\] and \[eq:errors\].
$$Final_{\lambda} = Img_{\lambda} - Bias_{\lambda} \label{eq:final}$$
$$\sigma^{TOT}_{\lambda} = \sqrt{\left( \sigma^{img}_{\lambda}\right)^{2} +
\left( \sigma^{bias}_{\lambda}\right)^{2} } \label{eq:errors}$$
$Img_{\lambda}$ is the median of the nine PSF subtracted images and $Bias_{\lambda}$ is the systematic bias from the mismatch in color and focus position. The two error terms in equation \[eq:errors\] are the statistical uncertainties of the image measurement and bias modeling respectively. The final row in Figure \[profile\] displays the azimuthally average of the bias corrected signal-to-noise (e.g., $Final_{\lambda} /
\sigma^{TOT}_{\lambda}$). Analysis of the F205W dataset is limited to the measurement of the binary star properties (§\[orbit\]) and integrated disk photometry (§\[colors\]), since it is significantly affected by the focus bias. Although this bias and its associated uncertainty have been modeled, the models were made using TinyTim PSFs that have been shown to deviate somewhat from the observed PSFs.
Results {#res}
=======
In this section we present the observed properties of the stellar system and the circumbinary disk. The binary orbital motion has been followed for the past 9 years and the newly estimated disk mass and inclination (G99) allow us to derive estimates of all the orbital parameters for the binary system using only minimal assumptions (§\[orbit\]). For the circumbinary disk (§\[cbdisk\]), we focus on the following observational quantities that have, in the past, been considered useful tools in comparing observations and disk models: in §\[1-D\], the disk morphology (Wood et al. 1999), in §\[colors\], the disk to star flux ratio (Wood et al. 1999) and color excesses (R96), and in §\[azvar\], the spatially resolved color-magnitude relationships (R96) and near side to far side flux ratios (Close et al. 1998; Wood et al. 1999). These properties are then compared to earlier measurements. Deviations from previous estimates of these quantities are most likely attributable to a combination of low SNR and over-deconvolution of the disk in the more challenging AO observations.
Central Binary Star Properties {#orbit}
------------------------------
Table \[astrom\] lists the values for the binary separation, position angle (P.A.) and magnitude of each component obtained from the PSF fitting of the central binary star. The uncertainties are the standard deviation of the estimates obtained from the nine individual measurements, plus additional contributions from the uncertainties of the spacecraft orientation (003; Holtzmann et al., 1995a) and the pixel scale (0.5%; NICMOS Instrument Handbook). The 5% uncertainty in the absolute photometry calibration, however, is not included as our analysis rests on the relative photometry of the disk compared to the central stars. The reported individual stellar flux density uncertainties are dominated by PSF variations. Wide aperture photometry was carried out with the assumption that the disk is a small contamination on the stellar light ($\sim$ 1%, see §\[colors\]); this both checks the PSF fitting measurements and provides a more precise value of the total stellar flux which is used in §\[colors\]. Using an aperture 26 in radius, we obtain a total binary flux density that is consistent to within 1$\sigma$ with the values found through PSF fitting (see Table \[astrom\]).
Over the 9 year period 1990 to 1998, the binary system was repeatedly observed (Leinert et al. 1993; Ghez et al. 1995, 1997; Roddier et al. 1996; Silber et al. 2000; White & Ghez 2001; Woitas, Köhler & Leinert 2001; Krist et al. 2002). Both the new HST photometric and astrometric measurements agree well with the other reported measurements. Furthermore, the astrometric collection of measurements has grown significantly such that estimates of the relative velocities in the plane of the sky can be greatly improved. Combining the HST measurements with those published previously[^3] and assuming constant linear motion, we find a velocity of $6.68 \pm 0.52$ mas yr$^{-1}$ at a position angle of $260^{\circ} \pm 5^{\circ}$, with an average relative position of $0\farcs2485 \pm 0\farcs0013$ at the position angle of $358\fdg2 \pm 0\fdg3$. This velocity has a factor of 5 smaller uncertainties than previous estimates (Ghez et al. 1995; Woitas et al. 2001), mostly resulting from including more measurements. With the greater precision in both the velocity reported here and the total mass of the system, derived from the rotation of the circumbinary disk (G99), it is now possible to solve for the whole orbit, assuming only that the orbit is coplanar with the circumbinary ring, and that the ring is intrinsically circular. Appendix \[getorb\] provides the details of how the orbital parameters, $e$, $a$, $P$, $T_{0}$, $\omega$ and their uncertainties are derived from the observables, $\overrightarrow{r}_{2D}$, $\overrightarrow{v}_{2D}$, $M$, $i$, $D$, $\Omega$.
We find that the stars are in an elliptic orbit with an eccentricity $e=0.32\pm0.20$ and a semi-major axis $a=35^{+22}_{-8}$AU. The corresponding orbital period is $185^{+195}_{-55}$yr. We emphasize that the uncertainties quoted here include the uncertainty on the distance to the Taurus molecular cloud, which must also be taken into account when considering the system mass (G99). Indeed, together with the uncertainty in the measured velocity, this is the dominant source of error in the quoted uncertainties. Although the orbital parameters derived here still have large uncertainties, this analysis shows the applicability of this method. Additional astrometric data over the next few years should help to significantly decrease these uncertainties and ascertain the exact orbit of the binary. We note that our orbital solution differs significantly from that obtained in the past, in the sense that we find that the stars are close to apoastron while R96 concluded that the stars must be close periastron. This contradiction is the result of our estimated velocity being much smaller than the earlier estimate, and smaller than the velocity one would observed if the orbit was circular. A similar conclusion, based on velocity comparison alone, was recently reached by Krist et al. (2002). Our derived orbit is shown in Figure \[orb\], where we have also plotted yearly weighted averages of the data points used in the fit.
In the framework of investigating interactions between the ring and the inner binary, it is interesting to compare the inner radius of the circumbinary disk to the semi-major axis of the system. The ratio of these two quantities lies in a 1$\,\sigma$ range $3.2 < R_{in}/a < 6.7$; the two stars have cleared a wide gap around them. Artymowicz & Lubow (1994) have used analytical and numerical approaches to study the formation of such a gap through gravitational resonances in geometrically thin disks around circular and eccentric binary systems. The orbital properties of GG Tau A provide us with the first direct test of these models in pre–main-sequence visual binary systems, although uncertainties prevent us from reaching definitive conclusions. Artymowicz & Lubow found that this gap clearing phenomenon is primarily driven by the orbital eccentricity, with a smaller dependence on the mass ratio, ($M_{B}/M_{A}\sim0.9$ in the case of GG Tau A; White et al. 1999). The results of Artymowicz & Lubow can be summarized as follows: while the $R_{in}/a$ ratio is on the order of 1.7 for circular orbits, it can grow up to $\sim3.3$ for highly eccentric binaries ($e=0.75$). The derived orbital parameters appear somewhat problematic, with a $R_{in}/a$ ratio that seems to be larger than those expected for a moderately eccentric system. This may mean that the dynamical evolution of the GG Tau circumbinary ring is more complex than previously thought.
Circumbinary Disk Properties {#cbdisk}
----------------------------
### Disk Detection & Morphology {#1-D}
Azimuthal averages of the PSF subtracted images (see Figure \[profile\]) show that the circumbinary disk is easily detected at radii greater than $\sim$1 at all three wavelengths. The circumbinary disk is detected with an average SNR $>$ 5 per pixel at radii of 09 to $\sim$19 (F110W and F160W) and 11 to 19 (F205W data) with an average peak surface brightness of 15.3, 14.1 and 14 mag/arcsec$^{2}$ in F110W, F160W and F205W respectively. At smaller radii the 1$\sigma$ subtraction noise level rises dramatically, from $\sim$14 mag/arcsec$^{2}$ at 05 to $\sim$11 mag/arcsec$^{2}$ at 02, with each filter having a slightly different noise level (see Figure \[profile\]). This prevents a meaningful investigation of gap material that has been posited in varying forms in earlier observations (R96; Silber et al. 2000; Krist et al. 2002).
A model of the two-dimensional near-infrared apparent disk geometry is constructed from the F110W and F160W data by analyzing the intensity profile of the disk as a function of azimuth. Note that what we are observing here is the geometry of the optically thick scattered light distribution, not the true disk geometry. For every 10 degree segment of the disk, an average radial profile is produced. The position of the radial profile peak value is found using two methods. The first calculates a weighted centroid over a 05 region centered on the maximum value of the radial profile, which works well on sections of the disk that have a distinct peak. The south side of the disk, however, has a much flatter radial profile, and the weighted centroid method is not as stable for these segments. The second method fits the radial profile with a 4 degree polynomial. The ‘center’ of the disk is then calculated by finding the midpoint between the two radii at which the disk value is equal to half the peak value. For the north side of the disk where the radial profile is almost gaussian in shape, this method finds the same disk peak. For the flatter southern profiles it provides a more robust method of finding the disk center. The final peak position values are taken from the results of the polynomial fit. This method of finding the disk peak location also provides an estimate of the width of the radial profile, which is taken to be the distance between the two half-peak radii. Uncertainties for these values are estimated by taking the largest of either the difference between the values found through the two methods or the standard deviation of the values found from applying the same analysis to 3 subsets of the data.
The overall apparent shape of the disk is explored by fitting an ellipse to the positions of the radial profile peak values. We find an ellipse with a semi-major axis of 142 $\pm$ 006 (200 AU) orientated at a position angle, $PA_{NIR}$ of 21$^{\circ}$ $\pm$ 9$^{\circ}$ and an eccentricity of 0.64 $\pm$ 0.02. Although only G99 provided uncertainties for their analysis, our measurements of the ring geometry are consistent with that found in both the previous near-infrared (R96) and millimeter observations (Dutrey et al. 1994; G99). If you assume that the disk is intrinsically circular and geometrically thin, the observed eccentricity corresponds to a disk inclination of 40$^{\circ} \pm 2^{\circ}$. While this is consistent with inclination measurements made from the optically thin millimeter images of the circumbinary ring ($i=37^{\circ} \pm 1^{\circ}$, $PA_{disk}=7^{\circ} \pm 2^{\circ}$; G99), the disk is known to be geometrically thick, which should bias the observed inclination towards slightly larger values. G99 also note that the geometrical thickness of the inclined ring will cause the northern inner edge of the scattered light ring to appear closer to the center of mass of the system than the inner edge observed in the optically thin millimeter images. This offset can be used to calculate the total height of the disk above the midplane, i.e., where the NIR light becomes optically thin. This height is where the disk is physically truncated in the Monte Carlo simulations (see §\[discus\]). In the NICMOS images, the northern inner edge (the peak of the radial profile) occurs $0\farcs88 \pm 0\farcs07$ away from the center of mass. From the disk model fits to the millimeter images (G99), the inner edge of the millimeter disk occurs at a projected distance of $R\cos i = 1\farcs03
\pm 0\farcs02$. The offset, $0\farcs15\pm0\farcs07$, corresponds to a total disk height of $35^{+21}_{-18}$AU (using the relation from G99). This is smaller than the 025 offset derived by G99 from the deconvolved adaptive optics images of R96. In addition, the disk height will also translate into an offset between the center of the ring and the center of mass of the system. The disk fitting routine finds that the center of the ring is offset from the center of mass of the binary by 021 $\pm$ 003 along a position angle of $162^{\circ} \pm 9^{\circ}$ degrees, consistent with the offset measured from the inner edge.
The width of the disk has a clear azimuthal dependence, with the North side being narrower than the South side, as noted by both Silber et al. (2000) and Krist et al. (2002). Figure \[width\] shows the measured width, which ranges from 03 to $\sim$1, as a function of position angle. We fit these measurements with a modified Henyey-Greenstein scattering phase function (described in $\S$\[azvar\]), which is physically meaningless in this context but provides an analytical description of the data. This analysis shows that the thinnest portion of the disk occurs at a position angle of 356$^{\circ}$, independent of wavelength (see Table \[azfit\]). Because the function we used for the fit is unphysical and provides only a moderately good fit, we do not estimate uncertainties.
At the lowest contour levels we observe a deviation away from the elliptical fit along the southern edge of the disk (see Figure \[kink\]). This deviation consists of a sharp elbow, or kink, in the south-east edge of the disk and a straightening of the isophotes along the southern edge which makes the disk outer edge appear boxy. This effect is present in all 3 filters, and coincides with the kink detected by Silber et al. (2000) and noted by Krist et al. (2002).
### Integrated Disk Photometry {#colors}
In order to assign flux densities to the disk, we define an annular aperture with an eccentricity of 0.64 centered at the position of the center of the ellipse with an inner semi-major axis at 1 and outer semi-major axis at 22, based on the results given in §\[1-D\]. The area affected by the diffraction spikes is particularly noisy and is excluded from the aperture by masking off these regions, reducing the effective aperture area from 9.2 to 5.82 square arcseconds. The spike mask comprises two diagonal stripes, 08 in width, centered on the position of the primary. Summing the counts over this area and correcting the flux for the masked off regions (by multiplying the area of the masked regions with the mean disk flux per pixel), provides the estimates of the total disk magnitudes given in Table \[astrom\]. Since the pixel scale is smaller than the diffraction limit, the estimated uncertainties are based on maps that have been averaged over regions corresponding to the diffraction limited beam sizes (01, 016 and 02 for F110W, F160W and F205W respectively). These measurements of the integrated disk intensity have an order of magnitude higher signal to noise ratios than earlier ground-based measurements.
The overall disk magnitudes are compared to those derived for the binary system (§\[orbit\]) to obtain disk to star flux ratios and color excesses (see Table \[nearfar\]). The NICMOS images presented here lead to a disk to star flux ratio of $\sim$1.5%, which is 2.5 times larger than that derived by R96. Likewise, the color of the circumbinary disk is comparable to that of the stars:
$\Delta(M_{F110W}-M_{F160W}) = 0.10 \pm 0.03 $\
$\Delta(M_{F160W}-M_{F205W}) = -0.04 \pm 0.06 $
where $\Delta(M_{1}-M_{2})$ is the color excess of the disk, or $(M_{1}-M_{2})_{disk} - (M_{1}-M_{2})_{star}$. This is in contrast to the large red excess suggested earlier.
### Spatially Resolved Disk Photometry {#azvar}
The spatially resolved color properties of the circumbinary disk are investigated by calculating the disk magnitudes within circular apertures which are 019 in diameter (roughly the size of the 2 diffraction limit) placed around the disk. For this comparative analysis, the F110W and F160W bias-subtracted images have been convolved to the resolution of the F205W image. Only areas of the disk used to calculate the integrated disk photometry are included in this analysis. Figure \[diskmag\] displays the resulting color-magnitude plot, which shows no significant trend in color with respect to magnitude. Unlike the results from R96, the NICMOS measurements provide no evidence for extinction within the disk.
As originally mentioned by R96, the disk exhibits azimuthal variations in intensity that are presumed to arise from angular variations in the scattering efficiency, with the brightest side of the ring corresponding to forward scattering from the edge of the disk that is nearest to our line of sight. Here we present a detailed quantitative analysis of these azimuthal variations, using normalized peak flux densities derived from the 10$^{\circ}$ azimuthal averages discussed in §\[1-D\]. To estimate the position angle of the brightest region, the data are fit, using a chi-square minimization technique, with a modified[^4] form of the Henyey-Greenstein scattering phase function (Henyey & Greenstein, 1941)
$$S(PA) \propto [1-g'^{2}][1+g'^{2} - 2g'cos(PA-PA_{0})]^{-3/2}$$
where $PA_{0}$ is the value of position angle that maximizes the function. While the use of this function is not physically meaningful, as the phase function depends on scattering angle, not position angle on the sky, it does provide a convenient method of characterization. The best-fit functions are displayed with the data in the left hand column of Figure \[phase\] while the values of $PA_{0}$ and the fitted peak-to-peak near side to far side flux ratios are listed in Table \[azfit\]. The peak near side to far side flux ratio is $\sim$4 with no significant wavelength dependence. The position angle of the brightest portion of the disk ($22^{\circ} \pm
4^{\circ}$) is roughly consistent with that measured by Krist et al. (2002) in I band images (25$^{\circ}$) and both the position angle of the semi-minor axis ($7^{\circ} \pm 2^{\circ}$, G99) and the position angle of the thinnest region ($-4^{\circ}$, derived in §\[1-D\]). This provides strong support for the hypothesis that the disk is geometrically thick.
The azimuthal variations in intensity arise from variations in dust grain scattering efficiencies as a function of scattering angle, rather than the observed position angle around the disk. Assuming a single scattering scenario in a geometrically thin disk, Close et al. (1998) derive scattering properties of the dust grain population by matching the intensity extrema to scattering angles of $90-i$ and $90+i$, which, with an assumed inclination, $i$, of 35$^{\circ}$, results in scattering angles of 55$^{\circ}$ and 125$^{\circ}$. The NICMOS images, however, contain much more information than just the intensity extrema, as the circumbinary disk samples a large range of scattering angles as a function of position angle around the disk. We therefore take the additional step of fitting the entire azimuthal variation of flux with a scattering phase function. Using the known disk geometry, the observed variation with position angle can be translated, or de-projected, into a variation with scattering angle, $\theta$, which is defined as the angle of deflection away from the forward direction of the incoming light. $\theta$ is geometrically related to the position angle around the disk, $PA$, the position angle which maximizes the azimuthal variation, $PA_{0}$, and the disk inclination, $i$, by:
$$cos(\theta+\phi_{open}) = \sqrt{1 - \frac{1}{1+cos^{2}(PA-PA_{0})tan^{2}(i)}} \times (-1)^{j}$$
where
$j = 1$ if $cos(PA) < 0$\
$j = 0$ if $cos(PA) > 0$
A derivation of this relation can be found in Appendix \[getscatang\]. $\phi_{open}$ is the opening angle of the disk as seen from the stars; including this factor converts the relation from a geometrically thin disk to a thick disk case. It is assumed here that, at all position angles on the sky, the intensity is dominated by the portion of the ring corresponding to the smallest scattering angle, which is true for forward-scattering grains. Using the values for the inner radius of the disk and a disk height of 35AU, we calculate $\phi_{open}$ to be 11$^{\circ}$. In this case, with an inclination of 37$^{\circ}$, the scattering angle ranges from $\sim$40$^{\circ}$ at the closest edge to $\sim$120$^{\circ}$ at the edge furthest from us. Using the above relationship to translate the observed position angle into scattering angles, the intensity variations are then fitted with a normalized Henyey-Greenstein scattering phase function, in order to estimate the value of the dust asymmetry parameter, $g$. This parameter is the averaged cosine of the scattering angle which ranges in value from -1 to 1. Given that we are using the one parameter Henyey-Greenstein scattering phase function we can only investigate forward throwing dust grains with $0 < g < 1$. In this case, if the light is scattered isotropically, then the amount of forward scattered light equals that being back-scattered and $g$=0. As the light becomes more strongly forward-scattered, the value of $g$ increases. The results of the fit for the dust asymmetry are summarized in Table \[azfit\]. We find that omitting the disk opening angle will slightly overestimate $g$. It should be recalled that these values are obtained under the assumption of single scattering (see §\[multiplescat\] for further discussion).
Since the system may be modulated by geometric factors, such as a possible shadowing by circumstellar disks, Wood et al. (1999) found it more convenient to work with an integrated quantity to describe the near side to far side flux ratio. Integrating the flux over the northern and southern halves of the disk, as defined by the position angle of the disk, we find the ratio of scattered light to be $\sim$1.4, with no significant wavelength dependence (see Table \[nearfar\]). This value is roughly a factor of two smaller than that seen in the ground-based AO images (Wood et al. 1999).
Discussion {#discus}
==========
In this section, we use the various quantities derived in §\[res\] to revisit the question of whether the existence of large dust grains or opaque circumstellar disks are required to explain the observed color excesses. Monte Carlo scattering simulations are presented, qualitatively reproducing the NICMOS data (§\[MCmodel\]) and emphasizing the key role played by grain properties, especially the dust asymmetry (§\[multiplescat\]). The scattering simulations show that neither circumstellar disks or large grains are absolutely needed to reproduce the data and that our fitted scattering phase function can be used to derive a lower limit on the median size of the scatterers.
Numerical Modeling of the GG Tau Ring {#MCmodel}
-------------------------------------
Since the NICMOS measurements of the GG Tau circumbinary disk provide different results from both those previously reported and modeled, we ran new Monte Carlo simulations to investigate whether the observed disk colors can be reproduced using a standard ISM grain model. We began, like Wood et al. (1999), with the simplest case: modeling the scattered light distribution from a thick torus of ISM-like dust, without the additional complications of either circumstellar disks or grain growth (their model 1).
We used a Monte Carlo code that was readily available to us; the same model which successfully reproduced the NICMOS polarization results (Silber et al. 2000). The Monte Carlo code is based on a program from Ménard (1990) that has been modified such that each photon interacts with a randomly sampled dust grain from the input grain size distribution. Details of the modifications can be found in Duchêne (1999). The disk is modeled as a thick torus, using a total disk height of 38 AU at the inner edge of the ring, close to that estimated from the observed inner edge offset (see §\[1-D\]). This torus has a total disk mass of 0.13 $M_{\odot}$ and a scale height of 21AU at the inner edge of the ring (180AU) that varies as $r^{1.05}$ (G99). Outcoming photons are sorted by inclination: all results presented here are for the range of inclinations closest to the actual inclination of the system, $37^{\circ} < i < 46^{\circ}$. The simulations follow 1 million photons per filter. The dust grain properties are taken from the Mathis & Whiffen (1989, hereafter known as MW) dust model. While the dust parameters (e.g., albedo and dust asymmetry) in the MW dust model differ significantly from the Kim, Martin & Hendry (1994; hereafter known as KMH) model[^5] used by Wood et al. (1999), both models successfully reproduce the ISM extinction curve. The simulated disk properties can be found in Table \[nearfar\]. The resulting Monte Carlo simulations (see model 1 in Table \[nearfar\]) roughly recreate the observed disk to star flux ratios and near side to far side flux ratios. They also produce a considerably red disk color excess, $\Delta(M_{F110W} - M_{F160W})=0.50$, $\Delta(M_{F160W} -
M_{F205W})=0.28$. This is distinctly larger than the color excess observed with NICMOS, and very different from the blue scattered light disk seen in the Wood et al. (1999) models.
The two main differences between the Wood et al. simulations and those presented here are the dust grain properties used[^6] and a slightly different disk geometry. Both the overall disk height and the degree of flaring differ (Wood et al. use a scale height that varies as $r^{1.25}$ from the earlier millimeter work of Dutrey et al. 1994). To fully compare our numerical results with those of Wood et al., we ran Monte Carlo simulations where the only difference is the dust grain properties. The results from these simulations (see model 2 in Table \[nearfar\]) show that the red disk color excesses remain and therefore are predominantly caused by the dust grain properties.
The Influence of Multiple Scattering and Grain Properties:\
Explaining the Different Modeled Colors {#multiplescat}
-----------------------------------------------------------
The apparently contradictory simulations presented here and in Wood et al. can be reconciled by investigating the effect the different dust grain properties have on the scattering process. In optically thick situations, photons are scattered multiple times. The resulting scattered light surface brightness is dependent on two key dust parameters, the albedo of the dust, $\omega$, and the azimuthal scattering dependence given by the dust asymmetry parameter, $g$. In terms of disk geometry, the inclination of the system is also an important factor. At any one scattering angle, the output scattered surface brightness is dependent on both the albedo of the dust and the number of scattering events $n$, such that $SB_{scat}
\propto \omega^{n(g_{\lambda})}$ (e.g., Witt & Oshel, 1977; Witt 1985). The greater the number of scatterings, which is a function of the dust asymmetry parameter, the greater reduction in the resulting scattered surface brightness.
The simulations presented here and those in Wood et al. (1999) are based on the same Monte Carlo scheme and have the same disk geometry. The main difference lies in grain properties used, and this alone is enough to explain the disk color reversal between the two sets of models. Table \[dustprops\] compares the dust properties used in the two simulations. The dust asymmetry of the median scatterer in our simulations is $\sim$2 times larger than that used by Wood et al., while the albedos are essentially the same. How does this affect the observed scattered light distribution? Consider first the far side (southern edge) of the ring where back-scattering is the dominant effect. For low $g$ values, such as those used in the Wood et al. simulations, the scattering is not far from being isotropic, which means that the photons leaving the disk toward the line of sight will in general have gone through a small number of scatterings ($\sim$ 1). Thus the resulting surface brightness distribution will not be strongly affected by multiple scattering effects and the scattered light colors will be similar to the wavelength dependence of the dust albedo, resulting in Wood et al.’s case in blue disk color excesses. For more strongly forward scattering dust particles (large $g$), back-scattering is rarer and in this region of the disk most of the scattering events will cause photons to move further into the disk, away from our line of sight. The photons that do escape from the surface of the disk toward our line of sight will have undergone a much larger number of scatterings and the resulting scattered light surface brightness will be significantly changed by the wavelength dependence of the dust asymmetry parameter. For the MW dust properties, not only is $g$ larger than those in the KMH model, they also have a stronger wavelength dependence. The scattered light surface brightness will be more reduced at 1 than it is at 2 producing significantly red disk colors. The same story applies to the nearest (northern) edge of the disk, where we are predominantly seeing forward scattering off the upper limb of the ring. This type of scattering is enabled by both low and high $g$ values (near-isotropic and strongly forward scattering); while the number of scattering events remains small in both simulations there will still be more scattering events for the photons in our simulation than those in Wood et al.’s due to the inclination of the system favoring low $g$ values slightly more. This, combined with the corresponding slightly longer pathlength in our simulation, enabling more extinction, is enough of an effect to compensate for the intrinsically blue color of the scattering, as seen in our simulations. Taking the disk as a whole, this explains why our simulations predict a red disk color, while those of Wood et al., being dominated by the flux from the far side of the disk, result in overall blue disk colors. The same $\omega^{n(g_{\lambda})}$ dependency is responsible for the reduced disk to star flux ratios and increased near to far flux ratios observed in our simulations, as photons undergo more scattering events in the far side of the disk than in Wood et al.’s (see Table \[nearfar\]). Given the range of colors that can be simulated using the two grain models, it could be expected that a grain population with properties intermediate of those in the KMH and MW models would result in neutral near-infrared colors, as observed.
An additional effect of multiple scattering observed in our simulations is that the disk inclination and optical thickness combine to modify the azimuthal intensity variation from that expected from single scattering. Given that the observed surface brightness is dominated by the least scattered photons ($\omega^n$ dependency), for grains that are predominantly forward-scattering, photons received from the back side of the ring are biased towards the back scattering part of the phase function. Statistically, most of the photons will be scattered towards the middle of the ring and will either not reach the observer or be of an extremely low intensity. Scattering from the front edge of the ring, however, will remain mostly unbiased. This biasing effect will cause the derived $g$ value measured in §\[azvar\] to underestimate the true dust asymmetry, and hence our value is only a lower limit. The same reasoning applies to the Close et al.’s analysis, as a consequence of considering only single scattering. As an example of this effect, in our simulations, the size of the median scattering grain is 0.56 and these have an asymmetry parameter $g\sim0.8$ at 1 . However, the measured $g$ from the simulations (obtained using the same method as in §\[azvar\]) is only $\sim0.3$.
The observed lower limit on g, measured in §\[azvar\] provides a lower limit on the median size of the scatterers as the asymmetry value is a sensitive function of the size parameter, $x$ ($=2\pi a /
\lambda$), for $g$ less than $\sim$0.8. From the MW grain properties used, we find that the measured asymmetry of $g > 0.39$ at 1 corresponds to a grain size, $a >$ 0.23 .
Implications and Summary {#implications}
========================
The GG Tau disk is clearly detected ($SNR_{pix} > 5$) in all three NICMOS filters between radii of 10 to 22, scattering approximately 1.5% of the stellar light. No discernible wavelength dependence of scattered light is observed, although azimuthal variations in both width and intensity are seen and modeled. We confirm the observation of a kink in the disk by Silber et al. (2000). Observed as a sharp elbow in the disk on the south-east edge, followed by a flattening of the isophotes along the southern edge of a disk, this feature is seen in all three filters. The projected distance of the northern edge from the center of mass, related to the total height of the ring, provides the first hint that the ring may not be as thick as previously estimated, with a total height of 35AU at the inner edge of the disk (180AU).
The presence of such a well-observed circumbinary ring around this system, combined with the 9 year baseline of binary observations has allowed us to constrain the orbital parameters of the binary. We find a slightly elliptical orbit $e=0.3 \pm 0.2$, with a semi-major axis $a=35^{+22}_{-8}$AU and a corresponding orbital period of $185^{+195}_{-55}$yr. The 1$\sigma$ range of estimated orbital semi-major axis is such that the ratio of semi-major axis to inner disk radius is larger than expected from previous SPH simulations (Artymowicz & Lubow, 1994). This suggests that the dynamical evolution of the circumbinary ring may be more complex than previously thought. Further astrometric observations of this close binary are needed to constrain the binary orbit further.
The Monte Carlo simulations summarized in §\[MCmodel\] show the wide range of colors one can ‘predict’ from the GG Tau disk, from blue to red, using different dust size distributions and grain properties ($\omega$, $g$). Specifically, our Monte Carlo simulations show that the observed neutral disk colors, ratio of disk to star light, and near side to far side flux ratio can probably be reproduced with some combination of an ISM-like grain size distribution and dust properties and does not necessarily require either the presence of large dust grains or prior extinction by circumstellar disks. However, this does not rule out that either is present. Because the disk is optically thick in the near-infrared, studies of scattered light are only sensitive to those grains that are located near the surface of the ring. Any conclusion about grain growth only applies to those regions and does not provide any information the size of grains deep in the ring. For instance, it has been shown that grains larger than $\sim
1\,\micron$ tend to segregate in the disk midplane because of their larger mass (e.g., Suttner & Yorke 2001) and possibly as the result of the enhanced grain-grain collision rate in higher density regions. The presence of such large grains in the midplane cannot be determined in our near-infrared observations. Circumstellar disks are known to surround both components of GG Tau A, however, the possible effects they have on the integrated circumbinary disk colors have been shown to be subtle and somewhat surprising. While Krist et al. (2002) suggest that the red disk colors observed in the optical could be caused by an $A_{V} \geq$ 1.2 mag of extinction, possibly in the form of circumstellar disks, Wood et al. (1999) find that the inclusion of small circumstellar disks can actually cause the integrated disk colors to become [*bluer*]{} rather than redder. Analysis of the disk colors at both optical and near-infrared wavelengths is planned for future work.
While the color of scattered light in optically thick disks cannot currently be used to constrain grain properties, the azimuthal intensity variations can be used to provide a lower limit on the median grain size, independent of grain model. This lower limit ($g >
0.39$ at 1 ) corresponds to a median grain size of $a > 0.23$ . Given that the ISM dust models produce a wide range of median grain sizes ranging from $\sim$0.16 (Mathis, Rumpl & Nordsieck 1977; KMH) to 0.56 (MW), this lower limit does not suggest that grain growth has occured in the surface layers of the disk in this young ($\sim$1 Myr old; White et al. 1999) system.
There is still considerable uncertainty regarding the ISM dust properties (e.g., Witt 2000), a point which is well illustrated here by the comparison of Monte Carlo scattering results using two of the more well known dust grain models. Both the KMH and MW grain models, like most of the ISM dust models to date, have been constrained using the ISM extinction curve, which they reproduce equally well. The results presented here suggest that using the extinction curve [*alone*]{} to constrain the dust properties introduces a significant level of uncertainty in the scattering properties of the grains. As an example, KMH and MW grain models predict V band $g$’s of $\sim0.5$ and $\sim0.9$ respectively. Observations of reflection nebulae in the same bandpass infer $g\sim0.7$, significantly different from either of these models (Witt, Oliveri & Schild, 1990). Additionally, models of T Tauri disks seen in scattered light with WFPC2 find that the observations are best fit by $g\sim0.65$ (Burrows et al. 1996; Stapelfeldt et al. 1998; Krist et al. 2002), although these values may be affected by grain growth. Combining additional sources of information, such as observations of dust scattering (e.g., Witt, et al. 1990), polarization (e.g., Zubko & Laor 2000), and dust thermal emission (e.g., Li & Draine 2001) will provide additional constraints. Without a more detailed knowledge of the ISM dust grain properties it is unlikely that we will be able to unambiguously determine whether grain growth is occuring in T Tauri disks through scattered light imaging.
Although dust properties in the disk cannot currently be constrained from either colors or dust asymmetry alone, this work provides an outline for future analysis and modeling. Observationally, further resolved intensity and polarization maps at other wavelengths are key in sampling a large enough grain size parameter ($x$) range to constrain the dust grain properties. For instance, at 3 to 5 the dust asymmetry is expected to be much more isotropic, allowing significant color variations (e.g., $J-L$) between the front and back sides, unless significant grain growth has already occurred. Such observations should be carried out in parallel with a full exploration of the parameter space (grain size distribution, grain properties, ring geometry) through Monte Carlo scattering simulations.
The authors thank Mike Jura, Alycia Weinberger, François Ménard, Kenny Woods, Lisa Prato, and Angelle Tanner for enlightening discussions. We also thank the anonymous referee for their constructive comments. Support for this work was provided by NASA through grant number GO-06735.01-95A from the Space Telescope Institute, the NASA AstroBiology Institute and the Packard Foundation. This research was based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc, under NASA contract NAS5-26555. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.
Determination of the orbital parameters of the binary {#getorb}
=====================================================
In this appendix, we present the method used to derive the orbital parameters of the binary from a single measurement of its separation and relative velocity with some additional knowledge available from previous studies.
The orbit of a binary system can be described with a set of 9 independent parameters: mass ratio ($q$), eccentricity ($e$), total mass ($M$), orbital period ($P$), time of periastron ($T_0$), inclination of the orbital plane ($i$), position angle on the sky of the ascending node of the orbit ($\Omega$), angle in the orbital plane between periastron and the ascending node ($\omega$), and distance to the system ($D$). Note also that $P$ and $M$ are linked to the semi-major axis of the orbit, $a$, through Kepler’s third law. When considering the [*relative*]{} motion of one component around the other one, $q$ is no longer a relevant parameter. Therefore, one must measure 8 independent quantities to solve the orbit unambiguously. In the case of GGTau, the millimeter interferometric measurement of the circumbinary ring by G99 yields precise estimates of three independent parameters: the inclination of the ring, $i'=37\degr\pm1\degr$, the position angle of the apparent semi-minor axis of the ring, $\Omega'=7\degr\pm2\degr$, and the total system mass $M=1.28\pm0.07\times (D/140\,{\rm pc}) M_\odot$. If the ring is intrinsically circular then $\Omega=\Omega'\pm90\degr$ (depending on the motion of the binary). Furthermore, if we assume that the orbital plane corresponds to the ring mid-plane, we have $i=i'$. These two assumptions (circular ring and coplanar orbit) are the only assumptions we make in the following analysis. Additionally, the HIPPARCOS determination of the distance to the Taurus star-forming region, $D=139\pm10$pc, by Bertout et al. (1999), allows us to convert the apparent separation and velocity of the binary into absolute values. Prior to our work, the problem could thus be simplified to a problem with only four unknowns.
The orbit of the system has been followed over the last ten years or so and, since all measurements so far are consistent with a uniform linear motion, we combined them all to obtain the average binary separation and linear velocity projected on the sky. These quantities can be represented for instance by the respective amplitude of the binary separation and of its velocity ($\rho$ and $\dot{\rho}$, measured in arcsec and arcsec.yr$^{-1}$) and by the angle between these two vectors in the plane of the sky ($\delta$). In the following derivation, a useful quantity is the angle $\theta$ between the apparent semi-minor axis of the ring and the binary separation, which represents the polar coordinate of the current measurement with respect to the disk itself. This angle can be expressed as $\theta=\Omega'-PA_{bin}$. Our data imply $\rho=0\farcs2485\pm0\farcs0013$, $\dot{\rho}=(6.68\pm0.52)\,10^{-3}\,\,\arcsec{\rm yr}^{-1}$, $\delta=96\fdg0\pm5\fdg5$ and $\theta=11\fdg2\pm2\fdg1$ (see §\[orbit\]). The orientation of the relative velocity implies that $\Omega=97\degr$. We have thus obtained the four last measurements needed to solve the whole orbit.
The two-body problem is a classical mechanics problem, the solution of which can be found in many textbooks. We are interested here in the case where only relative measurements of the 3D binary separation ($\overrightarrow{r}$) and velocity ($\overrightarrow{v}$) are available from the de-projection of the observed separation and motion in the orbital plane. The eccentricity and semi-major axis can then be derived from measurements of the angular momentum, $j=|\overrightarrow{r}\times\overrightarrow{v}|$, and total energy, $E=\frac{1}{2}v^2-\frac{GM}{r}$, [*per unit mass*]{}. We have the following relations: $e^2=1+\frac{2j^2E}{G^2M^2}$ and $a=-\frac{GM}{2E}$. These relations can be translated into the observables presented above as follows: $$e^2=1+\frac{\rho^2{\dot{\rho}}^2D^4\sin^2\delta}{G^2M^2\cos^2i}\left\{
{\dot{\rho}}^2D^2
\left[\sin^2(\delta+\theta)+\frac{\cos^2(\delta+\theta)}{\cos^2i}\right]
-2\frac{GM\cos i}{\rho D} \right\}$$ and $$a=GM\left\{2\frac{GM\cos
i}{\rho D}-{\dot{\rho}}^2D^2\left[
\sin^2(\delta+\theta)+\frac{\cos^2(\delta+\theta)}{cos^2 i}\right]
\right\}^{-1}$$
One can immediately derive the orbital period: $P({\rm yr}) =
\sqrt{\frac{a^3({\rm AU})}{M(M_\odot)}}$. Uncertainties for all of these quantities were obtained by allowing all seven observables to vary within 1$\,\sigma$ of their nominal value and retaining the extreme values for each orbital parameter. Special care must be taken when considering the uncertainties induced by the distance estimate, as the total mass of the system is also affected in a [*systematic*]{} manner by the distance. The distance and relative velocity of the binary estimates combine for the largest part of the total uncertainty for the orbital elements.
For completeness, we derive the remaining two orbital parameters that have not been addressed so far, $\omega$ and $T_0$. The former is the angle, in the orbital plane, between periastron and the ascending node. This can be derived from the angular distance between the current location of the companion and periastron. The latter angle can be determined using the general equation of the elliptical orbit, $$r=\frac{a(1-e^2)}{1+e\cos (\phi-\varpi)}$$ where $r$ and $\phi$ are the polar coordinates of a point running along the orbit and $\varpi$ is the position angle of periastron. Using the current de-projected separation of the system, 43.0AU, we find that the binary is about $(\phi-\varpi)=-145\degr$ away from periastron[^7]. Combining this information with the angle between the current binary location and the ascending node ($100\degr$), we conclude that $\omega=245\degr$. To estimate how much time it takes for the companion to go from one point to another along the orbit, we use Kepler’s second law and numerically integrate the area $A$ defined by the section of an ellipse between these two points. We can easily calculate the time derivative of the area, which is related to the angular momentum per unit mass by $\frac{{\rm
d}A}{{\rm d}t}=\frac{1}{2}j$. Applying this to the average position of the companion and periastron, we find that the next periastron passage will occur about 61yrs from now or, equivalently, that $T_0=2472467$JD. Both $\omega$ and $T_0$ depend heavily on the previously derived quantities, therefore, we emphasize that the uncertainties on these parameters are rather large.
Calculating Scattering Angles {#getscatang}
=============================
In this section we derive a relation between position angle around a disk and the angle a photon is scattered through. At each position angle around the disk, we assume that the observed photons are scattered off the closest disk edge which corresponds to the smallest available scattering angle (see §\[azvar\]).
Consider a circular ring that is inclined to the line of sight at an angle $i$. This ring is projected on the sky as an ellipse (see Figure \[model\]a), with normalized semi-major axis, $d=1$ and aspect ratio $c/d = cos i$. The position angle on the sky of a random point around the disk, $PA$, is defined as $tan(PA) = x/y$. In Figure \[model\]a we have rotated the image so that the semi-minor axis of the disk is now vertical. The amount of rotation, $PA_{0}$, is the $PA$ of the semi-minor axis of the disk, defined in the usual manner.
The scattering angle is defined as the angle between the pre- and post- scattering directions, $\vec{n}$ and $\vec{m}$ respectively, as shown in Figure \[model\]b. This angle, $\theta_{scat}$, is $\theta - \phi_{open}$, where $\phi_{open}$ is the angle subtended by the disk height. If the disk can be assumed to be flat, we have $\theta_{scat} = \theta$. We first consider this case before generalizing our results to $\phi_{open} \neq 0$. The coordinates of the unit vector $\vec{n}$, $(u,v,w)$ are along the photon’s initial path to a random point on the ring, $M (x,y)$, in the $(\vec{x},\vec{y},\vec{z})$ frame are:
$$\begin{aligned}
u = x \\
v = y \\
w = \sqrt{(1-x^2 -y^2)} \times sign(cos(PA))\end{aligned}$$
Writing the standard equation of an ellipse: $$(\frac{x}{d})^{2} + (\frac{y}{c})^{2} = 1$$
and introducing the definitions of $i$ and $PA$, we find: $$x^{2} \left( 1 + \frac{1}{tan^{2}(PA)cos^{2}(i)} \right) = 1$$
By definition, the dot product of the unit vectors, $\vec{n}$ and $\vec{m}$ is: $$\vec{n}\cdot\vec{m} = \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right) \left(\begin{array}{c} u \\ v \\ w \end{array}\right) = cos(\theta)$$
$$\begin{aligned}
cos(\theta) = \sqrt{(1-x^2 -y^2)} \times sign(cos(PA)) \\
cos(\theta) = \sqrt{1-x^2(1 + \frac{1}{tan^2(PA)})} \times sign(cos(PA))\\
cos^2(\theta) = 1 - \frac{x^2}{sin^2(PA)} \\\end{aligned}$$
Substituting the expression for $x^2$: $$\begin{aligned}
1-cos^2(\theta) = \frac{1}{sin^2(PA)(1 + \frac{1}{tan^{2}(PA)cos^2{i}} ))} \\
= \frac{1}{1 + cos^2(PA)\left(\frac{1}{cos^2(i)} -1 \right)} \\
= \frac{1}{1 + cos^2(PA)tan^2(i)} \\
cos(\theta) = \sqrt{ 1 - \frac{1}{1 + cos^2(PA) tan^2(i)}} \times sign(cos(PA))\end{aligned}$$
or $$cos(\theta) = \sqrt{ 1 - \frac{1}{1 + cos^2(PA) tan^2(i)}} \times (-1)^j$$ where
$j = 1$ if $cos(PA) < 0$\
$j = 0$ if $cos(PA) > 0$
Reintroducing the position angle on the sky of the semi-minor axis of the ellipse, $PA_{0}$ and the disk opening angle, $\phi_{open}$, we can generalize this as:
$$cos(\theta) = cos(\theta_{scat}+\phi_{open}) = \sqrt{ 1 - \frac{1}{1 + cos^2(PA-PA_{0}) tan^2(i)}} \times (-1)^j$$
For a face-on disk ($i =0^{\circ}$) the scattering angle is always 90$^{\circ} - \phi_{open}$. Note that this relation between scattering angle and disk inclination breaks down for edge-on disks ($i = 90^{\circ}$).
Artymowicz, P., & Lubow, S.H., 1994, , 421, 651 Beckwith, S. V. W., Sargent, A. I., Chini, R. S. & Guesten, R., 1990, , 99, 924 Bertout, C., Robichon, N., & Arenou, F., 1999, , 352, 574 Bohren & Huffman, 1983, Scattering of Light from Small Particles Bushouse, H., Skinner, C., & MacKenty, J., 1996 Instrument Science Report ISR NICMOS-009 Burrows, A., Stapelfeldt, K., Watson, A., Krist, J.E., Ballester, G.E. et al. 1996, , 473, 437 Close, L.M., Dutrey, A.M., Roddier, F., Guilloteau, S., Roddier, C.,Duvert, G., Northcott, M., Ménard, F., Graves, J.E. & Potter, D. 1998, , 499, 883 Duchêne, G. PhD, Université de Grenoble, 1999 Dutrey, A., Guilloteau, S., and Simon, M. 1994, , 286, 149 Ghez, A.M., Neugebauer, G. and Matthews, K. 1993, , 106, 2000 Ghez, A.M., Weinberger, A.J., Neugebauer, G., Matthews, K., McCarthy, D.W., Jr., 1995, , 110, 753 Ghez, A.M., White, R. and Simon, M. 1997, , 490, 353 Guilloteau, S., Dutrey, A., and Simon, M. 1999, , 348, 570 [**(G99)**]{} Henyey, L.G., & Greenstein, J.L., 1941, , 93, 70 Holtzmann, J. A. et al. 1995, , 107, 156 Jensen, E.L.N., Donar, A. X., & Mathieu, R. D., 2000, IAU Symp. 200, The Formation of Binary Stars, 85 Kawabe, R., Ishiguro, M., Omodaka, T. Kitamura, Y. & Miyama, S. M. 1993, , 404, L63 Kim, S., Martin, P.G., & Hendry, P.D., 1994, , 422, 164 [**(KMH)**]{} Krist, J. 1993, A.S.P. Conference Series, Astronomical Data Analysis Software and Systems II, Vol. 2, 536 Krist, J. & Hook, R.N., 1999, The 1997 HST Calibration Workshop with a New Generation of Instruments, 192 Krist, J.E., Stapelfeldt, K.R. & Watson, A.M., 2002, , in press Leinert, C., Haas, M., Mundt, R., Richichi, A., & Zinnecker, H., 1991, , 250, 407 Leinert, C., Zinnecker, H., Weitzel, N., Christou, J., Ridgway, S.T., Jameson, R., Haas, M. & Lenzen, R., 1993, , 278, 129 Li, A., & Draine, B.T., 2001, , 554, 778 Mathis, J.S., Rumpl, W., & Nordsieck, K.H., 1977, , 217, 425 Mathis, J.S., & Whiffen, G., 1989, , 341, 808 [**(MW)**]{} Ménard, F. PhD, Université de Montreal, 1990 Monin, J.-L., Ménard, F., & Duchêne, G., 1998, , 339, 113 Roddier, C., Roddier, F., Northcott, M.J., Graves, J.E., and Jim, K. 1996, , 463, 326 [**(R96)**]{} Silber, J., Gledhill, T., Duchêne, G., Ménard, F., 2000, , 536, L89 Simon, M., & Guilloteau, S., 1992, , 397, L47 Stapelfeldt, K.R., Krist, J.E., Ménard, F., Bouvier, J., Padgett, D.L., & Burrows, C.J., 1998, , 502, L65 Suttner, G. & Yorke, H.W., 2001, ApJ, 551, 461 Thompson, R., 1995, , 2475, 367 Walter, F., Brown, A., Mathieu, R.D., Myers, P.C., and Vrba, F.J. 1988, , 96, 297 White, R.J., Ghez, A.M., Reid, N., Schultz, G. 1999, , 520, 811 White, R.J., & Ghez, A.M., 2001, , 556, 265 Witt, A.N., & Oshel, E.R., 1977, , 35, 31 Witt, A.N., 1985, , 294, 216 Witt, A., Oliveri, M.V., & Schild, R.E., 1990, , 99, 888 Witt, A., 2000, IAU Symp. 197, Astrochemistry: From Molecular Clouds to Planetary Systems, 317 Woitas, J., Köhler, R., & Leinert, Ch., 2001, , 369, 249 Wood, K., Crosas, M. & Ghez, A., 1999, , 516, 335 Wood, K., Smith, D., Whitney, B., Stassun, K, Kenyon, S.J., Wolff, M.J., & Bjorkman, K.S., 2001, , 561, 299 Zubko, V.G., & Laor, A., 2000, , 128, 245
[cllcccc]{} F110W &0.248 $\pm$ 0.002 &353.9 $\pm$ 0.4 &9.37 $\pm$ 0.05 &10.27 $\pm$ 0.07 &8.98 $\pm$ 0.02 &13.59 $\pm$ 0.01\
F160W &0.251 $\pm$ 0.002 &353.8 $\pm$ 0.3 &8.39 $\pm$ 0.01 &9.08 $\pm$ 0.04 &7.91 $\pm$ 0.02 &12.42 $\pm$ 0.01\
F205W &0.257 $\pm$ 0.003 &353.6 $\pm$ 0.8 &7.94 $\pm$ 0.01 &8.61 $\pm$ 0.01 &7.46 $\pm$ 0.02 &12.01 $\pm$ 0.05\
[lccccc]{} F110W &2.3 &-1 &4.3 $\pm$ 0.7 &22 $\pm$ 4 & 0.39 $\pm$ 0.02\
F160W &3.1 &-8 &3.1 $\pm$ 0.4 &19 $\pm$ 7 & 0.30 $\pm$ 0.02\
[lll|llll]{} $R_{M_{F110W}}$ &0.014 $\pm$ 0.0003 &0.0043 $\pm$ 0.0012 &0.007 &0.012 &0.030 &0.008\
$\Delta(M_{F110W}-M_{F160W})$ &0.10 $\pm$ 0.03 &0.65 $\pm$ 0.32 &0.50 &0.47 &-0.07 &-0.31\
$\Delta(M_{F160W}-M_{F205W})$ &-0.04 $\pm$ 0.06 &-0.34 $\pm$ 0.45 &0.28 &0.20 &-0.21 &-0.20\
$NF_{F110W}$ &1.41 $\pm$ 0.03 &3.6 $\pm$ 1 &1.90 &2.35 &0.67 &4.11\
$NF_{F160W}$ &1.39 $\pm$ 0.02 &2.6 $\pm$ 0.25 &1.82 &2.03 &0.67 &4.07\
$NF_{F205W}$ &&1.5 $\pm$ 0.5 &1.70 &1.741 &0.67 &3.65\
[lcc|cc]{} 1.03 &[**0.47**]{} &[**0.85**]{} &&\
1.25 &0.44 &0.79 &0.46 &0.32\
1.55 &[**0.40**]{} &[**0.74**]{} &&\
1.6 &0.39 &0.73 &0.42 &0.29\
1.9 &[**0.35**]{} &[**0.69**]{} &&\
2.2 &0.31 &0.64 &0.35 &0.25\
[^1]: GG Tau A-B, Aa-b and Ba-b are separated by 10, 025 and 148 respectively (Leinert et al. 1991).
[^2]: The NICMOS F110W and F160W filters correspond well to the standard ground-based J and H band filters, with the F160W filter providing the closest match. The F205W filter is centered $\sim$0.3 shorter than the standard K band filter and is $\sim$0.2 wider.
[^3]: Average uncertainties have been assigned to all measurements obtained with the same method. As a result, decreasing weights are assigned to HST ($\sigma_{r} = 2.5 mas$), speckle interferometry ($\sigma_{r} = 5 mas$) and adaptive optics ($\sigma_{r} =
10 mas$) astrometric results, respectively. Furthermore, the astrometry from Leinert et al. (1993), obtained from three separate 1D measurements is not included in our fit.
[^4]: The following two small modifications have been made to the Henyey-Greenstein scattering phase function for fitting the position angle of the brightest region: (1) the albedo, which normally is included as a multiplicative factor, is omitted, since we have normalized the data and (2) the position angle of the brightest region, $PA_{0}$, is added.
[^5]: MW dust grains are composite and porous in nature, composed of silicate, graphite, amorphous carbon and vacuum. The grain size distribution follows $n(a) \propto a^{-3.7}$ between 0.03 and 0.9 (MW model A for an $R_{V}$ = 3.1). In contrast, the KMH dust grains are composed of separate silicate and graphite particle populations, with a size distribution of $n(a) \propto a^{-3.5}exp(-a/0.2\micron)$ between 0.005 and 1 .
[^6]: Note that there is also a slight difference in the way the dust parameters are calculated. Wood et al. calculate the mean dust parameters for the grain population and assign these to each dust particle, whereas the dust parameters for each photon-dust particle interaction in the simulations presented here are calculated on the fly after the grain size has been randomly sampled from the grain size distribution. While this method is more computationally expensive, we believe it to be more representative of the actual phenomenon. How large an effect this difference makes has not been investigated so far.
[^7]: The negative sign reflects the fact that the binary is slowly getting tighter, hence closer to periastron.
|
---
abstract: 'Beamforming is evidently a core technology in recent generations of mobile communication networks. Nevertheless, an iterative process is typically required to optimize the parameters, making it ill-placed for real-time implementation due to high complexity and computational delay. Heuristic solutions such as zero-forcing (ZF) are simpler but at the expense of performance loss. Alternatively, deep learning (DL) is well understood to be a generalizing technique that can deliver promising results for a wide range of applications at much lower complexity if it is sufficiently trained. As a consequence, DL may present itself as an attractive solution to beamforming. To exploit DL, this article introduces general data- and model-driven beamforming neural networks (BNNs), presents various possible learning strategies, and also discusses complexity reduction for the DL-based BNNs. We also offer enhancement methods such as training-set augmentation and transfer learning in order to improve the generality of BNNs, accompanied by computer simulation results and testbed results showing the performance of such BNN solutions.'
author:
- '[^1] [^2] [^3] [^4]'
title: 'Model-Driven Beamforming Neural Networks'
---
Introduction {#introduction .unnumbered}
============
The ever growing demand for mobile data as a result of new lifestyles and innovative applications has continuously pushed the limits of today’s mobile communication networks and created new challenges. 5G is the most recent and largest collective effort to bring the technology up to speed for the next 10 years or so. One core technology that has appeared in recent generations of mobile communication technology and will continue to have its presence in future generations is the multiuser multiple-input multiple-output (MIMO) system. This technology provides extraordinary spectral and energy efficiency by using the spatial degree of freedom that can be scaled up by having more antennas.
Beamforming in multiuser MIMO is a popular and excellent method for dealing with interference, especially in the downlink. There has been a rich body of literature on that, spanning from sum rate maximization [@shi2011an], to signal-to-interference-plus-noise ratio (SINR) balancing [@schubert2004solution], to quality-of-service (QoS) constrained base station (BS) transmit power minimization [@yu2007transmitter] and among others. The algorithms to find the optimal and even suboptimal solutions to these beamforming problems usually require iterative optimization procedures. The complexity and resulting latency make those techniques problematic for real-time applications because wireless channel fading changes rapidly in the order of milliseconds and expensive iterative procedures will render the obtained solutions invalid if the channel state information (CSI) becomes obsolete. Heuristic non-iterative solutions such as zero-forcing (ZF) and regularized ZF (RZF) beamforming exist but come at the price of performance loss.
Recent developments in deep learning (DL) have given this problem a new hope. DL is well known to be a generalizing technique that can produce an effective solution to a complex problem at relatively low complexity if it is sufficiently trained. The approach means that most complexity is shifted to offline training an artificial neural network (NN) with a large dataset [@chang2018learn]. An online solution can then be obtained by going through a trained NN generalizable from the dataset, with some simple linear and standard nonlinear operations [@xia2019deep]. Researchers have applied DL to network deployment and planning, resource management, and network operation and maintenance. Specific to the physical layer, DL applications include modulation recognition, channel estimation and detection, and encoding and decoding. These applications demonstrate the potential of applying DL in wireless communication networks.
Nevertheless, using DL for beamforming is not straightforward for many reasons. First of all, the number of variables for beamforming depends on the number of users and the number of antenna elements which are usually large. The beamforming problem’s high-dimensionality will cause issues in prediction complexity and errors. A common method in the existing works, such as [@long2018data], is codebook-based beam selection, but tends to suffer certain performance loss. Further, many beamforming optimization problems are non-convex, which makes it difficult to have high-quality training examples, if they are to be solved using a supervised learning approach. It is also unclear if unsupervised learning can be useful for beamforming. In addition, artificial NNs are often oversized, meaning that a lot of neurons are redundant, resulting in unnecessarily high complexity and memory cost. Also, the generality of DL-based solutions can be very limited and a new NN will need to be trained if the parameters of the wireless communication networks change. Understanding the limitations of [*data-driven*]{} DL for beamforming, this article introduces the [*model-driven*]{} beamforming NNs (BNNs) as a new means to utilize DL for beamforming optimization.
The remainder of this article is organized as follows. We will begin by reviewing and comparing the data-driven and model-driven BNNs based on the convolutional NN (CNN) structure. Then we will focus on the model-driven BNN framework in which a signal processing (SP) module is brought into the NN module to enhance the process of feature extraction. Rather than solely predicting the beamforming solution, the proposed BNN framework allows to predict the key features according to expert knowledge with much reduced dimension. We will use the SINR balancing problem as an example to illustrate the operation of this approach. Afterwards, several enhancement techniques, such as a hybrid learning strategy that combines unsupervised and supervised learning to deal with the issues for the lack of labelled examples, and NN trimming and compression that reduce the complexity and memory cost of the NN module, are proposed. Then we will discuss the use of training-set augmentation and transfer learning to improve the generality of DL-based BNNs before presenting simulation and testbed results and concluding the article.
BNNs {#bnns .unnumbered}
====
There are two main types of DL-based BNNs: data-driven and model-driven ones. The main difference is that the former takes the NN as a black box, while the latter introduces a specific SP module into the NN [@he2018model].
{width="75.00000%"}
Data-Driven Architecture {#data-driven-architecture .unnumbered}
------------------------
As shown in Fig. \[data\_driven\_model\_driven\]a, a data-driven BNN follows the structure of a CNN with an input layer, an output layer, and some hidden layers. The input layer takes real-valued channel coefficients as inputs to perform convolutional operations with kernels in the convolutional layers for feature extraction. The activation layers serve to introduce non-linearity into the NN, allowing it to capture complex functional mappings, and are also useful in mitigating the vanishing gradient problem for training the NN. The batch normalization layers are there to reduce the probability of over-fitting and enable a higher learning rate to accelerate convergence.
In the data-driven approach, the BNN acts like a black box and the functional establishment of the BNN relies heavily on both the quality and quantity of training samples. Moreover, the data-driven BNN is blind to any specialized signal structures, does not have the same computational efficiency and the performance is often inferior to that of traditional SP methods. This is because traditional SP methods are crafted according to the prior expert knowledge, such as solution structures, uplink-downlink duality, models, and the properties of signals. Such a priori expert knowledge acquired from extensive research in the literature over the past decades, is expected to be highly useful and should be utilized [@zappone2019model].
Crafting a complete SP solution using expert knowledge is however extremely difficult and sometimes impossible. It thus makes sense to combine the SP methods with the NN approach to reap the benefits of both sides [@zhang2018hybrid]. This then gives rise to the model-driven BNNs, which we will discuss next.
Model-Driven Architecture {#model-driven-architecture .unnumbered}
-------------------------
{width="75.00000%"}
Different from the data-driven version, the proposed model-driven BNN has a specific SP module to utilize prior expert knowledge, as illustrated in Fig. \[data\_driven\_model\_driven\](b). The SP module can be positioned either before or after the NN module as a pre-processing or post-processing block. Inside the SP module are the functional layers that are designed according to prior expert knowledge of beamforming problems, which is problem-specific and has no unified form [@xia2019deep]. It is also possible to replace one or more layers in the ordinary NN module by the SP module to achieve better feature extraction ability [@zhang2018hybrid]. The parameters in the SP module can also be tuned in the training phase.
The purpose of the SP module is to map/convert key features designated by the expert knowledge to the target beamforming matrix before updating the NN module. Let us use an example to explain the design process of the SP module, assuming, for convenience, a model-driven BNN framework where the SP module is inserted into the NN module and placed before the output layer, as shown in Fig. \[framwork\_example\]a.
The example considers the use of a BNN for a multiple-input-single-output (MISO) downlink, where the aim is to balance the SINR under a total power constraint, i.e., $$\textbf{P1:}\ \max_{{{\bf W}}}\min_{1\leq k\leq K} \gamma_k,~\text{s.t.}~\|{{\bf W}}\|^2_F\leq P_{\max},$$ where ${{\bf W}}\in \mathbb{C}^{N\times K}$ denotes the beamforming matrix, $K$ is the number of single-antenna users, $N$ is the number of BS antennas, $\gamma_k$ is the SINR of user $k$, $P_{\max}$ is the total power budget, and $\|\cdot\|_F$ denotes the Frobenius norm.
The optimal solution to problem **P1** can be obtained by the algorithm in [@schubert2004solution Table 1], using an iterative process that comes with high complexity and delay. Predicting the beamforming matrix ${{\bf W}}$ directly using a data-driven BNN approach on the other hand will lead to a high prediction error since the number of elements in ${{\bf W}}$ depends on both the number of users and the number of BS antennas. To circumvent this, expert knowledge regarding uplink-downlink duality can be specified through the functional layers of the SP module of the model-driven BNN.
According to the duality theory in [@schubert2004solution Theorems 1 and 3], both the uplink and downlink have the same achievable SINR region under the given total power constraint and the target SINRs are also achieved by the same set of normalized beamforming vectors. Thus, instead of solving problem **P1** directly, most existing works rightfully resort to its equivalent uplink problem which is easier to handle. In particular, suppose that we have for the equivalent uplink problem the optimal power allocation vector ${{\bf q}}^{\ast}$ and the optimal normalized beamforming matrix $\tilde{{{\bf W}}}^{\ast}$ with the same power budget $P_{\max}$. $\tilde{{{\bf W}}}^{\ast}$ is also a function of ${{\bf q}}^{\ast}$. It is then known that the optimal downlink beamforming matrix ${{\bf W}}^{\ast}$ is a function of $\tilde{{{\bf W}}}^{\ast}$. As a result, the optimal solution to problem **P1** is a function of ${{\bf q}}^{\ast}$, i.e., ${{\bf W}}^{\ast}=f_1({{\bf q}}^{\ast})$, where $f_1(\cdot)$ maps ${{\bf q}}$ to ${{\bf W}}$ based on the results in [@schubert2004solution]. Such expert knowledge suggests that instead of predicting the high dimensional ${{\bf W}}$ directly, we can predict the uplink power vector ${{\bf q}}$ with much less variables.
Based on the BNN framework in Fig. \[framwork\_example\]a, the model-driven BNN for the SINR balancing problem is shown in Fig. \[framwork\_example\](b), where the SP module is fulfilled with two functional layers: the scaling layer and the conversion layer. The scaling layer is used to ensure that the beamforming matrix meets the power constraint by multiplying the estimated $\hat{{{\bf q}}}$ by a scaling factor, whereas the conversion layer is used to execute the function $f_1(\cdot)$. After recovering from the uplink power vector via the function $f_1(\cdot)$, the resulting beamforming matrix is then used to calculate the loss function and update the parameters of the NN training module until convergence.
Supervised vs. Unsupervised Learning {#supervised-vs.-unsupervised-learning .unnumbered}
====================================
![a) Supervised learning and b) unsupervised learning.[]{data-label="supervised_unsupervised_learning"}](supervised_unsupervised_learning.eps){width="45.00000%"}
Supervised learning and unsupervised learning are two very different approaches to training NNs. Supervised learning is based on a ground truth generalizable from labelled training samples, while unsupervised learning finds natural patterns from unlabelled data. In other words, the objective of supervised learning is to learn a mapping function that can well approximate the relationship between the input and desired output in the training samples. Unsupervised learning on the other hand infers the potential structure in the training data.
Supervised Learning for BNN {#supervised-learning-for-bnn .unnumbered}
---------------------------
Supervised learning generalizes the mapping between input and output based on the training samples using an NN representation. More samples improve the mapping accuracy. As both the input and the target output are known, the learning process is interpreted as being done by a “supervisor”, see Fig. \[supervised\_unsupervised\_learning\]a. The NN module repeatedly makes predictions while the “supervisor” corrects the predictions based on the expected output in an iterative manner. Such a learning-correction process terminates until satisfactory performance is achieved.
For classification-type of beamforming problems, e.g., beam selection based on a codebook [@long2018data], cross entropy loss is a well-known metric to quantify the prediction error. For regression problems of beamforming optimization, e.g., problem **P1**, on the other hand, mean squared error (MSE) and mean absolute error (MAE) are two common loss functions, where the former is the average of squared distances and the latter is the average of absolute differences between the target and predicted outputs.
Due to the reliance on a large number of training samples, supervised learning is more suitable for cases where training samples are relatively easy to collect. For instance, for the SINR balancing problem under a total power constraint (i.e., problem **P1**) and the power minimization problems with QoS constraints, there exist computationally efficient algorithms in [@schubert2004solution] and [@yu2007transmitter] that can produce the optimal solutions as training samples, respectively. However, for some difficult beamforming problems, e.g., the sum-rate maximization problem under a total power constraint and SINR balancing problem under per-antenna power constraints, the acquisition of training samples with optimal solutions may be not easy or be at a much higher cost. Unsupervised learning provides an alternative for such beamforming optimization problems with limited samples.
Unsupervised Learning for BNN {#unsupervised-learning-for-bnn .unnumbered}
-----------------------------
Unsupervised learning aims to infer the underlying structure of data without any label, and hence it is particularly suitable for exploratory analysis. For some beamforming optimization problems, no known algorithms can find optimal solutions, and the target output is unknown. In this case, MSE or MAE will not be suitable to measure the loss for updating the parameters of the NN module. Instead, the problem’s original objective function may be used to construct the loss function. However, unsupervised learning with random initialization of parameters of BNNs usually suffers from a low convergence rate.
Hybrid Learning for BNN {#hybrid-learning-for-bnn .unnumbered}
-----------------------
Supervised and unsupervised learning can complement each other, when used together. Such hybrid learning is a promising technique to achieve both great performance and accelerate the convergence rate of unsupervised learning. In the first stage, supervised learning is used for pre-training and then unsupervised learning will be used for further improvement in the second stage [@lee2018deep]. Getting the best of both learning methods, hybrid learning is an attractive approach known to achieve performance that is better than most existing heuristics.
Take the sum-rate maximization problem with a total power constraint as an example. Though no known efficient algorithm can obtain the optimal solutions, we can adopt the weighted minimum MSE (WMMSE) algorithm in [@shi2011an] to generate training samples with locally optimal solutions, which can then be used for supervised learning and pre-training. After that, the learned NN parameters are reserved for unsupervised learning and the loss function can be replaced by the reciprocal function of the sum-rate. Thus, the convergence rate of unsupervised learning is accelerated and hybrid learning can achieve at least the same performance as the WMMSE algorithm [@xia2019deep].
Complexity Consideration {#complexity-consideration .unnumbered}
========================
Although BNNs are powerful, their computational complexity and memory cost can make them less attractive for resource-constrained hardware and equipment. Large-scale BNNs composed of massive neurons also have considerable energy consumption due to massive memory access and abundant computation. As a consequence, reducing the complexity is an important direction if the application of BNNs is to be practical. We next discuss ideas to reduce the complexity of DL-based BNNs in two aspects: the beamforming optimization problem itself and the NN module.
Complexity of the Optimization Problem {#complexity-of-the-optimization-problem .unnumbered}
--------------------------------------
{width="75.00000%"}
The input and output of the NN are specified according to the beamforming optimization problem. One way to reduce the prediction complexity is to shrink the dimensions of the input and output of the NN. A straightforward strategy which takes the wireless channel coefficients as input and the beamforming matrix as output may give rise to high prediction complexity because of their dependence on the numbers of users and BS antennas. To remove the redundant information carried by the input of the NN module, a promising scheme for beam selection problems is to take the locations of users and the BS as input instead of the channel coefficients. Another promising technique, which takes place at the output, is to predict the key features, but not the beamforming matrix, according to expert knowledge, which we elaborate in more details below.
It is possible to first predict some key features designated by prior expert knowledge and then recover the beamforming matrix from the predicted key features, as shown in Fig. \[reduce\_complexity\]a. Different from the BNN framework in Fig. \[framwork\_example\]a which predicts the beamforming matrix and updates the parameters of the NN module for minimizing the loss function of the predicted beamforming matrix, the BNN framework in Fig. \[reduce\_complexity\]a proposed by our previous work [@xia2019deep] only predicts the key features and calculates the loss function based on the predicted key features until convergence. The final step, which is executed only once, is to use the functional layers in the SP module to retrieve the beamforming matrix. In contrast, those SP layers in Fig. \[framwork\_example\]a are executed repeatedly during the entire training process. In addition, although the key features are abstract and problem-specific, the most important advantage is that the number of key features is often much less than the number of variables in the beamforming matrix.
Based on the BNN framework in Fig. \[reduce\_complexity\]a, the solution to the corresponding BNN problem **P1** can be found with reduced complexity using the approach shown in Fig. \[reduce\_complexity\]b [@xia2019deep].
Complexity of the NN module {#complexity-of-the-nn-module .unnumbered}
---------------------------
There is no formal way to obtain the best number of layers and the best number of neurons for each layer in the NN module. In many cases, numbers such as 64, 128, and 512 are typically used but such empirically designed NNs are usually oversized [@hu2016network]. In other words, many neurons may have very low activation regardless of the input and these weak neurons can be removed with little performance loss. Redundant neurons, if not removed, increase computational complexity, memory cost, and the probability of over-fitting, and are thus highly undesirable in terms of the NN performance.
To reduce the complexity of the NN module, we can first use trimming to prune all those connections with weights below a certain threshold and those with zero-activation neurons. Then we reduce the number of bits used to represent each weight via a compression technique and enforce weight sharing among different connections to reduce the number of weights. Finally, Huffman coding can be adopted to represent more common weights using symbols with fewer bits [@hu2016network].
Generality Improvement {#generality-improvement .unnumbered}
======================
In most existing works, the DL-based approaches to beamforming prediction can achieve very good performance but the NNs were trained with fixed wireless network parameters, meaning that the numbers of users and antennas, for example, are fixed. However, wireless networks are dynamic in nature. For example, user mobility means that users join and leave the network over time. Also, network operators may turn off/on a subset of BS antennas according to traffic load, leading to the variation of the number of serving antennas. The pre-trained model may suffer from serious performance degradation and even become unusable because the dimensions of input and output do not match as the size of the wireless network varies. This issue is commonly referred to as task mismatch [@shen2019transfer].
Ideally, a new NN model should be trained for prediction if one or more network parameters have changed. Thus, the time-varying nature of wireless networks does impose unique challenges in using the DL-based approaches. The generality of the trained BNN is key. If the generality is sufficient, this will mean that the trained BNN can cope with a variety of dynamic situations the wireless network may face. Here we introduce two heuristic methods that can improve the generality of the DL-based approaches. The first one is the training-set augmentation method, which collects enough training samples to cover the possible changes of the network size. A large-scale model is then trained based on the augmented training set. This method works well as long as the network size is within the training set. Nonetheless, in some problems, the acquisition of a large number of samples can be too expensive. In that case, transfer learning can be used by transferring knowledge from related scenarios with additional training and labeling efforts.
Training-set Augmentation {#training-set-augmentation .unnumbered}
-------------------------
Consider the BNN for problem **P1**, as shown in Fig. \[framwork\_example\]b, as an example which takes the channel coefficients as input and the beamforming matrix as output. With $N$ BS antennas and $K$ users, the BNN takes $2NK$ channel inputs to produce $2NK$ beamforming outputs and the factor of $2$ appears in order to handle the real and imaginary parts of the complex channel and beamforming matrices. Note that most DL tools (such as Keras and Tensorflow) only support real-valued inputs and outputs. In order to train a BNN suitable for different $N$ and $K$ values, we generate an augmented training set where the samples are diverse, i.e., the numbers of BS antennas and users can be different in different samples. To do so, the size of each sample is set to be $2N_0K_0$ for inputs and $2N_0K_0$ for outputs, where $K_0\ge K$ and $N_0\ge N$. Thus, for each training sample with specific $K$ and $N$, there will be redundant entries at the inputs and outputs, and these entries will be filled with $0$’s. In particular, the reductant $N_0-N$ rows (or $K_0-K$ columns) of the input channel matrix, as well as the corresponding positions in the output vector, are filled with $0$’s. To achieve good generality, the training samples for different combinations of $K\le K_0$ and $N\le N_0$ should be generated with equal probabilities.
The shortcoming of the training-set augmentation method is that training a large-scale model requires a large number of training samples and takes much longer processing time. Also, the trained BNN model is still not general enough to be able to handle the cases in which $N>N_0$ or $K>K_0$.
Transfer Learning {#transfer-learning .unnumbered}
-----------------
![a) Illustration of knowledge transfer and b) fine-tuning.[]{data-label="transfer_learning"}](transfer_learning.eps){width="47.00000%"}
Different network settings naturally lead to different training tasks, but these tasks share some knowledge in common about the underlying optimization problem [@pan2010asurvey]. As can be seen in Fig. \[transfer\_learning\]a, knowledge learned from one training task may be transferred to another training task and can help train a new model with a few additional samples [@shen2019transfer]. Fine-tuning is a well-known method to implement knowledge transfer in NNs. As shown in Fig. \[transfer\_learning\](b), both the input layer and output layer can be replaced by new ones suitable for the new task. Then one or more new layers are inserted into the pre-trained NN module. Fine-tuning aims to refine the pre-trained NN module with additional training samples by setting different learning rates for different layers. More specifically, we can fine-tune the newly added layers and set the learning rates of the layers from the pre-trained NN module as 0 or a very small learning rate since the initialization parameters of the pre-trained layers are expected to be a good starting point.
Performance Evaluation {#performance-evaluation .unnumbered}
======================
{width="95.00000%"}
To assess the performance of the DL-based BNNs, we here provide some numerical results for a downlink MISO system with $K$ users and an $N$-antenna BS using both simulations and experiments. The channel gains are generated by considering both Rayleigh fading and path loss which is modeled as $128.1 + 37.6 \log_{10}(d)~[{\rm dB}]$, with distance $d$ in km. Also, perfect CSI is assumed available at the BS in the simulation.
We first consider the power minimization problem with QoS constraints (e.g., problem **P2** in [@xia2019deep]). Different from the total power constraint in problem **P1**, the QoS constraints are nonlinear and even non-convex, with which the NNs are typically not good at dealing. Moreover, due to non-zero prediction errors, the predicted results cannot always satisfy the QoS constraints. This means that there is a certain probability of infeasibility of the BNN prediction for the power minimization problem. From the results in Figs. \[simulation\_testbed\]a and \[simulation\_testbed\]b, we compare the BNN solution to the power minimization problem with ZF and the optimal iterative algorithm in [@rashid1998transmit] in terms of transmit power performance and computational delay. Note that two convergence strategies for the optimal iterative algorithm are considered: the high convergence threshold case ($10^{-2}$) which can be reached with less iterations and the low convergence threshold case ($10^{-4}$) which requires more iterations. Besides, the results marked as squares in Fig. \[simulation\_testbed\]a correspond to the BNN solution using a generalized model with $N_0=8, K_0=7$ for the training-set augmentation method, whereas the other BNN solution is predicted based on individually trained models.
Results in Fig. \[simulation\_testbed\]a indicate that the BNN solutions achieve better performance than the ZF beamforming and the optimal iterative algorithm for the high convergence threshold case. Results also show that the performance of the optimal iterative algorithm can be improved with more iterations, but at the cost of higher computational delay (about two orders of magnitude, compared to the BNN solutions), as seen in Fig. \[simulation\_testbed\]b. According to the results in Figs. \[simulation\_testbed\]a and \[simulation\_testbed\]b, it is verified that the BNN solutions can achieve a good tradeoff between performance and complexity. Furthermore, we can find that the feasibility of the BNN solutions for the power minimization problem can reach above 99 per cent and that the training-set augmentation method can improve the generality of the DL-based BNNs.
For the experiment results, we further take per-antenna constraints into account for the fact that in practice each transmit antenna has its own power amplifier. The SINR balancing problem with per-antenna power constraints has been investigated in many works, e.g., [@wang2014transmit]. We set up the downlink MISO testbed system using software defined radio, where two NI’s USRP-2950 devices are combined together to form a 4-antenna transmitter and another two USRP-2950 devices are used to emulate 4 single-antenna users.
Figs. \[simulation\_testbed\](c) and (d) demonstrate the bit-error rate (BER) performance against the transmit power under static and dynamic channel conditions, respectively. Under the static condition, the BNN solution outperforms the ZF beamforming and RZF beamforming. Also, it can be seen that the performance of the proposed BNN solution is inferior but close to that of the optimal solution especially in the low power regime. This is expected since the optimal solution has enough time to execute its algorithm under the static channel condition. However, this is no longer the case under the dynamic channel condition. Note that the coherence time in the environment of the experiment is 10-20 ms. In this case, although the optimal solution can achieve the best beam weights, these weights are based on the outdated CSI and therefore no longer optimal for the current CSI after long computing delay, thus leading to the performance degradation. It can be also observed in Fig. \[simulation\_testbed\](d) that the BER performances of the ZF solution and RZF solution with less computing time are still much worse than the proposed BNN solution.
Conclusions and Open Issues {#conclusions-and-open-issues .unnumbered}
===========================
With the rise of DL, this article introduced the [*model-driven*]{} BNN solutions to beamforming optimization problems, where there is a SP module to utilize expert knowledge to empower the NN for enhanced convergence and prediction performance. We discussed the challenges of using DL for beamforming that include high dimensionality, difficulty of data acquisition for supervised learning, limited generality due to channel and network dynamics, and high prediction complexity. This article also provided methods that can improve the implementation of DL for beamforming. While there are inevitably omissions in this article, it is hoped that this article will spark interest in exploring the use of model-driven BNN for wireless communications.
It is also worth pointing out that there are some important open issues that deserve future study. For example, the first challenge is data acquisition, since generating real-world communication data is not straightforward and most existing works are based on artificial or simulated signals. It would be desirable to establish the datasets of some common problems, with which researchers can test their methods. The second challenge is how to make the DL-based BNN robust against corrupted data, which can cause inconsistency and failure of the BNN training. As the number of users increases, it is impossible to allocate each user an orthogonal pilot and non-orthogonal pilots cause pilot contamination and CSI estimation error. Finally, it is interesting to extend the proposed method to the multi-cell distributed scenarios, which are much more complicated than the single-cell scenarios.
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[^1]: Wenchao Xia and Hongbo Zhu are with the Jiangsu Key Laboratory of Wireless Communications, and also with the Engineering Research Center of Health Service System Based on Ubiquitous Wireless Networks, Ministry of Education, Nanjing University of Posts and Telecommunications, Nanjing 210003, China, e-mail: $\{\rm 2015010203, hbz\}@njupt.edu.cn$
[^2]: Gan Zheng is with the Wolfson School of Mechanical Electrical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, U.K., e-mail: $\rm g.zheng$@$\rm lboro.ac.uk$
[^3]: Kai-Kit Wong is with the Department of Electronic and Electrical Engineering, University College London, London WC1E 7JE, U.K., e-mail: $\rm kai$-$\rm kit.wong@ucl.ac.uk$
[^4]: ©20XX IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
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abstract: 'Based on molecular dynamics simulations of a lithium metasilicate glass we study the potential of bond valence sum calculations to identify sites and diffusion pathways of mobile Li ions in a glassy silicate network. We find that the bond valence method is not well suitable to locate the sites, but allows one to estimate the number of sites. Spatial regions of the glass determined as accessible for the Li ions by the bond valence method can capture up to 90% of the diffusion path. These regions however entail a significant fraction that does not belong to the diffusion path. Because of this low specificity, care must be taken to determine the diffusive motion of particles in amorphous systems based on the bond valence method. The best identification of the diffusion path is achieved by using a modified valence mismatch in the BV analysis that takes into account that a Li ion favors equal partial valences to the neighboring oxygen ions. Using this modified valence mismatch it is possible to replace hard geometric constraints formerly applied in the BV method. Further investigations are necessary to better understand the relation between the complex structure of the host network and the ionic diffusion paths.'
author:
- 'Christian Müller,$^{1}$ Egbert Zienicke,$^{1}$ Stefan Adams,$^{2}$ Junko Habasaki,$^{3}$ and Philipp Maass$^{1}$'
date: 'Dated: June 21, 2006'
title: |
Comparison of ion sites and diffusion paths in glasses obtained by\
molecular dynamics simulations and bond valence analysis
---
Introduction
============
On a coarse grained time scale, ionic conduction in glasses is commonly described by a thermally activated jump motion of the mobile ions in an irregular host framework formed by a network former (SiO$_2$, B$_2$O$_3$,...). Regions of long residence times in the network are associated with sites, and rare events, where ions move between neighboring sites, are associated with jumps. The problem how to identify and characterize sites and diffusion paths in such non-crystalline materials poses a challenging task of current research activities. Its solution can be expected to play a key role in approaching a more quantitative understanding of ion transport properties in glasses.[@Dieterich/Maass:2002] Methods developed in this field should be useful also for the description of thermally activated transport processes in other disordered systems.
In recent molecular dynamics simulations of modified network glasses [@Lammert/etal:2003; @Habasaki/Hiwatari:2004; @Vogel:2004; @Lammert/Heuer:2004; @Heuer/etal:2004; @Lammert/Heuer:2005; @Habasaki/Ngai:2005] sites were identified based on the number density $\rho({\bf x})$ of the mobile ions. In some of these works, the sites constitute those spatial regions, where $\rho({\bf x})$ exceeds a certain threshold $\rho_\star$. The threshold value was specified by requiring the number of these regions to be maximal (for details of the procedure and the question which of these regions are identified with sites, see ref. and the analysis in Sec. \[sec:mdresults\]). The sites form during the cooling process in the highly viscous melt [@Horbach:2004] and are stable on the time scale of the main structural rearrangements of the network, which, below the glass transition temperature, is much larger than the time scale needed for the mobile ions to reach the long-time diffusion regime. For the systems investigated,[@Lammert/etal:2003; @Habasaki/Hiwatari:2004; @Vogel:2004; @Lammert/Heuer:2004; @Heuer/etal:2004; @Lammert/Heuer:2005; @Habasaki/Ngai:2005] the number of sites exceeds the number of ions by 6-12% only, suggesting that a vacancy mediated hopping dynamics [@vacancy-comm] should be considered in a coarse grained description.[@Peibst/etal:2005; @Dyre:2003] Due to the interaction effects the ionic motion can be strongly cooperative in such situations, as reported by one of the authors, [@Habasaki/Hiwatari:2004; @Habasaki/Ngai:2005] and also in ref. .
Although the method of determining the local number density in MD simulations was by following the trajectories of the mobile ions and by registering their occupation times in small cells (over sufficiently long time periods), it should be noted that $\rho({\bf
x})$ can be viewed as an equilibrium quantity with respect to configurations belonging to the metastable attraction basin of the free energy reached after the cooling process. This means, that in the absence of aging effects, which involve transitions between different basins, $\rho({\bf x})$ can in principle be determined independently of the ion dynamics.
Experimentally, a measurement of the local number density of mobile ions with respect to the host structure of the network forming ions is currently not possible. Unlike in crystals, it is already difficult to obtain a reliable representation of the framework structure. The most successful procedure to date is to use x-ray and neutron diffraction data as input for a reverse Monte Carlo modeling. [@Roling/Swenson:2005] Structures obtained in this way are in agreement with the static structure factors and they fulfill certain constraints imposed by chemical requirements. Recently this structural modeling has been combined with bond valence (BV) sum calculations in order to identify diffusion paths for mobile ions in network glasses. [@Adams/Swenson:2000; @Swenson/Adams:2003; @Hall/etal:2004; @Adams/Swenson:2005] However, while for various crystalline systems it could be demonstrated that both sites and pathways predicted by the BV method match the results from detailed anharmonic crystal structure analyses,[@Garrett/etal:1982; @Brown:2002; @Adams:2000a; @Adams:2000b] the potential of the BV method for predicting diffusion pathways or sites in amorphous systems needs to be clarified.
Since ionic transport pathways in glasses cannot be directly inferred from experiment, we test the BV method in this work by MD simulations. The interaction potentials used in our MD simulations were derived from ab initio Hartree-Fock self-consistent calculations and were checked to keep the crystal structure stable under constant pressure conditions. [@Habasaki:1992] This is a severe test of the quality of the potential parameters. Many properties of ion conducting glasses, as e.g. structural information obtained from scattering experiments, vibrational spectra, values for ionic diffusion and various thermodynamic properties are successfully reproduced by the MD model. Using the MD simulations as a reference to test the BV method, we nevertheless have to bear in mind that the underlying Hamiltonian for the interactions between the ions and the way of vitrifying in the MD simulations (e.g. the high cooling rates) may not give a fully accurate representation of all properties of the real experimental system. In any case, even if the representation with respect to the sites and conduction pathways is not perfect, one should expect that the BV method is still applicable, since the MD model itself presents a valid physical system. For the model system, however, the optimal parameters used in the BV analysis could possibly be different from those applied to the corresponding real glass (cf.Sec. \[sec:bv-analysis\]).
Our results show that the BV paths entail the sites of the mobile ions in the MD simulations. The BV method yields a reasonable estimate of the number of sites for the mobile ions, but is not well suitable for locating the sites within the pathways. In order to evaluate the potential of the BV method with respect to diffusion pathways, the latter have to be specified from MD simulations. Such specification can be done based on $\rho({\bf x})$, by using the percolation threshold $\rho_{\rm perc}$ for attributing the regions with $\rho({\bf x})>\rho_{\rm perc}$ to the diffusion path. We will show that the BV path reaches a sensitivity up to 56%, i.e. it covers up to 56% of this diffusion path, and, conversely, the specificity lie in the range 50-60%, i.e. 50-60% of the BV path belong to the diffusion path. A variant of the BV method recently developed by one of the authors,[@Adams:2006] allows us, by introducing a penalty function for unfavorable asymmetric bonding situations, to achieve a sensitivity up to 90% and a specificity between 30-60%.
The paper is organized as follows. After a description of the MD simulations in Sec. \[sec:mdsimulation\], we perform an identification of the ion sites in Sec. \[sec:mdresults\] following the procedure suggested in ref. . We include a discussion of the role of the grid spacing used in this procedure and suggest an alternative criterion to finally assign the regions with $\rho({\bf x})>\rho_\star$ to sites. The diffusion path is determined from a percolation analysis. In Sec. \[sec:bv-analysis\] we perform a BV analysis based on time-averaged network configurations obtained from the MD simulations and evaluate in Sec. \[sec:comparison\] the potential of the BV method for identifying ion sites and the diffusion path by comparing the results with those obtained in Sec. \[sec:mdresults\]. Finally, we give in Sec. \[sec:conclusion\] a summary of the results and an outlook to further research.
Molecular Dynamics Simulations {#sec:mdsimulation}
==============================
We perform MD simulations of a Li$_2$SiO$_3$ glass in the NVE ensemble at a temperature of 700 K with periodic boundary conditions. Newton’s equations are solved using the Verlet algorithm with time step $\Delta
t=1$fs. The computational domain is a cubic box of side length $L=16.68$Å filled with 144 lithium, 72 silicon and 216 oxygen ions. The box size was determined by performing simulations in the NPT ensemble at atmospheric pressure. It corresponds to material densities that match the experimental ones within 5% of error.
Pair potentials of Gilbert-Ida type [@Gilbert:1968; @Ida:1976] describe the interactions between the species: $$\begin{aligned}
U_{ij}(r)&=& \frac{e^2}{4\pi\varepsilon_0} \, \frac{z_i z_j}{r}+ f_0
(b_i +b_j) \exp \left( \frac{a_i+a_j-r}{b_i+b_j} \right) \nonumber\\
&&{}-\frac{c_i c_j}{r^6},
\label{eq:uij}\end{aligned}$$ where the parameters listed in table \[params\] have been optimized [@Habasaki:1992] and shown to give good agreement with experimental data.[@Habasaki:1992; @Habasaki:1995; @Banhatti/Heuer:2001; @Heuer/etal:2002] The first term in eq. (\[eq:uij\]) is the Coulomb interaction with effective charge numbers $z_i$ for Li, Si and O. The long-range Coulomb interaction with the image charges in the periodically continued copies of the simulation box is taken into account by standard Ewald summation. A Born-Meyer type potential $A_{ij}\exp(-r/\lambda_{ij})$ in the second term of eq. (\[eq:uij\]) takes into account the repulsive short-range interactions, where the parameters $a_i$ and $b_j$ appearing in (\[eq:uij\]) decompose $A_{ij}$ and $\lambda_{ij}$into values assigned to the interacting species. The last term in eq. (\[eq:uij\]) is a dispersive interaction and present only for interactions between oxygen ions with distance larger than $r_{\rm c}=1.3$Å.[@dispersive-comm]
[|c||@c@|@c@|@c@|@c@|]{} Ion & $z$ & $a$ \[Å\] & $b$ \[Å\] & $c$ \[Å$^3\sqrt{\rm kJ/mol}$\]\
Li$^+$ & 0.87 & 1.0155 & 0.07321 & 22.24\
Si$^{4+}$ & 2.40 & 0.8688 & 0.03285 & 47.43\
O$^{2-}$ & -1.38 & 2.0474 & 0.17566 & 143.98\
\
We cooled down the system in the NVT ensemble using a velocity scaling to reach the final temperature of $T=700$ K and subsequently equilibrated the system in the NVE ensemble. For the analysis of the results we performed a simulation run over 40 ns. Figure \[fig:gr-msd\]a shows the pair correlation functions for Li-Li, Li-Si and Li-O that allow us to check if the minimal distances between pairs of ions used in the BV analysis (2.48 Å for Li-Si and 1.7 Å for Li-O, see below) fit to the potential model (\[eq:uij\]). The time-dependent mean square displacement of Li, O and Si ions is displayed in Figure \[fig:gr-msd\]b. The mean square displacement of the Li ions becomes normal diffusive at time scales larger than several hundred picoseconds, while that of the Si and O ions practically stays constant over the whole time interval. On the time scale of 2 ns, for which most of the calculations for the determination of sites and diffusion paths are carried out, aging effects of the network structure can be neglected.
Identification of ion sites and the diffusion path {#sec:mdresults}
==================================================
During their motion the Li ions explore only parts of the host network. To identify the regions encountered by the Li ions and their favorable sites, the local number density $\rho(\mathbf{x})$ of Li ions is calculated. Dependent on a threshold value $\rho_{\rm th}$, we define a path $\mathcal{P}(\rho_{\rm
th})=\{\mathbf{x}|\rho(\mathbf{x})>\rho_{\rm th}\}$ as the region with $\rho(\mathbf{x})>\rho_{\rm th}$, and perform a subsequent cluster analysis of the paths in dependence of $\rho_{\rm th}$. To determine $\rho(\mathbf{x})$ the simulation box is subdivided into a grid of cubic cells with spacing $\Delta$ and the time $t_i$ is registered, where it is occupied by a Li ion. The average density $\rho_{i}\simeq\rho({\bf x}_i)$ in cell $i$ is then calculated as $\rho_i=(t_i/t_{\rm sim})\Delta^{-3}$, where $t_{\rm sim}$ is the total simulation time.
In order to identify regions of high probability of occupation we determine clusters of connected (face sharing) cells $i$ with $\rho_i>\rho_{\rm th}$ by the Hoshen-Kopelman algorithm.[@Hoshen/Kopelman:1976] Figure \[fig:nr\_of\_clusters\] shows the number of clusters $N_{\rm cl}(\rho_{\rm th})$ in dependence of $\rho_{\rm th}$ for three grid spacings $\Delta$. All three curves have the same shape: Starting from large $\rho_{\rm th}$, $N_{\rm
cl}(\rho_{\rm th})$ increases with decreasing $\rho_{\rm th}$, since an increasing number of local maxima in $\rho(\mathbf{x})$ is identified. The strong dependence on the grid spacing at large $\rho_{\rm th}$ shows that the local maxima in $\rho(\mathbf{x})$ are sharp. For larger $\Delta$, less local maxima and consequently less clusters are found, since the average density $\rho_i$ in a cell containing a sharp local maximum of $\rho(\mathbf{x})$ becomes smaller and can fall below $\rho_{\rm th}$. As long as $\Delta$ is much smaller than the typical distance between the local maxima one expects that with decreasing $\rho_{\rm th}$ eventually all local maxima are resolved. This is indeed the case, as can be seen from the fact that the curves for different $\Delta$ pass the same maximum at $N_{\rm
cl}^{\rm max}\simeq 165$. By further decreasing $\rho_{\rm th}$ different clusters merge and $N_{\rm cl}(\rho_{\rm th})$ decreases.
The discussion makes clear that there exists a plateau $N_{\rm
cl}(\rho_{\rm th})=N_{\rm cl}^{\rm max}$ in the range $\rho_{\rm
min}<\rho_{\rm th}<\rho_{\rm max}$, which depends on the grid spacing (while the value $N_{\rm cl}^{\rm max}$ is not affected for sufficiently small $\Delta$). Above $\rho_{\rm max}$ some of the local maxima in $\rho({\bf x})$ are not resolved and below $\rho_{\rm min}$ clusters coalesce (which does not exclude that some of the $N_{\rm
cl}^{\rm max}$ clusters contain more than one local maximum of $\rho({\bf x})$). To identify clusters with possible Li sites one could choose any value of $\rho_{\rm th}$ in the threshold range $\rho_{\rm min}<\rho_{\rm th}<\rho_{\rm max}$. This would not change the identity of the possible sites but their size. In order to cover as much volume as possible we use $\rho_\star\equiv\rho_{\rm
min}\simeq0.56$Å$^{-3}$ for a unique definition of the clusters, which are candidates for sites (strictly speaking, one should use the value $\rho_{\rm min}$ in the limit $\Delta\to0$, since $\rho_{\rm
min}$ depends weakly on $\Delta$). The subsequent analysis is carried out for the smallest spacing $\Delta=0.139$.
Next we study properties of the clusters with respect to their assignment to sites. To this end we determine the volume $V_{\rm
cl}=n_{\rm cells}\Delta^3$ of the clusters and the mean number density $\rho_{\rm cl}=\sum_{i\in{\rm cl}}\rho_i/n_{\rm cells}$ of Li ions on them, and number the clusters in order of increasing $\rho_{\rm cl}$. $n_{\rm cells}$ denotes the number of cells in the corresponding cluster. Figures \[fig:cluster-properties\]a,b,c show $\rho_{\rm
cl}$, $V_{\rm cl}$, and the occupation probability $t_{\rm cl}/t_{\rm
sim}= \sum_{i\in{\rm cl}}t_i/t_{\rm sim}=\rho_{\rm cl}n_{\rm
cells}\Delta^3$ for the numbered clusters. The nearly constant increase of the density is interrupted by a jump after cluster number 17. Also the other two quantities show a strong increase in this region. The first clusters are very small and have very low occupation times. After the jump, the volume and the occupation time lie on a main branch between 0.6 and 0.35 Å$^3$ for the volume and between 50 and 90% of the ratio of occupation time to simulation time. In between there are some outliers with less volume or less occupation time, but above the values of the first 17 clusters. With these results one faces two problems: [*(i)*]{} Summing up the occupation probability of all clusters, one finds that the total occupation probability $\sum_{\rm cl}t_{\rm cl}/t_{\rm sim}$ of all clusters is 70% only. [*(ii)*]{} According to the behavior of all three shown quantities — very small occupation probability, volume and occupation time — the first clusters before the jump do not fit to the physical picture which one has of a site.
Problem [*(i)*]{} can be solved as follows. The criterion $\rho({\bf
x})>\rho_\star$ cuts off the outer fringe of a site, which is visited only a short time by a Li ion, but dynamically can be assigned to the site, since a Li ion in general returns to the cluster after entering its fringe. The clusters thus can be viewed to form the core of the sites. Problem [*(ii)*]{} is more subtle. As suggested in ref. , one can require that a cluster should only be assigned to a site if the mean occupation of Li ions on it exceeds a certain threshold. However, there can exist sites, which are visited only rarely but which still conform to the requirement of a sufficiently large ratio between the residence time of a Li ion on the site and the time for its transition to a neighboring site.
We therefore use a new criterion for the assignment of the clusters to sites by associating two times with each cluster, the total residence time $t_{\rm res}$ and the total hopping time $t_{\rm hop}$ to any of its neighbors. The hopping time is calculated as follows: When a Li ion leaves a cluster and enters a new one without returning, it performs a transition between two clusters. The hopping time then is the time interval between the departure from the initial cluster and the arrival at the target cluster. Summing over all transitions starting from a given cluster, one obtains the total hopping time from this cluster. The time interval between the entrance of a Li ion into a cluster and the onset of its transition to another cluster yields the residence time, and by taking the sum over all events we obtain $t_{\rm res}$.[@time-comm] Note that according to this definition the residence time includes events, where a Li ion leaves the cluster but returns to it before entering another cluster.[@rattling-comm]
Figure \[fig:res-hop-time\]a shows $t_{\rm res}$ and $t_{\rm hop}$ for the numbered clusters. While $t_{\rm hop}$ fluctuates from cluster to cluster but never exceeds one tenth (0.2 ns) of the simulation time (2 ns), $t_{\rm res}$ shows a more smooth variation with the cluster number as a consequence of a strong correlation to $\rho_{\rm cl}$. In particular, clusters with the smallest $\rho_{\rm cl}$ correspond to the one with lowest $t_{\rm res}$. The total residence probability $\sum_{\rm cl}t_{\rm res}/t_{\rm sim}$ of the Li ions on the clusters is now 98%, which confirms that the problem with the low total occupation probability $\sum_{\rm cl}t_{\rm cl}/t_{\rm sim}$ is resolved by taking into account the fringes of the clusters.
In order for a cluster to be assigned to a site, we now use the criterion that its residence time $t_{\rm sim}$ should be at least one order of magnitude larger than its total hopping time $t_{\rm
hop}$,[@site-definition-comm] $$\frac{t_{\rm res}}{t_{\rm hop}}>10\,.
\label{eq:site-cond}$$ Figure \[fig:res-hop-time\]b shows $t_{\rm res}/t_{\rm hop}$ for the numbered clusters. Most clusters fulfill condition (\[eq:site-cond\]), which demonstrates that the Li motion can be represented by a hopping dynamics on a coarse grained time scale larger than $t_{\rm hop}$. Only 17 fail to satisfy condition (\[eq:site-cond\]), and almost all of those have small numbers corresponding to low $\rho_{\rm cl}$. This shows that there is hardly any difference here between a criterion based on a threshold number density and (\[eq:site-cond\]). Accordingly, the sites are determined essentially by the equilibrium quantity $\rho({\bf x})$ (in the metastable basin after the cooling when disregarding slow non-equilibrium aging processes of the network structure). In total, we find 148 sites, which are only 2.8% more than the number of Li ions. This fraction of empty sites is slightly lower than that found in earlier studies [@Lammert/etal:2003; @Habasaki/Hiwatari:2004; @Vogel:2004] and supports the picture of an ion hopping with a small number of vacancies. Different from the usual situation in crystalline systems in thermal equilibrium, the concentration of vacancies here should be considered as a result of the freezing process and will not change significantly with temperature below the glass transition.
After having identified the sites, we determine the diffusion path. To this end we evaluate the percolation threshold[@Bunde/Havlin:1996] $\rho_{\rm perc}$ and the corresponding subset $\mathcal{P}_{\rm
perc}=\mathcal{P}(\rho_{\rm perc})$ of cells, on which the Li ions can diffuse across the system. We find $\rho_{\rm
perc}=0.030\,\mbox{\AA}^{-3}$, where $\mathcal{P}(\rho_{\rm perc})$ covers 7.0% of the volume of the system (cf.table \[tab:volume-fractions\]). Confining the Li motion to exactly the critical percolation path at $\rho=\rho_{\rm perc}$ would yield anomalous sub-diffusion. However, in our case the percolation threshold is not sharp due to the finite system size and the number density of the Li ions takes into account only the mean occupation probabilities of the ions but not their thermal fluctuations. Even more important, these issues are not of particular relevance here, since we are interested in the spatial overlap with the BV paths (see Sec. \[sec:bv-analysis\]). With respect to this overlap the criticality plays no decisive role (since connectivity properties are not important for the overlap).
Bond valence analysis {#sec:bv-analysis}
=====================
The development of the bond valence approach has its origin in the search of correlations between bond lengths, chemical valence and binding energies for the chemical bond,[@Pauling:1967] and has been widely applied to crystals with covalent and ionic bonds. The possibility to approximately describe both types of bonding using the same formalism makes it particularly useful for inorganic compounds with partially covalent bonds. A review of the BV method is given in refs. . We describe the method here in connection with our application, which is the determination of sites and diffusion paths for the Li ions based on the knowledge of the Si-O network structure. For a Li ion to be accommodated in some region, its valence should be close to its “natural values”, and additional constraints for its coordination number and minimal distance to neighboring ions are to be satisfied (see below).
According to Pauling,[@Pauling:1967] each bond in a structure induces, due to its polarity, bond valences of opposite sign (partial charges in units of the elementary charge $e$) at the two atoms that it connects. In an equilibrated structure, the bond valences of an ion induced by neighboring counterions should add to its ideal value (i.e. +1 for a Li ion). Since the effective overlap of electronic orbitals typically decreases exponentially with distance of the atomic nuclei, the partial valence $s_j({\bf x})$ of a Li ion at position ${\bf x}$ induced by a oxygen ion $j$ at position ${\bf x}_j$ is determined by[@Donnay/Allmann:1970] $$s_j({\bf x})=\exp\left[\frac{r_0-|{\bf x}-{\bf x}_j|}{\xi} \right],
\label{eq:sj}$$ where $r_0$ is the ideal bond length and $\xi$ is the so-called softness parameter that determines how fast the bond valence varies with distance. The total valence $V({\bf x})$ of a Li ion is computed as the sum of the bond valences $s_j({\bf x})$: $$V({\bf x})=\sum_j s_j({\bf x})=\sum_j\exp\left[\frac{r_0-|{\bf x}-{\bf
x}_j|}{\xi} \right]\,.
\label{eq:v}$$ Because of the exponential decay with distance the sum can be extended over all oxygen ions in the computational domain (taking into account the minimum image convention for periodic boundary conditions). The decomposition (\[eq:v\]) into bond valences allows one also to associate a coordination number $C({\bf x})$ of a Li number at position ${\bf x}$. This is defined by the number of oxygen ions contributing $s_j({\bf x})$ exceeding a threshold $s_{\rm min}$, $$C({\bf x})=\sum_j\theta\left(s_j({\bf x})-s_{\rm min}\right)\,,
\label{eq:c}$$ where $\theta(.)$ is the Heaviside jump function ($\theta(x)=1$ for $x>0$ and zero else).[@Adams/Swenson:2005] Here we choose $s_{\rm min}$ as suggested by Brown.[@Brown:2002] The BV parameters $r_{0}$, and $\xi$ are derived from a variety of crystalline phases,[@bv-param; @Adams:2001] and are given in table \[tab:bv-parameter\].
[|c|c|c|c|c|c|c|c|]{}$\xi$ \[Å\] & $r_0$ \[Å\] & $s_{\rm min}$ & $C_{\rm min}$ & $C_{\rm max}$ & $R_{\rm LiO}$ \[Å\] & $R_{\rm LiSi}$ \[Å\] & $\bar V$\
0.516 & 1.17096 & 0.04 & 4 & 6 & 1.70 & 2.48 & 0.814\
Figure \[fig:valence\_sum\]a shows the histogram of valences of the 144 Li ions for an instantaneous equilibrated configuration. While in a crystalline structure the Li ions $i$ at their equilibrium positions ${\bf x}_i$ have valences $V_i=V({\bf x}_i)$ close to the ideal value $V_{\rm id}=1$, we find a mean value $\bar V_{\rm inst.}=0.821$ significantly smaller than $V_{\rm id}$ in the (instantaneous) amorphous glass structure. An apparent under-bonding of the Li ions with average BV sums of ca. 0.9 has been found also in the BV analysis of RMC models obtained from scattering data[@Adams/Swenson:2005] and can be traced back to the non-crystalline structure of the glass. Since in the MD simulation the cooling rates are much larger than in experiments, the deviation from the ideal bonding situation in a crystal at thermal equilibrium is even larger.
With respect to the application of the BV analysis to structural models obtained from RMC simulations, one should take into account that the information provided by x-ray and neutron diffraction data corresponds to time-averaged positions of the network forming ions. This is due to the fact that the time for obtaining an evaluable signal is orders of magnitudes larger than a typical vibrational time. This means that a (non-unique) network configuration obtained in RMC modeling is a representative for a time-averaged density. We thus determined the mean positions of oxygen and silicon ions in a time interval of 2ns and calculated the valences of Li ions with respect to these mean positions. For a Li ion placed at the centers ${\bf
x}_i$ of the cells $i$ described in the previous section \[sec:mdresults\] (spacing $\Delta=0.139$Å), the valences $V_i=V({\bf x}_i)$ were calculated. Taking into account the probability of occupation $\rho({\bf x}_i)\Delta^3$ of the cells, we then determined the probability density $p(V)$ of valences shown in Fig. \[fig:valence\_sum\]b. Its mean $\bar V=0.814$ is even slightly smaller than that for the instantaneous configuration, confirming that the ideal value $V_{\rm id}=1$ is not the preferred one in the MD simulation of an amorphous glass structure. A small asymmetry is seen in $p(V)$ with a steeper decrease from its maximum on the side of large valences $V$. This reflects the asymmetry of the two-body interaction potentials of Li with O ions, where small distances corresponding to the repulsive part yield larger valences according to eq. (\[eq:sj\]), while large distances corresponding to the attractive part yield smaller valences. Since the asymmetry is not pronounced, the mean value $\bar V$ is close to the value where $p(V)$ attains its maximum.
In the following we analyze the data with respect to the time-averaged network structure and consider $\bar V$ as the “optimal value” of the bond valence sum. The valence mismatch of a Li ion at position ${\bf x}$ is then given by $$d({\bf x})=|V({\bf x})-\bar V|\,.
\label{eq:d}$$ The mean valence mismatch 12% calculated from $p(V)$ in Fig. \[fig:valence\_sum\]b is significantly larger than in crystalline structures (where typical values are less than 5%).
Following the BV method, a position ${\bf x}$ is accessible for a Li ion if the following conditions are fulfilled:
- The distances of the Li cation to silicon cations must exceed a minimum distance $$\min_j\{|{\bf x}-{\bf x}_j^{\rm Si}|\}>R_{\rm LiSi}\,,
\label{eq:cond1}$$ $R_{\rm LiSi}=2.48$ Å is chosen to be equal to (or a little less than) the sum of the radii of the two ions in agreement with the correlation hole of the pair correlation shown in Fig. \[fig:gr-msd\]a.
- According to the “equal valence rule”[@Brown:2002] among the conceivable states with matching BV sum, the one with more symmetric bonds is energetically preferable. The simplest way of excluding that a matching BV sum is achieved by an unphysical strong asymmetric coordination shell is to define a minimum acceptable bond distance, $$\min_j\{|{\bf x}-{\bf x}_j^{\rm O}|\}>R_{\rm LiO}\,.
\label{eq:cond2}$$ A value of $R_{\rm LiO}=1.7$ Å is in agreement with the correlation hole in Fig. \[fig:gr-msd\]a and restricts a partial Li-O bond valence to values smaller than 0.36.
- The coordination number has to lie between a minimal and maximal value, $$C_{\rm min}\le C({\bf x})\le C_{\rm max}\,,
\label{eq:cond3}$$ where $C_{\rm min}=4$ and $C_{\rm max}=6$. This condition has previously been assumed for sites only.[@Adams/Swenson:2005]
- The valence mismatch must be smaller than a threshold value $d_{\rm th}$, $$d({\bf x})=|V({\bf x})-\bar V|<d_{\rm th}\,.
\label{eq:cond4a}$$
As an alternative to the three conditions [*(ii-iv)*]{} one can use just one condition [*(v)*]{}, which is a modification of [*(iv)*]{}. It takes into account that a Li ion is better accommodated to the local network environment at position ${\bf x}$ if the valences $s_j({\bf
x})$ are more symmetrically distributed among the neighboring oxygen ions. The effect can be described by defining an “ideal partial valence” $s_{\rm id}$ by $C_{\rm min}s_{\rm id}=V_{\rm
id}$,[@Adams/Swenson:2005; @Adams:2006] corresponding to a Li ion that, when symmetrically connected to $C_{\rm min}$ oxygen ions, has the ideal total valence $V_{\rm id}$. The deviation from the symmetric situation is quantified by the penalty function $$p({\bf x})=\sqrt{\sum_j \left(\frac{s_j({\bf x})}{s_{\rm
id}}-1\right)^{2\mu}}\,,
\label{eq:penalty-function}$$ where the sum is taken over all oxygen ions and we choose $\mu=3$. A modified valence then is defined by $d'({\bf x})=d({\bf x})+p({\bf
x})$ and we require:
- The modified valence mismatch must be smaller than a threshold value $d_{\rm th}$, $$d'({\bf x})=|V({\bf x})-\bar V|+p({\bf x})<d_{\rm th}.
\label{eq:cond4b}$$
When applying eq. (\[eq:cond4b\]) instead of eq. (\[eq:cond4a\]) conditions [*(ii)*]{} and [*(iii)*]{} are no longer needed. As a consequence, instead of the four parameters $R_{\rm LiO}$, $C_{\rm
min}$, $C_{\rm max}$, and $s_{\rm min}$ the parameter $\mu$ enters the calculation.
To perform the BV analysis, the computational domain is divided into a grid of cells with spacing $\Delta$ analogous to the procedure used in sec. \[sec:mdresults\] ($\Delta=0.139$ Å). A cell is accessible if a Li ion placed at its center fulfills the requirements [*(i-iv)*]{}, or [*(i,v)*]{} in the modified BV method.[@sign-comm] The union of such cells is the BV path $\mathcal{P}_{\rm BV}(d_{\rm th})$ (for conditions [*(i-iv)*]{}) or the BV path $\mathcal{P}_{\rm BV}'(d_{\rm
th})$
Comparison of ion sites and diffusion paths {#sec:comparison}
===========================================
We first clarify the relevance of the purely geometric constraints [*(i,ii)*]{} in combination with the coordination number condition [*(iii)*]{}, without taking into account the bond valence condition [*(iv)*]{}. Accordingly we distinguish between the subsets defined by applying the constraints separately: $\mathcal{P}^{\rm BV}_{\rm ex}$ is the subset given by the geometric exclusions [*(i,ii)*]{}, while $\mathcal{P}^{\rm BV}_{\rm cn}$ refers to the condition imposed solely on the coordination number [*(iii)*]{}. The subset defined by the combined conditions [*(i-iii)*]{} is denoted as $\mathcal{P}^{\rm
BV}_{\rm ex+cn}$.
[|c|c||c|c|c|]{}$\mathcal{P}_{\rm sites}$ & $\mathcal{P}_{\rm perc}$ & $\mathcal{P}^{\rm BV}_{\rm ex}$ & $\mathcal{P}^{\rm BV}_{\rm cn}$ & $\mathcal{P}^{\rm BV}_{\rm ex+cn}$\
1.4 & 7.0 & 10.0 & 83.0 & 8.3\
[|@c@||@c@|@c@|@c@|]{} & $\mathcal{P}^{\rm BV}_{\rm ex}$& $\mathcal{P}^{\rm BV}_{\rm cn}$ & $\mathcal{P}^{\rm BV}_{\rm ex+cn}$\
$\mathcal{P}_{\rm sites}$ & 0.83 & 0.95 & 0.79\
$\mathcal{P}_{\rm perc}$ & 0.62 & 0.91 & 0.57\
$\mathcal{P}_{\rm sites}$ & 0.12 & 0.02 & 0.13\
$\mathcal{P}_{\rm perc}$ & 0.43 & 0.08 & 0.48\
The geometric conditions [*(i,ii)*]{} already limit the fraction of available space for the mobile ions to 10% for the time-averaged network structure, see table \[tab:volume-fractions\]. By contrast, constraint [*(iii)*]{} alone for $s_{\rm min}=0.04$ is a weak condition, which would allow a Li ion to occupy 83% of the space. The combined conditions [*(i-iii)*]{} reduce the accessible space to 8.3%, corresponding to an uncorrelated behavior. If conditions [*(ii-iv)*]{} are replaced by condition [*(v)*]{} in the limit $d_{\rm
th}\to\infty$ the available space is restricted to 21.3% of the volume of the system (see also Fig. \[fig:bvvolume\]). One should expect that this accessible space entails the ionic sites and the diffusion path identified in sec. \[sec:mdresults\].
To quantify the quality of agreement between the subsets $\mathcal{P}^{\rm BV}_\star$ ($\mathcal{P}^{\rm
BV}_\star=\mathcal{P}^{\rm BV}_{\rm ex}$, $\mathcal{P}^{\rm BV}_{\rm
cn}$, or $\mathcal{P}^{\rm BV}_{\rm ex+cn}$) with the subsets $\mathcal{P}_\star$ obtained from the MD simulations ($\mathcal{P}_\star=\mathcal{P}_{\rm sites}$ or $\mathcal{P}_{\rm
perc}$), we define two quantities, the sensitivity and specificity. The sensitivity is the conditional probability $\psi(\mathcal{P}^{\rm
BV}_\star|\mathcal{P}_\star)$ that a cell belonging to one of the subsets identified in the MD simulations belongs to one of the subsets of the BV analysis. Conversely, the specificity of the BV analysis is quantified by calculating the conditional probabilities $\psi(\mathcal{P}_\star|\mathcal{P}^{\rm
BV}_\star)$.[@specificity-comm]
Table \[tab:bv-test\] summarizes the results. Let us in particular consider the case where conditions [*(i-iii)*]{} are applied (corresponding to $\mathcal{P}^{\rm BV}_{\rm ex+cn}$). A high sensitivity of 79% is reached for the sites. However, from table \[tab:volume-fractions\] we see that the volume fraction of $\mathcal{P}^{\rm BV}_{\rm ex+cn}$ (8,3%) is by a factor of about 6 larger than that of $\mathcal{P}_{\rm sites}$ (1,4%). Accordingly, the specificity $\psi(\mathcal{P}_{\rm sites}|\mathcal{P}^{\rm
BV}_{\rm ex+cn})=13\%$ is rather low. The sensitivity for the diffusion path is 57% ($\mathcal{P}_{\rm perc}$). It is significantly lower than that for the sites, since the path contains regions with very low occupation probability (cf. Sec. \[sec:mdresults\]), which often violate the geometric constraints [*(i,ii)*]{}.
The question is, whether the specificity can be improved without significant reduction in the sensitivity by applying condition [*(iv)*]{} in addition to [*(i-iii)*]{}. The volume fraction $v_{\rm BV}$ of the BV path as a function of the threshold mismatch $d_{\rm th}$ is shown in Fig. \[fig:bvvolume\] (solid line). For small $d_{\rm th}$, $v_{\rm BV}$ increases linearly and for $d_{\rm th}\gtrsim0.3$ approaches the value 8.3% imposed by the constraints [*(i-iii)*]{}. Figures \[fig:sens-spec-bv\]a and b show how the sensitivity and specificity vary with $d_{\rm th}$ with respect to the sites and the diffusion path, respectively. The sensitivities in Figs. \[fig:sens-spec-bv\]a,b start to saturate for $d_{\rm
th}\gtrsim0.2-0.3$, a value in fair agreement with typical choices used in BV analyses of RMC models.[@Hall/etal:2006] Close to $d_{\rm th}\simeq0.1$, $v_{\rm BV}\simeq5\%$ is by about 40% smaller than the saturation value 8.3% for $d_{\rm th}\to\infty$ (see Fig. \[fig:bvvolume\]). However, this reduction is not strong enough to yield a significant improvement of the specificities. Even if one would take a large loss in the sensitivity by choosing $d_{\rm th}$ very small, the gain in the specificity remains low. In summary we find that the inclusion of criterion [*(iv)*]{} does not yield a substantial improvement and that the essential part of the obtained agreement is due to the geometric constraints [*(i,ii)*]{}.
To find an optimal value for $d_{\rm th}$ with respect to both sensitivity and specificity, we use the maximum of Cohen’s kappa value [@Fleiss:1981]. The kappa value for the sets ${\cal A}={\cal
P}^{\rm BV}(d)$ and ${\cal B}={\cal P}_\star$ is defined by $$\kappa=\frac{(p({\cal A} \cap {\cal B})+ p(\bar{\cal A}\cap \bar{\cal B}))
- (p({\cal A}) p({\cal B}) + p(\bar{\cal A})p(\bar{\cal B}))}
{1-(p({\cal A}) p({\cal B}) + p(\bar{\cal A})p(\bar{\cal B})) },
\label{eq:kappa}$$ where $p(.)$ denote the probabilities of the corresponding sets (for example, $p({\cal A} \cap {\cal B})$ is the probability that a cell belongs to both the sets ${\cal A}$ and ${\cal B}$). A value of $\kappa=1$ denotes complete agreement between the two sets, while $\kappa=0$ corresponds to a random overlap of the two sets (negative values indicate an anti-correlation). For the sites the maximum of $\kappa$ occurs at $d_{\rm th}=0.09$ and is only 25%. This is caused by the low specificity, which varies only slowly with $d_{\rm th}$. At $d_{\rm th}=0.09$ we find a specificity of 17%, while the sensitivity is 59%. For the path the maximum of $\kappa$ occurs at $d_{\rm
th}=0.23$ and is 48%. The sensitivity and specificity attain values of 54% and 50% at this maximum. In summary, the quality of agreement for the path is promising, while that for the sites is not yet satisfactory.
The sensitivity for the diffusion path can however be improved and the influence of the BV sum mismatch increased by using condition [*(v)*]{} instead of conditions [*(ii-iv)*]{}, i.e. by considering the path $\mathcal{P}_{\rm BV}'(d_{\rm th})$. For this path in comparison with $\mathcal{P}_{\rm perc}$ we plot in Fig. \[fig:sens-spec-kap\] the sensitivity and the specificity in dependence of $d_{\rm th}$ (see eq. (\[eq:cond4b\]). For large $d_{\rm th}$, the sensitivity reaches values up to 90%, which are significantly higher than the 57% in Fig. \[fig:sens-spec-bv\]b. The specificity for small $d_{\rm th}$ has values comparable to that in Fig. \[fig:sens-spec-bv\]b, and then decreases to smaller values for larger $d_{\rm th}$. As a function of $d_{\rm th}$, $\kappa$ reaches a maximum of 48% at $d_{\rm th}=0.22$. Further enhancements may be achieved by slight adjustments of the minimum distance $R_{\rm LiSi}$ or by replacing criterion [*(i)*]{} by a penalty function as well, which would mean that the oversimplifying hard sphere exclusion radius criteria [*(i,ii)*]{} may adversely affect the achievable level of agreement.
Finally, we test the potential of the BV method to identify sites (despite the low specificity). To this end we performed a cluster analysis with respect to $d_{\rm th}$, analogous to the one carried out in sec. \[sec:mdresults\] with respect to $\rho_{\rm cl}$. Different from the behavior found in Fig. \[fig:nr\_of\_clusters\], the number $N^{\rm BV}_{\rm cl}$ of BV clusters shown in Fig. \[fig:ncl-bv\] is very large already for small $d_{\rm th}$ and then decreases rapidly due to coalescence of clusters. Accessible cells start to percolate at $d_{\rm c}=0.09$. The reason for this behavior is that the valence mismatch $d(x)$ is a rather rapidly varying function. Already for small $d_{\rm th}$ there exists a large number of small disconnected clusters that soon merge together to form a percolating cluster. While the majority of $v_{\rm BV}$ belongs to this percolation cluster the fluctuations in $d(x)$ prevent $N^{\rm
BV}_{\rm cl}$ to assume comparatively small values for large $d_{\rm
th}$. Several hundred clusters are found for $d_{\rm th}\simeq0.1$. Almost all of these are very small: 80% consist of only a single cell and only 8 clusters contain more than 100 cells (i.e. are comparable to the size of sites). The percolation cluster includes 95% of $v_{\rm BV}(d_{\rm th}=0.1)$. As a consequence the cluster analysis is not successful for identifying ionic sites with respect to $d_{\rm
th}$.[@bv-sites-comm]
It is interesting, however, that the BV analysis can provide a good estimate of the [*number*]{} of sites (not their location in space). It turns out to be useful to include the coordination number constraint [*(iii)*]{} in addition to the conditions [*(i,v)*]{} in this case. For varying mismatch threshold $d_{\rm th}$ the cells belonging to the corresponding path $\mathcal{P}_{\rm BV}'(d_{\rm
th})$ are filled row by row under the condition that two centers have a distance larger than the minimal Li-Li distance $R_{\rm LiLi}=2.62$. The number of sites found in this way is shown in Fig. \[fig:site-number\] as a function of the mismatch threshold $d_{\rm th}$ (lower panel). For large $d_{\rm th}$, it approaches the number of clusters found by the Hoshen/Kopelman analysis in Sect. \[sec:mdresults\]. On the other hand, we can take the estimated number of sites at the optimal bond valence mismatch $d_{\rm
th}=0.22$ given by the maximum of Cohen’s $\kappa$ value with respect to the sets $\mathcal{P}_{\rm sites}$ and $\mathcal{P}_{\rm
BV}'(d_{\rm th})$. The result displayed in Fig. \[fig:sens-spec-kap\] yields 151 sites. This value is surprisingly close to the 148 sites found in Sec. \[sec:mdresults\] but studies of further systems are required to test whether such a prediction is generally possible.
In view of the success of the BV method to estimate the number of sites, we undertook further attempts to improve the level of agreement for the localization of the Li sites. If one distributes Li ions more randomly on the path $\mathcal{P}_{\rm BV}'(d_{\rm th})$ with the same minimal distance constraint $R_{\rm
LiLi}=2.62$ Å,[@Allen/Tildesley:1987] instead of doing it row by row as described above, the positioning of centers of sites is no longer biased to the rim of the BV path. In this way, we obtain 165 BV clusters for $d_{\rm th}=0.14$. Considering now as BV sites the spheres with ion radius $R_{\rm LiLi}/2=1.31$ Å, we found that 75% of these sites have at least one cell in common with one of the MD sites found in Sec. \[sec:mdresults\]. Still this is not enough for a reliable localization of Li sites.
Summary and Perspectives {#sec:conclusion}
========================
We have identified sites and diffusion paths for the mobile Li ions in molecular dynamics simulations of a Li$_2$SiO$_3$ glass. The identification was based on a cluster analysis of regions with high Li number density $\rho({\bf x})$. For the clusters to be assigned to sites, we chose the condition that the total residence time of a Li ion on it (including intermediate escape to fringe regions) is ten times larger than the total hopping time to any of its neighboring clusters. Using this criterion, a very low concentration 2,8% of vacant sites was found. An attempt to identify the diffusion path and sites by bond valence sums calculations was carried out. The comparison showed that the core of sites and parts of the diffusion path (up to about 50%) are captured. Conversely, the BV method was not suitable to distinguish between regions of high and low $\rho({\bf
x})$, and accordingly not suitable to determine the sites.
When dealing with the MD reference data of this work it proves to be necessary to adapt the BV parameters derived from experimental diffraction data to the MD force model. We showed that it is advantageous to optimize or replace the oversimplifying hard sphere exclusion radii for application of the BV analysis to the MD model. Moreover, an improvement of the BV analysis seems to be possible by adjusting the BV parameters $r_0$ and $\xi$ to the MD force model instead of only adjusting $V_{\rm id}$ to $\bar V$.
In searching for a powerful method to relate structural properties of the disordered network structure to transport properties of the mobile ions a further approach could be based on an effective potential $U_{\rm eff}({\bf x})\propto-k_{\rm B}T\ln\rho({\bf x})$. Taking this effective potential one could perform a critical path analysis to determine the activation energy for the long-range ion mobility (see e.g. ref. ). This approach would incorporate Coulomb interaction effects between the mobile ions in a mean field type approximation. Preliminary studies by us show that such Coulomb effects are important for the formation of the sites. With respect to the BV method we have also performed preliminary studies to evaluate the degree of correlation between $U_{\rm eff}({\bf x})$ and the BV mismatch $d({\bf x})$ (or $d'({\bf x})$). It turned out that BV method cannot clearly distinguish between regions of high and low $\rho({\bf
x})$ and accordingly it is not suitable to locate the sites. Besides the described shortcomings of a particular choice of parameter values this is mainly caused by neglecting the long-range Li-Li Coulomb repulsions.
The precise decomposition of the ion trajectories into residence and transition parts described in Sec. \[sec:mdresults\] moreover may allow one to bridge the time scale gap between molecular dynamics and coarse-grained Monte-Carlo simulations. Given two neighboring sites, many transitions of mobile ions can be followed and the average transition rate calculated. The elementary rates for the possible transitions can then be used in a subsequent Monte-Carlo simulation. However, such procedure is not particularly useful if one is not able to take into account the temperature dependence of the elementary rates. One possibility is to use MD data at various high temperatures, and to try to calculate from them activation energies for the elementary rates (if these follow an Arrhenius type behavior). With less effort, one could use again the density $\rho({\bf x})$ and the effective potential derived from it to determine the energetics of a coarse-grained hopping model. Further investigations in this direction have to be undertaken in order to establish a working multi-scale modeling of these complex amorphous systems.
Acknowledgments {#acknowledgments .unnumbered}
===============
Financial support to S. A. by the NUS ARF (R-284-000-029-112/133) and to the authors from TU Ilmenau by the HI-CONDELEC EU STREP project (NMP3-CT-2005-516975) is gratefully acknowledged.
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The $r^{-6}$ dependence can be motivated by the van-der-Waals interaction. However, since this term is used both for distances corresponding to covalent bonding as well as large distances, it in fact is not a van-der Waals interaction and often referred to as “dispersive interaction”. Nevertheless, the amplitude factors $c_i$, $c_j$ are usually assumed to scale with the polarizabilities. This gives reason to include this term here only for the large oxygen ions.
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The total hopping and residence time, of course, have to be normalized with respect to the simulation time to give an intrinsic cluster property. Alternatively, we could have defined averaged (with respect to the number of events) hopping and residence times. In order to calculate the occupation probabilities in Fig. \[fig:res-hop-time\]a we preferred to determine the total times.
This reentering mainly corresponds to a local non-activated ionic motion around a site on short time scales. It also includes rare (jump-like) events where a Li ions enters only the fringe of a neighboring site and returns without reaching its core. We note, the local motions cannot be simply accounted for by an oscillatory type of rattling motion in a potential minimum (see J. Habasaki, K. L. Ngai and Y. Hiwatari, J. Chem. Phys. [ **122**]{}, 054507 (2005)).
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This definition of specificity deviates from the usual one, as it is, for example, often applied in medical studies. In such studies the specificity is given by the conditional probability $\psi(\bar{\mathcal{P}}^{\rm
BV}_\star|\bar{\mathcal{P}}_\star)$ with $\bar{\mathcal{P}}^{\rm
BV}_\star$ and $\bar{\mathcal{P}}_\star$ being the complements of $\mathcal{P}^{\rm BV}_\star$ and $\mathcal{P}_\star$, respectively. Both definitions are related by $\psi(\mathcal{P}_\star|\mathcal{P}^{\rm
BV}_\star)=\psi(\mathcal{P}_\star)\psi(\mathcal{P}^{\rm
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BV}_\star|\bar{\mathcal{P}}_\star))]$, where $\psi((\mathcal{P}_\star)$ is the probability that a cell belongs to $\mathcal{P}_\star$ (i.e. equal to the volume fraction given in table \[tab:volume-fractions\]).
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min}=0.075$ in criterion [*(iii)*]{} to exclude strongly asymmetric coordinations, yields a slightly better separation of the pathway into clusters. In that case the number of clusters is reduced to a more plausible number 217, of which 61 contain only 1 cell for $d_{\rm th}=0.14$ (where clusters are still well separated).
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---
abstract: 'In this paper we define and analyze a generalization of the punctual Hilbert scheme of the plane which is associated to a simple complex Lie algebra ${\mathfrak{g}}$. Using this generalized punctual Hilbert scheme, we construct a new geometric structure on a surface, called a ${\mathfrak{g}}$-complex structure, generalizing complex structures and higher complex structures from [@FockThomas]. We investigate the geometry of the ${\mathfrak{g}}$-complex structure and define a moduli space which conjecturally is isomorphic to Hitchin’s component of the character variety for the real split group $G$.'
address: 'Université de Strasbourg, IRMA UMR 7501, 67084 Strasbourg, France'
author:
- Alexander Thomas
title: 'Generalized Punctual Hilbert Schemes and $\mathfrak{g}$-complex structures'
---
\[section\] \[Satz\][Remark]{} \[Satz\][Corollary]{} \[Satz\][Lemma]{}
Introduction {#introduction .unnumbered}
============
The main motivation for this paper is to get a geometric approach to Hitchin components. These components were constructed by Nigel Hitchin in his famous paper [@Hit.1] using analytic methods (Higgs bundles). Hitchin components are connected components of the character variety $\operatorname{Hom}(\pi_1(\Sigma), G)/G$ where $\Sigma$ is a smooth surface, closed and without boundary and $G$ is a adjoint group of a split real form of a complex simple Lie group.
For the group $G=PSL_2({\mathbb{R}})$, we obtain Teichmüller space which is also the moduli space of complex structures on $\Sigma$. For $G=PSL_n({\mathbb{R}})$, Vladimir Fock and the author defined in [@FockThomas] a new geometric structure, called higher complex structure, whose moduli space is conjecturally isomorphic to Hitchin’s component. The main ingredient to construct the higher complex structure is the punctual Hilbert scheme of the plane and its zero-fiber.
In this article, we pursue these ideas by defining a ${\mathfrak{g}}$-complex structure, for a complex simple Lie algebra ${\mathfrak{g}}$, using a generalization of the punctual Hilbert scheme, which we call ${\mathfrak{g}}$-Hilbert scheme.
Our strategy to define these new objects is twofold: on the one hand we use the various descriptions of the punctual Hilbert scheme, especially the matrix viewpoint, in order to generalize to an arbitrary ${\mathfrak{g}}$. On the other, we got inspiration from Hitchin’s original paper [@Hit.1] (section 5) where he starts with a principal nilpotent element and deforms it into an element of a principal slice, a generalized companion matrix.
We signal to the reader that our definition of the ${\mathfrak{g}}$-Hilbert scheme might get changed in the future, since as it is defined now, it is a non-Hausdorff space. There should be a way to get a nice topological space, without loosing any of its properties. This possible modification will not affect the ${\mathfrak{g}}$-complex structure since only the regular part of the ${\mathfrak{g}}$-Hilbert scheme plays a role in its construction.
The outline of the paper is the following: In the first section \[section1\], we define the generalized punctual Hilbert scheme, and some interesting subsets, for example the zero-fiber, the regular and the cyclic part. We analyze the relations between these parts and define a generalized Chow map. For classical ${\mathfrak{g}}$ we also define a map to a space of ideals and we analyze in detail the regular part of the ${\mathfrak{g}}$-Hilbert scheme. Finally, we study the topology of the ${\mathfrak{g}}$-Hilbert scheme and formulate some conjectures. In section \[section2\], we construct the ${\mathfrak{g}}$-complex structure and show how it induces in a natural way a complex structure. We give an equivalent construction using ideals for classical ${\mathfrak{g}}$. In section \[section3\], for ${\mathfrak{g}}$ of classical type, we define the notion of higher diffeomorphism of type ${\mathfrak{g}}$, which gives a definition of a moduli space of ${\mathfrak{g}}$-complex structures. We then explore its properties which show its similarity to Hitchin’s component. In particular we define a spectral curve in the complexified cotangent bundle $T^{*{\mathbb{C}}}\Sigma$. In the final part \[section4\] we give a larger conjectural picture which would give the link to Hitchin components. We include two appendices: In appendix \[appendix:A\], we review the main properties of the punctual Hilbert scheme of the plane. In the second appendix \[appendix:B\], we gather all properties we need in the paper of regular elements in semisimple Lie algebras and give precise references.
*Notations.* Throughout the paper, $\Sigma$ denotes a smooth surface, closed, without boundary and orientable. We denote by ${\mathfrak{g}}$ a complex simple Lie algebra, by ${\mathfrak{h}}$ a Cartan subalgebra, by $W$ its Weyl group and by $G$ its adjoint group (the unique Lie group $G$ with Lie algebra ${\mathfrak{g}}$ with trivial center).
*Acknowledgments.* The author wish to express his gratitude towards Vladimir Fock, for all the fruitful discussions. I also thank Loren Spice for answering a question of mine concerning connectedness of stabilizers on mathoverflow.
Generalized punctual Hilbert scheme {#section1}
===================================
In this section, we generalize the punctual Hilbert scheme to a ${\mathfrak{g}}$-Hilbert scheme and explore the properties of the new object. We analyze in detail its regular part in the case of a classical Lie algebra. The reader not familiar with punctual Hilbert schemes should consult Appendix \[appendix:A\].
Definitions and First Properties
--------------------------------
The punctual Hilbert scheme $\operatorname{Hilb}^n({\mathbb{C}}^2)$ has several descriptions:
- as a space of ideals (the *idealic viewpoint*)
- as a desingularization of the configuration space ${\mathfrak{h}}^2/W$ for ${\mathfrak{g}}={\mathfrak}{gl}_n$
- as a space of commuting matrices (the *matrix viewpoint*).
It is the matrix viewpoint which will be generalized. So let us recall it quickly here: $$\operatorname{Hilb}^n({\mathbb{C}}^2) \cong \{(A,B) \in {\mathfrak}{gl}_n^2 \mid [A,B]=0, (A,B) \text{ admits a cyclic vector}\} / GL_n.$$
The main difficulty is to find an intrinsic condition which generalizes the existence of a cyclic vector. Here is our proposal:
The generalized punctual Hilbert scheme, or **${\mathfrak{g}}$-Hilbert scheme**, denoted by $\operatorname{Hilb}({\mathfrak{g}})$, is defined by $$\operatorname{Hilb}({\mathfrak{g}}) = \{(A,B) \in {\mathfrak{g}}^2 \mid [A,B]=0, \dim Z(A,B) = \operatorname{rk}{\mathfrak{g}}\} /G$$ where $Z(A,B)$ denotes the common centralizer of $A$ and $B$, i.e. the set of elements $C \in {\mathfrak{g}}$ which commute with $A$ and $B$.
The condition on the dimension of the common centralizer does not come from nowhere: Proposition \[doublecomm\] of Appendix \[appendix:B\] shows that $\operatorname{rk}{\mathfrak{g}}$ is the minimal possible dimension for the centralizer of a commuting pair. Define the *commuting variety* by $\operatorname{Comm}({\mathfrak{g}})=\{(A,B)\in {\mathfrak{g}}^2 \mid [A,B]=0\}$. The ${\mathfrak{g}}$-Hilbert scheme is the set of all regular points of $\operatorname{Comm}({\mathfrak{g}})$ modulo $G$.
Ginzburg has defined the notion of a principal nilpotent pair in [@Ginzburg], which is more restrictive than ours. He calls “nil-pairs” elements of our ${\mathfrak{g}}$-Hilbert scheme, but he does not investigate them.
Let us give two examples of elements in the ${\mathfrak{g}}$-Hilbert scheme:
Let $A \in {\mathfrak{g}}$ be a regular element. Then by a theorem of Kostant (see \[thmKost\]), its centralizer $Z(A)$ is abelian. So for any $B\in Z(A)$, we have $Z(A) \subset Z(B)$, thus $Z(A,B) = Z(A) \cap Z(B) = Z(A)$ is of dimension $\operatorname{rk}{\mathfrak{g}}$. Therefore $[(A,B)] \in \operatorname{Hilb}({\mathfrak{g}})$.
If $A$ is principal nilpotent, then $B\in Z(A)$ is also nilpotent. So $[(A,B)]\in \operatorname{Hilb}_0({\mathfrak{g}})$, the zero-fiber defined below.
If $B=0$ then $[(A,0)]$ is in $\operatorname{Hilb}({\mathfrak{g}})$ iff $A$ is regular.
\[Young\] Let $(A,B)$ be a commuting pair of matrices in ${\mathfrak}{sl}_n$ admitting a cyclic vector, i.e. an element of the reduced Hilbert scheme. One way to get such a pair is the following construction: take a Young diagram (our convention is to put the origin in the upper left corner as for matrices) with $n$ boxes (see figure \[Youngdiag\]). Associate to each box a vector of a basis of ${\mathbb{C}}^n$. Define $A$ to be the matrix which translates to the left, i.e. sends a vector to the vector in the box to the left or to 0 if there is none. Let $B$ be the matrix which translates to the bottom. Then $A$ and $B$ clearly commute and are nilpotent. In proposition \[cycliccentralizer\] below, we show that $Z(A,B)$ is of minimal dimension in that case.
{height="2cm"}
\[Youngdiag\]
Guided by these examples, we define several subsets of the ${\mathfrak{g}}$-Hilbert scheme and explore their relations. First, we define the zero-fiber and the regular part which will both play a mayor role in the definition of a ${\mathfrak{g}}$-complex structure. We also define the cyclic part, which is not intrinsically defined since it uses a representation of ${\mathfrak{g}}$. The cyclic part will be used to define a map to a space of ideals, getting a generalization of the original description of the punctual Hilbert scheme.
\[partshilb\] The **zero-fiber** of the ${\mathfrak{g}}$-Hilbert scheme is defined by $$\operatorname{Hilb}_0({\mathfrak{g}})=\{[(A,B)] \in \operatorname{Hilb}({\mathfrak{g}}) \mid A \text{ and } B \text{ nilpotent}\}.$$
We define the **regular part** of the ${\mathfrak{g}}$-Hilbert scheme, denoted by $\operatorname{Hilb}^{reg}({\mathfrak{g}})$, to be those conjugacy classes $[(A,B)]$ in which $A$ or $B$ is a regular element of ${\mathfrak{g}}$.
Finally for classical ${\mathfrak{g}}$, let $\rho$ denote the natural representation of ${\mathfrak{g}}$ (i.e. ${\mathfrak}{sl}_n \subset {\mathfrak}{gl}_n, {\mathfrak}{so}_n \subset {\mathfrak}{gl}_n$ and ${\mathfrak}{sp}_{2n} \subset {\mathfrak}{gl}_{2n}$). Define the **cyclic part** of the ${\mathfrak{g}}$-Hilbert scheme by $$\operatorname{Hilb}^{cycl}({\mathfrak{g}}) = \{(A,B) \in {\mathfrak{g}}^2 \mid [A,B]=0, (\rho(A),\rho(B)) \text{ admits a cyclic vector}\}/G.$$
In the definition of the cyclic part, it would be more natural to consider the adjoint representation, but even in the case of ${\mathfrak}{sl}_2$, this would give a map to a space of ideals, which is not the one of $\operatorname{Hilb}^2_{red}({\mathbb{C}}^2)$.
Instead of the standard representation, one could also use a non-trivial representation of minimal dimension, which for classical ${\mathfrak{g}}$ is always the standard representation, apart from type $D_3$ and $D_4$. For $D_3$, the two spin representations are of minimal dimension, and they give an isomorphism between ${\mathfrak}{so}_6$ and ${\mathfrak}{sl}_4$. For type $D_4$, there are three representations of minimal dimension, the standard one and the two spin representations. All of them are linked by outer automorphisms coming from the symmetry of the Dynkin diagram. Thus the cyclic part of the ${\mathfrak{g}}$-Hilbert scheme is the same for all three representations.
Taking the representation of minimal dimension has the additional advantage to be well-defined for all ${\mathfrak{g}}$. We have not computed any example of an exceptional Lie algebra, so we do not know whether the use of the representation of minimal dimension is of interest.
The first relation between the various Hilbert schemes is the inclusion of the cyclic part in the ${\mathfrak{g}}$-Hilbert scheme, which justifies the name “cyclic part”:
\[cycliccentralizer\] For ${\mathfrak{g}}$ of classical type, we have $\operatorname{Hilb}^{cycl}({\mathfrak{g}}) \subset \operatorname{Hilb}({\mathfrak{g}})$.
Recall $\rho$ the natural representation of ${\mathfrak{g}}$ on ${\mathbb{C}}^m$. For simplicity, we write $A$ instead of $\rho(A)$ here.
Let $(A,B) \in {\mathfrak{g}}^2$ admitting a cyclic vector $v$. Let $C \in Z(A,B)$. Then $C$ is a polynomial in $A$ and $B$. Indeed, there is $P\in {\mathbb{C}}[x,y]$ such that $Cv = P(A,B)v$. Since $C$ commutes with $A$ and $B$, we then get for any polynomial $Q$ that $CQ(A,B)v = Q(A,B)Cv = Q(A,B)P(A,B)v= P(A,B)Q(A,B)v$, so $C=P(A,B)$.
Therefore the common centralizer of $(A,B)$ in ${\mathfrak}{gl}_m$ is ${\mathbb{C}}[A,B]/I$ where $I=\{P\in {\mathbb{C}}[x,y] \mid P(A,B) = 0\}$. We know from Appendix \[appendix:A\] that $I$ is of codimension $m$ since $(A,B)$ admits a cyclic vector. We have $Z(A,B) = Z_{{\mathfrak}{gl}_m}(A,B) \cap {\mathfrak{g}}$. One can easily check that for ${\mathfrak{g}}$ of type $A_n$, a polynomial $P(A,B)$ is in ${\mathfrak{g}}$ iff its constant term has a specific form, given by the other coefficients (to ensure trace zero). For type $B_n, C_n$ and $D_n$, $P(A,B)$ is in ${\mathfrak{g}}$ iff $P$ is odd. One checks in each case that the dimension of $Z(A,B)$ equals the rank of ${\mathfrak{g}}$.
In general, the inclusion of the cyclic Hilbert scheme is strict as shows the following example:
Consider $A=\left(\begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{smallmatrix} \right)$ and $B=\left(\begin{smallmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{smallmatrix}\right)$ in ${\mathfrak}{sl}_3$. One easily checks that the pair $(A,B)$ does not admit any cyclic vector, but that their common centralizer is of dimension 2. So $[(A,B)] \in \operatorname{Hilb}({\mathfrak}{sl}_3)\backslash \operatorname{Hilb}^{cycl}({\mathfrak}{sl}_3)$.
This example will be used in subsection \[topology\] to show that $\operatorname{Hilb}({\mathfrak{g}})$ is not Hausdorff.
In general, there is no link between regular and cyclic part. Example \[Young\] shows that cyclic elements are not always regular and the following example shows that regular element are not always cyclic:
\[regnotcyclic\] For ${\mathfrak{g}}$ of type $D_n$, let $f$ be a principal nilpotent element. Then one checks that $[(f,0)] \in \operatorname{Hilb}({\mathfrak}{so}_{2n})$ is regular but not cyclic (see also subsection \[Dn\]).
Let us turn to the regular part. It turns out that if one fixes a *principal slice* $f+Z(e)$ in ${\mathfrak{g}}$ (see Appendix \[appendix:B\]), there is a preferred representative for regular classes:
\[paramit\] Any class $[(A,B)] \in \operatorname{Hilb}^{reg}({\mathfrak{g}})$ where $A$ is regular can uniquely be conjugated to $(A \in f+Z(e), B \in Z(A))$.
By the property of the principal slice, there is a unique conjugate of $A$ which is in the principal slice $f+Z(e)$. Denote still by $A$ and $B$ these conjugates. The only thing to show is that $B$ is unique which is done in the next lemma.
If $A\in {\mathfrak{g}}$ is regular, $g\in G$ such that $Ad_g(A)=A$ and $B\in Z(A)$, then $Ad_g(B)=B$.
By Kostant’s theorem \[thmKost\], we know that $Z(A)$ is abelian. So the infinitesimal version of the lemma is true. We conclude by the connectedness of the stabilizer of $A$, given by the next lemma.
For a regular element $A \in {\mathfrak{g}}$, its stabilizer $\operatorname{Stab}(A) = \{g \in G \mid Ad_g(A)=A\}$ in the adjoint group $G$ is connected.
Decompose $A$ into Jordan form: $A=A_s+A_n$ with $A_s$ semisimple, $A_n$ nilpotent and $[A_s,A_n]=0$. So $A_n\in Z(A_s)$. The structure of the centralizer $Z(A_s)$ is well-known: it is a direct sum of a Cartan ${\mathfrak{h}}$ containing $A_s$ with all root spaces ${\mathfrak{g}}_{\alpha}$ where $\alpha$ is a root such that $\alpha(A_s)=0$. It is also known that $Z(A_s)$ is reductive, so a direct sum $Z(A_s)={\mathfrak}{c}\oplus {\mathfrak{g}}_s$ where ${\mathfrak}{c}$ is the center and ${\mathfrak{g}}_s$ is the semisimple part of $Z(A_s)$. In particular the center ${\mathfrak}{c}$ is included in ${\mathfrak{h}}$. So $A_n \in {\mathfrak{g}}_s$ since $A_n$ is nilpotent. Denote by $G_s$ the Lie group with trivial center with Lie algebra ${\mathfrak{g}}_s$.
We know that $A$ is regular is equivalent to $A_n$ being regular nilpotent in ${\mathfrak{g}}_s$ (see [@Kost2], proposition 0.4). We also know that the $G$-equivariant fundamental group of the orbit of $A$ (which is the space of connected components of $\operatorname{Stab}(A)$) is the same as the $\operatorname{Stab}(A_s)$-equivariant fundamental group of the $\operatorname{Stab}(A_s)$-orbit of $A_n$ (see Proposition 6.1.8. of [@Coll] adapted to the adjoint group). In other words, the connected components of $\operatorname{Stab}_G(A)$ are the same as the connected components of $\operatorname{Stab}_{G_s}(A_n)$ since the $\operatorname{Stab}(A_s)$-orbit of $A_n$ is equal to the $G_s$-orbit of $A_n$.
So we are reduced to the principal nilpotent case. Using the classification of simple Lie algebras, one can check explicitly in Collingwood-McGovern’s book [@Coll] the tables 6.1.6. for classical ${\mathfrak{g}}$ and the tables at the end of chapter 8 for exceptional ${\mathfrak{g}}$ that the stabilizer of a principal nilpotent element is always connected.
It is surprising that the last lemma has never been stated (at least not to our knowledge). It would be interesting to find a direct argument, without using the classification of simple Lie algebras.
The regular zero-fiber $\operatorname{Hilb}^{reg}_0({\mathfrak{g}}) = \operatorname{Hilb}^{reg}({\mathfrak{g}})\cap \operatorname{Hilb}_0({\mathfrak{g}})$ is an affine variety of dimension $\operatorname{rk}{\mathfrak{g}}$.
This follows directly from the previous proposition using the fact that $A \in f+Z(e)$ is nilpotent iff $A=f$. So $\operatorname{Hilb}^{reg}_0({\mathfrak{g}})$ is described by $Z(f)$ which is a vector space of dimension $\operatorname{rk}{\mathfrak{g}}$.
We know that both the regular and the cyclic part are in general strictly included in the ${\mathfrak{g}}$-Hilbert scheme. But they are dense subspaces:
\[density\] The regular part $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ is dense in $\operatorname{Hilb}({\mathfrak{g}})$. For classical ${\mathfrak{g}}$, the cyclic part is also dense in $\operatorname{Hilb}({\mathfrak{g}})$.
By a theorem of Richardson (see \[Richardson\]), the set of semisimple commuting pairs is dense in the commuting variety $\operatorname{Comm}({\mathfrak{g}})$. So the set of semisimple regular elements is also dense in $\operatorname{Comm}({\mathfrak{g}})$. Passing to the quotient by $G$, we get that the classes of semisimple regular pairs are dense in $\operatorname{Hilb}({\mathfrak{g}})$ since $\operatorname{Hilb}({\mathfrak{g}})\subset \operatorname{Comm}({\mathfrak{g}})/G$ and all semisimple regular pairs are in $\operatorname{Hilb}({\mathfrak{g}})$. Since the semisimple regular pairs are in the regular part, we get the density of $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ in $\operatorname{Hilb}({\mathfrak{g}})$.
For classical ${\mathfrak{g}}$, we have the same argument for the cyclic part since semisimple regular pairs are cyclic.
To end the section, we state an analogue of Kostant’s theorem about abelian subalgebras of centralizers:
For any commuting pair $(A,B) \in \operatorname{Comm}({\mathfrak{g}})$, there is an abelian subspace of dimension $\operatorname{rk}{\mathfrak{g}}$ in the common centralizer $Z(A,B)$.
The proof is completely analogous to Kostant’s proof for theorem \[thmKost\]: we use a limit argument. Let $(A_n,B_n)$ be a sequence of regular semisimple pairs converging to $(A,B)$ (exists since regular semisimple pairs are dense). We know that $Z(A_n,B_n)$ is a $\operatorname{rk}{\mathfrak{g}}$-dimensional abelian subspace of ${\mathfrak{g}}$. Since the Grassmannian $Gr(\operatorname{rk}{\mathfrak{g}}, \dim {\mathfrak{g}})$ is compact, there is a subsequence of $Z(A_n,B_n)$ which converges. It is easy to prove that the limit is included in $Z(A,B)$ and is commutative.
For $[(A,B)] \in \operatorname{Hilb}({\mathfrak{g}})$, the common centralizer $Z(A,B)$ is abelian.
For classical ${\mathfrak{g}}$, the corollary is easy for the cyclic part since $Z(A,B) = {\mathbb{C}}[x,y]/I \cap {\mathfrak{g}}$ which is abelian since ${\mathbb{C}}[x,y]$ is.
In the following sections, we generalize as far as possible the other viewpoints of the usual Hilbert scheme (resolution of configuration space and idealic viewpoint) to our setting.
Chow map
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We want to generalize the Chow map, which goes from $\operatorname{Hilb}^n({\mathbb{C}}^2)$ to the configuration space (see \[resofsing\]).
Fix a Cartan subalgebra ${\mathfrak{h}}$ in ${\mathfrak{g}}$. Recall the Jordan decomposition in a semisimple Lie algebra: for $x \in {\mathfrak{g}}$, there is a unique pair $(x_s, x_n)$ with $x=x_s+x_n$, $x_s$ semisimple, $x_n$ nilpotent and $[x_s,x_n]=0$. For a semisimple element $x$, denote by $x^*$ a conjugate in the Cartan ${\mathfrak{h}}$ (unique up to $W$-action).
The **Chow map** $ch: \operatorname{Hilb}({\mathfrak{g}}) \rightarrow {\mathfrak{h}}^2/W$ is defined by $$ch([(A,B)]) = [(A_s^*,B_s^*)]$$ where the brackets $[.]$ always denotes the equivalence class. For semisimple regular pairs, this map corresponds to a simultaneous diagonalization.
The Chow map $ch$ is well-defined and continuous.
Since $[A,B]=0$, we also have $[A_s,B_s]=0$ by a simultaneous Jordan decomposition in a faithful representation. Hence there is a conjugate of the pair $(A_s,B_s)$ which lies in ${\mathfrak{h}}^2$. Since the adjoint action of $G$ on ${\mathfrak{g}}$ restricts to the $W$-action on ${\mathfrak{h}}$, the map $ch$ is well-defined.
The map $x\mapsto x_s^*$ is continuous which simply follows from the continuity of eigenvalues. Hence the Chow map is continuous as well.
The Jordan decomposition $x\mapsto (x_s,x_n)$ is not continuous at all, since semisimple elements are dense in ${\mathfrak{g}}$ for which we have $x_n=0$ andt for all non-semisimple elements we have $x_n\neq 0$. But the map $x\mapsto x_s$ is continuous.
This map permits to think of a generic element of $\operatorname{Hilb}({\mathfrak{g}})$ as a point in ${\mathfrak{h}}^2/W$, or via a representation of ${\mathfrak{g}}$ on ${\mathbb{C}}^m$, as a set of $m$ points in ${\mathbb{C}}^2$ with a certain symmetry. For ${\mathfrak{g}}={\mathfrak}{sl}_n$ for example, these are $n$ points with barycenter 0.
Since $\operatorname{Hilb}({\mathfrak{g}})$ is even not Hausdorff (see subsection \[topology\]), it cannot be a non-singular variety. Nevertheless we conjecture the following:
\[conj1\] There is a modified version of $\operatorname{Hilb}({\mathfrak{g}})$, identifying some points, which is a smooth projective variety such that the Chow morphism is a resolution of singularities.
Idealic Map {#idealic}
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For ${\mathfrak{g}}$ of classical type, we can associate to any regular element of the ${\mathfrak{g}}$-Hilbert scheme an ideal, which we call *idealic map*. In this subsection, ${\mathfrak{g}}$ is a classical Lie algebra. Recall the natural representation $\rho$ of ${\mathfrak{g}}$ on ${\mathbb{C}}^m$ (see definition \[partshilb\]). We will write $A$ instead of $\rho(A)$.
We wish to define a map like in \[bijhilbert\]: $$\label{ideal}[(A,B)] \mapsto I(A,B)=\{P\in {\mathbb{C}}[x,y] \mid P(A,B) = 0\}.$$ If $[(A,B)] \in \operatorname{Hilb}^{cycl}({\mathfrak{g}})$ is cyclic, this ideal is of codimension $m$. But if the pair is not cyclic, there is no reason why the codimension should be $m$. In fact, there are examples for ${\mathfrak{g}}$ of type $D_n$ where the codimension is smaller.
We wish the idealic map to be continuous, so $I$ has to be of constant codimension. A strategy would be to define the idealic map $I$ on the cyclic part $\operatorname{Hilb}^{cycl}({\mathfrak{g}})$ (which is dense by proposition \[density\]) and to extend it by continuity. Unfortunately, the map can not be extended in a continuous way as shown in the following example:
\[idealnotcont\] Take ${\mathfrak{g}}$ of type $D_n$. Denote by $f$ a principal nilpotent element. The pair $[(f,0)] \in \operatorname{Hilb}({\mathfrak}{so}_{2n})$ is not cyclic (seen in example \[regnotcyclic\]). Using the matrix $S$ defined in equation , we can approach $(f,0)$ by $(f,tS)$ or by $(f+tS^T,0)$ for $t\in{\mathbb{C}}^{\times}$ going to 0. These pairs are all cyclic. In the first case, the ideal is $I=\langle x^{2n-1}, xy, y^2=t^2x^{2n-2}\rangle$ which converges as $t$ goes to 0 to $\langle x^{2n-1}, xy, y^2\rangle$. In the second case, the ideal is $I=\langle x^{2n}+t^2, y \rangle$ converging to $\langle x^{2n},y \rangle$.
Because of this difficulty, our strategy is to define a space of ideals $I_{{\mathfrak{g}}}({\mathbb{C}}^2)$, then a map $\operatorname{Hilb}^{cycl}({\mathfrak{g}})\rightarrow I_{{\mathfrak{g}}}({\mathbb{C}}^2)$ and to extent it over the regular part $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ (in a non-continuous way). The last step is only necessary for ${\mathfrak{g}}$ of type $D_n$ since for the other classical types the regular part is included in the cyclic part as we will see in the sequel. The extension for $D_n$ will be defined *ad hoc* in subsection \[Dn\].
The previous section taught us to think of a generic element of $\operatorname{Hilb}({\mathfrak{g}})$ as a $m$-tuple of points in ${\mathbb{C}}^2$ invariant under the Weyl group $W$. For type $A_n$ this means that the barycenter of the points is the origin. For the other classical types, this means that the set of points is symmetric with respect to the origin. Thus the defining ideal of these points is also invariant under the action of $W$. Hence the following definition.
We define the **space of ideals** of type ${\mathfrak{g}}$, denoted by $I_{{\mathfrak{g}}}({\mathbb{C}}^2)$, to be the set of ideals in ${\mathbb{C}}[x,y]$ which are of codimension $m$ and $W$-invariant. For type $B_n, C_n$ and $D_n$ this means that $I$ is invariant under $(x,y)\mapsto (-x,-y)$.
The map $I: \operatorname{Hilb}^{cycl}({\mathfrak{g}})\rightarrow I_{{\mathfrak{g}}}({\mathbb{C}}^2)$ given by equation above is well-defined. Indeed, the codimension is $m$ by cyclicity and the ideal is $W$-invariant since this is a closed condition and it is true on the dense subset of regular semisimple pairs.
Notice that $I_{{\mathfrak{g}}}({\mathbb{C}}^2)$ is the same for ${\mathfrak{g}}$ of type $C_n$ or $D_n$. But we will see that the idealic map $I$ has not the same image in the two cases. We will also see that for ${\mathfrak{g}}$ of type $A_n, B_n$ or $C_n$ the idealic map is injective. But for type $D_n$ it is not (it is generically 2 to 1). This comes from the fact that the Weyl group acting on the generic $2n$ points, coming in $n$ pairs $(P_i, P_{i+1}=-P_i)$, cannot exchange $P_1$ and $P_2$ while leaving all other points fixed.
As for the usual Hilbert scheme, there is a direct link between the idealic map and the Chow morphism:
The Chow map $ch$ is the composition of the idealic map with the map which associates to an ideal its support, seen as an element of ${\mathfrak{h}}^2/W$: $$ch([(A,B)]) = \operatorname{supp}I(A,B).$$
The statement is true on regular semisimple pairs which is a dense subset. For ${\mathfrak{g}}$ of type $A_n$, $B_n$ and $C_n$, it follows by continuity of both the Chow map and the idealic map. For $D_n$, our definition of the idealic map is to pick one of the various possible limits. In particular, the support of the ideal is still given by the Chow map.
Morphisms {#mu2}
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In this subsection, we analyze the functorial behavior of the ${\mathfrak{g}}$-Hilbert scheme. In particular we construct two maps linked to the zero-fiber of the Hilbert scheme of ${\mathfrak}{sl}_2$ which will lead in the construction of the moduli space $\hat{\mathcal{T}}_{{\mathfrak{g}}}\Sigma$ of ${\mathfrak{g}}$-complex structures to maps from and to Teichmüller space.
Let $\psi: {\mathfrak{g}}_1 \rightarrow {\mathfrak{g}}_2$ be a morphism of Lie algebras. For $[(A,B)] \in \operatorname{Hilb}({\mathfrak{g}}_1)$, we can associate $[(\psi(A), \psi(B))]$ which is a well-defined map to $\operatorname{Comm}({\mathfrak{g}}_2)/G_2$. But there is no reason why $\dim Z(\psi(A), \psi(B))$ should be minimal. If we accept conjecture \[conj1\], that there is a modified version of the ${\mathfrak{g}}$-Hilbert scheme which is a resolution of ${\mathfrak{h}}^2/W$, we have a functorial behavior:
Assuming conjecture \[conj1\], there is an induced map $\operatorname{Hilb}({\mathfrak{g}}_1) \rightarrow \operatorname{Hilb}({\mathfrak{g}}_2)$.
Choose Cartan subalgebras ${\mathfrak{h}}_1$ and ${\mathfrak{h}}_2$ such that $\psi({\mathfrak{h}}_1)={\mathfrak{h}}_2$. Consider the composition ${\mathfrak{h}}_1^2 \rightarrow {\mathfrak{h}}_2^2 \rightarrow {\mathfrak{h}}_2^2/W_2$ using $\psi$ for the first arrow. Since $\psi$ induces a homomorphism between the Weyl groups, we can factor the composition to get a map ${\mathfrak{h}}_1^2/W_1 \rightarrow {\mathfrak{h}}_2^2/W_2$. Finally, consider the composition $\operatorname{Hilb}(g_1) \rightarrow {\mathfrak{h}}_1^2/W_1 \rightarrow {\mathfrak{h}}_2^2/W_2$ where the first arrow comes from the minimal resolution. This is a continuous map and by the universal property of a minimal resolution, the map lifts to $\operatorname{Hilb}({\mathfrak{g}}_1) \rightarrow \operatorname{Hilb}({\mathfrak{g}}_2)$.
Let us study this induced map in the case of the reduced Hilbert scheme $\operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$, which is a minimal resolution (see appendix \[appendix:A\]). Take $\psi:{\mathfrak}{sl}_m\rightarrow {\mathfrak}{sl}_n$ inducing a map $\operatorname{Hilb}^m_{red}({\mathbb{C}}^2)\rightarrow \operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$. In the matrix viewpoint, this map is not given by $[(\psi(A),\psi(B))]$. Consider for example the map $\psi: {\mathfrak}{sl}_2\rightarrow {\mathfrak}{sl}_4$ given on the standard generators $(e,f,h)$ of ${\mathfrak}{sl}_2$ by $$\psi(e)=\left(\begin{smallmatrix} 0&&&1 \\&0&&\\&&0&\\&&&0\end{smallmatrix}\right), \psi(f)=\left(\begin{smallmatrix} 0&&& \\&0&&\\&&0&\\1&&&0\end{smallmatrix}\right) \text{ and } \psi(h)=\left(\begin{smallmatrix} 1&&& \\&0&&\\&&0&\\&&&-1\end{smallmatrix}\right).$$ The element $[(h,0)] \in \operatorname{Hilb}^2_{red}({\mathbb{C}}^2)$ corresponds to the ideal $I=\langle x^2-1,y\rangle$ which through $\psi$ goes to $\langle x^4-x^2,y\rangle$ which in turn gives the matrices $[(M,0)]$ where $M=\left(\begin{smallmatrix} 1&&& \\&0&1&\\&0&0&\\&&&-1\end{smallmatrix}\right)$. This is not $[(\psi(h), \psi(0))]$. It would be interesting to describe the induced map in the matrix viewpoint.
Despite this complication, there are two cases where a map between ${\mathfrak{g}}$-Hilbert schemes exists naturally.
The first one is linked to the principal map $\psi: {\mathfrak}{sl}_2 \rightarrow {\mathfrak{g}}$ which induces a map $$\label{teichcopy} \operatorname{Hilb}({\mathfrak}{sl}_2) \rightarrow \operatorname{Hilb}^{reg}({\mathfrak{g}}).$$ Indeed, any non-zero element of ${\mathfrak}{sl_2}$ is regular and cyclic. So if $[(A,B)] \in \operatorname{Hilb}({\mathfrak}{sl}_2)$ such that $A$ is non-zero, there is by proposition \[paramit\] a unique representative $(f+te,B\in Z(e+tf))$ where $(e,f,h)$ denotes the standard generators of ${\mathfrak}{sl}_2$ and $t\in {\mathbb{C}}$. So the image is $[(\psi(f)+t\psi(e),\psi(B))]$. Since $(\psi(e), \psi(f), \psi(h))$ is a principal ${\mathfrak}{sl}_2$-triple (property of the principal map), we know that $\psi(f)+t\psi(e)$ is in the principal slice, thus it is regular, so we land in $\operatorname{Hilb}^{reg}({\mathfrak{g}})$.
The second one is a sort of inverse map to the first one, but only on the level of the zero-fiber. Given $[(A,B)]\in \operatorname{Hilb}^{reg}_0({\mathfrak{g}})$ where $A$ is regular, there is a principal ${\mathfrak}{sl}_2$-subalgebra ${\mathcal}{S}$ with $A$ as nilpotent element. There is no reason why $B$ should be in ${\mathcal}{S}$ but there is a “best approximation” in the following sens:
\[mu2prop\] Let $A$ be a principal nilpotent element and $B \in Z(A)$. Then there is a unique $\mu_2 \in {\mathbb{C}}$ such that $B-\mu_2 A$ is not regular.
The strategy of the proof is to use Proposition \[prinnilp\] of the appendix which characterizes principal nilpotent elements $x$ as those nilpotent elements whose values $\alpha(x)$ for all simple roots $\alpha$ are non-zero. So the proposition is equivalent to the statement that $\alpha_1(B) = \alpha_2(B)$ for all simple roots $\alpha_1$ and $\alpha_2$.
Let $R$ be a root system in ${\mathfrak{h}}^*$ and denote by $R_+$ and $R_s$ the positive and respectively the simple roots. We can conjugate $A$ to the element given by $\alpha(A)=1$ if $\alpha \in R_s$ and $\alpha(A)=0$ otherwise.
For two simple roots $\alpha_1$ and $\alpha_2$ such that $\alpha_1+\alpha_2 \in R$, using $[A,B]=0$ we get: $$0=(\alpha_1+\alpha_2)([A,B])=\alpha_1(A)\alpha_2(B)-\alpha_2(A)\alpha_1(B) = (\alpha_2-\alpha_1)(B).$$
Since ${\mathfrak{g}}$ is simple, its Dynkin diagram is connected, so $\alpha_1(B) = \alpha_2(B)$ for all simple roots. The common value $\mu_2$ gives the unique complex number such that $B-\mu_2 A$ is not regular.
With this proposition, we can now define a map $$\label{mu} \mu: \operatorname{Hilb}^{reg}_0({\mathfrak{g}}) \rightarrow \operatorname{Hilb}_0({\mathfrak}{sl}_2)$$ given by $\mu([(A,B)])=[(e,\mu_2 e]$ or $[(\mu_2 e, e)]$ depending whether $A$ or $B$ is regular.
An equivalent way to define the map $\mu$ is the following: we can use the previous proposition \[mu2prop\] to show that the centralizer $Z(A)$ of a principal nilpotent element is a direct product $$Z(A) = \operatorname{Span}(A) \times Z(A)^{irreg}$$ where $Z(A)^{irreg}$ denotes the irregular elements of $Z(A)$. The map $\mu$ is nothing but the projection to the first factor.
We can describe the regular part of the ${\mathfrak{g}}$-Hilbert scheme $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ as those classes $[(A,B)]$ such that $\operatorname{Span}(A,B)$ intersects the regular part ${\mathfrak{g}}^{reg}$ non-trivially. This description is more symmetric since it does not prefer $A$ or $B$. From proposition \[mu2prop\] we see that the intersection of $\operatorname{Span}(A,B)$ with ${\mathfrak{g}}^{reg}$ is the whole two-dimensional $\operatorname{Span}(A,B)$ from which we have to take out a line. Hence, the intersection has two components.
In the following subsections, we study the regular part $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ and its zero-fiber case by case for classical ${\mathfrak{g}}$.
Case $A_n$
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Consider ${\mathfrak{g}}={\mathfrak}{sl}_n$ (of type $A_{n-1}$). We describe first $\operatorname{Hilb}^{reg}_0({\mathfrak}{sl}_n)$, its idealic map and then $\operatorname{Hilb}^{reg}({\mathfrak}{sl}_n)$ using proposition \[paramit\].
Fix the following principal nilpotent element (with 1 on the line just under the main diagonal): $$f=\begin{pmatrix} & & & \\ 1& & & \\ &\ddots & & \\ & &1 & \end{pmatrix}.$$
This element $f$ is cyclic, so we know from \[cycliccentralizer\] that the centralizer is given by polynomials: $Z(f)=\{\mu_2 f+\mu_3 f^2+...+\mu_n f^{n-1}\}$. So an element of $\operatorname{Hilb}^{reg}_0({\mathfrak}{sl}_n)$ can be represented by $(f,Q(f))$ where $Q$ is a polynomial without constant term of degree at most $n-1$. The coefficients $\mu_i$ are called *higher Beltrami coefficients*.
Since here we have $\operatorname{Hilb}^{reg}_0({\mathfrak}{sl}_n) \subset \operatorname{Hilb}^{cycl}({\mathfrak}{sl}_n)$ (already $f$ is cyclic), the idealic map is given by $$I(f,Q(f))=\{P\in {\mathbb{C}}[x,y] \mid P(f,Q(f))=0\} = \langle x^n, -y+Q(x) \rangle.$$ We recognize the big cell of the zero-fiber of the punctual Hilbert scheme.
To describe the whole regular part $\operatorname{Hilb}^{reg}({\mathfrak}{sl}_n)$, we take the following principal slice given by companion matrices: $$\begin{pmatrix} & & & t_n\\ 1& & & \vdots\\ &\ddots & &t_2\\ & &1 & \end{pmatrix}.$$
Let $A$ be a matrix of companion type. Notice that the characteristic polynomial of a companion matrix is given by $x^n+t_2x^{n-2}+...+t_n$. Since $A$ is still cyclic, its centralizer consists of polynomials in $A$ with constant term determined by the other coefficients (in order to ensure trace zero). Thus, a representative of $\operatorname{Hilb}^{reg}({\mathfrak}{sl}_n)$ is given by $(A,B=Q(A))$.
The idealic map is thus given by $$I(A,B)=\langle x^n+t_2x^{n-2}+...+t_n, -y+\mu_1+\mu_2x+...+\mu_nx^{n-1} \rangle$$ where $\mu_1$ is given by $\mu_1=\sum_{k=2}^{n-1}\frac{k}{n}t_k\mu_{k+1}$. One recognizes the big cell of the reduced punctual Hilbert scheme. Notive that the idealic map is injective here.
Case $B_n$ {#Bn}
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Consider ${\mathfrak{g}}={\mathfrak}{so}_{2n+1}$. Represent ${\mathfrak{g}}$ on ${\mathbb{C}}^{2n+1}$ using the metric given by $g(e_i,e_j)=\delta_{i,n-j}$ (where $e_i$ are standard vectors), i.e. $g=\left(\begin{smallmatrix} & & 1\\ & \udots & \\ 1& & \end{smallmatrix}\right)$. A matrix $A$ is in ${\mathfrak{g}}$ iff $\sigma(A)=-A$ where $\sigma$ is the involution consisting in a reflection along the anti-diagonal. In other words $A \in {\mathfrak{g}}$ iff $A_{i,j}=A_{n+1-j,n+1-i}$ for all $i,j$.
We fix the following principal nilpotent element: $$f=\begin{pmatrix} &&&&&& \\ 1&&&&&& \\ &\ddots &&&&&\\ &&1&&&&\\ &&&-1&&& \\ &&&& \ddots &&\\ &&&&&-1&\end{pmatrix}$$
This element is cyclic, so its centralizer by \[cycliccentralizer\] consists of all odd polynomials: $Z(f)=\{\mu_2f+\mu_4f^3+...+\mu_{2n}f^{2n-1}\}$. A representative of $\operatorname{Hilb}^{reg}_0({\mathfrak{g}})$ is thus given by $(f,Q(f))$ where $Q$ is an odd polynomial of degree at most $2n-1$. The coefficients $\mu_{2i}$ are called the higher Beltrami coefficients for $B_n$.
A principal slice is given by $$\begin{pmatrix} &&&&&t_{2n}& \\ 1&&&&\udots && -t_{2n}\\ &\ddots &&t_2&&\udots& \\ &&1&&-t_2&&\\ &&&-1&&& \\ &&&& \ddots &&\\ &&&&&-1&\end{pmatrix}.$$ Let $A$ be a matrix of this type. Its characteristic polynomial is given by $x^{2n+1}-2t_2x^{2n-1}+2t_4x^{2n-3}\pm ... +(-1)^n\times 2t_{2n}x$. So we can really think of the principal slice as a generalized companion matrix. Changing slightly $t_{2i}$ we can get rid of signs and the factor 2 in the characteristic polynomial, which we will do in the sequel.
The matrix $A$ is still cyclic, so we have the inclusion $\operatorname{Hilb}^{reg}({\mathfrak{g}}) \subset \operatorname{Hilb}^{cycl}({\mathfrak{g}})$. A representative of $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ is given by $(A,B=Q(A))$ where $Q$ is still an odd polynomial of degree at most $2n-1$. The idealic map is then given by $$I(A,B)=\langle x^{2n+1}+t_2x^{2n-1}+t_4x^{2n-3}+...+t_{2n}x, -y+\mu_2x+\mu_4x^3+...+\mu_{2n}x^{2n-1}\rangle.$$ This ideal is invariant under the map $(x,y)\mapsto (-x,-y)$. This is not surprising since a generic element of the ${\mathfrak{g}}$-Hilbert scheme is a pair of two diagonal matrices which for ${\mathfrak}{so}_{2n+1}$ are of the form $\operatorname{diag}(x_1,...,x_n,0,-x_n,...,-x_1)$ and $\operatorname{diag}(y_1,...,y_n,0,-y_n,...,-y_1)$. So they can be thought of as $2n+1$ points in ${\mathbb{C}}^2$ with one point being the origin and the other points being symmetric with respect to the origin. This set is invariant under the map $-\operatorname{id}$, so is its defining ideal.
The next type, $C_n$, is quite similar to $B_n$.
Case $C_n$
----------
Let ${\mathfrak{g}}={\mathfrak}{sp}_{2n}$. We use the symplectic structure $\omega=\sum_i e_i\wedge e_{n+i}$ of ${\mathbb{C}}^{2n}$ to represent ${\mathfrak{g}}$. So a matrix $$\left(\begin{array}{c|c}
A & B \\
\hline
C & D
\end{array}\right)$$ is in ${\mathfrak{g}}$ iff $D=-A^T$ and $B$ and $C$ are symmetric matrices.
Fix the principal nilpotent by
$$f=\left(\begin{array}{cccc|cccc}
&&&&&&& \\
1&&&&&&&\\
&\ddots&&&&&&\\
&&1&&&&&\\ \hline
&&&&&-1&&\\
&&&&&&\ddots&\\
&&&&&&&-1\\
&&&1&&&&
\end{array}\right)$$
This element is cyclic, so its centralizer is given by odd polynomials: $Z(f)=\{\mu_2f+\mu_4f^3+...+\mu_{2n}f^{2n-1}\}$. As for $B_n$ we call the $\mu_{2i}$ higher Beltrami coefficients.
A principal slice is given by
$$\left(\begin{array}{cccc|cccc}
&&&& t_{2n}&&& \\
1&&&&&t_{2n-2}&&\\
&\ddots&&&&&\ddots&\\
&&1&&&&&t_2\\ \hline
&&&&&-1&&\\
&&&&&&\ddots&\\
&&&&&&&-1\\
&&&1&&&&
\end{array}\right)$$
Let $A$ be an element of this form. Its characteristic polynomial is given by $x^{2n}-t_2x^{2n-2}+t_4x^{2n-4}\pm...+(-1)^nt_{2n}$. By changing signs in the $t_{2i}$ we can omit the minus signs in the characteristic polynomial.
The matrix $A$ is still cyclic so a representative of $\operatorname{Hilb}^{reg}({\mathfrak}{sp}_{2n})$ is given by $(A,B=Q(A))$ where $Q$ is an odd polynomial of degree at most $2n-1$.
The idealic map is given by $$I(A,B)= \langle x^{2n}+t_2x^{2n-2}+t_4x^{2n-4}+...+t_{2n}, -y+\mu_2x+\mu_4x^3+...+\mu_{2n}x^{2n-1} \rangle.$$
As for $B_n$, this ideal is invariant under $-\operatorname{id}$ which comes from the fact that two diagonal matrices in ${\mathfrak}{sp}_{2n}$ give $2n$ points in ${\mathbb{C}}^2$ which are symmetric with respect to the origin.
The last classical type, $D_n$, has some surprises.
Case $D_n$ {#Dn}
----------
Let ${\mathfrak{g}}={\mathfrak}{so}_{2n}$. We use the same representation as for $B_n$.
Fix the following principal nilpotent element: $$f=\begin{pmatrix}
&&&&&&& \\
1&&&&&&& \\
&\ddots&&&&&& \\
&&1&&&&& \\
&&1&0&&&& \\
&&&-1&-1&&& \\
&&&&&\ddots && \\
&&&&&&-1&\\
\end{pmatrix}$$
This elements is not cyclic, since $f^{2n-1}=0$. A direct computation shows that $Z(f)=\{\mu_2f+\mu_4f^3+...+\mu_{2n-2}f^{2n-3}\} \cup \{\sigma_n S\}$ where $S$ is the matrix
$$\label{matrixS}
S=\left(\begin{array}{@{}ccc|ccc@{}}
&&&&& \\
&&&&&\\
1&&&&& \\ \hline
-1&&&&&\\
&&&&&\\
&&1&-1&&
\end{array}\right)$$
We can give an intrinsic definition of the matrix $S$: let $R$ be a root system and $v_{\alpha}$ be a root vector in ${\mathfrak{g}}$ for the root $\alpha \in R$. Choose a base $\alpha_1, ..., \alpha_n$ of $R$ (the simple roots) such that $\alpha_{n-1}$ and $\alpha_n$ correspond to the two non-adjacent vertices in the Dynkin diagram of $D_n$ (see figure \[Dynkin\]). We can choose $f$ to be $\sum_i v_{\alpha_i}$. The matrix $S$ is then given by $$S=v_{\alpha_1+...+\alpha_{n-1}} \pm v_{\alpha_1+...+\alpha_{n-2}+\alpha_n}$$ where the sign depends on the choice of the root vectors.
{height="2cm"}
\[Dynkin\]
A representative of $\operatorname{Hilb}^{reg}_0({\mathfrak}{so}_{2n})$ is given by $(A=f,B=Q(f)+\sigma_nS)$ where $Q$ is an odd polynomial of degree at most $2n-3$. Such a pair is cyclic iff $\sigma_n \neq 0$.
Let us compute the ideal in the cyclic case. One checks easily that $fS=Sf$ and that $S^2=2f^{2n-2}$. Hence for $B=\mu_2f+...+\mu_{2n-2}f^{2n-2}+\sigma_nS$, we get $AB=fB=\mu_2f^2+...+\mu_{2n-2}f^{2n-2}$ and $B^2=(\mu_2f+...+\mu_{2n-2}f^{2n-3})^2+2\sigma_n^2f^{2n-2}$. Hence, the idealic map is given by $$I(A,B)=\langle x^{2n-1}, xy=\mu_2x^2+\mu_4x^4+...+\mu_{2n-2}x^{2n-2}, y^2=\nu_2x^2+\nu_4x^4+...+\nu_{2n-2}x^{2n-2} \rangle$$ where $\nu_{2k}=\sum_{i=1}^{k}\mu_{2i}\mu_{2k+2-2i}$ for $k=1,...,n-2$ and $\nu_{2n-2}=2\sigma_n^2+\sum_{i=1}^{n-1}\mu_{2i}\mu_{2n-2i}$. So we see that $(\mu_2, \mu_4, ..., \mu_{2n-2}, \nu_{2n-2})$ is a set of independent variables which we call higher Beltrami differentials for $D_n$. We will also call $\sigma_n$ a higher Beltrami differential. If $\sigma_n=0$, we define the idealic map to be the continuous extension of the above ideal which is still of the same form.
We have seen in example \[idealnotcont\] that inside $\operatorname{Hilb}^{cycl}({\mathfrak{g}})$ there is no well-defined continuous extension of the idealic map. But inside the zero-fiber, the limit is unique.
The Hilbert scheme is covered by charts indexed by partitions (see [@Haiman]). The chart in which $I$ is written corresponds to the partition $2n=(2n-1)+1$ which we write also $[2n-1,1]$. In fact, this is the highest partition of $2n$ of type $D_n$ (see [@Coll], chapter 5 for special types of partitions). A principal slice is given by
$$\left(\begin{array}{@{}cccc|cccc@{}}
&&&\tau_n&-\tau_n&&t_{2n-2}& \\
1&&&&&\udots&&-t_{2n-2}\\
&\ddots&&t_2&t_2& &\udots&\\
&&1&&&-t_2&&\tau_n\\ \hline
&&1&0&&-t_2&&-\tau_n\\
&&&-1&-1&&&\\
&&&&&\ddots&&\\
&&&&&&-1&
\end{array}\right).$$
Notice that the matrix for $\tau_n$ is $S^T$. Let $A$ be a matrix of this type. Its characteristic polynomial is given by $$\chi(A)=x^{2n}-4t_2x^{2n-2}+4t_4x^{2n-4}\pm... +(-1)^{n-1}\times 4t_{2n-2}x^2+(-1)^n\tau_n^2.$$ By changing signs and factors in $t_{2i}$ and $\tau_n$, we can omit signs and the factor 4 in the characteristic polynomial.
One can compute that the minimal polynomial of $A$ is equal to the characteristic polynomial iff $\tau_n\neq 0$. So $A$ is cyclic iff $\tau_n\neq 0$ (by proposition \[regularsln\]). In that case, the centralizer consists of all odd polynomials in $A$ of degree at most $2n-1$. If $\tau_n=0$, the centralizer is given by $$Z(A)=\{\mu_2A+\mu_4A^3+...+\mu_{2n-2}A^{2n-3}\} \cup \{\sigma_n S_t\}$$ where the matrix $S_t$ is given by $S_t= S+t_{2n-2}S^T$. The minimal polynomial is given by $\chi(x)/x$ (which is a polynomial since $\tau_n=0$).
The pair $(A,B)$ is cyclic iff either $\tau_n \neq 0$ or $\tau_n=0$ and $\sigma_n \neq 0$. In the first case, the idealic map is given by $$I=\langle x^{2n}+t_2x^{2n-2}+t_4x^{2n-4}+...+t_{2n-2}x^2+\tau_n^2, -y+\mu_2x+\mu_4x^3+...+\mu_{2n}x^{2n-1}\rangle.$$ In the second case, we need three generators for the ideal, like for the zero-fiber. We can compute that $$\begin{aligned}
I(A,B)= \; \langle x^{2n-1}=& u_2x+u_4x^3+...+u_{2n-2}x^{2n-3}+uy,\\
xy =& \; v_0+v_2x^2+...+v_{2n-2}x^{2n-2}, \\
y^2 =& \; w_0+w_2x^2+...+w_{2n-2}x^{2n-2}\rangle\end{aligned}$$ where the coordinates can be chosen to be $(u_2,u_4,...,u_{2n-2},u,v_2,...,v_{2n-2},w_{2n-2})$, i.e. all the other variables are functions of these. For a unified way to get coordinates in Hilbert schemes, see subsection \[spectralcurve\] or directly Haiman’s paper [@Haiman].
The second ideal is in the chart corresponding to the partition $[2n-1,1]$ whereas the first corresponds to the trivial partition $[2n]$. If $u \neq 0$ we can write the second ideal in the first chart, i.e. perform a coordinate change in the Hilbert scheme. The link between the coordinates is given by $$\left \{ \begin{array}{cl}
\tau_n^2 = uv_0 \\
\mu_{2n}=\frac{1}{u} \\
\mu_{2k} = -\frac{u_{2k}}{u} &\text{ for } 1\leq k < n\\
t_{2k} = u_{2n-2k}+uv_{2n-2k} & \text{ for } 1\leq k \leq n-1
\end{array}\right.$$
A regular pair $[(A,B)]$ which is not cyclic has both $\tau_n$ and $\sigma_n$ equal to 0. In that case, we define the idealic map $I(A,B)$ to be the limit of $I(A,B+tS_t)$ for $t\in{\mathbb{C}}$ goes to 0. So we stay in a chart associated to the partition $[2n-1,1]$.
Notice that the map from $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ to the space of ideals $I_{{\mathfrak{g}}}({\mathbb{C}}^2)$ is not injective, since for $\tau_n$ and $-\tau_n$ we get the same ideal. Even in the zero-fiber the map is not injective, since $\sigma_n$ and $-\sigma_n$ give the same ideal. In addition, the map is not surjective neither. Indeed the ideal $I=\langle x^5-y, xy,y^2 \rangle \in I_{{\mathfrak{g}}}({\mathbb{C}}^2)$ is not in the image since with the notations above we have $v_0=0$ and $u\neq 0$. Changing the chart, we can compute that $\tau_n^2=uv_0 = 0$. But for a matrix in $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ with $\tau_n=0$ we get $u=0$.
In the usual Hilbert scheme, there is only one cell of maximal dimension. Comparing type $C_n$ and type $D_n$, we see that the zero-fiber of $$\{I \text{ ideal of }{\mathbb{C}}[x,y] \mid \operatorname{codim}I=2n, I \text{ invariant under } -\operatorname{id}\}$$ has two components of maximal dimension, those corresponding to the zero-fibers $\operatorname{Hilb}^{reg}_0({\mathfrak}{sp}_{2n})$ and $\operatorname{Hilb}^{reg}_0({\mathfrak}{so}_{2n})$.
We notice the following analogue to Higgs bundles: the pair $[(f,0)] \in \operatorname{Hilb}({\mathfrak}{so}_{2n})$ corresponds to the Higgs field given by $\Phi = f$ on the bundle $V=K^2\oplus K\oplus K^0 \oplus K^{-2} \oplus K^{-1}\oplus K^0$. This Higgs bundle $(V,\Phi)$ is not stable, only polystable. This could explain why the idealic map can not be continuously extended to $[(f,0)]$, but the link between Higgs bundles and higher complex structures remains unclear, see also the perspective given in section \[section4\].
Topology of ${\mathfrak{g}}$-Hilbert schemes {#topology}
--------------------------------------------
It is clear that $\operatorname{Hilb}({\mathfrak{g}})$ is a topological space, as a quotient of a subset of ${\mathfrak{g}}^2$. In this section, we explore this topology of $\operatorname{Hilb}({\mathfrak{g}})$, especially for ${\mathfrak{g}}={\mathfrak}{sl}_n$. We then formulate some conjectures on its general structure.
For ${\mathfrak{g}}={\mathfrak}{sl}_2$, every non-zero element $A\in {\mathfrak{g}}$ is regular and cyclic. Since the centralizer of the pair $(0,0)$ is all of ${\mathfrak}{sl}_2$, this pair is not in $\operatorname{Hilb}({\mathfrak}{sl}_2)$. Thus we have $\operatorname{Hilb}({\mathfrak}{sl}_2) = \operatorname{Hilb}^{cycl}({\mathfrak}{sl}_2) = \operatorname{Hilb}^2_{red}({\mathbb{C}}^2)$ which is a smooth projective variety.
For ${\mathfrak{g}}={\mathfrak}{sl}_3$, a detailed analysis, putting $A$ into Jordan normal form, shows that $(A,B)$ has minimal centralizer and is not cyclic iff it is conjugated to a pair $P_1(b):=\left(\left(\begin{smallmatrix} 0&1&0 \\ 0&0&0 \\ 0&0&0\end{smallmatrix}\right), \left(\begin{smallmatrix} 0&b&1 \\ 0&0&0 \\ 0&0&0\end{smallmatrix}\right)\right)$. So $$\operatorname{Hilb}({\mathfrak}{sl}_3) = \operatorname{Hilb}^3_{red}({\mathbb{C}}^2) \cup \{P_1(b) \mid b\in{\mathbb{C}}\}.$$ At first sight, the topology seems to be a smooth variety (the reduced Hilbert scheme) and a complex line. But a closer look shows that each point of the extra line is infinitesimally close to a point in the variety, meaning that these two points cannot be separated by open sets, infringing the Hausdorff property. The pair $P_1(b)$ is infinitesimally close to $P_2(b):=\left(\left(\begin{smallmatrix} 0&1&0 \\ 0&0&0 \\ 0&0&0\end{smallmatrix}\right),\left( \begin{smallmatrix} 0&b&0 \\ 0&0&0 \\ 0&1&0\end{smallmatrix}\right)\right)$. Indeed any neighborhood of the first pair $P_1(b)$ contains $\left(\left(\begin{smallmatrix} 0&1&0 \\ 0&0&0 \\ 0&0&0\end{smallmatrix}\right), \left(\begin{smallmatrix} 0&b&1 \\ 0&0&0 \\ 0&s&0\end{smallmatrix}\right)\right)$ for some small $s\in {\mathbb{C}}$ which is conjugated to $\left(\left(\begin{smallmatrix} 0&1&0 \\ 0&0&0 \\ 0&0&0\end{smallmatrix}\right),\left( \begin{smallmatrix} 0&b&s \\ 0&0&0 \\ 0&1&0\end{smallmatrix}\right)\right)$ which lies in a neighborhood of the second pair$P_2(b)$. Since the idealic map is continuous and for ${\mathfrak}{sl}_n$ injective on the cyclic part, there cannot be another point of the cyclic part which is infinitesimally close to the first pair $P_1(b)$. Finally, two elements of the extra line can be separated by open sets. Hence, the space $\operatorname{Hilb}({\mathfrak}{sl}_3)$ is obtained from a smooth variety by adding “double points” (here in the sens of infinitesimally close points) along a complex line.
Since the idealic map is injective on the cyclic part $\operatorname{Hilb}^{cycl}({\mathfrak}{sl}_n)$, the same analysis holds for ${\mathfrak}{sl}_n$, i.e. $\operatorname{Hilb}({\mathfrak}{sl}_n)$ is obtained from a smooth variety (the reduced Hilbert scheme) by adding double points.
There should exist a procedure, like a GIT quotient, giving a modified ${\mathfrak{g}}$-Hilbert scheme which is a Hausdorff space. The GIT quotient does not apply here since $\{(A,B)\in {\mathfrak{g}}^2 \mid [A,B]=0, \dim Z(A,B)=\operatorname{rk}{\mathfrak{g}}\}$ is not a closed variety. In the language of GIT quotients, the pairs $P_1$ and $P_2$ above are both semistable, but there is no polystable element in their closure.
To give a feeling on what happens, consider the action of ${\mathbb{R}}_{>0}$ on ${\mathbb{R}}^2 \backslash \{(0,0)\}$ given by $\lambda.(x_1, x_2)=(\lambda x_1, \lambda^{-1} x_2)$. The orbits are drawn in figure \[nonhaus\]. The quotient space is a set of two lines $L_1$ and $L_2$ with origins $O_1$ and $O_2$ together with two extra points $O_3$ and $O_4$ such that the pairs $(O_1, O_3), (O_1, O_4), (O_2, O_3)$ and $(O_2, O_4)$ are infinitesimally close points (the four points $O_i$ correspond to the four half-axis). In the figure, the dashed lines indicate infinitesimally close points. From the GIT perspective, all points are semistable (take the constant function 1), the four half-axis are semistable and all other orbits are stable. The orbits of the half-axis are closed in ${\mathbb{R}}^2 \backslash \{(0,0)\}$ so they should be polystable, but in the quotient the points are still infinitesimally close.
{height="4cm"}
\[nonhaus\]
We conjecture the following:
There is a generalized GIT quotient procedure identifying infinitesimally close points in $\operatorname{Hilb}({\mathfrak{g}})$, giving a modified ${\mathfrak{g}}$-Hilbert scheme which is Hausdorff.
In particular one should find the reduced Hilbert scheme for ${\mathfrak{g}}={\mathfrak}{sl}_n$. See also conjecture \[conj1\] for a modified ${\mathfrak{g}}$-Hilbert scheme as a resolution of ${\mathfrak{h}}^2/W$.
Assume a smooth version of the ${\mathfrak{g}}$-Hilbert scheme exists. In the ${\mathfrak}{sl}_n$-case the reduced Hilbert scheme is covered by charts parametrized by partitions of $n$, which also parametrizes nilpotent orbits of ${\mathfrak}{sl}_n$. For ${\mathfrak{g}}$ of classical type, the nilpotent orbits are parametrized by special partitions (see [@Coll], chapter 5). In general, we conjecture the following for the zero-fiber of the ${\mathfrak{g}}$-Hilbert scheme:
The smooth version of $\operatorname{Hilb}_0({\mathfrak{g}})$ is covered by charts parametrized by nilpotent orbits and all these charts are necessary to cover $\operatorname{Hilb}_0({\mathfrak{g}})$. In particular for classical ${\mathfrak{g}}$, we conjecture that the modified version of $\operatorname{Hilb}_0({\mathfrak{g}})$ is isomorphic to the space of ideals of ${\mathbb{C}}[x,y]$ which are of codimension $m$, $W$-invariant, supported at 0 and which lie in a chart associated to a partition of type ${\mathfrak{g}}$.
In particular, for every nilpotent $A\in {\mathfrak{g}}$, there has to be an element in $\operatorname{Hilb}({\mathfrak{g}})$ containing the conjugacy class of $A$. More precisely, we conjecture:
Let ${\mathfrak{g}}$ be of rank at least 3. For a nilpotent element $A\in {\mathfrak{g}}$, there is $B\in Z(A)$ nilpotent such that $\dim Z(A,B) = \operatorname{rk}{\mathfrak{g}}$, i.e. $[(A,B)] \in \operatorname{Hilb}_0({\mathfrak{g}})$. We conjecture that this should be true for a generic element $B \in Z(A)$.
For ${\mathfrak}{sl}_n$ the conjecture is true: we can associate to a nilpotent element $A$ a partition $\nu$. To the transpose partition $\nu^T$ (using the transpose of the Young diagram) correspond a nilpotent element $B$ which satisfies the requirements since $(A,B)$ is cyclic. An equivalent way is to use example \[Young\] to produce $B$. For ${\mathfrak{g}}={\mathfrak}{sp}_{4}$ of type $C_2$, there is the following counterexample. That is why we formulate the conjecture only for Lie algebras of rank at least 3. Take the nilpotent element $$A = \left(\begin{array}{c|c}
0 & \operatorname{id}\\
\hline
0 & 0
\end{array}\right).$$ Its centralizer is given by $$Z(A) = \left(\begin{array}{cc|cc}
0&b & x&y \\
-b&0 &y&z \\\hline
0 & 0 &0 & b\\
0&0&-b&0
\end{array}\right).$$ An element $B$ of the centralizer is nilpotent iff $b= 0$. In that case the common centralizer $Z(A,B)$ is at least of dimension 3, so $[(A,B)]$ is not in $\operatorname{Hilb}_0({\mathfrak}{sp}_4)$.
In general, we cannot hope to find $B\in Z(A)$ such that $(A,B)$ is cyclic. For example take ${\mathfrak{g}}={\mathfrak}{sp}_{16}$ and $A$ a nilpotent element corresponding to the partition $[7,5,3,1]$ of 16. If there is $B\in Z(A)$ nilpotent and such that $(A,B)$ is cyclic, there would be an ideal $I$ of codimension 16 whose associated matrices are $A$ and $B$ (see example \[Young\]). Using $A$, we see that $I$ has to be of the form $$I=\langle x^4, x^3y, x^2y^3, xy^5, y^7=Q(x,y) \rangle$$ where $Q$ is a polynomial with monomial terms in the Young diagram $D$. A partition of type $C_n$ has all odd parts with even multiplicity and one can check that for all choices of the polynomial $Q$, the ideal $I$ is never in a chart with all odd parts with even multiplicity.
${\mathfrak{g}}$-complex structures {#section2}
===================================
Using the ${\mathfrak{g}}$-Hilbert scheme we are able to construct a new geometric structure on a smooth surface, generalizing complex structures and higher complex structures. The construction and methods are inspired by those used for higher complex structures in [@FockThomas]. We recall the ideas of constructing higher complex structures before defining the ${\mathfrak{g}}$-complex structure.
Complex and higher complex structures
-------------------------------------
A complex structure on a surface $\Sigma$ is completely encoded in the *Beltrami differential*.
This goes as follows: For surfaces, a complex structure is equivalent to an almost complex structure, i.e. an endomorphism $J(z)$ in $T^*_z\Sigma$ such that $J^2=-\operatorname{id}$ and varying smoothly with $z\in\Sigma$ ($J$ imitates the multiplication by $i$). We can diagonalize $J$ by complexifying the cotangent bundle. We get a decomposition into eigenspaces $T^{*{\mathbb{C}}}\Sigma = T^{*(1,0)}\Sigma \oplus T^{*(0,1)}\Sigma$. In addition $T^{*(1,0)}\Sigma$ is the complex conjugate of $T^{*(0,1)}\Sigma$, so one determines the other. Hence, the complex structure is completely encoded in a direction in each complexified cotangent space, i.e. in a section $s$ of $\mathbb{P}(T^{*{\mathbb{C}}}\Sigma)$ which is nowhere real (meaning $s$ and $\bar{s}$ are linear independent). The projectivization can also be obtained by the zero-fiber of the punctual Hilbert scheme of length 2: $\operatorname{Hilb}^2_0({\mathbb{C}}^2) \cong \mathbb{P}({\mathbb{C}}^2)$. In coordinates, we can write $T^{*(0,1)}_z\Sigma = \operatorname{Span}(\bar{p}-\mu_2(z)p)$ where $p$ and $\bar{p}$ are linear coordinates on $T^{*{\mathbb{C}}}\Sigma$. The coefficient $\mu_2(z)$ is the Beltrami differential. The condition that the section $s$ is nowhere real translates to $\mu_2(z)\bar{\mu}_2(z) \neq 1$ for all $z \in \Sigma$.
Generalizing this idea, we defined in [@FockThomas] the higher complex structure as a section $I$ of $\operatorname{Hilb}^n_0(T^{*{\mathbb{C}}}\Sigma)$ satisfying $I(z)\oplus \bar{I}(z) = \langle p,\bar{p}\rangle$ at every point $z\in \Sigma$. Here $p$ and $\bar{p}$ are linear coordinates on $T^{*{\mathbb{C}}}\Sigma$. The condition on $I$ generalizes the condition above of a nowhere real section. We call it the *reality constraint*.
We use exclusively the idealic viewpoint of the punctual Hilbert scheme in this definition. Since the ${\mathfrak{g}}$-Hilbert scheme uses the matrix viewpoint, we have to rewrite the definition of higher complex structure in that picture. So we replace the ideal $I(z)$ by a conjugacy class of commuting matrices $A(z)$ and $B(z)$. We can put them together in a gauge class of a ${\mathfrak}{sl}_n$-valued 1-form $\Phi(z)=A(a) dz+ B(z) d\bar{z}$. The commutativity of $A$ and $B$ translates to the fact that $\Phi$ satisfies $\Phi \wedge \Phi = 0$.
It is not surprising to use 1-forms since a generic point of the Hilbert scheme gives $n$ distinct points in each fiber $T^{*{\mathbb{C}}}_z\Sigma$ which can be put together to $n$ sections of $T^{*{\mathbb{C}}}\Sigma$, i.e. a $n$-tuple of complex 1-forms. Going to the zero-fiber of the Hilbert scheme means that all these 1-forms are collapsed to the zero-section $\Sigma \subset T^{*{\mathbb{C}}}\Sigma$.
Definition
----------
We are now ready to give the definition of a ${\mathfrak{g}}$-complex structure, but one difficulty stays: we have to incorporate the reality constraint in the matrix viewpoint. Recall from \[mu2\] equation the map $\mu_2: \operatorname{Hilb}^{reg}_0({\mathfrak{g}}) \rightarrow {\mathbb{C}}$ associating to $[(A,B)]$ the unique $\mu_2\in {\mathbb{C}}$ such that $B-\mu_2A$ is irregular.
\[def-g-complex-1\] A **${\mathfrak{g}}$-complex structure** is a gauge class of elements $$A(z) dz+ B(z) d\bar{z} \in \Omega^1(\Sigma, {\mathfrak{g}}) = \Omega^1(\Sigma,{\mathbb{C}})\otimes {\mathfrak{g}}$$ such that $$[(A(z), B(z))] \in \operatorname{Hilb}^{reg}_0({\mathfrak{g}})$$ and $\mu_2(z)\bar{\mu}_2(z) \neq 1$ for all $z\in \Sigma$.
Notice that for complex structures, the map $\mu_2(z)$ is nothing but the Beltrami differential. So our reality constraint coincides with the one for complex structures. In particular, for ${\mathfrak{g}}={\mathfrak}{sl}_2$, we get a usual complex structure. In the general case, we have the following:
\[inducedcomplex\] A ${\mathfrak{g}}$-complex structure induces a complex structure on $\Sigma$.
Recall from \[mu2\] equation the map $\mu:\operatorname{Hilb}^{reg}_0({\mathfrak{g}}) \rightarrow \operatorname{Hilb}_0({\mathfrak}{sl}_2)$ given by $\mu([(A,B)])=[(e,\mu_2 e)]$ or $[(\mu_2 e, e)]$ depending on whether $A$ or $B$ is regular. Since a ${\mathfrak}{sl}_2$-complex structure is a complex structure, the map $\mu$ induces a map from ${\mathfrak{g}}$-complex structures to complex structures.
To define the map $\mu$ in \[mu2\], we really need ${\mathfrak{g}}$ to be simple. Thus, we only get a unique complex structure out of a ${\mathfrak{g}}$-complex structure for ${\mathfrak{g}}$ simple.
In the definition of a higher complex structure in [@FockThomas], we use the zero-fiber $\operatorname{Hilb}^n_0({\mathbb{C}}^2)$, without imposing to be in the regular part. The fact that we actually are in the regular part follows from the reality constraint $I\oplus \bar{I}=\langle p,\bar{p} \rangle$. The same can be obtained for ${\mathfrak{g}}$ of classical type, where we can reformulate the definition of ${\mathfrak{g}}$-complex structures in a nicer way using the idealic map.
Idealic viewpoint
-----------------
Recall the space of ideals $I_{{\mathfrak{g}}}({\mathbb{C}}^2)$ constructed in \[idealic\]. Denote by $I_{{\mathfrak{g}},0}({\mathbb{C}}^2)$ the set of those ideals of $I_{{\mathfrak{g}}}({\mathbb{C}}^2)$ which are supported on the origin (the zero-fiber). We can rewrite the definition of a ${\mathfrak{g}}$-complex structure in the following way:
\[def-g-complex\] For classical ${\mathfrak{g}}$, a **${\mathfrak{g}}$-complex structure** is a section $I$ of $I_{{\mathfrak{g}},0}(T^{*{\mathbb{C}}}\Sigma)$ such that $$I(z) \oplus \bar{I}(z)= \left \{ \begin{array}{cl}
\langle p, \bar{p} \rangle & \text{ if } {\mathfrak{g}}\text{ of type } A_n, B_n, C_n \\
\langle p, \bar{p} \rangle^2 &\text{ if } {\mathfrak{g}}\text{ of type } D_n.
\end{array} \right.$$
Notice that the condition on the ideals does not depend on coordinates since $\langle p, \bar{p}\rangle$ is the maximal ideal associated to the origin.
We prove the equivalence of both definitions. For that recall that to an ideal $I$ one can associate a class of commuting matrices $[(A,B)]$ (see \[bijhilbert\]).
For classical ${\mathfrak{g}}$, the condition on $I\oplus \bar{I}$ given in definition \[def-g-complex\] is equivalent to $[(A(z),B(z))]$ being in the regular part $\operatorname{Hilb}^{reg}_0({\mathfrak{g}})$ and having $\mu_2\bar{\mu}_2\neq 1$, i.e. the condition in definition \[def-g-complex-1\].
The backwards direction is a direct computation using the preferred representatives for $\operatorname{Hilb}^{reg}_0({\mathfrak{g}})$ from proposition \[paramit\]. So we concentrate on the direct implication.
The case ${\mathfrak{g}}$ of type $A_n$ has been treated in [@FockThomas], appendix 5.1. The idea of the proof is similar to the case $D_n$ below.
For ${\mathfrak{g}}$ of type $B_n$ the standard representation gives ${\mathfrak}{so}_{2n+1} \hookrightarrow {\mathfrak}{sl}_{2n+1}$. By virtue of the case $A_n$, we know that $I\oplus \bar{I} =\langle p, \bar{p} \rangle$ implies $\mu_2\bar{\mu}_2\neq 1$ and $(A,B)$ regular for ${\mathfrak}{sl}_{2n+1}$, i.e. $$I(A,B)=\langle p^{2n+1}, -\bar{p}+\mu_2p+\mu_3p^2+...+\mu_{2n}p^{2n} \rangle.$$ Since we know that in case $B_n$, the ideal $I$ is invariant under the map $-\operatorname{id}$, we get $\mu_{2k+1}=0$ for all $k=1,...,n-1$. So $I$ corresponds to a pair $(f,Q(f))$ for $Q$ an odd polynomial of degree at most $2n-1$, which is precisely a representative of $\operatorname{Hilb}^{reg}_0({\mathfrak}{so}_{2n+1})$ (see subsection \[Bn\]).
This case is exactly analogous to $B_n$ via the injection ${\mathfrak}{sp}_{2n}\hookrightarrow {\mathfrak}{sl}_{2n}$.
We imitate the strategy of the proof for case $A_n$ in [@FockThomas] appendix 5.1 with only difference that we have to go further in the analysis, needing some computations. The main argument is an iteration process which always ends since $p^k\bar{p}^l = 0 \mod I$ for $k+l \geq 2n$.
Put $I_1 = (I \mod \langle p, \bar{p} \rangle^2)$, i.e. the set of all terms of degree at most 1 appearing in $I$. If $I_1$ is of dimension 2, then $I=\langle p, \bar{p}\rangle$ since both $p$ and $\bar{p}$ can be expressed by higher terms which by iteration become 0. If $I_1$ is of dimension 1, then we have a relation of the form $\bar{p}=\mu_2p+p^2R(p,\bar{p})$ where $R$ is a polynomial, which gives $\bar{p}$ as a polynomial in $p$ by iteration. We can then explicitly check that $I\oplus \bar{I}$ is either $\langle p, \bar{p} \rangle$ or $\langle p=\bar{p}, p\bar{p}, p^2\rangle$. Hence $I_1=\{0\}.$
Put $I_2 = (I \mod \langle p, \bar{p}\rangle^3)$. We have $I_2\oplus \bar{I}_2 = (I\oplus \bar{I})_2 = \langle p^2, p\bar{p}, \bar{p}^2\rangle$ by assumption on $I$. If $I_2$ is of dimension 3, then all of $p^2, p\bar{p}$ and $\bar{p}^2$ can be expressed by higher terms. By iteration, we get $I=\langle p^2, p\bar{p}, \bar{p}^2\rangle$ which is not of type $D_n$. If $\dim I_2 \leq 1$, then we also have $\dim \bar{I}_2 \leq 1$, so $2\geq \dim I_2+\dim \bar{I}_2 = \dim \langle p^2,p\bar{p}, \bar{p}^2 \rangle_2 = 3$, a contradiction. Hence $\dim I_2=2.$
There is a term containing $p\bar{p}$ in $I_2$ since if not, no such term would neither exist in $\bar{I}_2$, so neither in $I_2\oplus\bar{I}_2 = \langle p^2, p\bar{p}, \bar{p}^2 \rangle$, a contradiction. Without loss of generality, we can assume that there is another term containing $\bar{p}^2$ (if not change the role of $I$ and $\bar{I}$).
So there exist $\alpha, \beta, \gamma, \delta \in {\mathbb{C}}$ such that $$\left \{ \begin{array}{cl}
\bar{p}^2 = \alpha p^2+\beta p\bar{p} &\mod I_2\\
p\bar{p} = \gamma p^2 +\delta \bar{p}^2 &\mod I_2
\end{array}\right.$$
If $\beta\gamma \neq 1$, we can simplify by substitution one into the other to $$\left \{ \begin{array}{cl}
\bar{p}^2 = \alpha' p^2 &\mod I_2 \\
p\bar{p} = \gamma' p^2 &\mod I_2
\end{array}\right.$$ If $\beta\gamma = 1$, we have $p^2\in I_2$, so $p\bar{p}=\delta \bar{p}^2 \mod I_2$, so changing $I$ to $\bar{I}$ we are in the previous situation.
Iterating the substitution process we get that $\bar{p}^2$ and $p\bar{p}$ are polynomials in $p$. Using the invariance of $I$ under $-\operatorname{id}$, we see that these are polynomials in $p^2$, i.e. even polynomials. So the most generic ideal is given by $$I=\langle p^{2n-1}, p\bar{p}=\mu_2p^2+\mu_4p^4+...+\mu_{2n-2}p^{2n-2}, \bar{p}^2=\nu_2p^2+\nu_4p^4+...+\nu_{2n-2}p^{2n-2}\rangle$$ which corresponds to a regular element of $\operatorname{Hilb}^{reg}_0({\mathfrak}{so}_{2n})$. One checks that $I\oplus \bar{I}$ with $I$ of the form above equals $\langle p,\bar{p}\rangle ^2$ iff $\mu_2\bar{\mu_2} \neq 1$.
To end this section, we determine the geometric nature of the various higher Beltrami coefficients. Since $p$ and $\bar{p}$ are linear coordinates on $T^{*{\mathbb{C}}}\Sigma$, we can identify $p=\frac{\partial}{\partial z}= \partial$ and $\bar{p}=\frac{\partial}{\partial \bar{z}}=\bar{\partial}$. Denote by $K$ the canonical bundle, i.e. $K=T^{*(1,0)}\Sigma$, and by $\Gamma(B)$ the space of sections of a bundle $B$.
Analyzing the behavior under a coordinate change $z \mapsto w(z,\bar{z})$ analogous to the computation in [@FockThomas] section 3.1., we get $$\label{naturemu}\mu_i\in \Gamma(K^{1-i}\otimes \bar{K}) \text{ and } \nu_{2i} \in \Gamma(K^{-2i}\otimes \bar{K}^2).$$ Since $\sigma_n^2$ has the same nature as $\nu_{2n-2}$, we get $\sigma_n\in \Gamma(K^{1-n}\otimes \bar{K})$.
Moduli space {#section3}
============
In this section, we define the moduli space of ${\mathfrak{g}}$-complex structures and explore its properties. In the whole section ${\mathfrak{g}}$ is of classical type. We first have to define an equivalence relation on ${\mathfrak{g}}$-complex structures, which is accomplished by the notion of higher diffeomorphisms.
Higher diffeomorphisms
----------------------
In order to get a finite-dimensional moduli space, it is not sufficient to quotient by the diffeomorphisms of $\Sigma$ isotopic to the identity, as for Teichmüller space. The reason is that the ${\mathfrak{g}}$-complex structure is non-linear in the cotangent spaces $T^{*{\mathbb{C}}}_z\Sigma$. Diffeomorphisms act linearly on the cotangent space, so it cannot act much on ${\mathfrak{g}}$-complex structures.
For higher complex structures, in [@FockThomas] section 3.2 we defined higher diffeomorphisms to be Hamiltonian diffeomorphisms of $T^*\Sigma$ preserving the zero-section $\Sigma \subset T^*\Sigma$. This gives the higher diffeomorphisms for type $A_n$. We generalize this idea to other classical ${\mathfrak{g}}$. Recall the standard representation of ${\mathfrak{g}}$ on ${\mathbb{C}}^m$ (i.e. ${\mathfrak}{sl}_n \subset {\mathfrak}{gl}_n, {\mathfrak}{so}_n \subset {\mathfrak}{gl}_n$ and ${\mathfrak}{sp}_{2n} \subset {\mathfrak}{gl}_{2n}$).
As stated several times, one should think of a ${\mathfrak{g}}$-complex structure as a $m$-tuple of 1-forms with some symmetry, which collapses all to the zero-section. The space of higher diffeomorphisms which we are looking for has to preserve this symmetry. For example for ${\mathfrak}{sl}_n$, we have $n$ sections whose barycenter at every fiber is the origin, i.e. their sum gives the zero-section. That is why we have to impose that the Hamiltonian diffeomorphisms of $T^*\Sigma$ have to preserve the zero-section.
For ${\mathfrak{g}}$ of type $B_n, C_n$ or $D_n$, the set of $m$ points is symmetric with respect to the origin. Thus we define:
A **higher diffeomorphism** of type $B_n, C_n$ or $D_n$ is a Hamiltonian diffeomorphism of $T^*\Sigma$ invariant under the map $(z,p,\bar{p})\mapsto (z,-p,-\bar{p})$. We denote by $\operatorname{Symp}({\mathfrak{g}},\Sigma)$ the space of higher diffeomorphisms of type ${\mathfrak{g}}$.
In coordinates a Hamiltonian diffeomorphism is generated by a function $H(z,\bar{z},p,\bar{p})$ which can be Taylor developed to $\sum_{k,l} w_{k,l}(z,\bar{z})p^k\bar{p}^l$. The associated flow preserves the zero-section iff $w_{0,0}=0$. It is invariant under $-\operatorname{id}$ iff it has only odd terms, i.e. $w_{k,l}=0$ for all $k+l$ even.
Action on ${\mathfrak{g}}$-complex structures {#actiondiff}
---------------------------------------------
We can now analyze how higher diffeomorphisms act on ${\mathfrak{g}}$-complex structures.
Intuitively, Hamiltonian diffeomorphisms of $T^*\Sigma$ act on the space of 1-forms, so also on $m$-tuples of them. The invariance condition implies that the symmetry of the $m$ 1-forms is preserved. This action persists at the limit when the $m$-tuple of 1-forms is collapsed to the zero-section.
To compute the action, it is better to work in the idealic viewpoint. We imitate the steps from [@FockThomas] section 3.2.
Let $I$ be an ideal representing a ${\mathfrak{g}}$-complex structure. Write $I$ with generators $\langle f_1, ..., f_r \rangle$. Each $f_k$ can be considered as a function on $T^{*{\mathbb{C}}}\Sigma$, so its variation under a Hamiltonian $H$ is given by the Poisson bracket $\{H, f_k\}$. The tangent space at $I$ in the space of all ideals of codimension $m$ is the set of all ring homomorphisms from $I$ to $A/I$. Thus a Hamiltonian $H$ changes $I$ to $\langle f_1+\varepsilon\{H,f_1\} \mod I, ..., f_r+\varepsilon \{H,f_r\} \mod I \rangle$.
We restate a lemma from [@FockThomas] (lemma 4) which allows to simplify $H$:
\[simplification\] Let $I=\left\langle f_1, ..., f_r \right\rangle$ be an ideal of $\mathbb{C}[z,\bar{z},p,\bar{p}]$ such that $\{f_i, f_j\} = 0 \mod I$ for all $i$ and $j$. Then for all polynomials $H$ and all $k \in \{1,...,r\}$ we have $\{H, f_k\} \mod I = \{H \mod I, f_k\} \mod I$.
The only thing to show is that if we replace $H$ by $H+gf_l$ for some polynomial $g$ and some $l \in \{1,...,r\}$, the expression does not change. Indeed, $\{H+g f_l, f_k\}=\{H, f_k\}+g\{f_l,f_k\}+\{g,f_k\}f_l = \{H, f_k\} \mod I$ using the assumption.
The ideals of $\operatorname{Hilb}^{reg}_0({\mathfrak{g}})$ for ${\mathfrak{g}}$ classical satisfy the condition of the previous lemma.
For $A_n$, we have $I=\langle p^n, \bar{p}=\mu_2p+...+\mu_np^{n-1}=Q(p) \rangle$. We compute $\{p^n, -\bar{p}+Q(p)\} = np^{n-1}\partial Q = 0 \mod I$ since there is no constant term in $Q$.
The same argument holds for $B_n$ and $C_n$ since their ideals are special cases of the ideal of type $A_n$.
For $D_n$, the ideal $I$ is given by $$\begin{aligned}
\langle p^{2n-1}, p\bar{p} &= \mu_2p^2+\mu_4p^4+...+\mu_{2n-2}p^{2n-2}=Q(p)+\mu_{2n-2}p^{2n-2}, \\
\bar{p}^2 &= \nu_2p^2+\nu_4p^4+...+\nu_{2n-2}p^{2n-2}=R(p)+\nu_{2n-2}p^{2n-2}\rangle.\end{aligned}$$ As before the Poisson brackets with the first generator $p^{2n-1}$ vanishes modulo $I$ since $Q$ and $R$ have no constant terms. To compute the last Poisson bracket, define $\tilde{Q}=Q/p$. By the relations in $I$, we have $R=\tilde{Q}^2+p^{2n-2}\tilde{R}$ for some polynomial $\tilde{R}$ (see subsection \[Dn\]). Remark further that $\{a(z, \bar{z})p^k\bar{p}^l, b(z, \bar{z})p^{k'}\bar{p}^{l'}\}=0 \mod I$ whenever $k+l+k'+l' > n-1$ since any term of degree $n-1$ in $p$ and $\bar{p}$ is in $I$ and the Poisson bracket lowers this degree by 1. With all this, we compute $$\begin{aligned}
&\{-p\bar{p}+Q+\mu_{2n-2}p^{2n-2}, -\bar{p}^2+R+\nu_{2n-2}p^{2n-2}\} \\
=& \;\{-p\bar{p}+p\tilde{Q}+\mu_{2n-2}p^{2n-2}, -\bar{p}^2+\tilde{Q}^2+p^{2n-2}(\tilde{R}+\nu_{2n-2})\}\\
=& \;\{-p\bar{p}+p\tilde{Q}, -\bar{p}^2+\tilde{Q}^2\} & \text{ by degree argument} \\
=& \; 2\bar{\partial}\tilde{Q}(p\bar{p}-p\tilde{Q})-2\tilde{Q}\partial \tilde{Q}(\bar{p}-\tilde{Q}) \\
=& \; 2(p\bar{p}-Q)(\bar{\partial}\tilde{Q}-\frac{\tilde{Q}}{p}\partial \tilde{Q}) \\
=& \; 2\mu_{2n-2}p^{2n-2}(\bar{\partial}\tilde{Q}-\frac{\tilde{Q}}{p}\partial \tilde{Q}) &\mod I \\
=& \; 0 &\mod I \end{aligned}$$ where the last line comes from the fact that $p$ divides the polynomial $\bar{\partial}\tilde{Q}-\frac{\tilde{Q}}{p}\partial \tilde{Q}$.
As a consequence, when computing the action of a Hamiltonian $H$ on a ${\mathfrak{g}}$-complex structure, we can reduce it modulo $I$. In particular if $H \mod I = 0$, the higher diffeomorphism generated by $H$ does not act at all. For ${\mathfrak{g}}$ of type $A_n, B_n$ or $C_n$ we can reduce $H$ to a polynomial in $p$, and for $D_n$ we can reduce it to $H=w_{-}\bar{p}+\sum_{k=0}^{n-2} w_{2k+1}p^{2k+1}$.
Local theory
------------
Now, we can study the local theory of ${\mathfrak{g}}$-complex structures. Let $z_0$ be a point on $\Sigma$ and take a small chart around it which sends to the unit disk $\Delta$ in the complex plane (with $z_0$ send to the origin).
\[thm1\] For ${\mathfrak{g}}$ of type $A_n$, $B_n$ or $C_n$, the ${\mathfrak{g}}$-complex structure can be locally trivialized, i.e. there is a higher diffeomorphism of type ${\mathfrak{g}}$ which sends all higher Beltrami differentials to 0 for all small $z \in {\mathbb{C}}$.
For ${\mathfrak{g}}$ of type $D_n$, all ${\mathfrak{g}}$-complex structures with non-vanishing $\sigma_n$ on $\Delta$ are equivalent under higher diffeomorphisms. However, the zero locus of $\sigma_n$ on $\Delta$ is an invariant.
The proof for ${\mathfrak{g}}$ of type $A_n$ was done in [@FockThomas] appendix 5.2, using a method in the spirit of the proof of Darboux’s theorem in symplectic geometry.
If ${\mathfrak{g}}$ is of type $B_n$ or $C_n$, the standard representations realizes the ${\mathfrak{g}}$-complex structure as a substructure of type $A_n$. Since the last is trivializable, so is the ${\mathfrak{g}}$-complex structure in that case.
For ${\mathfrak{g}}$ of type $D_n$, we use the same method as for type $A_n$ by a Hamiltonian flow argument. We start with an ideal $I$ determined by higher Beltrami differentials $(\mu_2, \mu_4, ..., \mu_{2n-2}, \nu_{2n-2})$. The action on $\mu_{2i}$ is the same as for ${\mathfrak{g}}={\mathfrak}{sl}_{2n}$ so we can trivialize them using a Hamiltonian $H$ which is a polynomial in $p$. So we are left with $$I=\langle p^{2n-1}, p\bar{p}, -\bar{p}^2+\nu_{2n-2}p^{2n-2}\rangle.$$ We have seen at the end of subsection \[actiondiff\] that in the case $D_n$, any Hamiltonian can be reduced to $H=w_{-}\bar{p}+\sum_{k=0}^{n-2}w_{2k+1}p^{2k+1}$. The only part of this Hamiltonian acting on $\nu_{2n-2}$ is $H=w_{-}\bar{p}$, which also changes $\mu_{2n-2}$. So in order to assure that $\mu_{2n-2}$ stays zero, we use $$H=w_{-}\bar{p}+w_{2n-3}p^{2n-3}.$$
We compute the action of this Hamiltonian on the ideal $I$. For the second generator of $I$ we get: $$\{w_{-}\bar{p}+w_{2n-3}p^{2n-3}, -p\bar{p}\} = p^{2n-2}(\bar{\partial}w_{2n-3}+\partial w_{-}\nu_{2n-2}) \mod I.$$ For the third generator of $I$ we get $$\{w_{-}\bar{p}+w_{2n-3}p^{2n-3}, -\bar{p}^2+\nu_{2n-2}p^{2n-2}\} = p^{2n-2}(w_{-}\bar{\partial}\nu_{2n-2}+2\bar{\partial}w_{-}\nu_{2n-2}) \mod I.$$ Denote by $\mu_{2-2}^t$ and $\nu_{2n-2}^t$ the image of $\mu_{2n-2}$ and $\nu_{2n-2}$ under the flow generated by $H$ at time $t$. From the above computation we get $$\left \{\begin{array}{cl}
\frac{d}{dt} \mu_{2n-2}^t=& \bar{\partial}w_{2n-3}+\partial w_{-}\nu_{2n-2} \\
\frac{d}{dt} \nu_{2n-2}^t=& (w_{-}\bar{\partial}+2\bar{\partial}w_{-})\nu_{2n-2}
\end{array}\right.$$
Instead of keeping $\nu_{2n-2}$, we work with the higher Beltrami differential $\sigma_n$. Since all the $\mu_{2i}$ are zero in $I$, we have $\nu_{2n-2}=\sigma_n^2$. Therefore we get from the second equation above $\frac{d}{dt}(\sigma_n^2)=(w_{-}\bar{\partial}+2\bar{\partial}w_{-})(\sigma_n^2)$ which gives $$\label{varsigma}
\frac{d}{dt}\sigma_n^t = \bar{\partial}(w_{-}^t\sigma_n^t).$$
We wish to have $\frac{d}{dt} \mu_{2n-2}^t=0$ to stay with $\mu_{2n-2}=0$. For $\sigma_n$, we show that we can deform it to the constant function 1 on the unit disk, assuming $\sigma_n$ vanishes nowhere on $\Delta$. We choose the path $\sigma_n^t=(1-t)\sigma_n^0+t$ from the initial $\sigma_n^0$ to the constant function 1. If $\sigma_n^t=0$ for some $t$, we have to modify slightly the path. We get $\frac{d}{dt}\sigma_n^t=1-\sigma_n^0$.
Denote by $T$ the local inverse of the $\bar{\partial}$-operator, i.e. $\bar{\partial}(Tf)=f=T\bar{\partial}f$ for all $f\in L^2(\Delta)$. The operator $T$ is a pseudo-differential operator given by $$Tf(z)=\frac{1}{2\pi i}\int_{\mathbb{C}}\frac{f(\zeta)}{\zeta-z}d\zeta \wedge d\bar{\zeta}.$$
We can solve equation with $T$: $$w_{-}^t=\frac{1}{\sigma_n^t}T(1-\sigma_n^0).$$ Putting this solution into the equation for $\frac{d}{dt}\mu_{2n-2}^t$, we can solve for $w_{2n-3}$: $$w_{2n-3}^t=-T(\partial w_{-}^t\nu_{2n-2}^t).$$
Finally, we multiply $H$ by a bump function, a function on $\Delta$ which is 1 in a neighborhood of the origin and 0 outside a bigger neighborhood of the origin, which ensures that the Hamiltonian vector field is compactly supported, so it can be integrated to all times. In particular for $t=1$ we get $\sigma_n(z)=1$ for all $z$ near the origin.
The following argument shows that the zero locus of $\sigma_n$ can not be changed by a higher diffeomorphism: The ideal $\langle p^{2n-1}, p\bar{p}, -\bar{p}^2+\nu_{2n-2}p^{2n-2} \rangle$ is the deformation ideal of the singularity $-\frac{\nu_{2n-2}}{2n-1}p^{2n-1}+p\bar{p}^2$ which is the Kleinian singularity of type $D_{2n}$ if $\nu_{2n-2}\neq 0$. In particular, this singularity is robust under diffeomorphisms, so is its deformation ideal.
It is interesting to notice the appearance of Kleinian singularities, which have an $ADE$-classification. The fact that for ${\mathfrak{g}}$ of type $D_n$ the singularity is of type $D_{2n}$ is linked to the representation of ${\mathfrak}{so}_{2n}$ on ${\mathbb{C}}^{2n}$. There should be a more intrinsic way to link ${\mathfrak{g}}$-complex structures to singularities of type ${\mathfrak{g}}$.
An idea in this direction is the following: the singularity of type ${\mathfrak{g}}$ appears inside the Lie algebra ${\mathfrak{g}}$, more precisely inside the nilpotent variety along the subregular locus (see [@Steinberg]). A minimal resolution of this singularity is given by the Springer resolution. There should be a link between ${\mathfrak{g}}$-Hilbert schemes and the Springer resolution.
Since there are no local invariants for ${\mathfrak{g}}$-complex structures, only their global geometry is non-trivial.
Definition of the moduli space
------------------------------
To define the moduli space of ${\mathfrak{g}}$-complex structures, there is one more subtlety: in order to get one component, we have to fix an orientation on $\Sigma$. We then call a complex structure *compatible* if the induced orientation coincides with the given orientation on $\Sigma$. We call a ${\mathfrak{g}}$-complex structure *compatible* if the induced complex structure is.
The moduli space $\hat{{\mathcal}{T}}_{{\mathfrak{g}}}\Sigma$ is the space of all compatible ${\mathfrak{g}}$-complex structures modulo the action of higher diffeomorphisms of type ${\mathfrak{g}}$.
Notice that a ${\mathfrak{g}}$-complex structure is compatible iff $\mu_2(z)\bar{\mu}_2(z) < 1$. Reverting the orientation on $\Sigma$ we get another copy of $\hat{{\mathcal}{T}}_{{\mathfrak{g}}}\Sigma$ corresponding to those ${\mathfrak{g}}$-complex structures with $\mu_2(z)\bar{\mu}_2(z) > 1$. For ${\mathfrak{g}}={\mathfrak}{sl}_2$ we get Teichmüller space since we can reduce any Hamiltonian to $H=w(z,\bar{z})p$ which generates a linear diffeomorphism of $T^*\Sigma$, coming from a diffeomorphism on $\Sigma$ isotopic to the identity.
The moduli space has the following properties:
\[thm2\] For ${\mathfrak{g}}$ of classical type, and a surface $\Sigma$ of genus $g\geq 2$ the moduli space $\hat{\mathcal{T}}_{{\mathfrak{g}}}\Sigma$ is a contractible manifold of complex dimension $(g-1)\dim {\mathfrak{g}}$. In addition, its cotangent space at any point $I$ is given by $$T^*_{I}\hat{\mathcal{T}}_{{\mathfrak{g}}}\Sigma = \bigoplus_{m=1}^{r} H^0(\Sigma,K^{m_i+1})$$ where $(m_1,...,m_r)$ are the exponents of ${\mathfrak{g}}$ and $r=\operatorname{rk}{\mathfrak{g}}$ denotes the rank of ${\mathfrak{g}}$.
Notice that the differentials in $H^0(\Sigma,K^{m_i+1})$ are holomorphic with respect to the complex structure induced from the ${\mathfrak{g}}$-complex structure (see proposition \[inducedcomplex\]).
The case for $A_n$ has been treated in [@FockThomas], theorem 2. The cases $B_n$ and $C_n$ are exactly analogous:
One shows that at every point, the cotangent space exists and is of the form stated in the theorem. From this follows that $\hat{{\mathcal}{T}}_{{\mathfrak{g}}}\Sigma$ is a manifold. The only point to check is the appearance of the exponents of the Lie algebra. Since $\mu_{2i}$ is a section of $K^{1-2i}\otimes \bar{K}$ (see equation ) its dual $t_{2i}$ is a section of $K^{2i}$. Since the exponents for $B_n$ and $C_n$ are the same and equal to $(1,3,...,2n-1)$, we get the desired form stated in the theorem.
For ${\mathfrak{g}}$ of type $D_n$, suppose first that the higher Beltrami differential $\sigma_n$ vanishes nowhere [^1]. By the local theory, we know that there is a coordinate system in which $\mu_{2i}=0$ for all $i=1,..., n-1$. In that case, we know that the variation of $\mu_{2i}$ under a higher diffeomorphism generated by $H=w_{-}\bar{p}+\sum_{k=0}^{n-2}w_{2k+1}p^{2k+1}$ is given by $\delta \mu_{2i}=\bar{\partial}w_{2i-1}$ and equation gives $\delta \sigma_n=\bar{\partial}(w_{-}\sigma_n)$. The variation of $\mu_{2i}$ is the same as in the case of type $A_n$, so we know that these contribute to the cotangent bundle by a term $H^0(\Sigma, K^{2i})$. For the term $\sigma_n$ we use the pairing between differential of type $(1-n,1)$ and of type $(n,0)$ given by integration over the surface. We get $$\begin{aligned}
(\{\delta\sigma_n\} / \bar{\partial}(w_{-}\sigma_n))^* =& \; \{t_n \in \Gamma(K^n) \mid \smallint t_n \bar{\partial}(w_{-}\sigma_n) = 0 \; \forall \, w_{-} \in \Gamma(\bar{K}) \} \\
=& \; \{t_n \in \Gamma(K^n) \mid \smallint \bar{\partial}t_n w_{-}\sigma_n = 0 \; \forall \, w_{-} \in \Gamma(\bar{K}) \} \\
=& \; \{t_n \in \Gamma(K^n) \mid \bar{\partial}t_n = 0 \} \\
=& \; H^0(\Sigma,K^n)\end{aligned}$$ where we used that $\sigma_n$ vanishes nowhere.
Hence the cotangent bundle is given by $$T^*_I\hat{\mathcal{T}}_{{\mathfrak{g}}}\Sigma = \bigoplus_{m=1}^{n-1}H^0(\Sigma,K^{2m})\oplus H^0(\Sigma, K^n).$$ The exponents of ${\mathfrak}{so}_{2n}$ are precisely $(1,3,...,2n-3,n-1)$, so the cotangent bundle is of the form stated in the theorem.
If $\sigma_n$ vanishes at some places, we can get $\sigma_n$ by a limit of non-vanishing $\sigma_n^t$ such that for $t\rightarrow 0$ we get $\sigma_n$. Since holomorphicity is a closed condition, we still have the same result.
For the dimension of $\hat{\mathcal{T}}_{{\mathfrak{g}}}\Sigma$, we use $\dim H^0(K^{m_i+1})=(g-1)(2m_i+1)$ by Riemann-Roch (using $g \geq 2$). We get $$\dim \hat{{\mathcal}{T}}_{{\mathfrak{g}}}\Sigma=(g-1)\sum_{i=1}^r (2m_i+1)=(g-1)\dim {\mathfrak{g}}$$ using a well-known formula coming from the decomposition of ${\mathfrak{g}}$ as ${\mathfrak}{sl}_2$-module using the principal ${\mathfrak}{sl}_2$-triple.
Contractibility in all cases is easy: given an equivalence class of a ${\mathfrak{g}}$-complex structure by its set $S$ of higher Beltrami differentials, we can retract it in a direct linear way $tS$ to the structure where all Beltrami differentials are 0. We use that if $S$ and $S'$ are equivalent under a higher diffeomorphism, so are $tS$ and $tS'$.
From the previous theorem, we see that our moduli space $\hat{{\mathcal}{T}}_{{\mathfrak{g}}}\Sigma$ shares a lot of properties with Hitchin’s component, in particular the dimension and contractibility. There is another common property to notice:
There is a map from Teichmüller space into the moduli space $\hat{{\mathcal}{T}}_{{\mathfrak{g}}}\Sigma$.
The proposition follows from the map $\psi:\operatorname{Hilb}({\mathfrak}{sl}_2) \rightarrow \operatorname{Hilb}^{reg}({\mathfrak{g}})$ constructed in equation in section \[mu2\]. This map restricts to a map between the zero-fibers and extends over the surface $\Sigma$. Finally the map descends to the quotient by higher diffeomorphisms since for ${\mathfrak}{sl}_2$ we only quotient by diffeomorphisms of $\Sigma$.
The same property holds for $G$-Hitchin components which can be defined as the deformation space of representations of the form $\pi_1(\Sigma) \rightarrow PSL_2({\mathbb{R}}) \rightarrow G$ where the first map is a Fuchsian representation and the second one is the principal map. So inside the $G$-Hitchin component sits a copy of Teichmüller space. The same situation holds in our case, and the map $\psi$ is constructed using the principal map as well.
Of course, we conjecture the equivalence of Hitchin’s component and the moduli space of ${\mathfrak{g}}$-complex structures:
\[conjmaj\] The moduli space $\hat{\mathcal{T}}_{{\mathfrak{g}}}\Sigma$ is canonically isomorphic to Hitchin’s component in the character variety $\operatorname{Hom}(\pi_1(\Sigma),G)/G$ where $G$ is the real split Lie group associated to ${\mathfrak{g}}$.
Spectral curve {#spectralcurve}
--------------
In this final part, we construct a spectral curve in $T^{*{\mathbb{C}}}\Sigma$ associated to a cotangent vector to $\hat{\mathcal{T}}_{{\mathfrak{g}}}$, i.e. a ${\mathfrak{g}}$-complex structure and a set of holomorphic differentials. The case for ${\mathfrak{g}}$ of type $A_n$ was treated in [@FockThomas], section 4.
In *loc. cit.*, we proved that the zero-fiber $\operatorname{Hilb}^n_0({\mathbb{C}}^2)$ is Lagrangian in the reduced Hilbert scheme $\operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$. This stays true for all classical ${\mathfrak{g}}$:
The regular zero-fiber $\operatorname{Hilb}^{reg}_0({\mathfrak{g}})$ is a Lagrangian subspace of $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ for classical ${\mathfrak{g}}$.
Since we are in the regular part, proposition \[paramit\] gives a parametrization. For classical ${\mathfrak{g}}$, via the standard representation we can consider $\operatorname{Hilb}^{reg}({\mathfrak{g}})$ as subset of $\operatorname{Hilb}^m_{red}({\mathbb{C}}^2)$ which remains symplectic and we can explicitly check that the zero-fiber $\operatorname{Hilb}^{reg}_0({\mathfrak{g}})$ is Lagrangian.
More generally, Haiman in his paper [@Haiman] described a way to find coordinates of $\operatorname{Hilb}^n({\mathbb{C}}^2)$ in the chart associated to a Young diagram $D$. For each box $B_x \in D$ consider the rightmost box $B_r \in D$ in the same row as $B_x$ and the bottommost box $B_b \in D$ in the same column as $B_x$ (see figure \[Haimancoo\]). The box $B_{r+1}$ to the right of $B_r$ is not in $D$, so gives a linear combination of boxes in $D$. Denote by $b_{x,r}$ the coefficient of $B_b$ in this linear combination. Similarly, denote by $b_{x,b}$ the coefficient of $B_r$ in the linear combination associated to the box $B_{b+1}$ at the bottom of $B_b$. Haiman shows that the set $\{b_{x,r}, b_{x,b}\}_{x\in D}$ is a coordinate system.
{height="2cm"}
\[Haimancoo\]
We have even more:
Haiman’s coordinates $\{b_{x,r}, b_{x,b}\}_{x\in D}$ are canonical coordinates with respect to the symplectic structure of the punctual Hilbert scheme.
The symplectic structure of the punctual Hilbert scheme comes from the canonical symplectic structure of ${\mathbb{C}}^{2n}$ given by $\omega=\sum_i dx_i\wedge dy_i$. Consider the multiplication operators $M_x$ and $M_y$ in the quotient ${\mathbb{C}}[x,y]/I$ where $I$ is an element in the Hilbert scheme (idealic viewpoint). Diagonalizing these operators give diagonal matrices with entries $(x_1,...,x_n)$ and $(y_1,...,y_n)$. Changing to the base adapted to the Young diagram $D$ (basis generated by monomials $x^iy^j$ where $(i,j) \in D$), the matrix $M_x$ becomes a matrix $N_x$ with entries 1 on the line under the diagonal, apart from some columns where the linear combination associated to some $B_{r+1}$ is written. Similarly, the matrix $M_y$ becomes a matrix $N_y$ with entries 1 on the line under the diagonal, apart from some columns where the linear combination associated to some $B_{b+1}$ is written. Finally, we compute $$\begin{aligned}
\omega &= \operatorname{tr}d\left( \begin{smallmatrix}x_1&&\\ &\ddots &\\ &&x_n\end{smallmatrix}\right) \wedge d\left( \begin{smallmatrix} y_1&&\\ &\ddots &\\ &&y_n\end{smallmatrix}\right) \\
&= \operatorname{tr}dN_x \wedge dN_y \\
&= \sum_{x\in D} db_{x,r}\wedge db_{x,b}.\end{aligned}$$
As an application, we can use Haiman coordinates for the ideal with three generators for ${\mathfrak{g}}$ of type $D_n$ in subsection \[Dn\]. In particular, we see that the coordinates $u$ and $\nu_{2n-2}$ are canonically conjugated.
For a modified smooth version of the ${\mathfrak{g}}$-Hilbert scheme, we conjecture the following:
The modified smooth version of the ${\mathfrak{g}}$-Hilbert scheme is symplectic and the zero-fiber is a Lagrangian subspace.
If we assume conjecture \[conj1\] true, stating that the modified version of the ${\mathfrak{g}}$-Hilbert scheme is a minimal resolution of ${\mathfrak{h}}^2/W$, we get a symplectic structure. Indeed ${\mathfrak{h}}^2=T^*{\mathfrak{h}}$ has a canonical symplectic structure, which is invariant under the action of $W$. Hence it lifts to the minimal resolution.
Now we construct the spectral curve. First, we look at ${\mathfrak{g}}$ of type $A_n$, $B_n$ or $C_n$. We can write a cotangent vector in $T^*\hat{{\mathcal}{T}}_{{\mathfrak{g}}}$ as an equivalence class of higher Beltrami differentials $\mu_i$ and holomorphic differentials $t_i$. To write in a uniform way, set $\mu_i$ or $t_i$ to 0 whenever it does not appear for ${\mathfrak{g}}$. For example for type $B_n$ or $C_n$ all variables with odd index are 0.
Associate polynomials $P(p)=p^m+\sum_i t_ip^{m-i}$ and $Q(p, \bar{p})=-\bar{p}+\sum_i \mu_ip^{i-1}$ (where $m$ is the dimension of the standard representation of ${\mathfrak{g}}$). Put $I=\langle P, Q \rangle$. Define the **spectral curve** $\tilde{\Sigma} \subset T^{*{\mathbb{C}}}\Sigma$ by the zero set of $P$ and $Q$. It is a ramified cover over $\Sigma$ with $m$ sheets.
For ${\mathfrak{g}}$ of type $D_n$, a generic point in the cotangent bundle $T^*\hat{{\mathcal}{T}}_{{\mathfrak{g}}}$ corresponds to the ideal $$I=\langle p^{2n}+t_2p^{2n-2}+...+t_{2n-2}p^2+\tau_n^2, -\bar{p}+\mu_2p+...+\mu_{2n}p^{2n-1} \rangle$$ which can be seen as a special case of $A_n$. Thus we can proceed as above. In the case where $\tau_n = 0$ we have seen in \[Dn\] that the ideal changes to an ideal with three generators. These generators still define a spectral curve in $T^{*{\mathbb{C}}}\Sigma$.
The spectral curve $\tilde{\Sigma}$ is Lagrangian to order 1 in the holomorphic differentials $t$.
This is the precise analogue of proposition 5 in [@FockThomas].
In the case where the ideal has two generators $P$ and $Q$ this is equivalent to $\{P,Q\} = 0 \mod I \mod t^2$ for $I\in T^*\hat{{\mathcal}{T}}_{{\mathfrak{g}}}$. For $A_n$, the proof is given in *loc. cit*. For $B_n$ and $C_n$ it is completely analogous since the ${\mathfrak{g}}$-complex structure can be seen as a special case of $A_n$.
For ${\mathfrak{g}}$ of type $D_n$, a generic ideal has still two generators, so we have a special case of $A_n$. If the ideal has three generators, the spectral curve is still Lagrangian since it can be obtained as a limit of Lagrangian curves, and the property of being Lagrangian is closed.
Since the spectral curve is Lagrangian to order 1, the periods are well-defined up to this order. The ratios of these periods should give coordinates on $T^*\hat{{\mathcal}{T}}_{{\mathfrak{g}}}$ and also on $\hat{{\mathcal}{T}}_{{\mathfrak{g}}}$. For the trivial ${\mathfrak{g}}$-complex structure (where all higher Beltrami differentials are 0) we recover Hitchin’s spectral curve.
Finally, we can recover the same spectral data as Hitchin in his paper on stable bundles [@Hit3]. From a ${\mathfrak{g}}$-complex structure we get a bundle $V$ over the surface $\Sigma$ whose fiber at a point $z\in \Sigma$ is ${\mathbb{C}}[p,\bar{p}]/I(z)$ where we use the idealic viewpoint. We also get a line bundle $L$ on $\tilde{\Sigma}$ whose fiber is the eigenspace of $M_p$, the multiplication operator by $p$ in the quotient ${\mathbb{C}}[p,\bar{p}]/I$. This gives the spectral data for type $A_n$.
For ${\mathfrak{g}}$ of type $C_n$, we get in addition an involution $\sigma$ on the spectral curve $\tilde{\Sigma}$ given by $(p,\bar{p})\mapsto (-p, -\bar{p})$. For ${\mathfrak{g}}$ of type $D_n$, the spectral curve is singular, having a double point. The spectral data is given by a desingularization of $\tilde{\Sigma}$, the involution $\sigma$ as for $C_n$ and the line bundle $L$. For ${\mathfrak{g}}$ of type $B_n$, there is a canonical subbundle $V_0 \subset V={\mathbb{C}}[p,\bar{p}]/I$ generated by the span of the image of $1\in {\mathbb{C}}[p, \bar{p}]$ in the quotient ${\mathbb{C}}[p,\bar{p}]/I$ (since for $B_n$, we have $I\subset \langle p, \bar{p}\rangle$). Thus the vector bundle $V$ is an extension $V_0 \rightarrow V\rightarrow V_1$. The spectral data is given by $(V_0, V_1, \sigma, L, \tilde{\Sigma})$.
Perspectives {#section4}
============
In addition to the various conjectures in this paper, we wish to give a conjectural larger picture which would imply the main conjecture \[conjmaj\] about the equivalence between Hitchin’s component and our moduli space $\hat{\mathcal{T}}_{{\mathfrak{g}}}$.
Hitchin’s original construction in [@Hit.1] of components in character varieties uses Higgs bundles and the hyperkähler structure of its moduli space ${\mathcal}{M}_H$. In one complex structure, say $I$, ${\mathcal}{M}_H$ has the complex structure from Higgs bundles. In all combinations of $J$ and $K$, it is the moduli space of flat $G^{{\mathbb{C}}}$-connections. The non-abelian Hodge correspondence is equivalent to the twistor description of this hyperkähler manifold. Hitchin constructs a fibration of ${\mathcal}{M}_H$ over a space of holomorphic differentials, whose fibers via the non-abelian Hodge correspondence give flat connections with monodromy in the split real group $G$.
There is a similar conjectural picture for ${\mathfrak{g}}$-complex structures: a hyperkähler manifold ${\mathcal}{M}$, which in complex structure $I$ is the cotangent space to the moduli space of ${\mathfrak{g}}$-complex structures $T^*\hat{{\mathcal}{T}}_{{\mathfrak{g}}}\Sigma$ and in all combinations of $J$ and $K$ is the moduli space of flat $G^{{\mathbb{C}}}$-connections. The analogue of Hitchin’s fibration is simply the projection $\pi: T^*\hat{{\mathcal}{T}}_{{\mathfrak{g}}}\Sigma \rightarrow \hat{{\mathcal}{T}}_{{\mathfrak{g}}}\Sigma$. One has to prove an analogue of the non-abelian Hodge correspondence, i.e. a deformation of a pair (${\mathfrak{g}}$-complex structure, set of holomorphic differentials) to flat connections, and that the monodromy of the fibers of the projection $\pi$ lies in the split real group $G$.
The conception behind this analogy is the following: In Hitchin’s case, we have a fixed complex structure on $\Sigma$ and a holomorphic Higgs field $\Phi \in H^{(1,0)}(\Sigma,{\mathfrak{g}})$ which gives a flat connection ${\mathcal}{A}(\lambda)=\lambda \Phi +A+ \lambda^{-1}\Phi^*$. To get the Hitchin section, we choose a principal nilpotent element $f$ in the Lie algebra ${\mathfrak{g}}$ and deform it into the principal slice $f+Z(e)$. To avoid fixing a complex structure, we start with $\Phi = \Phi_1 dz+\Phi_2 d\bar{z}$. The flatness of ${\mathcal}{A}(\lambda)$ gives that $\Phi_1$ and $\Phi_2$ commute. We further impose $\Phi_1$ and $\Phi_2$ to be nilpotent. More specifically, we take $\Phi_1$ to be the principal nilpotent element $f$ and we choose $\Phi_2 \in Z(f)$. Thus we have the same number of degrees of freedom as in the Higgs bundle setting. A pair of commuting nilpotent matrices of this form is precisely a point in $\operatorname{Hilb}^{reg}_0({\mathfrak{g}})$ which we used to construct ${\mathfrak{g}}$-complex structures.
Punctual Hilbert schemes revisited {#appendix:A}
==================================
In this appendix, we review the punctual Hilbert scheme of the plane with its various viewpoints. Main references are Nakajima’s book [@Nakajima] and Haiman’s paper [@Haiman].
Definition
----------
To start, consider $n$ points in the plane $\mathbb{C}^2$ as an algebraic variety, i.e. defined by some ideal $I$ in $\mathbb{C}[x,y]$. Its function space $\mathbb{C}[x,y]/I$ is of dimension $n$, since a function on $n$ points is defined by its $n$ values. So the ideal $I$ is of codimension $n$. The space of all such ideals, or in more algebraic language, the space of all zero-subschemes of the plane of given length, is the punctual Hilbert scheme:
The **punctual Hilbert scheme** $\operatorname{Hilb}^n(\mathbb{C}^2)$ of length $n$ of the plane is the set of ideals of $\mathbb{C}\left[x,y\right]$ of codimension $n$: $$\operatorname{Hilb}^n(\mathbb{C}^2)=\{I \text{ ideal of } \mathbb{C}\left[x,y\right] \mid \dim(\mathbb{C}\left[x,y\right]/I)=n \}.$$ The subspace of $\operatorname{Hilb}^n(\mathbb{C}^2)$ consisting of all ideals supported at 0, i.e. whose associated algebraic variety is $(0,0)$, is called the **zero-fiber** of the punctual Hilbert scheme and is denoted by $\operatorname{Hilb}^n_0(\mathbb{C}^2)$.
A theorem of Grothendieck and Fogarty asserts that $\operatorname{Hilb}^n(\mathbb{C}^2)$ is a smooth and irreducible variety of dimension $2n$ (see [@Fogarty]). The zero-fiber $\operatorname{Hilb}^n_0(\mathbb{C}^2)$ is an irreducible variety of dimension $n-1$, but it is in general not smooth.
A generic element of $\operatorname{Hilb}^n({\mathbb{C}}^2)$, geometrically given by $n$ distinct points, is given by $$I=\left\langle x^n+t_1x^{n-1}+\cdots+t_n,-y+\mu_1+\mu_2x+...+\mu_nx^{n-1}\right\rangle.$$ The second term can be seen as the Lagrange interpolation polynomial of the $n$ points.
A generic element of the zero-fiber is given by $$I=\left\langle x^n,-y+\mu_2x+...+\mu_nx^{n-1}\right\rangle.$$
Resolution of singularities {#resofsing}
---------------------------
Given an ideal $I$ of codimension $n$, we can associate its support, the algebraic variety defined by $I$, which is a collection of $n$ points (counted with multiplicity). The order of the points does not matter, so there is a map, called the **Chow map**, from $\operatorname{Hilb}^n(\mathbb{C}^2)$ to $\operatorname{Sym}^n(\mathbb{C}^2) := (\mathbb{C}^2)^n/\mathcal{S}_n$, the configuration space of $n$ points ($\mathcal{S}_n$ denotes the symmetric group). A theorem of Fogarty asserts that the punctual Hilbert scheme is a *minimal resolution of the configuration space*.
In order to get a feeling for a general Lie algebra, notice that $n$ points of $\mathbb{C}^2$ is the same as two points in the Cartan $\mathfrak{h}$ of $\mathfrak{gl}_n$, and that the symmetric group is the Weyl group $W$ of $\mathfrak{gl}_n$. So the configuration space equals ${\mathfrak{h}}^2/W$ for ${\mathfrak{g}}={\mathfrak}{gl}_n$.
Matrix viewpoint
----------------
To an ideal $I$ of codimension $n$, we can associate two matrices: the multiplication operators $M_x$ and $M_y$, acting on the quotient ${\mathbb{C}}[x,y]/I$ by multiplication by $x$ and $y$ respectively. To be more precise, we can associate a conjugacy class of the pair: $[(M_x,M_y)]$.
The two matrices $M_x$ and $M_y$ commute and they admit a cyclic vector, the image of $1 \in {\mathbb{C}}[x,y]$ in the quotient (i.e. 1 under the action of both $M_x$ and $M_y$ generate the whole quotient).
\[bijhilbert\] There is a bijection between the Hilbert scheme and conjugacy classes of certain commuting matrices: $$\operatorname{Hilb}^n(\mathbb{C}^2) \cong \{(A,B) \in {\mathfrak}{gl}_n^2 \mid [A,B]=0, (A,B) \text{ admits a cyclic vector}\} / GL_n$$
The inverse construction goes as follows: to a conjugacy class $[(A,B)]$, associate the ideal $I=\{P \in {\mathbb{C}}[x,y] \mid P(A,B)=0\}$, which is well-defined and of codimension $n$ (using the fact that $(A,B)$ admits a cyclic vector). For more details see [@Nakajima].
It is this bijection which we use in the main text to generalize the punctual Hilbert scheme. Notice that the *zero-fiber of the Hilbert scheme corresponds to nilpotent commuting matrices*.
Reduced Hilbert scheme
----------------------
We wish to define a subspace of $\operatorname{Hilb}^n({\mathbb{C}}^2)$ corresponding to matrices in ${\mathfrak}{sl}_n$ in the matrix viewpoint. A generic point should be a pair of points in the Cartan ${\mathfrak{h}}$ of ${\mathfrak}{sl}_n$ modulo order. This corresponds to $n$ points in the plane with barycenter 0.
The **reduced Hilbert scheme** $\operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$ is the space of all elements of $\operatorname{Hilb}^n({\mathbb{C}}^2)$ whose image under the Chow map ($n$ points with multiplicity modulo order) has barycenter 0.
With this definition, we get
$$\operatorname{Hilb}^n_{red}(\mathbb{C}^2) \cong \{(A,B) \in {\mathfrak}{sl}_n^2 \mid [A,B]=0, (A,B) \text{ admits a cyclic vector}\} / SL_n.$$
Finally, it can be proven that the reduced Hilbert scheme is symplectic and that the zero-fiber $\operatorname{Hilb}^n_0({\mathbb{C}}^2)$ is a Lagrangian subspace of $\operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$.
Regular elements in semisimple Lie algebras {#appendix:B}
===========================================
In this appendix, we gather all properties we need in the main text of regular elements in semisimple Lie algebras and we give precise references for these results. The main references are the books of Collingwood and McGovern [@Coll], Steinberg [@Steinberg] and Humphreys [@Hum], as well as the papers [@Kost] and [@Kost2] by Kostant.
An element $x \in {\mathfrak{g}}$ is called **regular** if the dimension of its centralizer $Z(x)$ is equal to the rank of the Lie algebra $\operatorname{rk}({\mathfrak{g}})$. A regular nilpotent element is called **principal nilpotent**.
Notice that in older literature, regular elements are defined in another way, using the characteristic polynomial of the adjoint map. The “old” notion includes only semisimple regular (in the sens above) elements.
The condition that the dimension of the centralizer has to be equal to the rank, does not come from nowhere: in fact it is the minimal possible dimension.
For any $x \in {\mathfrak{g}}$, we have $\dim Z(x) \geq \operatorname{rk}({\mathfrak{g}})$.
See for example lemma 2.1.15. in [@Coll].
For the Lie algebras ${\mathfrak}{gl}_n$ and ${\mathfrak}{sl}_n$, we have the following characterization of regular elements from Steinberg [@Steinberg], proposition 2 in section 3.5:
\[regularsln\] For ${\mathfrak{g}}={\mathfrak}{gl}_n$ or ${\mathfrak}{sl}_n$ and $x\in {\mathfrak{g}}$, we have the following equivalence: $$x \text{ is regular} \Leftrightarrow \mu_x = \chi_x \Leftrightarrow x \text{ admits a cyclic vector}$$ where $\mu_x$ and $\chi_x$ denote respectively the minimal and the characteristic polynomial of $x$, seen as a matrix.
Let us turn to the study of regular elements which are nilpotent.
There is a unique open dense orbit in the nilpotent variety consisting of principal nilpotent elements.
The original proof is due to Kostant, see corollary 5.5. in [@Kost]. See also theorem 4.1.6. in [@Coll].
There is a useful characterization of principal nilpotent elements in coordinates. For this, fix a root system $R$, fix a direction giving positive roots $R_+$. Denote by ${\mathfrak}{n}$ the positive nilpotent elements (upper triangular for ${\mathfrak}{sl}_n$).
\[prinnilp\] Let $A \in {\mathfrak}{n}$. Then $A$ is principal nilpotent iff $\alpha(A)\neq 0$ for all simple roots $\alpha$.
The group version of this can be found in section 3.7. of [@Steinberg].
For a principal nilpotent element $f$, its centralizer $Z(f)$ has properties quite analogous to a Cartan, the centralizer of a regular semisimple element:
\[thmKost\] For $f$ a principal nilpotent element, its centralizer $Z(f)$ is abelian and nilpotent.
Kostant proves even more, using a limit argument: for any element $x\in {\mathfrak{g}}$, there is an abelian subalgebra of $Z(x)$ of dimension $\operatorname{rk}{\mathfrak{g}}$, see [@Kost], theorem 5.7. The nilpotency of $Z(f)$ can be found in [@Steinberg], corollary in section 3.7. The more precise structure of $Z(x)$ for any nilpotent $x$ is described in [@Coll], section 3.4.
A principal nilpotent element permits to give a preferred representative of a conjugacy class of regular elements. Given $f$ principal nilpotent, denote by $e$ the other nilpotent element in a principal ${\mathfrak}{sl}_2$-triple constructed from $f$ (see Kostant [@Kost]). Then we get
Any regular orbit intersects $f+Z(e)$ in a unique point. So we have ${\mathfrak{g}}^{reg}/G \cong f+Z(e)$.
This follows from Lemma 10 of [@Kost2]. The set $f+Z(e)$ is called a *principal slice* of ${\mathfrak{g}}$ (also *Kostant section*).
We are now going to “double” the previous setting. Define the commuting variety to be $\operatorname{Comm}({\mathfrak{g}}):=\{(A,B)\in {\mathfrak{g}}^2 \mid [A,B]=0\}$.
\[Richardson\] The set of commuting semisimple elements is dense in the commuting variety $\operatorname{Comm}({\mathfrak{g}})$.
See the paper of Richardson [@Richardson] for a proof. As a consequence, $\operatorname{Comm}({\mathfrak{g}})$ is an irreducible variety, but highly singular.
With this, we can explore the minimal dimension of a centralizer of a commuting pair:
\[doublecomm\] For $(A,B) \in \operatorname{Comm}({\mathfrak{g}})$, we have $\dim Z(A,B) \geq \operatorname{rk}{\mathfrak{g}}.$
Consider the set $M$ of elements with centralizer of minimal dimension. Since $$M=\{(A,B) \in \operatorname{Comm}({\mathfrak{g}}) \mid \operatorname{rk}(ad_A, ad_B) \text{ maximal}\}$$ we see that $M$ is Zariski-open. By the theorem of Richardson it intersects the space of semisimple pairs for which the common centralizer is a Cartan ${\mathfrak{h}}$, so of dimension $\operatorname{rk}{\mathfrak{g}}$.
[11111]{}
David H. Collingwood and William M. McGovern: *Nilpotent Orbits in Semisimple Lie Algebras*, Van Nostrand Reinhold Math. Series, New York, 1993
Vladimir V. Fock and Alexander Thomas: *Higher complex structures*, accepted at IMRN journal, https://arxiv.org/abs/1812.11199
John Fogarty: *Algebraic families on an algebraic surface*, Amer. J. Math. Vol. 90 No. 2 (1968), p. 511 - 521
Victor Ginzburg: *Principal Nilpotent pairs in a semisimple Lie algebras I*, https://arxiv.org/abs/math/9903059
Mark Haiman: *$t,q$-Catal numbers and the Hilbert scheme*, Discrete Mathematics 193 (1998), pp. 201-224
Nigel Hitchin: *Lie Groups and Teichmüller Space*, Topology Vol. 31, No.3 (1992), p. 449 - 473
Nigel Hitchin: *Stable Bundles and Integrable Systems*, Duke Math. J. Vol. 54 No. 1 (1987), p. 91-114
James E. Humphreys: *Conjugacy Classes in Semisimple Algebraic Groups*, Math. Surveys and Monographs Vol. 43 (1995), AMS
Anthony Iarrobino: *Punctual Hilbert schemes*, Bull. AMS Vol. 78, No. 5 (1972), p. 819 - 823
Bertram Kostant: *The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group*, Amer. J. Math. 81 (1959), p. 973-1032
Bertram Kostant: *Lie group representations on polynomial rings*, Amer. J. Math. 85 (1963), p. 327-404
Hiraku Nakajima: *Lectures on Hilbert Schemes of Points on Surfaces*, University Lecture Series 18, AMS, 1999
R. William Richardson:*Commuting varieties of semisimple Lie algebras and algebraic groups*, Comp. Math., Tome 38 (1979) no. 3, p. 311-327
Robert Steinberg: *Conjugacy Classes in Algebraic Groups*, Lecture Notes in Mathematics 366, Springer-Verlag, New York, 1974
[^1]: This exists since $\sigma_n$ is a smooth section of $K^{1-n}\otimes \bar{K}$ which is of negative degree since $n>2$ and $g\geq 2$.
|
---
abstract: |
In this paper we study a linearized eigenvalue problem derived from a a free boundary problem modeling the growth of a tumor containing two species of cells: proliferating cells and quiescent cells. The reduced form of this eigenvalue problem is a $2$-system of a first-order nonlocal singular differential-integral equation in a ball coupled by a third-order elliptic pseudo-differential equation in the unit sphere. The singularity joined with non-localness of the first-order equation causes the main difficulty of this problem. By using Fourier expansion via a basis of spherical harmonic functions and some techniques for solving singular differential integral equations developed in some previous literature, we prove that there exists a null sequence $\{\gamma_k\}_{k=2}^{\infty}$ for the surface tension coefficient $\gamma$, with each of them being an eigenvalue of the linearized problem, i.e., if $\gamma=\gamma_k$ for some $k\geq 2$ then the linearized problem has extra nontrivial solutions besides the standard nontrivial solutions, and if $\gamma\neq\gamma_k$ for all $k\geq 2$ then the linearized problem does not have other nontrivial solutions than the standard nontrivial solutions. Invertibility of some linear operators related to the linearized problem in suitable function spaces is also studied.
[**Key words and phrases**]{}: Free boundary problem, tumor growth, linearization, eigenvalue problem, nontrivial solution.
[**2000 mathematics subject classifications**]{}: 34B15, 35C10, 35Q80.
author:
- 'Shangbin Cui[^1]'
date: |
[Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275,]{}\
\[-0.05cm\] [People’s Republic of China]{}
title: 'Linearized eigenvalues for a free boundary problem modeling two-phase tumor growth[^2]'
---
Introduction
============
2em It has long been observed that under a constant circumstance, a solid tumor will finally evolve into a dormant or stationary state. In a dormant state, the tumor’s macrostructure such as size, shape and etc. does not vary in time, while cells inside the tumor are alive and keep undergoing the process of proliferation and movement before they die. In 1972 Greenspan established the first mathematical model in the form of a free boundary problem of a system of partial differential equations to illustrate this phenomenon [@Green1; @Green2]. Since then an increasing number of tumor models in similar forms have appeared in the literature; see the reviewing articles [@tumrev1; @tumrev2; @Fried1; @Fried3; @tumrev3] and the references cited therein. Rigorous mathematical analysis of such models has drawn great attention during the past twenty years, and many interesting results have been obtained, cf., [@ChenCuiF] – [@EM2], [@tumrev2] – [@FriRei2], [@HHH], [@WZ], [@ZEC] and references cited therein.
This paper is concerned with the following free boundary problem modeling the dormant state of a solid tumor with two species of cells — proliferating cells and quiescent cells (see [@PPM]): $$\Delta\sigma=F(\sigma) \quad \mbox{for}\;\; x\in\Omega,$$ $$\sigma=1 \quad \mbox{for}\;\; x\in\partial\Omega,$$ $$\nabla\cdot(\vec{v}p)=[K_B(\sigma)-K_Q(\sigma)]p+K_P(\sigma)q \quad
\mbox{for}\;\; x\in\Omega,$$ $$\nabla\cdot(\vec{v}q)=K_Q(\sigma)p-[K_P(\sigma)+K_D(\sigma)]q \quad
\mbox{for}\;\; x\in\Omega,$$ $$p+q=1 \quad \mbox{for}\;\; x\in\Omega,$$ $$\vec{v}=-\nabla\varpi \quad \mbox{for}\;\; x\in\Omega,$$ $$\varpi=\gamma\kappa \quad \mbox{for}\;\; x\in\partial\Omega,$$ $$V_n\equiv\vec{v}\cdot\vec{n}=0 \quad \mbox{for}\;\; x\in\partial\Omega.$$ Here $\Omega$ is the domain occupied by the dormant tumor, $c=c(x)$, $p=p(x)$ and $q=q(x)$ are the concentration of nutrient, the density of proliferating cells and the density of quiescent cells, respectively, $\vec{v}=\vec{v}(x)$ is the velocity of tumor cell movement, $\varpi=\varpi(x)$ is the pressure distribution in the tumor, $\kappa$ is the mean curvature of the tumor surface whose sign is designated by the convention that $\kappa\geq 0$ at points where $\partial\Omega$ is convex, $\vec{n}$ is the unit outward normal vector of $\partial\Omega)$, and $V_n$ is the normal velocity of the tumor surface. Besides, $F(\sigma)$ is the consumption rate of nutrient by tumor cells, $K_B(\sigma)$ is the birth rate of tumor cells, $K_P(\sigma)$ and $K_Q(\sigma)$ are respectively the transferring rates of tumor cells from quiescent state to proliferating state and from proliferating state to quiescent state, and $K_D(\sigma)$ is the death rate of quiescent cells. Finally, $\gamma$ is a positive constant and is referred as surface tension coefficient. For illustration of biological implications of each equation in the above model, we refer the reader to see [@Fried1; @Fried3; @PPM] and references therein.
A main feature of the above model compared with various other models describing the growth of tumors consisting of only one species of cells, or one-phase tumor model in short, is that it contains conservation laws, i.e., the equations (1.3) and (1.4). This determines that the above model is much more difficult to make analysis than one-phase tumor models. Indeed, for one-phase tumor models of the stationary form, we know that they contains only elliptic equations (cf. [@CuiEsc1; @CuiEsc2; @CuiEsc3; @EM1; @EM2; @FonFri1; @FriHu1; @FriRei1; @FriRei2; @HHH; @WZ; @ZEC]). But in the above two-phase model, the system contains both elliptic equations and hyperbolic equations. Since hyperbolic equations have quite different and much worse properties compared with elliptic equations, such a system is much harder to tackle. For instance, as far as radial stationary solution is concerned, existence and uniqueness is very easy to prove for the one-phase tumor model (cf. [@FriRei1]); but for the above two-phase model the same topic needs a lot of work (cf. [@ChenCuiF; @CuiFri1]). The same situation occurs in the analysis of asymptotic stability of the radial stationary solution (cf. [@FriRei1] and [@ChenCuiF; @Cui3; @Cui4]).
In [@CuiFri1] and [@ChenCuiF] it was proved that the above model has a unique radial (i.e. spherically symmetric) solution under the following assumptions: $$F,\;\;K_B,\;\;K_D,\;\;K_P\;\;{\rm and}\;\;K_Q\;\;\mbox{are
$C^{\infty}$-functions};$$ $$F(0)=0\;\;{\rm and}\;\; F'(c)>0 \quad {\rm for}\;\;0\leq c\leq 1;$$ $$\left\{
\begin{array}{l}
K_B'(c)>0\;\;\mbox{and}\;\;K_D'(c)<0\;\;{\rm for}\;\;0\leq c\leq 1,\;\;
K_B(0)=0\;\;{\rm and}\;\;K_D(1)=0;\\
K_P\;\;\mbox{and}\;\;K_Q\;\;\mbox{satisfy the same conditions as}
\;\; K_B\;\;\mbox{and}\;\;K_D,\;\;\mbox{respectively};\\
K_B'(c)+K_D'(c)>0\;\;{\rm for}\;\; 0\leq c\leq 1.
\end{array}
\right.$$ Naturally, we may ask: Does this model has any non-radial solutions? This is a very difficult question to which we have not a satisfactory answer up to now. As a first step toward finding an answer to this question, in this paper we make a systematic study to the linearized problem of the above model around its radial stationary solution.
Let $(\sigma_s,p_s,q_s,\varpi_s,v_s,\Omega_s)$, where $\Omega_s=\{x\in {\mathbb R}^n:
\; r<R_s\}$, be the unique radial stationary solution of the system (1.1)–(1.8) ensured by [@CuiFri1] and [@ChenCuiF]. After simplification, the linearized system of (1.1)–(1.8) at $(\sigma_s,p_s,q_s,\varpi_s,v_s,\Omega_s)$ is as follows (see the next section): $$\begin{aligned}
\Delta\chi&=&F'(\sigma_s(r))\chi, \quad x\in\Omega_s,
\\[0.1cm]
\chi|_{r=R_s} &=&-\sigma_s'(R_s)\eta(\omega), \quad
\omega\in\mathbb{S}^{n-1},
\\[0.1cm]
v_s(r)\varphi_r&=&p_s'(r)\psi_r+f_\sigma^*(r)\chi+f_p^*(r)\varphi,
\quad x\in\Omega_s,
\\[0.1cm]
\vec{w}&=&-\nabla\psi, \quad x\in\Omega_s,
\\[0.1cm]
-\Delta\psi &=&g_\sigma^*(r)\chi+g_p^*(r)\varphi, \quad x\in\Omega_s,
\\[0.1cm]
\psi|_{r=R_s} &=&-{\gamma\over R_s^2}[\eta(\omega)
+{1\over n\!-\!1}\Delta_\omega\eta(\omega)], \quad
\omega\in\mathbb{S}^{n-1},
\\[0.1cm]
\psi_r|_{r=R_s}&=&g(1,1)\eta(\omega), \quad \omega\in\mathbb{S}^{n-1}.\end{aligned}$$ Here $\chi=\chi(r,\omega)$, $\varphi=\varphi(r,\omega)$, $\psi=\psi(r,\omega)$, $\vec{w}=\vec{w}(r,\omega)$ and $\eta=\eta(\omega)$, where $r=|x|$ and $\omega=
x/|x|$, are new unknown functions, the subscript $r$ denotes the derivative in radial direction (e.g., $\varphi_r=\frac{\partial\varphi}{\partial r}=
\frac{x}{r}\cdot\nabla\varphi$ etc.), $\Delta_\omega$ denotes the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^{n-1}$, and $$\begin{array}{c}
f_\sigma^*(r)=f_\sigma(\sigma_s(r),p_s(r)), \quad f_p^*(r)=f_p(\sigma_s(r),p_s(r)),
\\
g_\sigma^*(r)=g_\sigma(\sigma_s(r),p_s(r)), \quad g_p^*(r)=g_p(\sigma_s(r),p_s(r)).
\end{array}$$ where $$\left\{
\begin{array}{rcl}
f(\sigma,p)&=&K_P(\sigma)\!+\big[K_M(\sigma)\!-\!K_N(\sigma)\big]p-\!K_M(\sigma)p^2,\\
g(\sigma,p)&=&K_M(\sigma)p-K_D(\sigma),
\end{array}
\right.$$ where $$K_M(\sigma)=K_B(\sigma)+K_D(\sigma), \qquad K_N(\sigma)=K_P(\sigma)+K_Q(\sigma).$$ Note that for all $0\leq r\leq 1$ (see Lemma 3.1 of [@Cui4]), $$f_p^*(r)<0, \quad f_\sigma^*(r)>0, \quad g_p^*(r)>0
\quad \mbox{and}\quad g_\sigma^*(r)>0.$$ For any $\gamma\in\mathbb{R}$, the system (1.12)–(1.18) has the following family of nontrivial solutions: $$\left\{
\begin{array}{l}
\chi(r,\omega)=\sigma_s'(r) z\cdot\omega, \quad
\varphi(r,\omega)=p_s'(r) z\cdot\omega, \quad
\psi(r,\omega)=-v_s(r) z\cdot\omega,
\\
\vec{w}(r,\omega)=\displaystyle\frac{v_s(r)}{r}
[ z-( z\cdot\omega)\omega]+v_s'(r)( z\cdot\omega)\omega,
\quad \eta(\omega)=- z\cdot\omega,
\end{array}
\right.$$ where $z$ is an arbitrary nonzero vector in $\mathbb{R}^n$. This is actually a reflection to the system (1.12)–(1.18) of the property of translation invariance of the system (1.1)–(1.8). Indeed, since $(\sigma_s,p_s,v_s,\varpi_s,\Omega_s)$ is a solution of an equivalent system of (1.1)–(1.8) (see (2.2)–(2.8) in the next section), translation invariance implies that for any $z\in\mathbb{R}^n$ and any $\varepsilon\in\mathbb{R}$ with $|\varepsilon|$ sufficiently small, $(\sigma_\varepsilon,p_\varepsilon,\vec{v}_\varepsilon,\varpi_\varepsilon,\Omega_s
-\varepsilon z)$ is also a solution of that system, where $$\sigma_\varepsilon(x)=\sigma_s(|x+\varepsilon z|), \quad
p_\varepsilon(x)=p_s(|x+\varepsilon z|), \quad
\varpi_\varepsilon(x)=\varpi_s(|x+\varepsilon z|)$$ and $\vec{v}_\varepsilon(x)=v_s(|x+\varepsilon z|)(x+\varepsilon z)/|x+
\varepsilon z|$. Differentiating $(\sigma_\varepsilon,p_\varepsilon,
\vec{v}_\varepsilon,\varpi_\varepsilon,\Omega_s-\varepsilon z)$ in $\varepsilon$ at $\varepsilon=0$, we obtain the above nontrivial solutions of the system (1.12)–(1.18). The purpose of this paper is to investigate for what values of $\gamma$, the system (1.12)–(1.18) has nontrivial solutions different from (1.20), and study invertibility and ranges of some linear operators related to the system (1.12)–(1.18) in certain function spaces.
To state the main result of this paper, we first recall some basic notion of analysis in the unit sphere ${\mathbb S}^{n-1}$. For every $k\in\mathbb{Z}_+=\{0,1,2,
\cdots\}$, let $\lambda_k$ be the $k\!+\!1$-the eigenvalue of the operator $-\Delta_\omega$ and $d_k$ be the dimension of the space $\mathcal{H}_k$ of all spherical harmonics of degree $k$, i.e. (cf. [@SW; @T]) $$\lambda_k=(n+k-2)k \quad \mbox{and} \quad d_k=\dim\mathcal{H}_k, \quad k=0,1,2,\cdots,$$ where $$\mathcal{H}_k=\{\phi\in C^{\infty}({\mathbb S}^{n-1}):
\Delta_\omega\phi=-\lambda_k\phi\}, \quad k=0,1,2,\cdots.$$ Recall that (cf. [@SW]) $$d_0=1, \quad d_1=n \quad\hbox{and}\quad
d_k={n\!+\!k\!-\!1\choose k}-{n\!+\!k\!-\!3\choose k\!-\!2}
\quad\hbox{for}\quad k\geq 2.$$ For every $k\in\mathbb{Z}_+$, let $Y_{kl}(\omega)$, $l=1,2,\cdots,d_k$, be a normalized orthogonal basis of the space $\mathcal{H}_k$, i.e. $$\Delta_\omega Y_{kl}(\omega)=-\lambda_k Y_{kl}(\omega),$$ $$\int_{\mathbb{S}^{n-1}}Y_{kl}(\omega)Y_{kl'}(\omega)d\omega=0\;\; (l\neq l'),
\qquad
\int_{\mathbb{S}^{n-1}}Y_{kl}^2(\omega)d\omega=1,$$ where $d\omega$ is the induced element on $\mathbb{S}^{n-1}$ of the Lebesque measure $dx$ in $\mathbb{R}^n$. Note that in particular, $$Y_{01}(\omega)=\frac{1}{\sqrt{\sigma_n}} \quad \mbox{and} \quad
Y_{1l}(\omega)=\frac{\sqrt{n}\omega_l}{\sqrt{\sigma_n}}, \quad l=1,2,\cdots,n,$$ where $\sigma_n$ denotes the surface area of $\mathbb{S}^{n-1}$, i.e. $\sigma_n=
\displaystyle\frac{2\pi^{n/2}}{\Gamma(n/2)}$, and $\omega_l$ denotes the $l$-th component of $\omega\in\mathbb{S}^{n-1}$ regarded as a vector in $\mathbb{R}^n$. We note that the $\eta$-component of the nontrivial solution given by (1.20) ranges over all nonzero functions in $\mathcal{H}_1$.
The main result of this paper is as follows:
[**Theorem 1.1**]{} [*There exists a null sequence $\{\gamma_k\}_{k=2}^{\infty}$, which is strictly monotone decreasing for sufficiently large $k$ and satisfies the property $\gamma_k\sim ck^{-3}$ as $k\to\infty$, where $c$ is a positive constant independent of $k$, such that if $\gamma=\gamma_k$ for some $k\geq 2$ then the system $(1.12)$–$(1.18)$ has a family of nontrivial solutions with the $\eta$-component ranging over all nonzero functions in $\displaystyle
\bigoplus_{\gamma_{k'}=\gamma_k}\mathcal{H}_{k'}$, so that they are different from $(1.20)$. If $\gamma\not=\gamma_k$ for any $k\geq 2$ then $(1.12)$–$(1.18)$ does not have other nontrivial solutions than $(1.20)$.*]{}
The exact expression of $\gamma_k$ ($k=2,3,\cdots$) will be given in Section 3; see (3.14). The idea for the proof of the above result is as follows: By solving (1.12)–(1.13) and (1.16)–(1.17) in terms of $\eta$ and $\varphi$, we get $\chi$ and $\psi$ as functionals of $\eta$ and $\varphi$. It follows that the system (1.12)–(1.18) reduces into a $2$-system containing only the unknown functions $\eta$ and $\varphi$. In such a reduced system, the equation obtained from (1.14) is a non-local singular differential-integral equation: Singularity comes from the fact that $v_s(0)=v_s(R_s)=0$ (see (2.14) and (2.17) in the next section), and non-localness is caused by the term $\psi_r$ in (1.14) because $\psi$ is the solution of an elliptic boundary value problem containing $\varphi$. This is the main difficulty encountered in the proof of the above theorem. We shall appeal to Fourier expansions of functions in ${\mathbb S}^{n-1}$ via the sequence of spherical harmonics $\{Y_{kl}(\omega):
k=0,1,2,\cdots; l=1,2,\cdots, d_k\}$ and some techniques for solving singular differential equations developed in [@ChenCuiF; @Cui2; @CuiFri1] to overcome this difficulty; see Sections 4 and 5 for details.
In addition to the above result, we shall also study invertibility and ranges of some linear operators related to the system (1.12)–(1.18) in certain function spaces. This has potential applications in the study of non-radial solutions of the original system (1.1)–(1.8). Since the exact statements of such results require a big number of new notations, we leave them for later presentation; see Theorems 6.2 and 6.3 in the last section.
The structure of the rest part is as follows. In the next section we compute the linearization of the system of (1.1)–(1.8) around its radial solution $(\sigma_s,
p_s,q_s,\varpi_s,v_s,\Omega_s)$ and reduce the linearized system into a $2$-system. In Section 3 we use Fourier expansions of functions in ${\mathbb S}^{n-1}$ via spherical harmonics to further reduce the PDE $2$-system into a sequence of ODE systems, and use them to derive the eigenvalues $\gamma_k$, $k=2,3,\cdots$, by assuming existence and uniqueness of a solution to a nonlocal singular differential-integral equation. In Section 4 we give the proof of the assertion stated in the last sentence. Section 5 aims at studying properties of the eigenvalues $\gamma_k$. In the last section we study invertibility and ranges of some linear operators related to the system (1.12)–(1.18) in certain function spaces.
Linearization
=============
2em In this section we derive the system (1.12)–(1.18) and make some basic reduction to it.
We first make a basic simplification to the system (1.1)–(1.8). Firstly, by summing up (1.3) and (1.4) and using (1.5), we get $$\nabla\cdot\vec{v}=K_M(\sigma)p-K_D(\sigma) \quad \mbox{for}\;\; x\in\Omega.$$ Substituting this relation into (1.3) and using (1.5) we get (recall that $\partial_tp
=0$) $$\vec{v}\cdot\nabla p=f(\sigma,p) \quad \mbox{for}\;\; x\in\Omega.$$ Moreover, substituting (1.6) into (1.12) and (1.8) (recall that $V_n=0$) we respectively get $$-\Delta\varpi=g(\sigma,p) \quad \mbox{for}\;\; x\in\Omega,$$ $$\frac{\partial\varpi}{\partial\vec{n}}=0 \quad \mbox{for}\;\;
x\in\partial\Omega.$$ Hence, the time-independent version of the system (1.1)–(1.8) reduces into the following system of equations: $$\begin{aligned}
\Delta\sigma&=&F(\sigma) \quad \mbox{for}\;\; x\in\Omega,
\\
\sigma&=&1 \quad \mbox{for}\;\; x\in\partial\Omega,
\\
\vec{v}\cdot\nabla p&=&f(\sigma,p) \quad \mbox{for}\;\; x\in\Omega,
\\
\vec{v}&=&-\nabla\varpi \quad \mbox{for}\;\; x\in\Omega,
\\
-\Delta\varpi&=&g(\sigma,p) \quad \mbox{for}\;\; x\in\Omega,
\\
\varpi&=&\gamma\kappa \quad \mbox{for}\;\; x\in\partial\Omega,
\\
\displaystyle\frac{\partial\varpi}{\partial\vec{n}}&=&0 \quad
\mbox{for}\;\; x\in\partial\Omega.\end{aligned}$$
Let $(\sigma_s,p_s,\varpi_s,v_s,\Omega_s)$, where $\Omega_s=\{x\in {\mathbb R}^n:\;
r<R_s\}$, be the unique radial stationary solution of (2.2)–(2.8), i.e., $(\sigma_s,p_s,\varpi_s,v_s,R_s)$ is the unique solution of the following system of equations: $$\sigma_s''(r)+\frac{n\!-\!1}{r}\sigma_s'(r)=F(\sigma_s(r)), \quad 0<r<R_s,$$ $$\sigma_s'(0)=0,\quad \sigma_s(R_s)=1,$$ $$v_s(r)p_s'(r)=f(\sigma_s(r),p_s(r)), \quad 0<r<R_s,$$ $$v_s'(r)+\frac{n\!-\!1}{r}v_s(r)=g(\sigma_s(r),p_s(r)), \quad 0<r<R_s,$$ $$v_s(r)=-\varpi_s'(r), \quad 0<r<R_s,$$ $$v_s(0)=0,\quad v_s(R_s)=0,$$ Later on we shall also use the following simplified notations: $$\begin{array}{c}
f^*(r)=f(\sigma_s(r),p_s(r)), \qquad g^*(r)=g(\sigma_s(r),p_s(r)).
\end{array}$$ As we mentioned before, existence and uniqueness of the above system has been proved in [@CuiFri1; @ChenCuiF] in the $3$-dimension case. Moreover, this solution satisfies the following properties (cf. [@CuiFri1]): $$0<\sigma_s(r)<1\;\; \mbox{for}\;\; 0\leq r<R_s, \quad \sigma_s'(r)>0\;\; \mbox{for}\;\; 0<r\leq R_s,$$ $$0<p_s(r)<1\;\; \mbox{for}\;\; 0\leq r<R_s, \quad p_s'(r)>0\;\; \mbox{for}\;\; 0<r\leq R_s,$$ and there exist positive constants $c_1$, $c_2$ such that $$-c_1r(R_s-r)\leq v_s(r)\leq-c_2r(R_s-r)\;\; \mbox{for}\;\; 0\leq r\leq R_s.$$ For the general $n$-dimension case ($n\geq 2$), the argument is quite similar so that we omit it here. Note that the above properties are also valid in the general $n$-dimension case.
Consider a perturbation of $(\sigma_s,p_s,v_s,\varpi_s,\Omega_s)$ of the following form: $$\left\{
\begin{array}{l}
\sigma(x)=\sigma_s(r)+\varepsilon\chi(r,\omega), \qquad
p(x)=p_s(r)+\varepsilon\varphi(r,\omega),\quad
\\
\varpi(x)=\varpi_s(r)+\varepsilon\psi(r,\omega), \quad\;\;
\vec{v}(x)=v_s(r)\omega+\varepsilon\vec{w}(r,\omega), \quad
\\
\Omega=\{x\in {\mathbb R}^n:\; r<R_s+\varepsilon\eta(\omega)\},
\end{array}
\right.$$ where $r=|x|$, $\omega=x/|x|$, $\varepsilon$ is a small parameter and $\chi$, $\varphi$, $\psi$, $\vec{w}$, $\eta$ are new unknown functions. Substituting these expressions into (2.2)–(2.8), making the first-order Taylor expansions to all nonlinear functions containing $\varepsilon$, subtracting the corresponding equations in (2.9)–(2.14), then dividing both sides of all equations with $\varepsilon$ and finally letting $\varepsilon\to 0$, we obtain the system (1.12)–(1.18).
Indeed, deductions of the equations (1.12), (1.13), (1.15), (1.16) and (1.18) are quite standard, see [@CuiEsc1; @CuiEsc2] for instance. To get (1.17) we need to use the following asymptotic formula for the mean curvature $\kappa$ of the hypersurface $r=R_s+\varepsilon\eta(\omega)$ (cf. [@FriRei3]): $$\kappa=\frac{1}{R_s}-\frac{\varepsilon}{R_s}[\eta(\omega)
+{1\over n\!-\!1}\Delta_\omega\eta(\omega)]+o(\varepsilon).$$ Here we only give the deduction of the equation (1.14). Substituting the relations $\sigma(x)=\sigma_s(r)+\varepsilon\chi(x)$, $p(x)=
p_s(r)+\varepsilon\varphi(x)$ and $\vec{v}(x)=v_s(r)\omega+\varepsilon\vec{w}(x)$ into the third equation in (2.4), we get $$[v_s(r)\omega+\varepsilon\vec{w}]\cdot
[\nabla p_s(r)+\varepsilon\nabla\varphi]
=f(\sigma_s(r)+\varepsilon\chi,p_s(r)+\varepsilon\varphi).$$ By (2.10) we have $$v_s(r)\omega\cdot\nabla p_s(r)=v_s(r)p_s'(r)=f(\sigma_s(r),p_s(r)).$$ Subtracting both sides of (2.18) with the left and the right terms in (2.19), respectively, next dividing both sides with $\varepsilon$, using the first-order Taylor expansion of the function $f$ at the point $(\sigma_s(r),p_s(r))$ and finally letting $\varepsilon\to 0$, we get $$v_s(r)\omega\cdot\nabla\varphi+\vec{w}\cdot\nabla p_s(r)
=f_\sigma(\sigma_s(r),p_s(r))\chi+f_p(\sigma_s(r),p_s(r))\varphi.$$ Note that $\omega\cdot\nabla\varphi=\varphi_r$ and, by virtue of (1.15), $$\vec{w}\cdot\nabla p_s(r)=-\nabla\psi\cdot p_s'(r)\omega=-p_s'(r)\psi_r.
$$ Substituting these expressions into (2.20), we see that (1.14) follows.
Since all the rest equations in (1.12)–(1.18) can be decoupled from (1.15), in what follows we neglect (1.15). This system can be reduced into a 2-system of linear equations in the unknowns $\varphi$ and $\eta$ only. To see this we denote by $\mathscr{J}$, $\mathscr{J}_0$ and $\mathscr{G}$ respectively the following operators: Given $\eta\in C^2(\mathbb{S}^{n-1})$, we let $u=\mathscr{J}(\eta)\in
C^{2*}(\overline{\Omega}_s)$ and $v=\mathscr{J}_0(\eta)\in C^{2*}(\overline{\Omega}_s)$, where $C^{2*}(\overline{\Omega}_s)$ denotes the second-order Zygmund space on $\overline{\Omega}_s$, be respectively solutions of the following elliptic boundary value problems: $$\left\{
\begin{array}{l}
\Delta u=F'(\sigma_s(r))u, \quad x\in\Omega_s,
\\
u|_{x=R_s\omega}=\eta(\omega), \quad \omega\in\mathbb{S}^{n-1};
\end{array}
\right.$$ $$\left\{
\begin{array}{l}
\Delta v=0, \quad x\in\Omega_s,
\\
v|_{x=R_s\omega}=\eta(\omega), \quad \omega\in\mathbb{S}^{n-1}.
\end{array}
\right.$$ Next, given $h\in C(\overline{\Omega}_s)$, we let $w=\mathscr{G}(h)\in
C^{2*}(\overline{\Omega}_s)$ be the solution of the following elliptic boundary value problem: $$\left\{
\begin{array}{l}
\Delta w=h, \quad x\in\Omega_s,
\\
w=0, \quad x\in\partial\Omega_s.
\end{array}
\right.$$ Then from (1.12), (1.13), (1.16) and (1.17) we have $$\chi=-\sigma_s'(R_s)\mathscr{J}(\eta),\qquad
\psi=\Phi+\Upsilon+\Psi,$$ where $$\left\{
\begin{array}{l}
\Phi=-\mathscr{G}[g_p^*(r)\varphi],
\\
\Upsilon=-\mathscr{G}[g_\sigma^*(r)\chi]
=\sigma_s'(R_s)\mathscr{G}[g_\sigma^*(r)\mathscr{J}(\eta)],
\\
\Psi=\displaystyle-{\gamma\over R_s^2}\mathscr{J}_0(\eta+{1\over n\!-\!1}\Delta_\omega\eta).
\end{array}
\right.$$ Substituting these expressions into (1.14) and (1.18), we see that the system (1.12)–(1.18) reduces into the following $2$-system: $$\left\{
\begin{array}{l}
\mathscr{A}_{\gamma}(\varphi,\eta)=0,
\\
\mathscr{B}_{\gamma}(\varphi,\eta)=0,
\end{array}
\right.$$ where $$\begin{aligned}
\mathscr{A}_{\gamma}(\varphi,\eta)&=&-v_s(r)\partial_r\varphi+f_p^*(r)\varphi+p_s'(r)\partial_r\Phi
+p_s'(r)\partial_r\Upsilon+p_s'(r)\partial_r\Psi+f_\sigma^*(r)\chi
\nonumber\\
&=&-v_s(r)\partial_r\varphi+f_p^*(r)\varphi-p_s'(r)\partial_r\mathscr{G}[g_p^*(r)\varphi]
+\sigma_s'(R_s)p_s'(r)\partial_r\mathscr{G}[g_\sigma^*(r)\mathscr{J}(\eta)]
\nonumber\\
&&\displaystyle-{\gamma\over R_s^2}p_s'(r)\partial_r\mathscr{J}_0(\eta
+{1\over n\!-\!1}\Delta_\omega\eta)
-\sigma_s'(R_s)f_\sigma^*(r)\mathscr{J}(\eta),
\\
\mathscr{B}_{\gamma}(\varphi,\eta)&=&\displaystyle-\partial_r\Phi|_{r=R_s}
-\partial_r\Upsilon|_{r=R_s}-\partial_r\Psi|_{r=R_s}+g(1,1)\eta
\nonumber\\
&=&\displaystyle \partial_r\mathscr{G}[g_p^*(r)\varphi]|_{r=R_s}
-\sigma_s'(R_s)\partial_r\mathscr{G}[g_\sigma^*(r)\mathscr{J}(\eta)]|_{r=R_s}
\nonumber\\
&&+\displaystyle{\gamma\over R_s^2}\partial_r\mathscr{J}_0(\eta
+{1\over n\!-\!1}\Delta_\omega\eta)|_{r=R_s}+g(1,1)\eta.\end{aligned}$$
Hence, to get nontrivial solutions of the system (1.12)–(1.18) we only need to find nontrivial solutions of the system (2.21). This is the task of the next two sections.
We note that the operator $\varphi\mapsto\mathscr{A}_{\gamma}(\varphi,\eta)$ (for fixed $\eta$) is a first-order nonlocal singular differential-integral operator. Since the Dirichlet-Neumann operator $\eta\mapsto\partial_r\mathscr{G}(\eta)|_{r=R_s}$ is a first-order elliptic pseudo-differential operator in $\mathbb{S}^{n-1}$ (cf. [@ES]), and $\Delta_\omega$ is a second-order elliptic partial differential operator in $\mathbb{S}^{n-1}$, we see that the operator $\eta\mapsto\mathscr{B}_{\gamma}
(\varphi,\eta)$ (for fixed $\varphi$) is a third-order elliptic pseudo-differential operator in the unit sphere $\mathbb{S}^{n-1}$. Main difficulty for solving the system (2.21) comes from the singularity and non-localness of the operator $\mathscr{A}_{\gamma}$.
Expansion via spherical harmonics
=================================
2em Recall that in the polar coordinate $(r,\omega)$ the Laplacian $\Delta$ on $R^{n}$ has the following expression (cf. [@SW; @T]): $$\Delta=\frac{\partial^2}{\partial r^2}
+\frac{n\!-\!1}{r}\frac{\partial}{\partial r}
+\frac{1}{r^2}\Delta_\omega.$$ Let $Y_{kl}$, $k=0,1,2,\cdots$, $l=1,2,\cdots,d_k$, be the basis of spherical harmonics introduced in Section 1. We expand $\varphi$ and $\eta$ in (2.21) via $Y_{kl}$’s: $$\varphi(r,\omega)=\displaystyle\sum_{k=0}^\infty
\displaystyle\sum_{l=1}^{d_k}\varphi_{kl}(r)Y_{kl}(\omega),
\qquad
\eta(\omega)=\displaystyle\sum_{k=0}^\infty
\displaystyle\sum_{l=1}^{d_k}y_{kl}Y_{kl}(\omega).$$ Convergence of the first series is considered in $\mathscr{D}'(\mathbb{B}(0,R_s))=
\mathscr{D}'((0,R_s),\mathscr{D}'(\mathbb{S}^{n-1}))$, and the second one is considered in $\mathscr{D}'(\mathbb{S}^{n-1})$. A simple computation shows that $$\left\{
\begin{array}{l}
\mathscr{A}_{\gamma}(\varphi,\eta)=\displaystyle\sum_{k=0}^\infty\sum_{l=1}^{d_k}
\Big[\mathscr{L}_k(\varphi_{kl})+b_k(r,\gamma)y_{kl}\Big]Y_{kl}(\omega),
\\
\mathscr{B}_{\gamma}(\varphi,\eta)=\displaystyle\sum_{k=0}^\infty\sum_{l=1}^{d_k}
\Big[J_k(\varphi_{kl})+\alpha_k(\gamma)y_{kl}\Big]Y_{kl}(\omega),
\end{array}
\right.$$ where $$\begin{aligned}
\alpha_k(\gamma)&=&\Big(1-\frac{\lambda_k}{n\!-\!1}\Big)
\frac{k\gamma}{R_s^3}+g(1,1)-\frac{\sigma_s'(R_s)}{R_s^{n+2k-1}}
\int_0^{R_s}\rho^{n+2k-1}g_\sigma^*(\rho)u_k(\rho)d\rho,
\\
b_k(r,\gamma)
&=&\displaystyle-\Big(1-{\lambda_k\over n\!-\!1}\Big)\gamma
kR_s^{-k-2}r^{k-1}p_s'(r)-\sigma_s'(R_s)R_s^{-k}f_\sigma^*(r)r^ku_k(r)
\nonumber\\
&&\displaystyle-\sigma_s'(R_s)R_s^{-k}r^{k-1}p_s'(r)\Big[\theta_k
\int_r^{R_s}\rho g_\sigma^*(\rho)u_k(\rho)d\rho-\frac{1-\theta_k}{r^{n+2(k-1)}}
\int_0^r\rho^{n+2k-1}g_\sigma^*(\rho)u_k(\rho)d\rho
\nonumber\\
&&\displaystyle-\frac{\theta_k}{R_s^{n+2(k-1)}}
\int_0^{R_s}\rho^{n+2k-1}g_\sigma^*(\rho)u_k(\rho)d\rho\Big],\end{aligned}$$ where $\theta_k=\displaystyle\frac{k}{n\!+\!2(k\!-\!1)}$, and for $\phi=\phi(r)$, $$\begin{aligned}
\mathscr{L}_k(\phi)
&=&\displaystyle-v_s(r)\phi'(r)+f_p^*(r)\phi(r)+r^{k-1}p_s'(r)
\Big[{\theta_k}\int_r^{R_s}\rho^{-k+1}g_p^*(\rho)\phi(\rho)d\rho
\nonumber\\
&&\displaystyle-\frac{1-\theta_k}{r^{n+2(k-1)}}
\int_0^r\rho^{n+k-1}g_p^*(\rho)\phi(\rho)d\rho-\frac{\theta_k}{R_s^{n+2(k-1)}}
\int_0^{R_s}\rho^{n+k-1}g_p^*(\rho)\phi(\rho)d\rho\Big],
\\
J_k(\phi)&=&\frac{1}{R_s^{n+k-1}}\int_0^{R_s}\rho^{n+k-1}g_p^*(\rho)\phi(\rho)d\rho.\end{aligned}$$ [**Lemma 3.1**]{} [*Given $\gamma\in\mathbb{R}$, the system (2.21) has a nontrivial solution if and only if there exists a nonnegative integer $k$ such that the following system has a nontrivial solution:* ]{} $$\begin{aligned}
\left\{
\begin{array}{l}
\mathscr{L}_k(\phi_k)+b_k(r,\gamma)y_k=0 \quad \mbox{for}\;\; 0<r<R_s,
\\
J_k(\phi_k)+\alpha_k(\gamma)y_k=0.
\end{array}
\right.\end{aligned}$$ [*Proof*]{}: Indeed, if $(\phi_k,y_k)$ is a nontrivial solution of the above system, then from (3.3) we see that for any $1\leq l\leq d_k$, $(\varphi(r,\omega),\eta(\omega))
=(\phi_k(r)Y_{kl}(\omega),y_kY_{kl}(\omega))$ is a nontrivial solution of the system (2.12). Conversely, if $(\varphi(r,\omega),\eta(\omega))$ is a nontrivial solution of the system (2.12), then by expanding $\varphi(r,\omega)$ and $\eta(\omega))$ into the expressions in (3.2), there must be a pair of $k$ and $l$ such that $(\varphi_{kl},
y_{kl})\neq (0,0)$. By (3.3), we see that $(\phi_k,y_k)=(\varphi_{kl},y_{kl})$ is a nontrivial solution of (3.8). This proves the lemma. $\quad\Box$
For every $k\in\mathbb{Z}_+$ we denote by $\tilde{\mathscr{L}}_k$ the following linear differential-integral operator in $(0,R_s)$: for $\phi=\phi(r)$, $$\begin{aligned}
\tilde{\mathscr{L}}_k(\phi)&=&\displaystyle\mathscr{L}_k(\phi)
+R_s^{-(k-1)}r^{k-1}p_s'(r)J_k(\phi)
\nonumber\\ [0.3cm]
&=&\displaystyle-v_s(r)\phi'(r)+f_p^*(r)\phi(r)
+r^{k-1}p_s'(r)\Big[{\theta_k}
\int_r^{R_s}\rho^{-k+1}g_p^*(\rho)\phi(\rho)d\rho
\nonumber\\ [0.3cm]
&&\displaystyle+\frac{1-\theta_k}{R_s^{n+2(k-1)}}
\int_0^{R_s}\rho^{n+k-1}g_p^*(\rho)\phi(\rho)d\rho
-\frac{1-\theta_k}{r^{n+2(k-1)}}
\int_0^r\rho^{n+k-1}g_p^*(\rho)\phi(\rho)d\rho\Big],\qquad\end{aligned}$$ and let $$\begin{aligned}
\tilde{b}_k(r)&=&b_k(r,\gamma)+R_s^{-(k-1)}r^{k-1}p_s'(r)\alpha_k(\gamma)
\nonumber\\
&=&\displaystyle \frac{g(1,1)}{R_s^{k-1}}r^{k-1}p_s'(r)
-\frac{\sigma_s'(R_s)}{R_s^k}r^kf_\sigma^*(r)u_k(r)-\frac{\sigma_s'(R_s)}{R_s^k}r^{k-1}p_s'(r)
\Big[\theta_k\int_r^{R_s}\rho g_\sigma^*(\rho)u_k(\rho)d\rho
\nonumber\\
&&\displaystyle+\frac{1-\theta_k}{R_s^{n+2(k-1)}}
\int_0^{R_s}\rho^{n+2k-1}g_\sigma^*(\rho)u_k(\rho)d\rho
-\frac{1-\theta_k}{r^{n+2(k-1)}}\int_0^r\rho^{n+2k-1}g_\sigma^*(\rho)u_k(\rho)d\rho\Big].\end{aligned}$$ [**Lemma 3.2**]{} [*For fixed $\gamma\in\mathbb{R}$ and $k\in\mathbb{Z}_+$, the system (3.8) has a nontrivial solution $(\phi_k,y_k)$ if and only if the following system has a solution $\psi_k$: $$\begin{aligned}
\tilde{\mathscr{L}}_k(\psi_k)+\tilde{b}_k(r)&=&0,
\\ [0.3cm]
J_k(\psi_k)+\alpha_k(\gamma)&=&0.\end{aligned}$$ More precisely, if $\psi_k$ is a solution of the above system then for any nonzero constant $c$, $(\phi_k,y_k)=(c\psi_k,c)$ is a nontrivial solution of $(3.8)$, and conversely, if $(\phi_k,y_k)$ is a nontrivial solution of $(3.8)$ then $y_k\neq 0$ and $\psi_k(r)=y_k^{-1}\phi_k(r)$ is a solution of the above system.*]{}
[*Proof*]{}: Later we shall see that the system of equations $\mathscr{L}_k(\phi)
=0$ and $J_k(\phi)=0$ has only the trivial solution $\phi=0$ (see the remark following Lemma 4.4). It follows that if $(\phi_k,y_k)$ is a nontrivial solution of the system (3.8), then $y_k\neq 0$. Let $\psi_k(r)=y_k^{-1}\phi_k(r)$. Then the system (3.8) reduces into the equation $$\begin{aligned}
\mathscr{L}_k(\psi_k)+b_k(r,\gamma)&=&0 \quad \mbox{for}\;\; 0<r<R_s\end{aligned}$$ coupled by the equation (3.12). Multiplying (3.12) with $R_s^{-(k-1)}r^{k-1}p_s'(r)$ and adding it into (3.13), we get (3.11). Conversely, it is easy to check that if $\psi_k$ is a solution of the system (3.11)–(3.12) then for any nonzero constant $c$, $(\phi_k,y_k)=(c\psi_k,c)$ is a nontrivial solution of (3.8). This proves the lemma. $\quad\Box$
We note that for fixed $\gamma\in\mathbb{R}$ and $k\in\mathbb{Z}_+$, (3.11)–(3.12) is an over-determined system. Hence, later on for fixed $k\in\mathbb{Z}_+$ we shall regard (3.8) as an eigenvalue problem by regarding $\gamma$ as the eigenvalue variable. In the next section we shall prove that for every $k\in\mathbb{Z}_+$, the equation (3.11) has a unique solution $\psi_k\in C[0,R_s]$. It follows that the system (3.11)–(3.12) has a solution if and only if $\gamma$ satisfies the equation (3.12). For each $k\geq 2$ we let $$\begin{array}{rl}
\gamma_k=&\displaystyle {(n\!-\!1)R_s^3\over (\lambda_k\!-\!n\!+\!1)k}
\Big[g(1,1)-\frac{\sigma_s'(R_s)}{R_s^{n+2k-1}}
\int_0^{R_s}\xi^{n+2k-1}g_\sigma^*(\xi)u_k(\xi)d\xi
\\ [0.3cm]
&\displaystyle +\frac{1}{R_s^{n+k-1}}
\int_0^{R_s}\xi^{n+k-1}g_p^*(\xi)\psi_k(\xi)d\xi\Big].
\end{array}$$ Then $$J_k(\psi_k)+\alpha_k(\gamma)=-\frac{(\lambda_k\!-\!n\!+\!1)k}{(n\!-\!1)R_s^3}
(\gamma-\gamma_k).$$ Hence, we have the following result:
[**Lemma 3.3**]{} [*For $k\geq 2$, the system $(3.8)$ has a nontrivial solution if and only if $\gamma=\gamma_k$.*]{}
[*Proof*]{}: See Corollary 4.6 in the next section. $\quad\Box$
For $k=0,1$ it is clear that $\alpha_0$, $\alpha_1$, $b_0$ and $b_1$ are independent of $\gamma$, so that the system (3.8) does not contain $\gamma$ in these cases.
[**Lemma 3.4**]{} [*For $k=1$ we have $\psi_1(r)=-p_s'(r)$ and $J_1(\psi_1)
+\alpha_1=0$.*]{}
[*Proof*]{}: Indeed, since $u_1(r)=\displaystyle\frac{R_sc_s'(r)}{rc_s'(R_s)}$ (see Lemma 4.1 in the next section), by using the equations (2.10), (2.12), (2.14) and the equality $v_s(r)=\displaystyle\frac{1}{r^{n-1}}\int_0^r\rho^{n-1}g^*(\rho)
d\rho$ implied by (2.12), we see that $$\begin{aligned}
\tilde{\mathscr{L}}_1[-p_s'(r)]+\tilde{b}_1(r)&=&v_s(r)p_s''(r)-f_p^*(r)p_s'(r)
-f_\sigma^*(r)\sigma_s'(r)+g(1,1)p_s'(r)
\\
&&-p_s'(r)\Big[\theta_1\!\!\int_r^{R_s}\!\!\frac{d}{d\rho}g^*(\rho)d\rho
+\frac{1\!-\!\theta_1}{R_s^n}\!\int_0^{R_s}\!\!\rho^n\frac{d}{d\rho}g^*(\rho)d\rho
-\frac{1\!-\!\theta_1}{r^n}\!\int_0^r\!\!\rho^n\frac{d}{d\rho}g^*(\rho)d\rho\Big]
\\
&=&v_s(r)p_s''(r)-f_p^*(r)p_s'(r)-f_\sigma^*(r)\sigma_s'(r)+v_s'(r)p_s'(r)
\\
&=&[v_s(r)p_s'(r)-f^*(r)]'=0.\end{aligned}$$ Hence $\psi_1(r)=-p_s'(r)$. Consequently, we have $$\begin{aligned}
J_1(\psi_1)+\alpha_1&=&-\frac{1}{R_s^n}\!\int_0^{R_s}\!\!\rho^ng_p^*(\rho)p_s'(\rho)d\rho
+g(1,1)-\frac{1}{R_s^n}\!\int_0^{R_s}\!\!\rho^ng_\sigma^*(\rho)\sigma_s'(\rho)d\rho=0.\end{aligned}$$ This proves the lemma. $\quad\Box$
The above lemma implies that in the case $k=1$, the system (3.8) has nontrivial solutions for all $\gamma\in\mathbb{R}$. This is actually a restatement of the fact that (1.20) are nontrivial solutions of the system (1.12)–(1.18) for all $\gamma\in\mathbb{R}$.
[**Lemma 3.5**]{} [*For $k=0$ the system $(3.8)$ does not have a nontrivial solution.*]{}
[*Proof*]{}: Since as a stationary solution of the corresponding time-dependent system of (1.1)–(1.8), $(\sigma_s,p_s,q_s,v_s,\varpi_s,\Omega_s)$ is asymptotically stable under radial perturbations, it follows that in the case $k=0$ the system (3.8) cannot have a nontrivial solution. This is an implicit proof. We can also give an explicit proof by repeating some arguments in [@ChenCuiF]. To save spaces we omit it here. $\quad\Box$
It remains to prove existence and uniqueness of a solution for (3.11). This is the task of the next section.
Existence and uniqueness of the solution of (3.11)
==================================================
2em In this section we prove existence and uniqueness of the solution of (3.11). We need the following preliminary lemma:
[**Lemma 4.1**]{}
*Let $u_k(r)$ be the solution of the problem (3.1). We have the following assertions:*
$(1)$ $u_k\in C^{\infty}[0,R_s]$, and $0<u_k(r)\leq 1$ for $0\leq r\leq R_s$.
$(2)$ There exists a constant $C>0$ independent of $k$ such that $$1-\frac{C}{n+2k}(R_s-r)\leq u_k(r)\leq 1 \quad \mbox{for}\;\;
0\leq r\leq R_s,$$ $$0\leq u_k'(r)\leq\frac{Cr}{n+2k} \quad \mbox{for}\;\;
0\leq r\leq R_s.$$ $(3)$ $u_k(r)$ is monotone non-decreasing in $k$, i.e., $u_k(r)\geq u_l(r)$ for $0\leq r\leq R_s$ and $k>l$.
$(4)$ $u_1(r)=\displaystyle\frac{R_sc_s'(r)}{rc_s'(R_s)}$.
[*Proof*]{}: See Lemma 3.3 of [@Cui4]. $\quad\Box$
In the next lemma we shall use the following notations: $$\alpha_0=\frac{f_p^*(0)}{v_s'(0)}, \qquad \alpha_1=-{f^*_p(R_s)\over v_s'(R_s)}.$$ Note that from (1.19) and (2.17) we have $\alpha_0,\alpha_1>0$.
[**Lemma 4.2**]{} [*For any $h\in C[0,R_s]$, the equation $$-v_s(r)\varphi'(r)+f_p^*(r)\varphi(r)=h(r) \quad \mbox{for}\;\;0<r<R_s$$ has a unique solution $\varphi\in C[0,R_s]\cap C^1(0,R_s)$, with boundary values $$\varphi(0)=\frac{h(0)}{f_p^*(0)} \quad \mbox{and} \quad
\varphi(R_s)=\frac{h(R_s)}{f_p^*(R_s)}.$$ Moreover, there exists a constant $C>0$ independent of $h$ such that $$\max_{0\leq r\leq R_s}|\varphi(r)|\leq C\max_{0\leq r\leq R_s}|h(r)|.$$ If furthermore $h(r)=O(r^{\mu})$ as $r\to 0^+$ for some constants $\mu>0$, then $$|\varphi(r)|\leq Cm_{\mu}(r) \quad \mbox{for}\;\; 0<r<R_s,$$ where $$m_{\mu}(r)=
\left\{
\begin{array}{ll}
r^{\alpha_0}, &\quad \mbox{if}\;\; \mu>{\alpha_0},
\\
r^{{\alpha_0}}\ln(\frac{2R_s}{r}), &\quad \mbox{if}\;\; \mu={\alpha_0},
\\
r^{\mu}, &\quad \mbox{if}\;\; \mu<{\alpha_0}.
\end{array}
\right.$$ Moreover, if $h\in C^{\infty}(0,R_s]$ then also $\varphi\in C^{\infty}(0,R_s]$.*]{}
[*Proof*]{}: The first two assertions follow from Lemma 4.1 of [@Cui4]. Here we only give the proof of the last two assertions. Choose an $r_0\in (0,R_s)$ and set $$W(r)=\exp\Big(-\int^r_{r_0}{f^*_p(\rho)\over v_s(\rho)}d\rho\Big)
\quad \mbox{for}\quad 0<r<R_s.$$ It is easy to see that $W\in C^{\infty}(0,R_s)$, $W(r)>0$ for $0<r<R_s$, and $$\begin{aligned}
W(r)&=&C_0r^{-\alpha_0}\big(1+o(1)\big)\;\;\; {\rm as}\;\; r\to 0^+,
\\
W(r)&=&C_1(R_s-r)^{\alpha_1}\big(1+o(1)\big)\;\;\; {\rm as}\;\; r\to R_s^-,\end{aligned}$$ where $C_0,C_1$ are positive constants depending on the choice of $r_0$. From the proof of Lemma 4.1 of [@Cui4] we see that the unique solution of the equation (4.3) in the class $C[0,R_s]\cap C^1(0,R_s)$ is given by (4.4) and $$\varphi(r)={1\over W(r)}\int^{R_s}_r{h(\eta)W(\eta)\over v_s(\eta)}d\eta
\quad \mbox{for}\;\;0<r<R_s.$$ From (2.17), (4.8), (4.9) and the hypothesis that $h(r)=O(r^{\mu})$ as $r\to 0^+$ we have $$\Big|{h(r)W(r)\over v_s(r)}\Big|\leq Cr^{\mu-\alpha_0-1}(R_s-r)^{\alpha_1-1}
\quad \mbox{for}\;\;0<r<R_s.$$ This implies that $$\Big|\int^{R_s}_r{h(\eta)W(\eta)\over v_s(\eta)}d\eta\Big|\leq
\left\{
\begin{array}{ll}
C, &\quad \mbox{if}\;\; \mu>{\alpha_0},
\\
C\ln(\frac{2R_s}{r}), &\quad \mbox{if}\;\; \mu={\alpha_0},
\\
Cr^{\mu-\alpha_0}, &\quad \mbox{if}\;\; \mu<{\alpha_0}.
\end{array}
\right.
\quad \mbox{for}\;\;0<r<R_s.$$ Hence, using (4.8) once again we obtain the estimate (4.6).
Next we assume that $h\in C^1(0,R_s]$. Then clearly the unique solution of (4.3) obtained above satisfies $\varphi\in C^2(0,R_s)$. To show that $\varphi(r)$ is continuously differentiable at $r=R_s$ we differentiate both sides of (4.3) to get $$-v_s(r)[\varphi'(r)]'+[f_p^*(r)-v_s'(r)]\varphi'(r)=h_1(r) \quad \mbox{for}\;\;0<r<R_s,$$ where $h_1(r)=h'(r)-f_p^{*'}(r)\varphi(r)$. It follows that $$\varphi'(r)={1\over W_1(r)}\Big[c_1-\int_{r_0}^r{h_1(\eta)W_1(\eta)\over v_s(\eta)}d\eta\Big]
\quad \mbox{for}\;\;0<r<R_s,$$ where $c_1=\varphi'(r_0)$ and $W_1(r)=\displaystyle\exp\Big(-\int^r_{r_0}{f^*_p(\rho)-
v_s'(\rho)\over v_s(\rho)}d\rho\Big)$. It is easy to see that $$W_1(r)=C(R_s-r)^{\alpha_1+1}\big(1+o(1)\big)\;\;\; {\rm as}\;\; r\to R_s^-$$ for some constant $C>0$. It follows that if $c_1\neq\displaystyle\int_{r_0}^{R_s}
{h_1(\eta)W_1(\eta)\over v_s(\eta)}d\eta$ then $$\varphi'(r)=C'(R_s-r)^{-\alpha_1-1}\big(1+o(1)\big)\;\;\; {\rm as}\;\; r\to R_s^-$$ for some nonzero constant $C'$, which will lead to the absurd conclusion that $|\varphi(r)|\to\infty$ as $r\to R_s^-$. Hence we must have $c_1=\displaystyle
\int_{r_0}^{R_s}{h_1(\eta)W_1(\eta)\over v_s(\eta)}d\eta$ and, consequently, $$\lim_{r\to R_s^-}\varphi'(r)=-\lim_{r\to R_s^-}\frac{1}{W_1'(r)}\cdot
\frac{h_1(r)W_1(r)}{v_s(r)}=\frac{h_1(R_s)}{f_p^*(R_s)-v_s'(R_s)},$$ i.e., $\varphi(r)$ is continuously differentiable at $r=R_s$. Using an induction method we can finally prove that if $h\in C^{\infty}(0,R_s]$ then also $\varphi\in
C^{\infty}(0,R_s]$. This completes the proof of Lemma 4.2. $\quad\Box$
For every integer $k\geq 2$, we introduce a differential-integral operator $\tilde{\mathscr{L}}_{k}^0$ in $(0,R_s)$ as follows: For $\varphi\in C(0,R_s]\cap C^1(0,R_s)$, $$\begin{array}{rcl}
\tilde{\mathscr{L}}_{k}^0(\varphi)&=&\displaystyle -v_s(r)\varphi'(r)
+f_p^*(r)\varphi(r)+r^{k-1}p_s'(r)\Big[
\theta_k\int_r^{R_s}\xi^{-k+1}g_p^*(\xi)\varphi(\xi)d\xi
\\ [0.3cm]
&&\displaystyle+\frac{1-\theta_k}{r^{n+2(k-1)}}
\int_r^{R_s}\xi^{n+k-1}g_p^*(\xi)\varphi(\xi)d\xi\Big]
\quad \mbox{for}\;\;0<r<R_s.
\end{array}$$
[**Lemma 4.3**]{}
*Let $k\geq 2$, $h\in C(0,R_s]$ and consider the equation $$\tilde{\mathscr{L}}_k^0(\varphi)=h \quad \mbox{in}\;\;(0,R_s).$$ We have the following assertions:*
$(1)$ The above equation has a solution $\varphi\in C(0,R_s]\cap C^1(0,R_s)$ which is unique in the class $L^{\infty}_{\rm loc}(0,R_s]$, and $\displaystyle\varphi(R_s)
=\frac{h(R_s)}{f_p^*(R_s)}$.
$(2)$ If $h\in C^{\infty}(0,R_s]$ then also $\varphi\in C^{\infty}(0,R_s]$.
$(3)$ If $h(r)\geq 0$ for $0<r\leq R_s$ then $\varphi(r)\leq 0$ for $0<r\leq R_s$.
$(4)$ If $|h(r)|\leq Cr^{-a}$ for $0<r\leq R_s$ for some $a<n+k$, then $\displaystyle\int^{R_s}_0\xi^{n+k-1}|\varphi(\xi)|d\xi<\infty$ or more precisely, $$\int^{R_s}_0\xi^{n+k-1}|\varphi(\xi)|d\xi\leq C\int^{R_s}_0\!\!\int^{R_s}_{\xi}
{\xi^{n+k-1}W(\eta)|h(\eta)|\over W(\xi)|v_s(\eta)|}d\eta d\xi<\infty.$$ Here $C$ is a positive constant independent of $k$.
[*Proof*]{}: The proof uses some similar arguments as in the proof of Lemma 4.4 of [@Cui4]; but for completeness we write it below.
The equation (4.11) can be explicitly rewritten as follows: $$\begin{aligned}
\displaystyle -v_s(r)\varphi'(r)&+&f_p^*(r)\varphi(r)
+\theta_kr^{k-1}p_s'(r)\int_r^{R_s}\xi^{-k+1}g_p^*(\xi)\varphi(\xi)d\xi
\nonumber\\
&+&\displaystyle\frac{(1-\theta_k)p_s'(r)}{r^{n+k-1}}
\int_r^{R_s}\xi^{n+k-1}g_p^*(\xi)\varphi(\xi)d\xi=h(r).\end{aligned}$$ Let $W(r)$ be as before. By rewriting the above equation in the form $$\begin{array}{c}
\displaystyle\frac{d}{dr}\Big(W(r)\varphi(r)\Big)
={W(r)\over v_s(r)}\Big[-h(r)+\theta_kr^{k-1}p_s'(r)
\int_r^{R_s}\xi^{-k+1}g_p^*(\xi)\varphi(\xi)d\xi
\\ [0.3cm]
+\displaystyle\frac{(1-\theta_k)p_s'(r)}{r^{n+k-1}}
\int_r^{R_s}\xi^{n+k-1}g_p^*(\xi)\varphi(\xi)d\xi\Big],
\end{array}$$ we can apply a similar argument as in the proof of Theorem 5.3 (1) of [@ChenCuiF] to show that, as far as solutions which are bounded near $r=R_s$ are concerned, the differential-integral equation (4.13) is equivalent to the following integral equation: $$\begin{array}{c}
\displaystyle\varphi(r)=-{1\over W(r)}\int^{R_s}_r{W(\eta)\over v_s(\eta)}\Big[
-h(\eta)+\theta_k\eta^{k-1}p_s'(\eta)
\int_{\eta}^{R_s}\xi^{-k+1}g_p^*(\xi)\varphi(\xi)d\xi
\\ [0.3cm]
\displaystyle+\frac{(1-\theta_k)p_s'(\eta)}{\eta^{n+k-1}}
\int_\eta^{R_s}\xi^{n+k-1}g_p^*(\xi)\varphi(\xi)d\xi\Big]d\eta.
\end{array}$$ It then follows from the standard contraction mapping argument that there exists a sufficiently small $\delta>0$ such that (4.13) has a unique bounded solution in the interval $(R_s-\delta,R_s)$, such that $\varphi\in C(R_s-\delta,R_s]\cap
C^1(R_s-\delta,R_s)$, and $$\varphi(R_s)=\lim_{r\to R_s^-}{1\over W(r)}\int^{R_s}_r{W(\eta)\over v_s(\eta)}
h(\eta)d\eta=\frac{h(R_s)}{f_p^*(R_s)}.$$ Since $v_s(r)\neq 0$ for $0<r<R_s$, by standard ODE theory we can uniquely extend the solution to the whole interval $(0,R_s)$. This proves the assertion (1). The assertion (2) follows from a similar argument as in the proof of Lemma 4.2. The assertion (3) follows from (4.14) and a standard continuity argument; cf. the proof of Lemma 7.1 of [@ChenCuiF]. To prove the assertion (4) we note that from (4.14) we have $$\begin{array}{rl}
\displaystyle |\varphi(r)|\leq &
\displaystyle{1\over W(r)}\int^{R_s}_r{W(\eta)\over |v_s(\eta)|}\Big[
|h(\eta)|+C\eta^{k-1}p_s'(\eta)\int_{\eta}^{R_s}\xi^{-k+1}|\varphi(\xi)|d\xi
\\ [0.3cm]
&\displaystyle +\frac{Cp_s'(\eta)}{\eta^{n+k-1}}\int_\eta^{R_s}\xi^{n+k-1}|\varphi(\xi)|d\xi\Big]d\eta
\\ [0.3cm]
\leq &\displaystyle{1\over W(r)}\int^{R_s}_r{W(\eta)\over |v_s(\eta)|}\Big[
|h(\eta)|+\frac{Cp_s'(\eta)}{\eta^{n+k-1}}\int_\eta^{R_s}\xi^{n+k-1}|\varphi(\xi)|d\xi\Big]d\eta.
\end{array}$$ It follows that for any $0<r<r'\leq R_s$ we have $$\begin{array}{rl}
\displaystyle\int^{r'}_r\rho^{n+k-1}|\varphi(\rho)|d\rho\leq &
\displaystyle\int^{r'}_r\!\!\int^{R_s}_{\rho}
{\rho^{n+k-1}W(\eta)|h(\eta)|\over W(\rho)|v_s(\eta)|}d\eta d\rho
\\ [0.3cm]
&\displaystyle+C\int^{r'}_r\!\!\int^{R_s}_{\rho}\!\!\int_{\eta}^{R_s}
{\rho^{n+k-1}W(\eta)p_s'(\eta)\over\eta^{n+k-1}W(\rho)|v_s(\eta)|}
\xi^{n+k-1}|\varphi(\xi)|d\xi d\eta d\rho
\\ [0.3cm]
\leq &\displaystyle\int^{r'}_r\!\!\int^{R_s}_{\rho}
{\rho^{n+k-1}W(\eta)|h(\eta)|\over W(\rho)|v_s(\eta)|}d\eta d\rho
\\ [0.3cm]
&\displaystyle+C\Big(\int^{r'}_r\!\!\int^{R_s}_{\rho}
{W(\eta)p_s'(\eta)\over W(\rho)|v_s(\eta)|}
d\eta d\rho\Big)\Big(\int_{r}^{R_s}\xi^{n+2k-1}|\varphi(\xi)|d\xi\Big).
\end{array}$$ By Lemma 5.2 of [@ChenCuiF] we have $$p_s'(r)=c_0r^{\sigma}\big(1+o(1)\big)\;\;\; {\rm as}\;\; r\to 0^+,$$ where $c_0>0$ and $-1<\sigma\leq 1$. Using (4.8), (4.9) and (4.15) we easily see that $$\int^{R_s}_0\!\!\int^{R_s}_{\rho}{W(\eta)p_s'(\eta)
\over W(\rho)|v_s(\eta)|}d\eta d\rho<\infty.$$ Hence there exists a constant $\delta>0$ independent of $k$ such that if $0<r'-r\leq\delta$ then $$C\int^{r'}_r\!\!\int^{R_s}_{\rho}{W(\eta)p_s'(\eta)\over
W(\rho)|v_s(\eta)|}d\eta d\rho\leq\frac{1}{2},$$ which implies that $$\int^{r'}_r\rho^{n+k-1}|\varphi(\rho)|d\rho
\leq 2\int^{r'}_r\!\!\int^{R_s}_{\rho}{\rho^{n+k-1}W(\eta)|h(\eta)|\over W(\rho)|v_s(\eta)|}d\eta d\rho
+\int_{r'}^{R_s}\rho^{n+k-1}|\varphi(\rho)|d\rho.$$ Hence, by dividing the interval $[0,R_s]$ into finite number (independent of $k$) of subintervals and using an iteration argument, we see that there exists a constant $C>0$ independent of $k$ such that $$\int^{R_s}_r\rho^{n+k-1}|\varphi(\rho)|d\rho
\leq C\int^{R_s}_r\!\!\int^{R_s}_{\rho}{\rho^{n+k-1}W(\eta)|h(\eta)|\over W(\rho)|v_s(\eta)|}d\eta d\rho
\quad \mbox{for any}\;\; 0<r<R_s.
$$ From (4.8) and (4.9) we have $$C_1r^{-{\alpha_0}}(R_s-r)^{\alpha_1}\leq W(r)\leq C_2r^{-{\alpha_0}}(R_s-r)^{\alpha_1}
\quad \mbox{for}\;\; 0<r<R_s,$$ where $0<C_1<C_2$. By this fact it is not hard to prove that if $|h(r)|\leq Cr^{-a}$ for $0<r\leq R_s$ for some $a<n+k$, then $\displaystyle\int^{R_s}_0\!\!
\int^{R_s}_{\rho}{\rho^{n+k-1}W(\eta)|h(\eta)|\over W(\rho)|v_s(\eta)|}d\eta d\rho<\infty$. Hence we have the assertion (4). The proof of Lemma 4.3 is complete. $\quad\Box$
[**Lemma 4.4**]{} [*Let $k\geq 2$. For any $h\in C(0,R_s]$ such that $|h(r)|\leq Cr^{-a}$ for $0<r\leq R_s$ for some $a<n+k$, the equation $$\tilde{\mathscr{L}}_k(\varphi)=h \quad \mbox{in}\;\;(0,R_s)$$ has a solution $\varphi\in C(0,R_s]\cap C^1(0,R_s)$ such that $J_k(|\varphi|)<\infty$, and the solution is unique in the class $\{\varphi\in L^{\infty}_{\rm loc}(0,R_s]:J_k(|\varphi|)
<\infty\}$.*]{}
[*Proof*]{}: It is clear that $$\tilde{\mathscr{L}}_k(\varphi)=\tilde{\mathscr{L}}_{k}^0(\varphi)-e_k(r)J_k(\varphi),$$ where $$e_k(r)=\displaystyle\frac{n\!+\!k\!-\!2}{n\!+\!2(k\!-\!1)}
\frac{(R_s^{n+2(k-1)}-r^{n+2(k-1)})p_s'(r)}{R_s^{k-1}r^{n+k-1}}.$$ Hence, the equation (4.17) is equivalent to the following system of equations for $\varphi$ and $\nu$: $$\begin{aligned}
\tilde{\mathscr{L}}_k^0(\varphi)&=&h(r)+\nu e_k(r),
\\
J_k(\varphi)&=&\nu.\end{aligned}$$ Let $\psi_k$ and $\phi_k$ be respectively solutions of the following equations: $$\tilde{\mathscr{L}}_k^0(\psi_k)=h(r),$$ $$\tilde{\mathscr{L}}_k^0(\phi_k)=e_k(r).$$ By Lemma 4.3, these solutions exist, belong to $C(0,R_s]\cap C^1(0,R_s)$, satisfy $J_k(|\psi_k|)<\infty$ and $J_k(|\phi_k|)<\infty$, and are unique in the class $\{\varphi\in L^{\infty}_{\rm loc}(0,R_s]:J_k(|\varphi|)<\infty\}$. Moreover, the assertion (3) of Lemma 4.3 ensures that $\phi_k(r)<0$ for $0<r<R_s$. Let $\varphi=\psi_k+\nu\phi_k$, where $$\nu=\frac{J_k(\psi_k)}{\displaystyle 1-J_k(\phi_k)}
=\frac{J_k(\psi_k)}{\displaystyle 1+J_k(|\phi_k|)}.$$ Then a simple computation shows that $(\varphi,\nu)$ satisfies the equations (4.18) and (4.19), so that $\varphi$ is a solution of the equation (4.17). This proves existence. To prove uniqueness we assume that $\varphi$ is a solution of (4.17) in the class $\{\varphi\in L^{\infty}_{\rm loc}(0,R_s]:J_k(|\varphi|)<\infty\}$ and set $\nu=J_k(\varphi)$. Then from (4.17) we see that $\varphi$ is a solution of the equation (4.18). By uniqueness of the solution of this equation in the class $\{\varphi\in L^{\infty}_{\rm loc}(0,R_s]:
J_k(|\varphi|)<\infty\}$, we conclude that $\varphi=\psi_k+\nu\phi_k$ and, consequently, $\nu=
J_k(\varphi)=J_k(\psi_k)+\nu J_k(\phi_k)$, which implies that (4.22) holds. Hence $\varphi$ coincides with the solution we constructed above. The proof is complete. $\quad\Box$
[*Remark*]{}. As a corollary of the above lemma we see that the system of equations $\mathscr{L}_k(\phi)=0$ and $J_k(\phi)=0$ does not have a nontrivial solution. Indeed, from the first equality in (3.9) we see that any solution of this system is also a solution of the equation $\tilde{\mathscr{L}}_k(\phi)=0$. Hence, by the uniqueness of the solution for this equation ensured by Lemma 4.4, we obtain the desired assertion.
By applying Lemma 4.4 to $h(r)=-\tilde{b}_k(r)$, we see that the equation (3.11) has a unique solution in the class $C(0,R_s]\cap C^1(0,R_s)\cap\{\varphi\in
L^{\infty}_{\rm loc}(0,R_s]: J_k(|\varphi|)<\infty\}$. However, apparently, the solution obtained in this approach might be unbounded at $r=0$, or more precisely, we cannot exclude the possibility that the solution obtained above is unbounded at $r=0$. In what follows we use a different approach to reconsider the equation (3.11). This new approach relies on the uniqueness assertion in Lemma 4.4.
We denote by $B$ the following operator in $C[0,R_s]$: For any $h\in C[0,R_s]$, $$Bh=\mbox{the right-hand side of (4.10)}.$$ By (4.5), this is a bounded linear operator in $C[0,R_s]$. Next let $K$ be the following operator in $C[0,R_s]$: For any $\phi\in C[0,R_s]$, $$\begin{aligned}
K\phi(r)&=&\displaystyle r^{k-1}p_s'(r)\Big[{\theta_k}
\int_r^{R_s}\rho^{-k+1}g_p^*(\rho)\phi(\rho)d\rho+\frac{1-\theta_k}{R_s^{n+2(k-1)}}
\int_0^{R_s}\rho^{n+k-1}g_p^*(\rho)\phi(\rho)d\rho
\nonumber\\ [0.3cm]
&&\displaystyle -\frac{1-\theta_k}{r^{n+2(k-1)}}
\int_0^r\rho^{n+k-1}g_p^*(\rho)\phi(\rho)d\rho\Big].\end{aligned}$$ Using (4.15) we can easily prove that $K$ is a bounded linear operator in $C[0,R_s]$ and is compact. We rewrite the equation (3.11) as follows: $$-v_s(r)\psi_k'(r)+f_p^*(r)\psi_k(r)+K\psi_k(r)+\tilde{b}_k(r)=0 \quad \mbox{for}\;\;0<r<R_s.$$ Clearly, if $w_k\in C[0,R_s]$ is a solution of the equation $$w_k(r)+KBw_k(r)+\tilde{b}_k(r)=0 \quad \mbox{for}\;\;0<r<R_s,$$ then $\psi_k=Bw_k$ is a solution of (4.23). Note that $KB$ is a compact operator in $C[0,R_s]$ and $\tilde{b}_k\in C[0,R_s]$. Now, by uniqueness of the solution of (4.17) in the class $\{v\in
L^{\infty}_{\rm loc}(0,R_s]:J_k(|v|)<\infty\}$ we easily see that the equation $v+KBv=0$ has only the trivial solution $v=0$ in $C[0,R_s]$. It follows by a well-known theorem for Fredholm operators that the equation (4.24) has a unique solution $w_k\in C[0,R_s]$. Letting $\psi_k=Bw_k$, we get a solution of (4.23) in the class $C[0,R_s]$. This proves the existence assertion of the following result:
[**Theorem 4.5**]{} [*For any $k\geq 2$, the equation $(3.11)$ has a unique solution $\psi_k\in C[0,R_s]$. Moreover, $\psi_k\in C^{\infty}(0,R_s]$, and there exists $0<\mu_k\leq 1$ such that $\psi_k\in C^{\mu_k}[0,R_s]$.*]{}
[*Proof*]{}: The equation (3.11) can be rewritten as follows: $$\tilde{\mathscr{L}}_k^0(\psi_k)=-\tilde{b}_k(r)+J_k(\psi_k)e_k(r).$$ Since $\tilde{b}_k,e_k\in C^{\infty}(0,R_s]$, by the assertion (3) of Lemma 4.3 we see that $\psi_k\in C^{\infty}(0,R_s]$. Next, since $$|K\psi_k(r)|\leq \tilde{b}_krp_s'(r)\leq \tilde{b}_kr^{1+\sigma}\quad \mbox{and} \quad
|\tilde{b}_k(r)|\leq \tilde{b}_kr^{k-1}p_s'(r)+\tilde{b}_kr^k\leq \tilde{b}_kr^{1+\sigma}$$ for $0<r\leq R_s$ (recall that $-1<\sigma\leq 1$ and $k\geq 2$), using Lemma 4.2 to the equation (4.23) we see that $|\psi_k(r)|\leq \tilde{b}_kr^{\mu_k}$ for $0<r\leq
R_s$ for some constant $0<\mu_k\leq 1+\sigma$. Again by (4.23), it follows that $|\psi_k'(r)|\leq \tilde{b}_kr^{\mu_k-1}$ for $0<r\leq R_s$. Using this fact we easily deduce that $|\psi_k(r)-\psi_k(s)|\leq \tilde{b}_k|r-s|^{\min\{\mu_k,1\}}$ for $r,s\in
[0,R_s]$. This completes the proof. $\quad\Box$
[*Remark*]{}. A more delicate analysis shows that if we denote by $m_k(r)$ the function $m_{\mu}(r)$ given by (4.7) for $\mu=k-1+\sigma$, then the solution of (3.8) satisfies $|\psi_k(r)|\leq C_km_k(r)$ for $0<r\leq R_s$. To prove this assertion we only need to consider the equation (4.24) in the class $$\Big\{v\in C[0,R_s]: |v(r)|\leq Cm_k(r)\;\, \mbox{for some}\;\,C>0,\;\;
\mbox{and}\;\,\frac{v(r)}{m_k(r)}\in C[0,R_s]\Big\}.$$ Then a similar argument as before yields the desired assertion. Since we do not need this result later on, we omit the details of the proof.
[**Corollary 4.6**]{} [*Let $k\geq 2$ and $\gamma_k$ be defined by $(3.16)$. For $\gamma=\gamma_k$ the system $(3.8)$ has a nontrivial solution $(\phi_k,y_k)\in
(C[0,R_s]\cap C^1(0,R_s))\times\mathbb{R}$, which is unique up to a nonzero factor. Moreover, $\phi_k\in C^{\infty}(0,R_s]$, and there exists $0<\mu_k\leq 1$ such that $\phi_k\in C^{\mu_k}[0,R_s]$. For $\gamma\not=\gamma_k$ the system $(3.8)$ does not have a nontrivial solution.*]{}
Estimates of the nonlinear eigenvalues $\gamma_k$
=================================================
2em In this section we study properties of the eigenvalues $\gamma_k$, $k=2,3,\cdots$.
Let $\psi_k$ be the solution of the equation (3.8) and set $$v_k(r)=\psi_k(r)-\frac{c_s'(R_s)}{R_s^k}\frac{g_c^*(r)}{g_p^*(r)}r^ku_k(r).$$ A simple computation shows that $v_k$ satisfies the following equation: $$\tilde{\mathscr{L}}_k(v_k)=d_k(r),$$ where $$d_k(r)=-\frac{g(1,1)}{R_s^{k-1}}r^{k-1}p_s'(r)
+\frac{c_s'(R_s)}{R_s^k}v_s(r)\Big(\frac{g_c^*(r)}{g_p^*(r)}r^ku_k(r)\Big)'
+\frac{c_s'(R_s)}{R_s^k}\frac{f_c^*(r)g_p^*(r)-f_p^*(r)g_c^*(r)}{g_p^*(r)}r^ku_k(r).$$ Since $\tilde{\mathscr{L}}_k(v_k)=\tilde{\mathscr{L}}_{k}^0(v_k)
-e_k(r)J_k(v_k)$, by letting $\tilde{\nu}_k=J_k(v_k)$, from (5.2) we get $$\tilde{\mathscr{L}}_k^0(v_k)=d_k(r)+\tilde{\nu}_k e_k(r).$$ Hence, by letting $\tilde{\psi}_k$ be the solution of the equation $$\tilde{\mathscr{L}}_k^0(\tilde{\psi}_k)=d_k(r),$$ we have $$v_k=\tilde{\psi}_k+\tilde{\nu}_k\phi_k,$$ where $\phi_k$ is as before, i.e., $\tilde{\phi}_k$ is the solution of the equation (4.21). Note that by Lemma 4.3, the equation (5.4) has a unique solution $\tilde{\psi}_k
\in C^{\infty}(0,R_s]$.
[**Lemma 5.1**]{}
*Let $k\geq 2$. For $\tilde{\psi}_k$ defined above we have the following assertions:*
$(1)$ $\tilde{\psi}_k(R_s)=\displaystyle -\frac{c_s'(R_s)g_c^*(R_s)}{g_p^*(R_s)}
-p_s'(R_s)$.
$(2)$ $J_k(|\tilde{\psi}_k|)\leq\displaystyle Ck^{-1}$, where $C$ is a constant independent of $k$.
[*Proof*]{}: By the assertion (2) of Lemma 4.3 we have $$\tilde{\psi}_k(R_s)=\frac{d_k(R_s)}{f_p^*(R_s)}=-\frac{c_s'(R_s)g_c^*(R_s)}{g_p^*(R_s)}
-\frac{g(1,1)p_s'(R_s)-c_s'(R_s)f_c^*(R_s)}{f_p^*(R_s)}.$$ Note that $$\begin{array}{rl}
& g(1,1)p_s'(R_s)-c_s'(R_s)f_c^*(R_s)
\\ [0.3cm]
=&\displaystyle g(1,1)p_s'(R_s)-\frac{d}{dr}[f(c_s(r),p_s(r))]\Big|_{r=R_s}
+f_p^*(R_s)p_s'(R_s)
\\ [0.3cm]
=&\displaystyle g(1,1)p_s'(R_s)-\frac{d}{dr}[v_s(r)p_s'(r))]\Big|_{r=R_s}
+f_p^*(R_s)p_s'(R_s)
\\ [0.2cm]
=&\displaystyle g(1,1)p_s'(R_s)-[v_s'(R_s)p_s'(R_s))+v_s(R_s)p_s''(R_s))]
+f_p^*(R_s)p_s'(R_s)
\\
=&f_p^*(R_s)p_s'(R_s).
\end{array}$$ Here we have used the fact that $v_s(R_s)=0$ and $v_s'(R_s)=g(1,1)$. Hence the assertion (1) follows. Next, using (4.15) we easily see that $$|d_k(r)|\leq Cp_s'(r)+Ck|v_s(r)|+Cr\leq Cr^{\sigma}+Ck|v_s(r)|.$$ Using (4.12), the above estimate and (4.16), we see that $$\begin{array}{rl}
J_k(|\tilde{\psi}_k|)\leq &
\displaystyle\frac{C}{R_s^{n+k-1}}\int^{R_s}_0\!\!\int^{R_s}_{\xi}
{\xi^{n+k-1}W(\eta)|d_k(\eta)|\over W(\xi)|v_s(\eta)|}d\eta d\xi
\\ [0.3cm]
\leq &\displaystyle\frac{C}{R_s^{n+k-1}}\int^{R_s}_0\!\!\!\!\int^{R_s}_{\xi}
{\xi^{n+k-1+{\alpha_0}}(R_s-\eta)^{\alpha_1-1}
\over\eta^{{\alpha_0}-\sigma+1}(R_s-\xi)^{\alpha_1}}d\eta d\xi
+\frac{Ck}{R_s^{n+k-1}}\int^{R_s}_0\!\!\!\!\int^{R_s}_{\xi}
{\xi^{n+k-1+{\alpha_0}}(R_s-\eta)^{\alpha_1}
\over\eta^{{\alpha_0}}(R_s-\xi)^{\alpha_1}}d\eta d\xi
\\ [0.3cm]
\leq &\displaystyle\frac{C}{R_s^{n+k-1}}\int^{R_s}_0\!\!\!\!\int^{R_s}_{\xi}
{\xi^{n+k+\sigma-2}(R_s-\eta)^{\alpha_1-1}\over (R_s-\xi)^{\alpha_1}}d\eta d\xi
+\frac{Ck}{R_s^{n+k-1}}\int^{R_s}_0\!\!\!\!\int^{\eta}_0
{\xi^{n+k-1+{\alpha_0}}(R_s-\eta)^{\alpha_1}
\over\eta^{{\alpha_0}}(R_s-\xi)^{\alpha_1}}d\xi d\eta
\\ [0.3cm]
\leq &\displaystyle\frac{C}{R_s^{n+k-1}}\int^{R_s}_0\!\xi^{n+k+\sigma-3}d\xi
+\frac{Ck}{R_s^{n+k-1}}\int^{R_s}_0\!\!\!\!\int^{\eta}_0
{\xi^{n+k-1+\alpha_0}\over\eta^{\alpha_0}}d\xi d\eta
\\ [0.3cm]
\leq &\displaystyle\frac{C}{k}+\frac{Ck}{(n+k+\alpha_0)(n+k)}
\\ [0.3cm]
\leq &\displaystyle\frac{C}{k} \quad \mbox{for}\;\; k\geq 2.
\end{array}$$ This completes the proof. $\quad\Box$
[**Lemma 5.2**]{}
*Let $k\geq 2$. For $\phi_k$, the solution of $(4.21)$, we have the following assertions:*
$(1)$ $\phi_k(R_s)=0$, and $\phi_k(r)<0$ for $0<r<R_s$.
$(2)$ $J_k(|\phi_k|)\leq\displaystyle Ck^{-\min\{\alpha_1,\frac{1}{2}\}+\varepsilon}$, where $C$ is a positive constant independent of $k$, and $\varepsilon$ represents an arbitrarily small positive number.
[*Proof*]{}: The assertion (1) follows from the fact that $e_k(R_s)=0$ and $e_k(r)>0$ for $0<r<R_s$. Next, by using (4.12), (4.16) and the fact that $$0\leq e_k(r)\leq\frac{R_s^{n+k-1}p_s'(r)}{r^{n+k-1}}\leq CR_s^{n+k-1}r^{-n-k+1+\sigma}$$ we have $$\begin{array}{rl}
J_k(|\phi_k|)\leq &
\displaystyle\frac{C}{R_s^{n+k-1}}\int^{R_s}_0\!\!\int^{R_s}_{\xi}
{\xi^{n+k-1}W(\eta)e_k(\eta)\over W(\xi)|v_s(\eta)|}d\eta d\xi
\\ [0.3cm]
\leq &\displaystyle C\int^{R_s}_0\!\!\int^{\eta}_0
{\xi^{n+k-1+\alpha_0}(R_s-\eta)^{\alpha_1-1}\over
\eta^{n+k+\alpha_0-\sigma}(R_s-\xi)^{\alpha_1}}d\xi d\eta
\\ [0.3cm]
= &\displaystyle C\Big(\int^{\frac{R_s}{2}}_0\!\!\int^{\eta}_0
+\int^{R_s}_{\frac{R_s}{2}}\!\!\int^{\frac{R_s}{2}}_0
+\int^{R_s}_{\frac{R_s}{2}}\!\!\int^{\eta}_{\frac{R_s}{2}}\Big)
{\xi^{n+k-1+\alpha_0}(R_s-\eta)^{\alpha_1-1}\over\eta^{n+k+\alpha_0-\sigma}(R_s-\xi)^{\alpha_1}}d\xi d\eta
\\ [0.3cm]
\leq &\displaystyle C\int^{\frac{R_s}{2}}_0\!\!\int^{\eta}_0
{\xi^{n+k-1+\alpha_0}\over\eta^{n+k+\alpha_0-\sigma}}d\xi d\eta
+C\Big(\frac{2}{R_s}\Big)^{n+k+\alpha_0-\sigma}\int^{R_s}_{\frac{R_s}{2}}\!\!\int^{\frac{R_s}{2}}_0
\xi^{n+k-1+\alpha_0}(R_s-\eta)^{\alpha_1-1}d\xi d\eta
\\ [0.3cm]
&\displaystyle
+C\int^{R_s}_{\frac{R_s}{2}}\!\!\int^{R_s}_{\xi}
{\xi^{n+k-1+\alpha_0}(R_s-\eta)^{\alpha_1-1}\over\eta^{n+k+\alpha_0-\sigma}(R_s-\xi)^{\alpha_1}}d\eta d\xi
\\ [0.3cm]
&\displaystyle=I+I\!I+I\!I\!I.
\end{array}$$ It is immediate to see that $$I\leq\frac{C}{k}, \quad I\!I\leq\frac{C}{k} \quad \mbox{for}\;\; k\geq 2.$$ For $I\!I\!I$ we let $$p=\frac{1}{1-\min\{\alpha,\frac{1}{2}\}+\varepsilon} \quad \mbox{and} \quad
q=\frac{1}{\min\{\alpha,\frac{1}{2}\}-\varepsilon},$$ where $\varepsilon$ is a sufficiently small positive number. Then by the Hölder inequality we have $$I\!I\!I\leq \Big(\int^{R_s}_{\frac{R_s}{2}}\!\!\!\int^{R_s}_{\xi}\!\!
{\xi^{nq+kq+{\alpha_0} q-q}\over\eta^{nq+kq+{\alpha_0} q-\sigma q}}d\eta d\xi\Big)^{\frac{1}{q}}
\Big(\int^{R_s}_{\frac{R_s}{2}}\!\!\!\int^{R_s}_{\xi}\!\!
{(R_s\!-\eta)^{(\alpha-1)p}\over(R_s\!-\xi)^{\alpha p}}d\eta d\xi\Big)^{\frac{1}{p}}
\leq Ck^{-\frac{1}{q}}.$$ Hence the assertion (2) follows. This completes the proof. $\quad\Box$
[**Theorem 5.3**]{}
*Let $k\geq 2$. We have the following assertions:*
$(1)$ $\gamma_k=\displaystyle\frac{C_n}{k^3}\Big[1+O\Big(\frac{1}{k}\Big)\Big]$ as $k\to\infty$, where $C_n$ is a positive constant independent of $k$.
$(2)$ $\gamma_k>0$ and $\gamma_{k+1}<\gamma_k$ for $k$ sufficiently large.
[*Proof*]{}: From (3.12) and (5.1) we see that $$\gamma_k={(n\!-\!1)R_s^3\over (\lambda_k\!-\!n\!+\!1)k}[g(1,1)+J_k(v_k)]
={(n\!-\!1)R_s^3\over (\lambda_k\!-\!n\!+\!1)k}[g(1,1)+\tilde{\nu}_k].$$ From (5.5) we have $$\tilde{\nu}_k=J_k(v_k)=J_k(\psi_k)+\tilde{\nu}_kJ_k(\phi_k).$$ Hence $$\tilde{\nu}_k=\frac{J_k(\psi_k)}{\displaystyle 1-J_k(\phi_k)}
=\frac{J_k(\psi_k)}{\displaystyle 1+J_k(|\phi_k|)}.$$ By Lemmas 5.1 and 5.2, it follws that $$|\tilde{\nu}_k|\leq Ck^{-1}.$$ Hence $$\gamma_k={(n\!-\!1)R_s^3g(1,1)\over (\lambda_k\!-\!n\!+\!1)k}
\Big[1+O\Big(\frac{1}{k}\Big)\Big]
=\frac{C_n}{k^3}\Big[1+O\Big(\frac{1}{k}\Big)\Big] \quad \mbox{as}\;\;k\to\infty,$$ where $C_n=(n\!-\!1)R_s^3g(1,1)$. This proves the assertion (1). The assertion (2) is an immediate consequence of the assertion (1). $\quad\Box$
By now, we have finished proving Theorem 1.1. Indeed, that theorem follows from Lemmas 3.1, 3.2, 3.3 and Theorems 4.5 and 5.3.
Invertibility of some operators
===============================
2em In this section we study invertibility of the linear operator $(u,\eta)\mapsto
(\mathscr{A}_{\gamma}(u,\eta),\mathscr{B}_{\gamma}(u,\eta))$ in suitable function spaces, or equivalently, solvability of the system of equations $$\left\{
\begin{array}{l}
\mathscr{A}_{\gamma}(u,\eta)=h(x) \quad \mbox{for}\;\; x\in\mathbb{B}(0,R_s)
\\
\mathscr{B}_{\gamma}(u,\eta)=\rho(\omega) \quad \mbox{for}\;\; \omega\in\mathbb{S}^{n-1}
\end{array}
\right.$$ for given functions $h$ and $\rho$ defined in $\mathbb{B}(0,R_s)$ and $\mathbb{S}^{n-1}$, respectively.
In view of the Fourier expansion (3.3) of the operators $\mathscr{A}_{\gamma}$ and $\mathscr{B}_{\gamma}$, we see that the above system is equivalent to the following series of systems of equations: $$\left\{
\begin{array}{l}
\mathscr{L}_k(u_{kl})+b_k(r,\gamma)y_{kl}=h_{kl}(r) \quad \mbox{for}\;\; 0<r<R_s
\\
J_k(u_{kl})+\alpha_k(\gamma)y_{kl}=z_{kl}
\end{array}
\right.$$ ($k=0,1,2,\cdots$, $l=1,2,\cdots,d_k$), where $u_{kl}=u_{kl}(r)$, $y_{kl}$, $h_{kl}=h_{kl}(r)$ and $z_{kl}$ are the Fourier coefficients of the functions $u=(x)$, $\eta=\eta(\omega)$, $h=h(x)$ and $\rho=\rho(\omega)$, respectively, with respect to the basis spherical harmonic functions $\{Y_{kl}(\omega):k=0,1,\cdots,
l=1,2,\cdots,d_k\}$.
We first consider the case $\gamma\neq\gamma_k$ for all $k\geq 2$. Since for $k=1$ the homogeneous version of the system (6.2) has nontrivial solutions, so that for $k=1$ the system (6.2) is not generally solvable, in what follows we only consider the cases $k=0$ and $k\geq 2$. Hence, in what follows we study the following system of equations $$\left\{
\begin{array}{l}
\mathscr{L}_k(\varphi)+b_k(r,\gamma)y=\zeta(r) \quad \mbox{for}\;\; 0<r<R_s
\\
J_k(\varphi)+\alpha_k(\gamma)y=z
\end{array}
\right.$$ for $k=0$ and $k=2,3,\cdots$. Here $\zeta$ is a given continuous function in $[0,R_s]$, $z$ is a given real constant, and $\varphi$, $y$ are unknown variables. Note that from the expression of $b_k(r,\gamma)$ (see (3.5)) we see that for $k\neq 1$, we have $b_k(\cdot,\gamma)\in C[0,R_s]$.
[**Lemma 6.1**]{} [*Let $k\in\mathbb{Z}_+$, $k\neq 1$, and assume that $\gamma
\neq\gamma_j$ for all $j\geq 2$. For any $(\zeta,z)\in C[0,R_s]\times\mathbb{R}$, the system $(6.3)$ has a unique solution $(\varphi,y)\in (C[0,R_s]\cap C^1(0,R_s))\times
\mathbb{R}$. Moreover, there exists a constant $C>0$ independent of $k$ and $(\zeta,z)$ such that the following estimate holds:*]{} $$\max_{0\leq r\leq R_s}|\varphi(r)|+\max_{0\leq r\leq R_s}|r(R_s-r)\varphi'(r)|
+(1+k)^3|y|\leq C[\max_{0\leq r\leq R_s}|\zeta(r)|+|z|].$$ [*Proof*]{}: Let $L$ be the following unbounded linear operator in $C[0,R_s]$ with domain $C_\vee^1[0,R_s]=\{\phi\in C[0,R_s]\cap C^1(0,R_s):r(R_s-r)\phi'(r)
\in C[0,R_s]\}$: $$L\phi(r)=-v_s(r)\phi'(r)+f_p^*(r)\phi(r) \quad
\mbox{for}\;\;\phi\in C_\vee^1[0,R_s].$$ For each $k\in\mathbb{Z}_+$ let $B_k$ be the following bounded linear operator in $C[0,R_s]$: $$\begin{aligned}
B_k\phi(r)&=&\displaystyle r^{k-1}p_s'(r)
\Big[{\theta_k}\int_r^{R_s}\rho^{-k+1}g_p^*(\rho)\phi(\rho)d\rho
-\frac{1-\theta_k}{r^{n+2(k-1)}}
\int_0^r\rho^{n+k-1}g_p^*(\rho)\phi(\rho)d\rho
\nonumber\\
&&\displaystyle -\frac{\theta_k}{R_s^{n+2(k-1)}}
\int_0^{R_s}\rho^{n+k-1}g_p^*(\rho)\phi(\rho)d\rho\Big] \quad
\mbox{for}\;\;\phi\in C[0,R_s].\end{aligned}$$ Then we have $\mathscr{L}_k=L+B_k$. By Lemma 4.2, the operator $L:
C_\vee^1[0,R_s]\to C[0,R_s]$ is invertible, and its inverse $L^{-1}$ is a bounded linear operator in $C[0,R_s]$. Clearly, for $k\geq 2$, $B_k$ is a compact linear operator in $C[0,R_s]$. For $k=0$, $B_0$ has the following form: $$\begin{aligned}
B_0\phi(r)&=&\displaystyle -rp_s'(r)\cdot
\frac{1}{r^{n}}\int_0^r\rho^{n-1}g_p^*(\rho)\phi(\rho)d\rho \quad
\mbox{for}\;\;\phi\in C[0,R_s].\end{aligned}$$ From this expression it is clear that $B_0$ is also a compact linear operator in $C[0,R_s]$. Now, by letting $\tilde{\zeta}(r)=L^{-1}\zeta(r)$ and $\tilde{b}_k(r,\gamma)
=L^{-1}b_k(r,\gamma)$, we see that the system (6.1) is equivalent to the following one: $$\left\{
\begin{array}{l}
\varphi(r)+L^{-1}B_k\varphi(r)+\tilde{b}_k(r,\gamma)y=\tilde{\zeta}(r),
\quad \mbox{for}\;\; 0<r<R_s
\\
J_k(\varphi)+\alpha_k(\gamma)y=z.
\end{array}
\right.$$ Since $L^{-1}B_k$ is a compact operator in $C[0,R_s]$, $J_k$ is a continuous functional in $C[0,R_s]$, and $\tilde{b}_k(\cdot,\gamma)\in C[0,R_s]$, it follows that the operator $$(\varphi,y)\mapsto (\varphi+L^{-1}B_k\varphi+\tilde{b}_k(\cdot,\gamma)y,
J_k(\varphi)+\alpha_k(\gamma)y)$$ from $C[0,R_s]\times\mathbb{R}$ to itself is a Fredholm operator of index zero. Hence, solvability of the system (6.5) in $C[0,R_s]\times\mathbb{R}$ for any given $(\tilde{\zeta},z)\in C[0,R_s]\times\mathbb{R}$ is equivalent to uniqueness of the solution of this system. By equivalence of the two systems (6.3) and (6.5), we infer that solvability of the system (6.3) in $C[0,R_s]\times\mathbb{R}$ for any given $(\zeta,z)\in C[0,R_s]\times\mathbb{R}$ is equivalent to uniqueness of the solution of this system. Now, since $\gamma\neq\gamma_j$ for all $j\geq 2$ and by assumption we have $k=0$ or $k\geq 2$, by Lemmas 3.3 and 3.5 it follows that the system (6.3) with $(\zeta,z)=(0,0)$ does not have a nontrivial solution so that its solution is unique. Hence, the system (6.3) is uniquely solvable for any given $(\zeta,z)\in C[0,R_s]\times\mathbb{R}$ and, furthermore, there exists a constant $C_k>0$ such that the following estimate holds: $$\max_{0\leq r\leq R_s}|\varphi(r)|+|y|\leq C_k[\max_{0\leq r\leq R_s}|\zeta(r)|+|z|].$$ In what follows we prove that the constant $C_k$ can be chosen to be independent of $k$.
For $k\geq 2$, we make a transformation of unknown variables $(\varphi,y)\mapsto
(\psi,y)$ as follows: $$\psi(r)=\varphi(r)+R_s^{-(k-1)}r^{k-1}p_s'(r)y.$$ Note that since $k\geq 2$, we have that $r^{k-1}p_s'(r)\in C[0,R_s]$. Multiplying both sides of the second equation in (6.3) with $R_s^{-(k-1)}r^{k-1}p_s'(r)$ and adding them into the respective sides of the first equation in (6.3), we see that the system (6.3) reduces into the following equivalent one: $$\left\{
\begin{array}{l}
\tilde{\mathscr{L}}_k(\psi)+c_k(r)y=\hat{\zeta}(r) \quad \mbox{for}\;\; 0<r<R_s
\\
J_k(\psi)+\tilde{\alpha}_k(\gamma)y=z,
\end{array}
\right.$$ where $\tilde{\mathscr{L}}_k$ is as before, i.e., $\tilde{\mathscr{L}}_k(\psi)=
\mathscr{L}_k(\psi)+R_s^{-(k-1)}r^{k-1}p_s'(r)J_k(\psi)$ (see (3.14)), $$\begin{aligned}
c_k(r)&=&b_k(r,\gamma)+\alpha_k(\gamma)R_s^{-(k-1)}r^{k-1}p_s'(r)
-R_s^{-(k-1)}\tilde{\mathscr{L}}_k[r^{k-1}p_s'(r)]
\nonumber\\ [0.3cm]
&=&\displaystyle\frac{r^{k-1}}{R_s^{k-1}}\Big\{[g(1,1)-g^*(r)]p_s'(r)
+\frac{n\!+\!k\!-\!2}{r}f^*(r)+f_c^*(r)
\Big[c_s'(r)-c_s'(R_s)R_s^{-1}r u_k(r)\Big]
\nonumber\\ [0.3cm]
&&\displaystyle -p_s'(r)\Big[{\theta_k}
\int_r^{R_s}v_k(\rho)d\rho+\frac{1-\theta_k}{R_s^{n+2(k-1)}}
\int_0^{R_s}\rho^{n+2(j-1)}v_k(\rho)d\rho
\nonumber\\ [0.3cm]
&&\displaystyle -\frac{1-\theta_k}{r^{n+2(k-1)}}
\int_0^r\rho^{n+2(k-1)}v_k(\rho)d\rho\Big]\Big\},\end{aligned}$$ where $$v_k(r)=g_p^*(r)p_s'(r)+c_s'(R_s)R_s^{-1}g_c^*(r)r u_k(r),$$ $$\begin{aligned}
\widetilde{\alpha}_k(\gamma)&=&\alpha_k(\gamma)-R_s^{-(k-1)}J_k(r^{k-1}p_s'(r))
\nonumber\\
&=&\displaystyle\Big(1-\frac{\lambda_k}{n\!-\!1}\Big)\frac{k\gamma}{R_s^3}
+g(1,1)-\frac{1}{R_s^{n+2(k-1)}}\int_0^{R_s}\rho^{n+2(j-1)}v_k(\rho)d\rho,\end{aligned}$$ and $$\hat{\zeta}(r)=\zeta(r)+R_s^{-(k-1)}r^{k-1}p_s'(r)z.$$ Note that $c_k,\hat{h}\in C[0,R_s]$. By using Lemma 4.1, it is easy to see that $$\max_{0\leq r\leq R_s}|c_k(r)|\leq C(1+k), \quad k=0,1,2,\cdots,$$ where $C$ is positive constant independent of $k$. Besides, from (6.9) we see that there exists integer $k_0=k_0(\gamma)\geq 2$ and constant $C(\gamma)>0$ such that for $k\geq k_0$ we have $$|\widetilde{\alpha}_k(\gamma)|\geq C(\gamma)k^3.$$ In particular, this implies that $\widetilde{\alpha}_k(\gamma)\neq 0$ for sufficiently large $k$. Using this fact, we deduce from (6.8) the following equation for $\psi$: $$\tilde{\mathscr{L}}_k(\psi)-\frac{c_k(r)}{\widetilde{\alpha}_k(\gamma)}
J_k(\psi)=\hat{\zeta}(r)-\frac{c_k(r)}{\widetilde{\alpha}_k(\gamma)}z.$$ This equation can be rewritten as follows: $$L\psi(r)+\tilde{B}_k\psi(r)=\hat{\zeta}(r)-\frac{c_k(r)}{\widetilde{\alpha}_k(\gamma)}z,$$ where $\tilde{B}_k$ is the following bounded linear operator in $C[0,R_s]$: $$\tilde{B}_k\psi(r)=B_k\psi(r)+\Big(\frac{r}{R_s}\Big)^{k-1}p_s'(r)J_k(\psi)
-\frac{c_k(r)}{\widetilde{\alpha}_k(\gamma)}J_k(\psi).$$ It is easy to see that for $k\geq 2$, $$\max_{0\leq r\leq R_s}|B_k\phi(r)|+\max_{0\leq r\leq R_s}|J_k\phi(r)|
\leq Ck^{-1}\max_{0\leq r\leq R_s}|\phi(r)|
\quad \mbox{for} \;\; \phi\in C[0,R_s],$$ where $C$ is a positive constant independent of $k$. Moreover, from (6.13) and (6.14) we see that $|c_k(r)/\widetilde{\alpha}_k(\gamma)|$ is bounded by a constant independent of $k$ and, since $k\geq 2$, $(r/R_s)^{k-1}p_s'(r)=(r/R_s)^{k-2}R_s^{-1}
rp_s'(r)$ is also bounded by a constant independent of $k$. Hence, for sufficiently large $k$ we have $$\max_{0\leq r\leq R_s}|\tilde{B}_k\phi(r)|\leq Ck^{-1}\max_{0\leq r\leq R_s}|\phi(r)|
\quad \mbox{for} \;\; \phi\in C[0,R_s].$$ Using this estimate and the boundedness of $L^{-1}$ in $C[0,R_s]$ we easily deduce from (6.15) that for sufficiently large $k$, $$\max_{0\leq r\leq R_s}|\psi(r)|\leq C[\max_{0\leq r\leq R_s}|\zeta(r)|+|z|],$$ where $C$ is a positive constant independent of $k$. Now, since $y=
[z-J_k(\psi)]/\widetilde{\alpha}_k(\gamma)$ (by the second equation in (6.8)), from (6.7), (6.14) and (6.17) we see that there exists a constant $C>0$ such that $$\max_{0\leq r\leq R_s}|\varphi(r)|+|y|\leq C[\max_{0\leq r\leq R_s}|\zeta(r)|+|z|].$$ for sufficiently large $k$. Since (6.6) ensures that this estimate also holds for $k$ in any finite interval and $k\neq 1$, we see that (6.17) holds for all $k\geq 0$ and $k\neq 1$.
We now prove (6.4). Indeed, from (3.4) we see that a similar estimate as (6.14) also holds for $\alpha_k(\gamma)$. It follows from the second equation in (6.3) and (6.17) that $$(1+k)^3|y|\leq C[\max_{0\leq r\leq R_s}|\zeta(r)|+|z|]$$ for $k\neq 1$. By (3.5) we see that $|b_k(r,\gamma)|$ is bounded by $C(\gamma)(1+k)^3$. Hence from the first equation in (6.3) and (6.17), (6.18) we get $$\max_{0\leq r\leq R_s}|r(R_s-r)\varphi'(r)|\leq C[\max_{0\leq r\leq R_s}|\zeta(r)|+|z|].$$ Combining (6.17), (6.18) and (6.19) together, we see that (6.4) follows. This completes the proof of Lemma 6.1. $\quad\Box$
For any $1\leq\alpha<\infty$, we denote by $X_{\alpha}$ the space of all measurable functions $u(x)$ in the ball $\mathbb{B}(0,R_s)\subseteq\mathbb{R}^n$ satisfying the following conditions: $$u(x)=\sum_{k=0}^{\infty}\sum_{l=1}^{d_k}u_{kl}(r)Y_{kl}(\omega)\;\;
\mbox{in}\;\; C([0,R_s],\mathscr{D}'(\mathbb{S}^{n-1})),
$$ $$\|u\|_{X_{\alpha}}=\Big[\sum_{k=0}^{\infty}\sum_{l=1}^{d_k}
\Big(\max_{0\leq r\leq R_s}|u_{kl}(r)|\Big)^{\alpha}
\Big]^{\frac{1}{\alpha}}<\infty.$$ The notations $X_{\infty}$ denotes the space defined by modifying the above definition in conventional sense. It is clear that for any $1\leq\alpha\leq\infty$, $X_{\alpha}$ is a Banach space. We also introduce the Banach space $$X^1_{\alpha}=\{u\in X_{\alpha\beta}:r(R_s-r)\partial_ru\in
X_{\alpha}\},$$ with norm $\|u\|_{X^1_{\alpha}}=\|u\|_{X_{\alpha}}+\|r(R_s-r)\partial_ru\|_{X_{\alpha}}$. Note that for $u$ given by (6.18) we have $$\|u\|_{X^1_{\alpha}}\approx\Big[\sum_{k=0}^{\infty}\sum_{l=1}^{d_k}
\Big(\max_{0\leq r\leq R_s}|u_{kl}(r)|+\max_{0\leq r\leq R_s}r(R_s-r)
|u_{kl}'(r)|\Big)^{\alpha}\Big]^{\frac{1}{\alpha}}.$$
Next, for any $1\leq\alpha<\infty$, we denote by $Y_{\alpha}$ the space of all measurable functions $\varphi(\omega)$ on the sphere $\mathbb{S}^{n-1}$ satisfying the following conditions: $$\varphi(\omega)=\sum_{k=0}^{\infty}\sum_{l=1}^{d_k}a_{kl}Y_{kl}(\omega)\;\;
\mbox{in}\;\; \mathscr{D}'(\mathbb{S}^{n-1}), \quad
\|\varphi\|_{Y_{\alpha}}=\Big(\sum_{k=0}^{\infty}\sum_{l=1}^{d_k}|a_{kl}|^{\alpha}
\Big)^{\frac{1}{\alpha}}<\infty.$$ The notation $Y_{\infty}$ denotes the space by replacing the summation over $k,l$ with supremum. It is clear that for any $1\leq\alpha\leq\infty$, $Y_{\alpha}$ is a Banach space. We also denote by $Y^3_{\alpha}$ the Banach space made by functions $\varphi(\omega)$ on the sphere $\mathbb{S}^{n-1}$ with the expansion (6.19) satisfying the following condition: $$\|\varphi\|_{Y^3_{\alpha}}=
\left\{
\begin{array}{l}
\displaystyle\Big\{\sum_{k=0}^{\infty}\sum_{l=1}^{d_k}[(1+k)^3|a_{kl}|]^{\alpha}
\Big\}^{\frac{1}{\alpha}}<\infty \quad \mbox{if}\;\; 1\leq\alpha<\infty,
\\ [0.3cm]
\displaystyle\sup_{k,l}(1+k)^3|a_{kl}|<\infty \quad \mbox{if}\;\; \alpha=\infty.
\end{array}
\right.$$ It is clear that $Y^3_{\alpha}$ ($1\leq\alpha\leq\infty$) are also Banach spaces.
Moreover, for every $k\in\mathbb{Z}_+$ we denote by $X_{\alpha,k}$ and $Y_{\alpha,k}$ the following closed subspaces of $X_{\alpha}$ and $Y_{\alpha}$, respectively: $$\begin{array}{c}
X_{\alpha,k}=\{u\in X_{\alpha}: \mbox{the coefficients}\;\,u_{kl}(r)\;
(l=1,2,\cdots,d_k)\;\mbox{in (6.20) are identically zero}\},\\
Y_{\alpha,k}=\{\varphi\in Y_{\alpha}: \mbox{the coefficients}\;\,a_{kl}\;
(l=1,2,\cdots,d_k)\;\mbox{in (6.21) are identically zero}\},
\end{array}$$ and denote by $X^1_{\alpha,k}$ and $Y^3_{\alpha,k}$ similar closed subspaces of $X^1_{\alpha}$ and $Y^3_{\alpha}$, respectively.
It is easy to see that the linear operator $(u,\eta)\mapsto
(\mathscr{A}_{\gamma}(u,\eta),\mathscr{B}_{\gamma}(u,\eta))$ maps $X^1_{\alpha}
\times Y^3_{\alpha}$ into $X_{\alpha}\times Y_{\alpha}$ boundedly, and when restricted to $X^1_{\alpha,1}\times Y^3_{\alpha,1}$, it maps this space into $X_{\alpha,1}\times Y_{\alpha,1}$ boundedly. From Lemma 6.1 we immediately get:
[**Theorem 6.2**]{} [*Assume that $\gamma\neq\gamma_k$ for all $k\geq 2$ and let $1\leq\alpha\leq\infty$ be given. For any $(h,\rho)\in X_{\alpha,1}\times Y_{\alpha,1}$, the system $(6.1)$ has a unique solution $(u,\eta)\in X^1_{\alpha}\times Y^3_{\alpha}$. Moreover, there exists a constant $C>0$ depending on $\gamma$ such that the following estimate holds:*]{} $$\|u\|_{X^1_{\alpha}}+\|\eta\|_{Y^3_{\alpha}}\leq
C[\|h\|_{X_{\alpha}}+\|\rho\|_{Y_{\alpha}}].$$
Using a similar argument, we can also prove the following result:
[**Theorem 6.3**]{} [*Assume that $\gamma=\gamma_k$ for some $k\geq 2$ and let $1\leq\alpha\leq\infty$ be given. Let $$\tilde{X}_{\alpha,k}\times \tilde{Y}_{\alpha,k}
=\bigcap_{\gamma_j=\gamma_k}X_{\alpha,j}\times Y_{\alpha,j}, \qquad
\tilde{X}^1_{\alpha,k}\times \tilde{Y}^3_{\alpha,k}
=\bigcap_{\gamma_j=\gamma_k}X^1_{\alpha,j}\times Y^3_{\alpha,j}.$$ For any $(h,\rho)\in(\tilde{X}_{\alpha,k}\times \tilde{Y}_{\alpha,k})\bigcap
(X_{\alpha,1}\times Y_{\alpha,1})$, the system $(6.1)$ has a unique solution $(u,\eta)\in(\tilde{X}^1_{\alpha,k}\times\tilde{Y}^3_{\alpha,k})\bigcap
(X^1_{\alpha}\times Y^3_{\alpha})$. Moreover, there exists a constant $C_k>0$ such that the following estimate holds:*]{} $$\|u\|_{X^1_{\alpha}}+\|\eta\|_{Y^3_{\alpha}}\leq
C_k[\|h\|_{X_{\alpha}}+\|\rho\|_{Y_{\alpha}}].$$
We omit the proof of this result.
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[^1]: E-mail: cuishb@mail.sysu.edu.cn
[^2]: This work is supported by the China National Natural Science Fundation under grant number 11171357.
|
---
author:
- |
Long Wang, Yi-Zheng Fan[^1], Yi Wang\
[*School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China*]{}
title: 'The triangle-free graphs with rank $6$[^2]'
---
[**Abstract:**]{} The rank of a graph $G$ is defined to be the rank of its adjacency matrix $A(G)$. In this paper we characterize all connected triangle-free graphs with rank $6$.
[**MR Subject Classifications:**]{} 05C50
[**Keywords**]{}: Graphs; rank; nullity
Introduction
============
Throughout this paper we consider simple graphs. The [*rank of a graph*]{} $G=(V(G),E(G))$, denoted by $r(G)$, is defined to be the rank of its adjacency matrix; and the [*nullity of $G$*]{}, denoted by $\eta(G)$, is defined to be the multiplicity of zero eigenvalues of its adjacency matrix. It is easy to see that $r(G)+\eta(G)=|V(G)|$. The graph $G$ is called [*singular*]{} or [*nonsingular*]{} if $\eta(G)>0$ or $\eta(G)=0$.
In chemistry, a conjugated hydrocarbon molecule can be modeled by its molecular graph. It was known in $\cite{cvet}$ or $\cite{lon}$, if the molecule represented by a graph $G$ is chemically stable, then $G$ is nonsingular. In 1957 Collatz and Sinogowitz $\cite{colla}$ posed the problem of characterizing all singular graphs. The problem is very hard and only some particular results are known (see $\cite{che2}$, $\cite{guo}$, $\cite{hu}$, $\cite{tang}$), although it has received a lot of attention.
We review some known results related to this topic. In [@fio] and [@li] the smallest rank among $n$-vertex trees in which no vertex has degree greater than a fixed value was determined, and the corresponding trees were constructed. The singular line graphs of trees were described in [@gut] and [@sci]. The rank set of bipartite graphs of fixed order and bipartite graphs of rank $4$ were determined in [@fan]. It was shown in [@che1] that all connected graphs with rank $2$ (respectively, $3$) are complete bipartite graphs (respectively, complete tripartite graphs). In [@cha1] and [@cha2], the authors proved that each connected graph with rank $4$ (or $5$) is obtained from one of the $8$ (or $24$) reduced graphs by multiplication of vertices.
Now a natural problem is left open: [*To characterize graphs with rank $6$.*]{} Because a graph of rank $6$ can be obtained by multiplication of vertices from its reduced form, the problem is equivalent to characterize reduced graphs with rank $6$. We also note that regular bipartite graphs with rank $6$ has been characterized in $\cite{fan}$. In the present paper, we focus our attention on reduced connected triangle-free graphs with rank $6$, and characterize all such graphs.
Preliminaries
=============
We first introduce two graph operations: [*multiplication and reduction of vertices*]{}. Given a graph $G$ on vertices $v_{1}, v_{2},\ldots,v_{n}$. Let $m=(m_{1},m_{2},\ldots,m_{n})$ be a list of positive integers. Denote by $G\circ m$ the graph obtained from $G$ by replacing each vertex $v_{i}$ of $G$ with an independent set of $m_{i}$ vertices $v_{i}^{1},v_{i}^{2},\ldots,v_{i}^{m_{i}}$, and joining $v_{i}^{s}$ with $v_{j}^{t}$ if and only if $v_{i}$ and $v_{j}$ are adjacent in $G$. The resulting graph $G\circ m$ is said to be obtained from $G$ by [*multiplication of vertices*]{}; see [@cha1; @cha2].
Define a relation $\approx$ in $V(G)$ in the way that $x\approx y$ if and only if $N_G(x)=N_G(y)$, where $N_G(x)$ denotes the neighborhoods of a vertex $x$ in $G$. Obviously, the relation is an equivalence one, and each equivalence class $\bar v=\{x:\ N_G(x)=N_G(v)\}$ is an independent set. Now construct a new graph $R(G)$ obtained from $G$ by taking the vertex set to be all equivalence classes $\bar v$ and joining $\bar v$ with $\bar u$ if and only if $v, u$ are adjacent in $G$. The graph $R(G)$ is called to be obtained from $G$ by [*reduction of vertices*]{}, and is a [*reduced form* ]{} of $G$. One can see the above two operations are inverse to each other, and preserve the rank of graphs.
A graph is called [*reduced*]{} if itself is a reduced form, i.e. the neighborhoods of distinct vertices are distinct. Denote by $H \subseteq G$ if $H$ is a subgraph of $G$, and $H \lhd G$ if $H$ is an induced subgraph of $G$. For $H \subseteq G$ and $v\in V(G)$, denote by $N_{H}(v)$ the neighborhoods of $v$ in $H$. If two vertices $u$ and $v$ are adjacent in $G$, then we write $u\sim v$; otherwise write $u \nsim v$. The distance of two vertices $u, v$ in $G$ is denoted as $\dist_{G}(u,v)$. For $w\in V(G)\backslash V(H)$ and $H\lhd G$, the distance between $w$ and $H$ is denoted and defined by $\dist_{G}(w,H)=\min_{v\in V(H)}\{\dist_{G}(w,v)\}$. For later use we now introduce some basic results.
\[tree-nul\][*[@cvet]*]{} Let $G$ be a tree, then $r(G)=2\mu(G)$, where $\mu(G)$ is the matching number of $G$.
\[pend-nul\] [*[@cvet]*]{} Let $G$ be a graph containing a pendant vertex, and let $H$ be the induced subgraph of G by deleting the pendant vertex and the vertices adjacent to it. Then $\eta(G)=\eta(H)$, or equivalently, $r(G)=r(H)+2$.
\[bi-nulset\] [*[@fan]*]{} Let $\mathcal{B}_{n}$ be the set of bipartite graphs of order $n$. The nullity set of $\mathcal{B}_{n}$ is $\{n-2k:k=0,1,2,\ldots, \lfloor\frac{n}{2}\rfloor$}.
\[neigh-nonaj\][*[@wong]*]{} Let $H$ be an induced subgraph of a reduced graph $G$ for which $r(H)=r(G)$. If $u,v\in V(G)\backslash V(H)$ are not adjacent in $G$, then $N_{H}(u)\neq N_{H}(v)$.
\[neigh-outside\][*[@wong]*]{} Let $H$ be an induced subgraph of a reduced graph $G$ for which $r(H)=r(G)$, and let $v$ be a vertex not in $H$. Then $N_{H}(v)\neq N_{H}(u)$ for any $u\in V(H)$.
\[dist\] [*[@wong]*]{} Let $H$ be a proper induced subgraph of a connected graph $G$ for which $r(H)\geq r(G)-1$. Then $\dist_{G}(v,H)=1$ for each vertex $v\in V(G)\backslash V(H)$.
\[op1\] Let $H$ be a nonsingular induced subgraph of a reduced graph $G$ for which $r(H)=r(G)$. Assume $G$ has exactly two distinct vertices $u,v$ outside $H$. Let $\tilde{G}$ be obtained from $G$ by adding a new edge $uv$ if $uv \notin E(G)$ or deleting the edge $uv$ otherwise. Then $r(\tilde{G})=r(G)+2$.
[**Proof.**]{} The adjacency matrix of $G$ and $\tilde{G}$ can be written as: $$A(G)=\left(
\begin{array}{ccc}
0 & \theta & \alpha \\
\theta & 0 & \beta \\
\alpha^{T} & \beta^{T} & B \\
\end{array}
\right), \ A(\tilde{G})=\left(
\begin{array}{ccc}
0 & 1-\theta & \alpha \\
1-\theta & 0 & \beta \\
\alpha^{T} & \beta^{T} & B \\
\end{array}
\right),$$ where the first two rows of both matrices correspond to $u,v$ respectively, $\theta$ equals $1$ or $0$ if $uv \in E(G)$ or not, $B$ is the adjacency matrix of $H$. Let $Q=\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
-B^{-1}\alpha^{T} & -B^{-1}\beta^{T} & I_{k} \\
\end{array}
\right).$ Then [ $$Q^{T}A(G)Q=\left(
\begin{array}{ccc}
-\alpha B^{-1} \alpha^T & \theta-\alpha B^{-1}\beta^{T} & 0 \\
\theta-\beta B^{-1}\alpha^{T} & -\beta B^{-1}\beta^{T} & 0 \\
0 & 0 & B \\
\end{array}
\right)
,Q^{T}A(\tilde{G})Q=\left(
\begin{array}{ccc}
-\alpha B^{-1} \alpha^T & 1-\theta-\alpha B^{-1}\beta^{T} & 0 \\
1-\theta-\beta B^{-1}\alpha^{T} & -\beta B^{-1}\beta^{T} & 0 \\
0 & 0 & B \\
\end{array}
\right).$$]{} Since $r(G)=r(H)=r(B)$, the left-upper submatrix of order $2$ of $Q^{T}A(G)Q$ is zero. So $$Q^{T}A(\tilde{G})Q=\left(
\begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & A(H) \\
\end{array}
\right)
,$$ whose rank is $k+2$.$\blacksquare$
\[op2\] Let $H$ be a nonsingular induced subgraph of a reduced graph $G$ for which $r(H)=r(G)$. Assume that $V(G)\backslash V(H)=\{v_{1},v_{2},\ldots,v_{k}\}$, $k\geq 2$. Let $\bar{G}$ be obtained from $G$ by adding some edges if these edges are not in $G$ or deleting some edges if they are in $G$. Then $r(\bar{G})\geq r(G)+2$.
[**Proof.**]{} Without loss of generality, the adjacency relation of $v_1,v_2$ is different between $G$ and $\bar{G}$. Taking the subgraph $G'$ of $G$ and $\bar{G}'$ of $\bar{G}$ both induced by $V(H)\cup \{u,v\}$, we have $r(\bar{G}') \ge r(G')+2$ by Lemma \[op1\]. The result now follows as $r(\bar{G}) \ge r(\bar{G}')$ and $ r(G')=r(G)$.$\blacksquare$
Characterizing nonsingular triangle-free graphs of order $6$
============================================================
Let $\P_{k}$ denote the path of order $k$, and let $\C_{k}$ denote the cycle of order $k$ (to avoid the confusion of $C_k$ used in Fig. \[picC\]). The disjoint union of two graphs $G$ and $H$ is written as $G\cup H$. Denote by $kH$ the disjoint union of $k$ copies of $H$. In this section, we obtain the following main result by proving a series of lemmas below.
\[nons-6\] If $G$ is a nonsingular triangle-free graph of order $6$, then $G$ is one graph in Fig. \[pic-nons\].
![$8$ nonsingular graphs $A,B,C,D,E,F,3\P_{2},\P_{2}\cup \P_{4}$[]{data-label="pic-nons"}](2.eps "fig:")\
For a nonsingular bipartite graph $G$ of order $6$, if $G$ is disconnected, by Lemma \[bi-nulset\], the rank of each component must be $2$ or $4$. Note that connected nonsingular bipartite graphs of order $2$ must be $\P_{2}$, and connected nonsingular bipartite graphs of order $4$ must be $\P_{4}$; see $\cite{cha1}$ or $\cite{che1}$ for details. So we have the following result immediately.
\[nons-6-disc\] Let $G$ be a disconnected nonsingular bipartite graph of order $6$. Then $G$ is $3\P_{2}$ or $\P_{4}\cup \P_{2}$.
Denote by $g(G)$ the [*girth*]{} of a graph $G$ (i.e., the shortest length of cycles in $G$). If $G$ is a nonsingular triangle-free graph of order $6$, then either $g(G)=4,5,6$, or $G$ is a tree. If $g(G)=6$, clearly $G=C_6$.
\[nons-6-g5\] If $G$ is a connected nonsingular graph of order $6$ with $g(G)=5$, then $G=F$ in Fig. \[pic-nons\].
[**Proof.**]{} Let $v$ be the unique vertex of $G$ outside $\C_{5}$. If $|N_{\C_{5}}(v)|\geq 3$, then $G$ contains triangles. If $|N_{\C_{5}}(v)|=2$, say $N_{\C_{5}}(v)=\{u_{1},u_{2}\}$, then $\dist_{\C_{5}}(u_{1},u_{2})= 2$; otherwise $G$ contains triangles. However, in this case $G$ is not reduced and hence has rank at most $5$. Thus $|N_{\C_{5}}(v)|=1$ and the result follows. $\blacksquare$
\[nons-6-g4\] If $G$ is a connected nonsingular graph of order $6$ with $g(G)=4$, then $G$ is the graph $B$ or $C$ in Fig. \[pic-nons\].
[**Proof.**]{} Note that $r(C_{4})=2$, $r(G)=6$, and deleting a vertex from $G$ reduces $r(G)$ at most $2$. There exists a graph $H \lhd G$ such that $|V(H)|=5$ and $r(H)=4$. As shown in $\cite{cha1}$, $H$ is obtained from $\C_{4}$ by attaching a pendant vertex. For the reconstruction of $G$, one need to add a vertex together with some edges. It is easily to check that $G$ is the graph $B$ or $C$.$\blacksquare$
\[nons-6-tree\] If $G$ is a nonsingular tree of order $6$, then $G$ is the graph $D$ or $E$ in Fig. \[pic-nons\].
[**Proof.**]{} By Lemma \[tree-nul\], $\mu(G)=3$. Thus $G \supseteq 3\P_{2}$. Since a tree of order $6$ must contain $5$ edges, we have $|E(G\backslash 3\P_{2})|=2$. One can easily see that $G$ is the graph $D$ or $E$. $\blacksquare$
Characterizing reduced triangle-free graphs with rank $6$
=========================================================
Let $G$ be a connected reduced triangle-free graph with rank $6$. We know that $G$ contains an induced subgraph of order $6$ and rank $6$, which are listed in Fig. \[pic-nons\] by Theorem \[nons-6\]. In order to reconstruct $G$ we are left to consider how to add vertices to the graphs in Fig. \[pic-nons\]. The main result of this paper is as follows.
\[rank6\] Let $G$ be a connected reduced triangle-free graph. Then $r(G)=6$ if and only if there exist two graphs $H_{1}$ and $H_{2}$ such that $H_{1}\rhd G\rhd H_{2}$, where $H_1$ is one graph in Fig. \[pic-nons\] and $H_2$ is one graph in Fig. \[pic-rank-6\].
![$6$ reduced graphs with rank $6$: $G_{1},G_{2},G_{3},G_{4},G_{5},G_{6}$[]{data-label="pic-rank-6"}](3.eps "fig:")\
It is easy to check each graph in Fig. \[pic-rank-6\] has rank $6$. So the sufficiency of Theorem \[rank6\] follows. We first discuss the necessity for the case of $G$ being bipartite, and then the case of $G$ being non-bipartite. For $H\lhd G$, Denote by $H(k)$ the set of vertices $v \in V(G) \backslash V(H)$ such that $|N_{H}(v)|=k$.
Reduced bipartite graphs of rank $6$
------------------------------------
\[degree\] Let $G$ be a connected reduced bipartite graph with rank $6$.\
(1) If $G\supseteq H_{1}\supseteq \P_{6}$, where $|V(H_{1})|=6$, then $|N_{H_{1}}(v)|\leq 3$ for each $v\in V(G)\backslash V(H_{1})$;\
(2) If $G\supseteq H_{2}\supseteq C$ of Fig. \[pic-nons\], where $|V(H_{2})|=6$, then $|N_{H_{2}}(v)|\leq 2$ for each $v\in V(G)\backslash V(H_{2})$.
[**Proof.**]{} Suppose that $v\in V(G)\backslash V(H_{1})$ and $N_{H_{1}}(v)=\{v_{1},v_{2},\ldots,v_{k}\} \ne \emptyset$. Then $\dist_{H_{1}}(v_{i},v_{j})$ is even; otherwise $G$ is not bipartite. As $H_{1}\supseteq \P_{6}$, there exist at most three vertices whose pairwise distance is even in $H_{1}$, which implies that $k\leq 3$. The proof for the next result is similar and omitted.$\blacksquare$
\[bi-A\] Let $G$ be a connected reduced bipartite graph with rank $6$, which contains $A$ (or $\C_6$) in Fig. \[pic-nons\] as an induced subgraph. Then $G$ is an induced subgraph of $G_{1}$, $G_{2}$ or $G_{3}$ in Fig. \[pic-rank-6\]
[**Proof.**]{} By Lemma \[dist\] and Lemma \[degree\], $1\leq |N_{A}(v)|\leq 3$ for $v\in V(G)\backslash V(A)$. If $A(2)\neq\emptyset$, let $v\in A(2)$ and $v_{1}$, $v_{2}$ be its two neighbors in $A$. If $\dist_{A}(v_{1},v_{2})$ is odd, then $G$ contains an odd cycle. If $\dist_{A}(v_{1},v_{2})=2$, then $G$ is not reduced by Lemma \[neigh-outside\]. So $A(2)=\emptyset$. We divide the remaining proof into some cases:
[*Case 1:*]{} $A(3)=\emptyset$. If $|A(1)|=1$, then $G\lhd G_{1}$ of Fig. \[pic-rank-6\] obviously. Now we assume $|A(1)|\geq 2$. Suppose $v_{1},v_{2},\ldots,v_{k}\in A(1)$ and $v_{1}',v_{2}',\ldots,v_{k}'$ are their unique neighbors in $A$, respectively. Note that $v_{i}'\neq v_{j}'$ for $i\neq j$ as $G$ is reduced. Observing the graph $A_1,A_2$ in Fig. \[picA\] both have rank $8$, so $\dist_{A}(v_{i}',v_{j}')\neq 1$ for $i\neq j$, which implies that $\dist_{A}(v_{i_{1}}',v_{j_{1}}')=2$ and $\dist_{A}(v_{i_{2}}',v_{j_{2}}')=3$ can not hold at the same time for some $v_{i_{1}}',v_{i_{2}}',v_{j_{1}}',v_{j_{2}}'$. If $\dist_{A}(v_{i}',v_{j}')=2$ for some $i\neq j$, then $|A(1)|\leq 3$. Since $\C_6\lhd A_3 \lhd G_1$, we have $r(A_{3})=6$, where $A_3$ is listed in Fig. \[picA\]. It follows that $v_{i} \nsim v_{j}$ in $G$; otherwise $r(G) \ge 8$ by Lemma \[op1\]. Hence $G\lhd A_{3}\lhd G_{1}$. If $\dist_{A}(u_i',v_j')=3$, then obviously $|A(1)|= 2$. Since $r(G_{2})=6$, $v_{i} \sim v_{j}$ in $G$ also by Lemma \[op1\], which implies that $G=G_{2}$, where $G_2$ is listed in Fig. \[pic-rank-6\].
[*Case 2:*]{} $A(1)=\emptyset$. Suppose $u,v\in A(3)$, $N_{A}(u)=\{u_{1},u_{2},u_{3}\}$ and $N_{A}(v)=\{v_{1},v_{2},v_{3}\}$. Then $\dist_{A}(u_{i},u_{j})$ and $\dist_{A}(v_{i},v_{j})$ must be even; otherwise $G$ is not bipartite. So, if $N_{A}(v)\cap N_{A}(u)\neq \emptyset$, then $N_{A}(u)=N_{A}(v)$, which implies $|A(3)|\leq 2$. If $|A(3)|=1$, then $G \lhd G_{3}$ of Fig. \[pic-rank-6\]. If $|A(3)|=2$, let $u,v\in A(3)$. If $N_A(u)=N_A(v)$, then $u \sim v$ as $G$ is reduced by Lemma \[neigh-nonaj\]; but in this case $G$ would have triangles. So $N_A(u) \cap N_A(v)=\emptyset$. Noting that $r(G_{3})=6$, $u \nsim v$ in $G$ by Lemma \[op1\], which implies that $G= G_{3}$, where $G_3$ is listed in Fig. \[pic-rank-6\].
[*Case 3:*]{} $ A(1)\neq \emptyset$ and $A(3)\neq \emptyset$. Assume $u\in A(3)$ and $v\in A(1)$. Noting that $r(A_{4})=r(A_{5})=8$ while $r(G_{1})=6$, we have $\dist_{G}(u,v)=2$, where $A_4,A_5$ are listed in Fig. \[picA\]. Then $G \lhd G_{1}$.$\blacksquare$
![Illustration for Lemma \[bi-A\], where $r(A_{1})=r(A_{2})=r(A_{4})=r(A_{5})=8$, $r(A_{3})=6$[]{data-label="picA"}](4.eps "fig:")\
\[bi-B\] Let $G$ be a connected reduced bipartite graph of rank $6$, which contains $B$ of Fig. \[pic-nons\] as an induced subgraph. Then $G$ is an induced subgraph of $G_{1}$, $G_{3}$, $G_{4}$ or $G_{5}$.
[**Proof.**]{} By Lemma \[dist\] and Lemma \[degree\], $1\leq |N_{B}(v)|\leq 2$ for $v\in V(G)\backslash V(B)$.
[*Case 1:*]{} $B(2)\neq \emptyset$. Let $v\in B(2)$ and let $v_{1}$ and $v_{2}$ be two neighbors of $v$ in $B$. As $G$ is bipartite, $\dist_{B}(v_{1},v_{2})$ should be even, which implies $\dist_{B}(v_{1},v_{2})=2$. As $G$ is reduced, $G \rhd A $ (or $\C_6$). By Lemma \[bi-A\], $G$ is an induced subgraph of $G_{1}$, $G_{2}$ or $G_{3}$. Noting that $B$ is not an induced subgraph of $G_{2}$, so $G\lhd G_{1}$ or $G\lhd G_{3}$.
[*Case 2:*]{} $ B(2)=\emptyset$. One can check that $r(B_{i})=8$ for $i=1,2,3,4$, where $B_i$’s are listed in Fig. \[picB\], which shows $|B(1)| \le 3$ by the rank of graphs $B_1,B_3$. If $|B(1)|=1$, the result holds obviously.
If $|B(1)|=2$, let $u,v \in B(1)$ and let $u',v'$ be their unique neighbors in $B$ respectively. As $r(B_3)=r(B_4)=8$, $\dist_{B}(u',v') \le 2$. We have four subcases:
\(a) $\dist_{B}(u',v')=1$ and $|N_{B}(u')|=|N_{B}(v')|=2$. As $r(G_{5})=6$, $u \nsim v$ by Lemma \[op1\]. Then $G\lhd G_{5}$.
\(b) $\dist_{B}(u',v')=1$ and one of $u',v'$ has $3$ neighbors in $B$. As $r(B_1)=r(B_2)=8$, exactly one of $u',v'$ has $3$ neighbors in $B$. As $r(G_{5})=6$, $u\sim v$ by Lemma \[op1\]. Then $G\lhd G_{5}$.
\(c) $\dist_{B}(u',v')=2$ and $|N_{B}(u')|=|N_{B}(v')|=2$. As $r(B_{5})=6$, $u \nsim v$. Then $G\lhd B_{5}\lhd G_{1}$.
\(d) $\dist_{B}(u',v')=2$ and exactly one of $\{u',v'\}$ has $3$ neighbors in $B$. As $r(G_{5})=6$, $u \nsim v$. Then $G\lhd G_{5}$.
If $|B(1)|=3$, let $u,v,w \in B(1)$ and $u',v',w'$ be their unique neighbors in $B$ respectively. There are three possible subcases:
\(a) $u'\sim v'$ and $v' \sim w'$, $|N_{B}(v')|=3$ and $|N_{B}(u')|=|N_{B}(w')|=2$. As $r(G_{4})=6$, $G\lhd G_{4}$.
\(b) $u'\sim v'$ and $v' \sim w'$, $|N_{B}(u')|=3$ and $|N_{B}(v')|=|N_{B}(w')|=2$. As $r(G_{5})=6$, $G\lhd G_{5}$.
\(c) $G=B_5$. As $r(B_{5})=6$, $G\lhd G_{1}$.$\blacksquare$
![Illustration for Lemma \[bi-B\], where $r(B_{1})=r(B_{2})=r(B_{3})=r(B_{4})=8; r(B_{5})=6$[]{data-label="picB"}](5.eps "fig:")\
\[bi-C\] Let $G$ be a connected reduced bipartite graph with rank $6$, which contains $C$ of Fig. \[pic-nons\] as an induced subgraph. Then $G$ is an induced subgraph of $G_{1}$, $G_{3}$, $G_{4}$ or $G_{5}$.
[**Proof.**]{} We have $1\leq |N_{C}(v)|\leq 2$ for each $v\in V(G)\backslash V(C)$.
[*Case 1:*]{} $C(2)\neq \emptyset$. Let $v\in C(2)$, and let $v_{1},v_{2}$ be its two neighbors in $C$. As $G$ is bipartite, $\dist_{C}(v_{1},v_{2})$ should be even, and $\dist_{C}(v_{1},v_{2})=2$. As $G$ reduced, one of $\{v_{1},v_{2}\}$ is a pendant vertex of $C$. Thus $G\rhd B$ and the result follows by Lemma \[bi-B\].
[*Case 2:*]{} $C(2)=\emptyset$. If $|C(1)|=1$, the result holds obviously. Now we suppose $|C(1)|\geq 2$. If there exist $u,v\in C(1)$ and $u\sim v$. Let $u'$, $v'$ be their unique neighbors in $C$ respectively. Then $\dist_{C}(u',v')$ should be odd by the bipartiteness of $G$. Note that $|N_{C}(v')|\neq 3$ and $|N_{C}(u')|\neq 3$; otherwise $G$ is not reduced. If $\dist_{C}(u',v')=1$, then $G\rhd B$. If $\dist_{C}(v',u')=3$, noting that $r(C_{1})=r(C_{2})=8$, this case cannot occur, where $C_1,C_2$ are listed in Fig. \[picC\].
Now assume $v_{1},v_{2},\ldots,v_{k}\in C(1)$ and $v_{i} \nsim v_{j}$ for all $1\leq i< j\leq k$. Similarly, none of them has a neighbor with degree $3$ in $C$. Note that $r(C_{3})=r(C_{4})=8$ and $r(C_{5})=r(C_{6})=6$. We conclude that $G\lhd C_{5}\lhd G_{5}$ or $G\lhd C_{6}\lhd G_{1}$. $\blacksquare$
![Illustration for Lemma \[bi-C\], where $r(C_{1})=r(C_{2})=r(C_{3})=r(C_{4})=r(C_{5})=8;r(C_{6})=6$[]{data-label="picC"}](6.eps "fig:")\
\[bi-D\] Let $G$ be a connected reduced bipartite graph with rank $6$, which contains $D$ of Fig. \[pic-nons\] as an induced subgraph. Then $G$ is an induced subgraph of $G_{1}$, $G_{3}$, $G_{4}$ or $G_{5}$.
[**Proof.**]{} We have $1\leq |N_{C}(v)|\leq 3$ for each $v\in V(G)\backslash V(C)$.
[*Case 1:*]{} $D(2)\neq \emptyset$. Let $v\in D(2)$, and let $v_{1}$ and $v_{2}$ be its two neighbors in $D$. The $\dist_{D}(v_{1},v_{2})$ should be even. If $\dist_{D}(v_{1},v_{2})=2$, then $G$ is not reduced. If $\dist_{D}(v_{1},v_{2})=4$, then $G\rhd A$. Thus the result follows from Lemma \[bi-A\].
[*Case 2:*]{} $D(3)\neq \emptyset$. Let $v\in B(3)$, and let $v_{1},v_{2}$ and $v_{3}$ be its three neighbors in $D$. Since $\dist_{D}(v_{i},v_{j})$ must be even, we have $G\rhd B$. Then the result follows from Lemma \[bi-B\].
[*Case 3:*]{} $D(2)=D(3)=\emptyset$. Let $v_{1},v_{2},\ldots,v_{k}\in D(1)$ and let $v_{1}',v_{2}'\ldots,v_{k}'$ be their unique neighbors in $D$ respectively. As $G$ is reduced, $v_{i}'$ cannot be the quasi-pendant vertex of $D$ (i.e., the vertex adjacent to the pendant vertex). If $k=1$, then $G\lhd G_{1}$ and the result follows easily.
If $v_{i}\sim v_{j}$ for some $1\leq i\leq j\leq k$, then $\dist_{D}(v_{i}',v_{j}')$ should be odd. We have the following subcases:
\(a) If $\dist_{D}(v_{i}',v_{j}')=1$, then $v_{i}'$ and $v_{j}'$ must be the two vertices in the middle of the path $D$. Thus $G\rhd C$ and the result follows.
\(b) If $\dist_{D}(v_{i}',v_{j}')=3$, then one of $\{v_{i}',v_{j}'\}$ must be the pendant vertex of $D$. Thus $G\rhd A$ and the result follows.
\(c) If $\dist_{D}(v_{i}',v_{j}')=5$, then $G\rhd D_{1}$. Noting that $r(D_{1})=6$ and $r(D_{2})=8$, we have $G\lhd D_{1}\lhd G_{4}$, where $D_1,D_2$ are listed in Fig. \[picD\].
If $v_{i} \nsim v_{j}$ for any $1\leq i< j\leq k$. Noting that $r(D_{3})=6$ and $r(D_{4})=r(D_{5})=r(D_{6})=8$, We have $G\lhd D_{3}\lhd G_{1}$, where $D_3,D_4,D_5,D_6$ are listed in Fig. \[picD\]. $\blacksquare$
![Illustration for Lemma \[bi-D\], where $r(D_{1})=r(D_{3})=6;r(D_{2})=r(D_{4})=r(D_{5})=r(D_{6})=8$[]{data-label="picD"}](7.eps "fig:")\
\[bi-E\] Let $G$ be a connected reduced bipartite graph with rank $6$, which contains $E$ of Fig. \[pic-nons\] as an induced subgraph. Then $G$ is an induced subgraph of $G_{1}$, $G_{2}$ $G_{3}$, $G_{4}$ or $G_{5}$.
[**Proof.**]{} We have $1\leq |N_{E}(v)|\leq 3$ for each $v\in V(G)\backslash V(E)$.
[*Case 1:*]{} $E(2)\neq \emptyset$. Let $v\in E(2)$ and let $v_{1}$ and $v_{2}$ be its two neighbors in $E$. The $\dist_{E}(v_{1},v_{2})$ is even. If $\dist_{E}(v_{1},v_{2})=4$, then $G\rhd A$. Suppose $\dist_{E}(v_{1},v_{2})=2$. Noting that $G$ is reduced, $G\rhd A$ or $G\rhd C$. The result follows from Lemmas \[bi-A\] and \[bi-C\].
[*Case 2:*]{} $E(3)\neq \emptyset$. Let $v\in E(3)$ and let $v_{1},v_{2}$ and $v_{3}$ be the neighbors of $v$ in $E$. The $\dist_{E}(v_{i},v_{j})$ is even for $1\leq i< j\leq 3$. By Lemma \[neigh-outside\], $G\rhd B$ and the result follows.
[*Case 3:*]{} $E(2)=E(3)=\emptyset$. For each $v\in E(1)$, it must be adjacent to a pendant vertex of $E$ since $G$ is reduced, which implies $|E(1)|\leq 3$. Let $v\in E(1)$, and let $v'$ be its unique neighbor in $E$ (i.e. a pendant vertex of $E$). If $|E(1)|=1$, then $G\lhd G_{1}$ and the result follows. If $|E(1)|\geq 2$, noting that $r(E_{1})=r(E_{2})=8$ and $r(E_{3})=6$, we have $G\lhd E_{3}\lhd G_{1}$, where $E_1,E_2,E_3$ are listed in Fig. \[picE\]. $\blacksquare$
![Illustration for Lemma \[bi-E\], where $r(E_{1})=r(E_{2})=8;r(E_{3})=6$[]{data-label="picE"}](8.eps "fig:")\
\[p4p2\] Let $G$ be a connected reduced bipartite graph with rank $6$, which contains $\P_{4}\cup \P_{2}$ of Fig. \[pic-nons\] as an induced subgraph. Then $G$ is an induced subgraph of $G_{1},G_{2},G_{4}$ or $G_{5}$.
[**Proof.**]{} Note that $G$ is connected and bipartite, and $\dist_{G}(v, \P_{4}\cup \P_{2})=1$ for each vertex $v\in V(G)\backslash V(\P_{4}\cup \P_{2})$. We have two cases:
[*Case 1:*]{} There exists a vertex $v\in V(G)\backslash V(\P_{4}\cup \P_{2})$ such that $\dist_{G}(v,\P_{4})=\dist_{G}(v,\P_{2})=1$. Then $|N_{\P_{2}}(v)|=1$ and $|N_{\P_{4}}(v)|\leq 2$. If $|N_{\P_{4}}(v)|=1$, then $G\rhd D$; if $|N_{\P_{4}}(v)|=2$, the $G\rhd C$. The result follows by Lemmas \[bi-D\] and \[bi-C\].
[*Case 2:*]{} There exist two vertices $u,v\in V(G)\backslash V(\P_{4}\cup \P_{2})$ such that $\dist_{G}(u,\P_{4})=\dist_{G}(v,\P_{2})=1$ and $u\sim v$. Then $G\rhd D$ and the result also follows.$\blacksquare$
\[3p2\] Let $G$ be a connected reduced bipartite graph with rank $6$, which contains $3\P_{2}$ of Fig. \[pic-nons\] as an induced subgraph. Then $G$ is an induced subgraph of $G_{1},G_{2},G_{3},G_{4}$ or $G_{5}$.
[**Proof.**]{} We denote the three disjoint paths as $\P_{2}^{1},\P_{2}^{2},\P_{2}^{3}$ respectively. Then we have two cases:
[*Case 1:*]{} There exist two vertices $u,v\in V(G)\backslash V(3\P_{2})$ such that $\dist_{G}(u,\P_{2}^{1})=\dist_{G}(v,\P_{2}^{2} )=1$ and $u\sim v$. Then $G\rhd D$ and the result follows by Lemma \[bi-D\].
[*Case 2:*]{} There exists a vertex $v\in V(G)\backslash V(3\P_{2})$ such that $\dist_{G}(v,\P_{2}^{1})=\dist_{G}(v,\P_{2}^{2})=1$. If $\dist_{G}(v,\P_{2}^{3})=1$, then $G\rhd E$; if $\dist_{G}(v,\P_{2}^{3})\geq 2$, then $G\rhd (\P_{4}\cup \P_{2})$. The result follows by Lemmas \[bi-E\] and \[p4p2\] .$\blacksquare$
Reduced triangle-free and non-bipartite graphs with rank $6$
------------------------------------------------------------
Observe that if a graph $G$ contains an induced odd cycle $\C_{2k+1}$, then $r(G) \ge r(\C_{2k+1})=2k+1$. So, if $G$ is triangle-free, non-bipartite, and has rank $6$, then $G$ contains an induced $\C_5$.
\[nonbi-rank6\] Let $G$ be a triangle-free and non-bipartite graph with rank $6$. Then $G\rhd F$ of Fig. \[pic-nons\].
[**Proof.**]{} As discussed above, $G$ contains an induced cycle $\C_5$. Let $v$ be an arbitrary vertex not in $\C_{5}$. Note that $r(\C_{5})=5$, thus $d_{\C_{5}}(v)\geq 1$ by Lemma $\ref{dist}$. If $d_{\C_{5}}(v)=1$, then $G\rhd F$. If $d_{\C_{5}}(v)\geq 3$, then $G$ contains triangles; a contradicition. Now suppose $d_{\C_{5}}(v)=2$ for each $v\notin V(\C_{5})$. Let $V(G)\backslash V(\C_{5})=\{v_{1},v_{2},\ldots,v_{k}\}$. If $v_{i_{0}}\sim v_{j_{0}}$ for some $i_{0},j_{0}$, then $N_{\C_{5}}(v_{i_{0}})\cap N_{\C_{5}}(v_{j_{0}})=\emptyset$; otherwise $G$ contains triangles. Thus for any $i \ne j$, either $v_{i}$ is not adjacent to $v_{j}$, or $v_{i}\sim v_{j}$ and $N_{\C_{5}}(v_{i})\cap N_{\C_{5}}(v_{j})=\emptyset$. However, in any case $G$ can be obtained from $\C_{5}$ by several steps of multiplication of vertices, which implies $r(G)=r(\C_{5})=5$.$\blacksquare$
![Illustration for Lemma \[red-nonbi-rank6\], where $r(F_{i})\geq 7$, $i=1,2,\ldots,8$[]{data-label="picF"}](9.eps "fig:")\
\[red-nonbi-rank6\] Let $G$ be a connected reduced triangle-free graph with rank $6$, which contains $F$ of Fig. \[pic-nons\] as an induced subgraph. Then $G$ is an induced subgraph of $G_{6}$ of Fig. \[pic-rank-6\].
[**Proof.**]{} If $|N_{F}(v)|\geq 4$ for some $v\in V(G)\backslash V(F)$, then $G\rhd C_{3}$. If $|N_{F}(v)|=3$ for some $v\in V(G)\backslash V(F)$, say $N_{F}(v)=\{v_{1},v_{2},v_{3}\}$, then $\dist_{F}(v_{i},v_{j})\geq 2$ for $1\leq i<j\leq 3$. Thus one of $\{v_{1},v_{2},v_{3}\}$ is the pendant vertex in $F$. So, $G\rhd F_{1}$ or $G$ is obtained from $F$ by multiplication of the vertex with degree $3$. However, $r(F_{1})=7$, where $F_1$ is listed in Fig. \[picF\]. Now we conclude that $1\leq |N_{F}(v)|\leq 2$ for $v\in V(G)\backslash V(F)$.
[*Case 1:*]{} $F(2)=\emptyset$. Let $F(1)=\{v_{1},v_{2},\ldots,v_{k}\}$, and $N_{F}(v_{i})=\{u_{i}\}$ for $i=1,2,\ldots,k$. Then $u_{i}$ cannot be adjacent to the pendant or the quasi-pendant vertex of $F$; otherwise $r(G)\geq 7$ by Lemma \[pend-nul\] or $G$ is not reduced by Lemma \[neigh-outside\]. Thus $|F(1)|\leq 4$. One can check that $r(F_{2})=r(F_{3})=8$, and $r(G_{6})=6$. By Corollary \[op2\] we have $G\lhd G_{6}$.
[*Case 2:*]{} $F(1)=\emptyset$. Let $w_{1}$ be the vertex with maximum degree in $F$, and $N_{F}(w_{1})=\{w_{2},w_{3},w_{4}\}$, where $w_{2}$ is the pendant vertex of $F$. Let $F(2)=\{v_{1},v_{2},\ldots,v_{k}\}$ and $N_{F}(v_{i})=\{u_{i}^{1},u_{i}^{2}\}$ for $i=1,2,\ldots,k$. Since $G$ is triangle-free, we have $2\leq \dist_{F}(u_{i}^{1},u_{i}^{2})\leq 3$. If $u_{i}^{1}=w_{3}$ and $u_{i}^{2}=w_{4}$ for some $i$, then $r(G)\geq r(F_{4})=7$, where $F_4$ is listed in Fig. \[picF\]. As $G$ is reduced, one of $\{u_{i}^{1},u_{i}^{2}\}$ must be $w_{2}$. Set $F(2)=F(2)_{1}\cup F(2)_{2}$, where $F(2)_{1}=\{v_{i}|\dist_{F}(u_{i}^{1},u_{i}^{2})=2\}$ and $F(2)_{2}=\{v_{i}|\dist_{F}(u_{i}^{1},u_{i}^{2})=3\}$. As $r(F_{5})=7$, then $F(2)_{2}=\emptyset$, where $F_5$ is listed in Fig. \[picF\]. Note that $|F(2)_{1}|\leq 2$. Suppose that $F(2)_{1}=\{v_{1},v_{2}\}$. We have $N_{F}(v_{1})=\{w_{2},w_{3}\}$ and $N_{F}(v_{2})=\{w_{2},w_{4}\}$. If $v_{1}\sim v_{2}$, $G\rhd C_{3}$; otherwise $G\rhd C_{7}$ which implies $r(G)\geq 7$. Thus $|F(2)_{1}|\leq 1$ and $G\lhd G_{6}$.
[*Case 3:*]{} $ F(1)\neq\emptyset$ and $F(2)\neq\emptyset$. By the above discussion, we know $|F(2)_{1}|=1$ and $F(2)_{2}=\emptyset$. By a simple checking, $r(F_{i})=8$ for $i=6,7,8$. So $G\lhd G_{6}$.$\blacksquare$
[90]{}
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[^1]: Corresponding author. E-mail address: fanyz@ahu.edu.cn (Y.-Z. Fan), wangy@ahu.edu.cn (Y. Wang), wanglongxuzhou@126.com (L. Wang)
[^2]: Supported by National Natural Science Foundation of China (11071002), Program for New Century Excellent Talents in University, Key Project of Chinese Ministry of Education (210091), Specialized Research Fund for the Doctoral Program of Higher Education (20103401110002), Anhui Provincial Natural Science Foundation (10040606Y33), Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University(KJJQ1001), Academic Innovation Team of Anhui University Project (KJTD001B), Fund for Youth Scientific Research of Anhui University(KJQN1003).
|
---
author:
- |
Adrien Poulenard\
LIX, Ecole Polytechnique\
[adrien.poulenard@inria.fr]{}
- |
Marie-Julie Rakotosaona\
LIX, Ecole Polytechnique\
[mrakotos@lix.polytechnique.fr]{}
- |
Yann Ponty\
LIX, Ecole Polytechnique\
[yann.ponty@lix.polytechnique.fr]{}
- |
Maks Ovsjanikov\
LIX, Ecole Polytechnique\
[maks@lix.polytechnique.fr]{}
- |
Adrien Poulenard\
LIX, Ecole Polytechnique\
[adrien.poulenard@inria.fr]{}
- |
Marie-Julie Rakotosaona\
LIX, Ecole Polytechnique\
[mrakotos@lix.polytechnique.fr]{}
- |
Yann Ponty\
LIX, Ecole Polytechnique\
[yann.ponty@lix.polytechnique.fr]{}
- |
Maks Ovsjanikov\
LIX, Ecole Polytechnique\
[maks@lix.polytechnique.fr]{}
bibliography:
- 'egbib.bib'
title: 'Effective Rotation-invariant Point CNN with Spherical Harmonics Kernels'
---
|
---
abstract: |
This paper generalizes an earlier result by the author based on well-established embedding theorems that connect the classical theory of relativity to higher-dimensional spacetimes. In particular, an $n$-dimensional Riemannian space is said to be of class $m$ if $m+n$ is the lowest dimension of the flat space in which the given space can be embedded. To study traversable wormholes, we concentrate on spacetimes that can be reduced to embedding class one by a suitable transformation. It is subsequently shown that the extra degrees of freedom from the embedding theory provide the basis for a complete wormhole solution in the sense of obtaining both the redshift and shape functions.\
**Keywords.** Traversable wormholes; Embedding class one\
\
**PACS No.** **04.20.-q, 04.20.Jb, 04.50.-h**
author:
- |
PETER K F KUHFITTIG\*\
[^1] Department of Mathematics, Milwaukee School of Engineering,\
Milwaukee, Wisconsin 53202-3109, USA
title: '[Spherically symmetric wormholes of embedding class one]{}'
---
Introduction {#S:Introduction}
============
Wormholes are handles or tunnels in spacetime that are able to connect widely separated regions of our Universe and may even connect entirely different universes [@MT88]. Such wormholes can be described by the static and spherically symmetric line element $$\label{E:wormhole}
ds^{2}=-e^{\nu(r)}dt^{2}+\frac{dr^2}{1-b(r)/r}
+r^{2}(d\theta^{2}+\text{sin}^{2}\theta\,
d\phi^{2}),$$ using units in which $c=G=1$. Here $\nu=\nu(r)$ is called the *redshift function*, which must be everywhere finite to avoid an event horizon. The function $b=b(r)$ is called the *shape function*. The spherical surface $r=r_0$ is the *throat* of the wormhole. Here $b(r)$ must satisfy the following conditions: $b(r_0)=r_0$, $b(r)<r$ for $r>r_0$, and $b'(r_0)<1$, called the *flare-out condition*. We also wish to assume that $b'(r)>0$ due to the field equation $8\pi\rho(r)=b'(r)/r^2$, where $\rho$ is the energy density, normally considered positive. The flare-out condition can only be satisfied by violating the null energy condition, discussed in Sec. \[S:other\]. For a Morris-Thorne wormhole, this violation requires the use of “exotic matter."
The discussion in Ref. [@MT88] is based on the following strategy: specify the geometric conditions required for a traversable wormhole and then either manufacture or do a search for matter or fields that can produce the desired energy-momentum tensor. The main goal of this paper is to reverse this strategy by showing that the conditions discussed are sufficient for producing a complete solution, i.e., for producing both the redshift and shape functions. The approach in this paper differs significantly from that in Ref. [@pK18], which discusses charged wormholes admitting a one-parameter group of conformal motions, together with a new model to explain the flat galactic rotation curves without the need for dark matter.
The embedding {#S:embedding}
=============
Unlike Ref. [@pK18], the conditions discussed in this paper are derived directly from the assumption that the spacetime is of embedding class one. Here we need to recall that an $n$-dimensional Riemannian space is said to be of embedding class $m$ if $m+n$ is the lowest dimension of the flat space in which the given space can be embedded [@MG17; @M1; @M2; @M3; @M4; @sM]. It is well known that the exterior Schwarzschild solution is a Riemannian space of embedding class two. Following Ref. [@MG17], we start with the static and spherically symmetric line element $$\label{E:line1}
ds^{2}=e^{\nu(r)}dt^{2}-e^{\lambda(r)}dr^{2}
-r^{2}\left(d\theta^{2}+\sin^{2}\theta \,d\phi^{2}
\right).$$ (For physical reasons, it is generally assumed that $\nu(r)$ is finite and that $\text{lim}_{r\rightarrow \infty}\nu(r)=0$.) It is shown in Ref. [@MG17] that this metric of class two can be reduced to a metric of class one and can therefore be embedded in a five-dimensional flat spacetime. The following transformation can accomplish this reduction: $z^1=r\,\text{sin}\,\theta\,\text{cos}\,\phi$, $z^2=
r\,\text{sin}\,\theta\,\text{sin}\,\phi$, $z^3=r\,\text{cos}\,\theta$, $z^4=\sqrt{K}\,e^{\frac{\nu}{2}}
\,\text{cosh}{\frac{t}{\sqrt{K}}}$, and $z^5=\sqrt{K}
\,e^{\frac{\nu}{2}}\,\text{sinh}{\frac{t}{\sqrt{K}}}$. The result is [@MG17] $$\label{E:line3}
ds^{2}=e^{\nu}dt^{2}-\left(\,1+\frac{K\,e^{\nu}}{4}\,
{\nu'}^2\,\right)\,dr^{2}-r^{2}\left(d\theta^{2}
+\sin^{2}\theta\, d\phi^{2} \right).$$ Metric (\[E:line3\]) is therefore equivalent to metric (\[E:line1\]) if $$\label{E:lambda1}
e^{\lambda}=1+\frac{K\,e^{\nu}}{4}\,{\nu'}^2,$$ where $K>0$ is a free parameter. The condition is equivalent to the following condition due to Karmarkar [@kK48]: $$\label{E:Kar}
R_{1414}=\frac{R_{1212}R_{3434}-R_{1224}R_{1334}}
{R_{2323}},\quad R_{2323}\neq 0.$$ (See Ref. [@pB16] for further details.) So while Eq. (\[E:Kar\]) provides the justification for the above embedding process, Eq. (\[E:line3\]) yields a useful mathematical model, helped by the free parameter $K$. Moreover, this model is consistent with the induced-matter theory in Ref. [@pW92], discussed further in Sec. \[S:other\].
The solution
============
To produce the desired wormhole solution, we prefer the opposite signature in line element (\[E:line1\]) in order to be consistent with line element (\[E:wormhole\]): $$\label{E:line4}
ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}
+r^{2}\left(d\theta^{2}+\sin^{2}\theta \,d\phi^{2}
\right).$$ Most importantly, no additional assumptions will be made regarding $\nu=\nu(r)$.
The shape function $b=b(r)$ has to incorporate Eq. (\[E:lambda1\]) on account of the embedding. An entire class of such shape functions can be readily obtained by inspection: $$\label{E:shape3}
b(r)=r\left(1-\frac{1}
{1+\frac{1}{4}Ke^{\nu(r)}[\nu'(r)]^2}
\right)+\frac{r^n/r_0^{n-1}}
{1+\frac{1}{4}Ke^{\nu(r_0)}[\nu'(r_0)]^2}.$$ Observe that $b(r_0)=r_0$ for all $n$. Our main task is to show that these shape functions satisfy all the other requirements for shape functions. That is the topic of the next section.
The condition $0<b'(r_0)<1$ {#S:condition}
===========================
As noted in the Introduction, we need to examine the condition $0<b'(r_0)<1$. So we start with $$\begin{gathered}
\label{E:bprime}
b'(r_0)=1-\frac{1}
{1+\frac{1}{4}Ke^{\nu(r_0)}[\nu'(r_0)]^2}
+\frac{n}
{1+\frac{1}{4}Ke^{\nu(r_0)}[\nu'(r_0)]^2}\\
+r_0\left
(1+\frac{1}{4}Ke^{\nu(r_0)}[\nu'(r_0)]^2
\right)^{-2}\frac{1}{4}Ke^{\nu(r_0)}
\left(2\nu'(r_0)\nu''(r_0)
+[\nu'(r_0]^3\right).\end{gathered}$$ To simplify the analysis, let us introduce the following notations: $$\label{E:A}
A=\frac{1}{4}e^{\nu(r_0)}[\nu'(r_0)]^2$$ and $$\label{E:Omega}
\Omega=
e^{\nu(r_0)}\left(2\nu'(r_0)\nu''(r_0)
+[\nu'(r_0]^3\right).$$ In view of Eq. (\[E:bprime\]), the condition $0<b'(r_0)<1$ now yields $$\label{E:master}
\frac{\left(\frac{1-n}{1+AK}-1\right)
(1+AK)^2}{\frac{1}{4}r_0K}<\Omega<
\frac{(1-n)(1+AK)}{\frac{1}{4}r_0K}.$$ In the trivial case $\nu'(r_0)=0$, the condition $0<b'(r_0)<1$ is satisfied provided that $0<n<1$. Accordingly, we need to concentrate on the nontrivial case $\nu'(r_0)\neq 0$; as a result, $A$ is positive but $\Omega$ can be positive or negative. So the right-hand side of Inequality (\[E:master\]) is equivalent to the flare-out condition $b'(r)<1$ at or near the throat, while the left-hand side is equivalent to $b'(r_0)>0$. We will consider the two cases separately.
The condition $b'(r_0)<1$
-------------------------
To analyze the flare-out condition, we need to consider the two cases, $\Omega>0$ and $\Omega<0$. To this end, let us rewrite the right-hand side of Inequality (\[E:master\]) as follows: $$\label{E:K}
K\left(\frac{1}{4}r_0\Omega-A(1-n)\right)
<1-n.$$ If $\Omega>0$, then $n$ must be less than 1 to keep $K$ positive. It also becomes apparent that $r_0$ is another free parameter. So we can choose $r_0$ large enough so that $$\label{E:free1}
\frac{1}{4}r_0\Omega>A(1-n).$$ As a result, $$\label{E:K1a}
K<\frac{1-n}{\frac{1}{4}r_0\Omega-A(1-n)},
\quad n<1.$$
In Inequality (\[E:K\]), if $\Omega<0$, then we must have $n>1$ to keep $K$ positive. This time we need to choose $r_0$ sufficiently large so that $$\label{E:free2}
\frac{1}{4}r_0|\Omega|>-A(1-n).$$ The result is $$\label{E:K1b}
K>\frac{1-n}{\frac{1}{4}r_0\Omega-A(1-n)},
\quad n>1.$$
It should be noted that Conditions (\[E:K1a\]) and (\[E:K1b\]) for the free parameter $K$ can always be met by increasing the throat size of the wormhole. Observe also that $n\neq 1$.
The condition $b'(r_0)>0$
-------------------------
The left-hand side of Inequality (\[E:master\]) is more difficult to analyze since, after simplifying, we get the quadratic inequality $$\label{E:quadratic}
A^2K^2+K\left(\frac{1}{4}r_0\Omega
+A+nA\right)+n>0.$$ Once again, we need to consider the two cases $\Omega>0$, $n<1$, and $\Omega<0$, $n>1$.
$\boldsymbol{\Omega >0, n<1}:$ If $\Omega>0$ and $0\le n<1$, Inequality (\[E:quadratic\]) is automatically satisfied and we have $b'(r_0)>0$.
If $n<0$, we first need to solve the quadratic inequality to obtain $$\label{E:K2}
K<
\frac{-\left(\frac{1}{4}r_0\Omega +A +nA\right)
-\sqrt{\left(\frac{1}{4}r_0\Omega +A +nA\right)^2
-4A^2n}}{2A^2}$$ or $$\label{E:K3}
K>
\frac{-\left(\frac{1}{4}r_0\Omega +A +nA\right)
+\sqrt{\left(\frac{1}{4}r_0\Omega +A +nA\right)^2
-4A^2n}}{2A^2}.$$ Algebraically, the solution is valid for both $\Omega>0$ and $\Omega<0$. Because of the “or," only one of the inequalities is actually needed. (Since $K$ has to be positive, the first inequality is unphysical anyway.) For the second inequality, $K>0$ since $n<0$. We conclude that for the case $\Omega>0$, $n<0$, the parameter $K$ must satisfy the following inequality: $$\label{E:K4}
\frac{-\left(\frac{1}{4}r_0\Omega +A +nA\right)
+\sqrt{\left(\frac{1}{4}r_0\Omega +A +nA\right)^2
-4A^2n}}{2A^2}\\<K<
\frac{1-n}{\frac{1}{4}r_0\Omega-A(1-n)},
\quad n<0,$$ referring back to Inequality (\[E:K1a\]). So if $\Omega >0$ and $n<0$, then $K$ must lie between two positive values. We therefore have a solution for the case $\Omega >0, n<1$.
$\boldsymbol{\Omega <0, n>1}:$ For the case $\Omega<0$, $n>1$, the real difficulty is that solutions (\[E:K2\]) and (\[E:K3\]) may not be real. To avoid this problem, let us choose the free parameter $r_0$ sufficiently large to start with, i.e., choose $r_0$ so that $\frac{1}{4}r_0\Omega=-bA$ for some sufficiently large positive constant $b$ to obtain $$\label{E:real}
(-bA+A+nA)^2-4nA^2>0,$$ thereby resulting in a real solution. Consequently, Inequality (\[E:K2\]) yields $$\label{E:K5}
K<\frac{-(-b+1+n)-\sqrt{(-b+1+n)^2-4n}}
{2A}$$ while Inequality (\[E:K1b\]) gives $$\label{E:K6}
K>\frac{1-n}{\frac{1}{4}r_0\Omega-A(1-n)}=
\frac{2(1-n)/(-b-1+n)}{2A}.$$ (Inequality (\[E:K3\]) is not needed.)
The significance of the conditions on $K$ can best seen graphically. Fig. 1 shows that for any fixed $n$,
{width="80.00000%"}
$$f_1(b)=-(-b+1+n)-\sqrt{(-b+1+n)^2-4n}>f_2(b)\
=\frac{2(1-n)}{-b-1+n},$$
referring to Inequalities (\[E:K5\]) and (\[E:K6\]). So once again, $K$ must lie between two positive values. We therefore have a solution for the case $\Omega<0$, $n>1$, as well.
Other conditions {#S:other}
================
Having shown that the flare-out condition $b'(r_0)<1$ has been met, let us return to the violation of the null energy condition (NEC), which states that for the energy-momentum tensor $T_{\alpha\beta}$, $$T_{\alpha\beta}\mu^{\alpha}\mu^{\beta}\ge 0$$ for all null vectors. So given the radial outgoing null vector $(1,1,0,0)$, we have that $\rho(r_0) +p_r(r_0)<0$ whenever the condition is violated. By Ref. [@MT88], this violation is equivalent to the condition $$\frac{b'(r_0)-b(r_0)/r_0}{2[b(r_0)]^2}<0,$$ which holds whenever $b'(r)<1$ at or near the throat. As noted in the Introduction, for a Morris-Thorne wormhole, the violation of the NEC requires the use of “exotic matter," since ordinary matter normally satisfies the NEC. We have seen, however, that the shape functions and subsequent flare-out conditions were obtained from the embedding theory, which may be viewed as part of the induced-matter theory [@pW92] in the following sense: according to Ref. [@pW15], the field equations for the five-dimensional flat embedding space yield the Einstein field equations in four dimensions *containing matter*. The induced-matter theory therefore implies that the matter in our Universe actually comes from geometry and this may very well include exotic matter. So while exotic matter cannot be avoided, it may be less problematical in the present context.
Our final observation concerns asymptotic flatness. Since $\nu(r)\rightarrow
0$ as $r\rightarrow\infty$, we also have $\text{lim}_{r\rightarrow \infty}\nu'(r)=0$. So if $n<1$, we see from Eq. (\[E:shape3\]) that $b(r)/r\rightarrow 0$ (in addition to $e^{\nu(r)}\rightarrow 1$), resulting in an asymptotically flat spacetime.
Unfortunately, this conclusion does not hold for $n>1$. So the wormhole spacetime has to be cut off at some $r=a$ and joined to an external Schwarzschild spacetime $$ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}
+\frac{dr^2}{1-2M/r}
+r^{2}(d\theta^{2}+\text{sin}^{2}\theta\,
d\phi^{2})$$ in the usual way. From $e^{\nu(a)}=
1-2M/a$, we have $2M=a\left(1-e^{\nu(a)}
\right)$. But $2M=b(a)$; so the cut-off at $r=a$ is implicitly determined by the equation $b(a)=\left(1-e^{\nu(a)}\right)$, provided, of course, that such a solution exists.
Conclusions
===========
An $n$-dimensional Riemannian space is said to be of embedding class $m$ if $m+n$ is the lowest dimension of the flat space in which the given space can be embedded. Following Ref. [@MG17], we assume a spherically symmetric metric of embedding class two that can be reduced to class one by a suitable transformation.
These ideas were applied toward obtaining a complete wormhole solution without the usual engineering considerations, i.e., without being required to find or to manufacture matter or fields that produce the desired energy-momentum tensor. The free parameters $K$ and $r_0$ provided the extra degrees of freedom to obtain both the redshift and shape functions from the embedding theory and may even account for exotic matter.
[20]{}
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[^1]: E-mail: kuhfitti@msoe.edu
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abstract: 'Understanding the nature of multipartite entanglement is a central mission of quantum information theory. To this end, we investigate the question of tripartite entanglement convertibility. We find that there exists no easy criterion to determine whether a general tripartite transformation can be performed with a nonzero success probability and in fact, the problem is NP-hard. Our results are based on the connections between multipartite entanglement and tensor rank (also called Schmidt rank), a key concept in algebraic complexity theory. Not only does this relationship allow us to characterize the general difficulty in determining possible entanglement transformations, but it also enables us to observe the previously overlooked fact that [*the Schmidt rank is not an additive entanglement measure*]{}. As a result, we improve some best known transformation rates between specific tripartite entangled states. In addition, we find obtaining the most efficient algorithm for matrix multiplication to be precisely equivalent to determining the optimal rate of conversion between the Greenberger-Horne-Zeilinger state and a triangular distribution of three Einstein-Podolsky-Rosen states.'
author:
- Eric Chitambar$^1$
- 'Runyao Duan$^{2}$'
- 'Yaoyun Shi$^{3}$'
bibliography:
- 'QuantumBib.bib'
title: Tripartite entanglement transformations and tensor rank
---
One of the greatest discoveries in quantum physics [@EPR] is that a multipartite quantum system can be in a so-called entangled state. There are an uncountable number of entangled states realizable by any quantum system and a natural question is how they are related to each other — specifically, if two given states can be converted to each other through [*local operations and classical communications (LOCC)*]{}, i.e. a protocol in which no quantum information is exchanged among the subsystems. Ideally, one would like to have an efficient way to decide whether one entangled state can be transformed into another via LOCC as this question has utmost physical relevance. By nature, entangled states are fragile and highly susceptible to [*decoherence*]{}, a process in which quantum powers are lost. LOCC protocols describe the cheapest and experimentally easiest ways to convert entanglement while minimizing decoherence among parties separated by arbitrary distances.
As the notion of probability is inherent to quantum mechanics, the more natural question is with what probability $p$ can $|\phi\rangle$ be converted into $|\psi\rangle$ under LOCC? For $p=1$, the LOCC transformation is called [*deterministic*]{} and for a general nonzero $p$, the protocol is called [*stochastic (SLOCC)*]{}. Transformations of the latter form are written as $|\phi\rangle{\stackrel{\textrm{\small SLOCC}}{\longrightarrow}}|\psi\rangle$. For bipartite systems, the problem is completely solved. Nielsen has provided necessary and sufficient conditions for whether two states are deterministically convertible [@nielsen-99-83]. Probabilistically, any bipartite state $|\phi\rangle$ can be transformed into $|\psi\rangle$ if and only if the matrix rank of the reduced density operator of $|\phi\rangle$ is greater than that of $|\psi\rangle$. Furthermore, Vidal [@vidal-99-83] has derived a simple formula that gives the optimal probability for conversion.
When the number of subsystems is greater than two, the situation becomes much more complicated. No longer can SLOCC convertibility be determined by examining the ranks of the reduced density matrices of the initial and final states. For example, a system of three qubits can be partitioned into six equivalence classes defined by SLOCC convertibility between states in the same class [@DVC-2000]. However, two of the classes are indistinguishable by examining the ranks of each subsystem’s density matrix. The states $|GHZ\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$ and $|W\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$ are representatives of each class respectively. Progress toward understanding SLOCC convertibility among four qubit states has also been made as the state space for these systems can be partitioned into nine different families of equivalence classes [@verstraete-2002-65]. Nevertheless, for both three and four qubit systems, the separation into SLOCC equivalence classes does not provide a solution for determining whether two states are SLOCC related simply because one must first determine to which classes the states belong.
In this letter, we ask whether there is some relatively simple criterion for determining the convertibility of arbitrary tripartite states like there is for bipartite states. As a complete solution to the convertibility problem should be able to determine whether one state can be transformed into another with a nonzero probability, we focus our attention on the class of SLOCC protocols to judge the difficulty of the complete problem. Ultimately we find that no simple criterion exists for testing the possibility of a general tripartite entanglement transformation. In addition, through the course of investigating this problem many other interesting results are obtained concerning specific tripartite transformation rates. The novel conversion rates are derived in part from our observation that the Schmidt measure (to be defined below) is [*not an additive quantity*]{}, something previously thought to be true [@eisert-2001-64]. We now summarize our main findings:
Denote by ${|\Phi^3\rangle}$ the unnormalized tripartite state where any two parties share an (unnormalized) EPR state $|\Phi\rangle=|00\rangle+|11\rangle$: $$\begin{aligned}
|\Phi^3\rangle&=|\Phi\rangle_{AB}|\Phi\rangle_{AC}|\Phi\rangle_{BC}\notag\\&=\Big(|00\rangle_A|00\rangle_B+|10\rangle_A|10\rangle_B\Big)|00\rangle_C\notag\\&+\Big(|00\rangle_A|01\rangle_B+|10\rangle_A|11\rangle_B\Big)|01\rangle_C\notag\\&+\Big(|01\rangle_A|00\rangle_B+|11\rangle_A|10\rangle_B\Big)|10\rangle_C\notag\\&+\Big(|01\rangle_A|01\rangle_B+|11\rangle_A|11\rangle_B\Big)|11\rangle_C.\end{aligned}$$
\[thm:main\]
- For any general tripartite $|\phi\rangle$ and $|\psi\rangle$, determining whether $|\phi\rangle$ can be obtained from $|\psi\rangle$ by SLOCC is NP-hard.
- ${|\textrm{GHZ}\rangle}^{\otimes 3}{\stackrel{\textrm{\small SLOCC}}{\longrightarrow}}{|W\rangle}^{\otimes 2}$.
- ${|\textrm{GHZ}\rangle}^{\otimes 17}{\stackrel{\textrm{\small SLOCC}}{\longrightarrow}}|\Phi^3\rangle^{\otimes 6}$.
- Let $\lambda=inf\{u:|GHZ\rangle^{\otimes \lfloor un\rfloor}{\stackrel{\textrm{\small SLOCC}}{\longrightarrow}}|\Phi^3\rangle^{\otimes n}$ [*for sufficiently large*]{} $n\}$. Then $\lambda$ is precisely the [*exponent for matrix multiplication*]{}, i.e., the smallest real number $\omega$ such that two $N$ by $N$ matrices can be multiplied with $O(N^{\omega})$ number of multiplications between linear functions on entries of the first matrix and linear functions on entries of the second matrix.
Previously, only one copy of the W state is known to be convertible from three copies of GHZ and result (b) provides an improvement to this rate. Transformation (c) is important because it reveals that the three-party EPR extraction rate from GHZ is greater than one, a previously unknown possibility. Result (d) shows that existing lower and upper bounds for matrix multiplication translate to lower and upper bounds on the optimal conversion rate between groups of the given states. This connection implies that finding the GHZ to three-party EPR conversion rate is highly difficult since the complexity of multiplying matrices is one of the most challenging open problems in computation theory.
Our main technical tool is tensor rank, a key concept in algebraic complexity theory [@Burg97] that has also been used to measure multipartite entanglement under the synonymous names of Schmidt rank and Schmidt measure [@eisert-2001-64]. The [*tensor rank*]{} of a multipartite state $|\phi\rangle\in H_1\otimes H_2\otimes \cdots \otimes H_n$, denoted by $rk(|\phi\rangle)$, is the minimum number $r$ such that there exists $|\phi_j\rangle_i\in H_i$, $1\le j\le r$ and $$|\phi\rangle=\sum_{j=1}^r \bigotimes_{i=1}^n |\phi_j\rangle_i.$$ The quantity $log_2(rk(|\phi\rangle)$ is called the [*Schmidt measure*]{} of $|\phi\rangle$, denoted by $sch(|\phi\rangle)$.
Tensor rank has been used in algebraic complexity theory as it captures the complexity of computing a set of bilinear maps [@Burg97] and in particular the multiplicative complexity of multiplying two matrices. A set of bilinear maps are polynomials with respect to two distinct groups of indeterminates. The [*multiplicative complexity*]{} of the set is the minimum number of multiplications between the two groups required to evaluate all the polynomials. The multiplication of two $N\times N$ matrices produces a set of $N^2$ bilinear maps, one for each entry in the $N\times N$ product. The complexity of $N\times N$ matrix multiplication is denoted by $\mu(N,N)$ and the current best upper and lower bounds for $\mu(N,N)$ are $O(N^{2.36})$ and $\frac{5}{2}N^2-3N$ respectively . The complexity of matrix multiplication is also expressed as $\mu(N,N)=O(N^\omega)$ where $\omega$ is called the [*exponent for matrix multiplication*]{} and defined as the smallest real number such that an algorithm exists for multiplying two $N\times N$ matrices using $O(N^\omega)$ multiplications. While $\omega$ is hypothesized to be two, determining the validity of this conjecture is a major open problem in computational science. For more details, a good reference is chapter 28 of [@CLR-algorithms].
Tensor rank analysis has already shown to be valuable in quantum information as it is the distinguishing property between the $|GHZ\rangle$ and $|W\rangle$ equivalence classes of three qubits [@DVC-2000; @brylinski-2000]. It has also been useful in characterizing the entanglement in graph states [@hein-2004-69] as well as studying the distinguishability of states by separable operations [@duan-2007]. An important property of the tensor rank is that it cannot increase under SLOCC:
\[prop:monotone\] *If $|\phi\rangle{\stackrel{\textrm{\small SLOCC}}{\longrightarrow}}|\psi\rangle$ then ${\textrm{rk}}(|\phi\rangle)\ge{\textrm{rk}}(|\psi\rangle)$*.
Through Proposition \[prop:monotone\], the monotonic nature of the tensor rank makes studying it physically worthwhile. Unfortunately, determining the rank of an arbitrary state is a very difficult problem [@Haastad-1990] which is ultimately why there is no simple convertibility test applicable to all tripartite transformations. However, in some special cases it is possible to calculate the tensor rank or at least determine some useful bounds. In this Letter, we establish our main results described above by examining the ranks of certain tripartite states. We prove the following where each statement is in one-to-one correspondence with the main results stated earlier.
\[lm:main\]
- $|\phi\rangle\in H_A\otimes H_B\otimes H_C$ can be SLOCC converted from state $\frac{1}{\sqrt{n}}\sum_{i=1}^N|i\rangle_A|i\rangle_B|i\rangle_C$ if and only if $rk(|\phi\rangle)\leq N$.
- ${\textrm{rk}}({{|W\rangle}^{\otimes2}})\le 8$.
- ${\textrm{rk}}(|\Phi^3\rangle)=7$.
- $rk({|\Phi^3\rangle}^{\otimes n})$ is the multiplicative complexity for multiplying two $2^n\times 2^n$ matrices.
These results immediately indicate that when many copies of a state are considered, the tensor rank does not necessarily scale proportionately. As a result, impossible SLOCC transformations between individual sates may be possible when bulk quantities are considered.
Extending Lemma 1 to prove Theorem 1 is straightforward. It follows from item (a’) that, given a tripartite tensor $|\phi\rangle$ and a number $k$, deciding if ${\textrm{rk}}(|\phi\rangle)\le k$ can be reduced to the question of whether $\sum_{i=1}^k|i\rangle_A|i\rangle_B|i\rangle_C{\stackrel{\textrm{\small SLOCC}}{\longrightarrow}}|\phi\rangle$. The former problem is shown to be NP-hard by H[å]{}stad [@Haastad-1990], thus the latter is also NP-hard (item (a)).
Results (b), (c), and (d) follow directly from applying (a’) to (b’), (c’), and (d’) respectively. The 17 to 6 conversion ratio of (c’) is important because 6 copies of $|\Phi^3\rangle$ is a total of 18 EPR pairs. Thus, the stochastic EPR distillation rate from multiple copies of $|GHZ\rangle$ is greater than 1. In fact (d) shows that this rate can be further improved as the upper bound for $\omega$ is lowered. However, the distillation is specific in that the EPR pairs must be shared among all three parties. Indeed, if the EPR pairs are held by just two parties, $rk(|\Phi\rangle^{\otimes n})=2^n$ so the EPR distillation rate from $n$ copies of $|GHZ\rangle$ equals 1. The related problem of EPR distillation from the W state has recently been studied in [@Lo-2007]. There, the authors show that for a single W state, the probability of extracting an EPR state via LOCC is not only higher if one does not specify which two parties share the state, but it can also be made arbitrarily close to one.
From (d) and the lower bound on $\mu(2^n,2^n)$, it follows that $2n$ copies of GHZ cannot be converted into $n$ copies of $|\Phi^3\rangle$ with a nonzero probability. This result is stronger than the one derived in [@linden-1999] where the authors demonstrate the impossibility of $|GHZ\rangle^{\otimes 2n}\rightarrow|\Phi^3\rangle^{\otimes n}$ under deterministic LOCC. What is most interesting is that the authors prove the impossibility strictly through entropy arguments. Here, we obtain the same conclusion using tools of algebraic complexity theory. On the surface these two lines of attack appear to be unrelated, but the similarity in both results suggests that the two may be deeply connected.
Now we turn to prove Lemma 1. We will work with unnormalized states below since any overall factor does not affect the tensor rank. For any $|\phi\rangle\in H_A\otimes H_B\otimes H_C$, let $\rho_{AB}$ denote Alice and Bob’s subsystem obtained by taking the partial trace $Tr_C(|\phi\rangle\langle\phi|)$. As $\rho_{AB}$ is a positive operator, it has a spectral decomposition $\rho_{AB}=\sum_{k=1}^mp_k|\psi_k\rangle\langle\psi_k|$ where $0<p_k\le 1$. The vector span of $\{|\psi_k\rangle:1\le k\le m\}$ is called the support of $\rho_{AB}$ and denoted by $supp(\rho_{AB})$. To proceed, we need the following simple equivalent characterization of a tripartite state’s tensor rank.
\[lm2\] Suppose $|\phi\rangle\in H_A\otimes H_B\otimes H_C$. The tensor rank of $|\phi\rangle$ equals the minimum number of product states in $H_A\otimes H_B$ whose linear span contains the support of $\rho_{AB}=Tr_C(|\phi\rangle\langle\phi|)$.
Let $k$ denote $rk(|\phi\rangle)$. Suppose that the span of $r$ product states $\{|\alpha_j\rangle|\beta_j\rangle:1\le j\le r\}$ contain $supp(\rho_{AB})$. Let $|\phi\rangle=\sum_{i=1}^m|i\rangle_{AB}|i\rangle_C$ be a Schmidt decomposition of $|\phi\rangle$. Each $|i\rangle_{AB}$ belongs to $supp(\rho_{AB})$ and thus $|i\rangle_{AB}=\sum_{j=1}^r\lambda_{i,j}|\alpha_j\rangle|\beta_j\rangle$. Regrouping the $|i\rangle_C$ according to the $r$ product states gives $r\ge k$. On the other hand, consider a “minimal” decomposition $|\phi\rangle=\sum_{i=1}^k|a_i\rangle|b_i\rangle|c_i\rangle$. Then $\rho_{AB}=\sum_{i,j=1}^k|a_i\rangle|b_i\rangle\langle c_j|c_i\rangle\langle a_j|\langle b_j|$ and hence $k\ge r$.
Using Lemma \[lm2\], the general procedure for determining tensor rank is now straightforward. Write $|\phi\rangle=\sum_{i=1}^m|i\rangle_{AB}|i\rangle_C$ where the $\{|i\rangle_C:1\le 1\le m\}$ are orthonormal and then determine the minimum number of product states needed to contain the $\{|i\rangle_{AB}:1\le i\le m\}$. This question can be rephrased in another way by mapping each $|i\rangle_{AB}$ to a bilinear form $f_i$ from the ring of indeterminates $C[\{a_j\},\{b_j\}]$ where each $a_j$ $(b_j)$ is in a one-to-one correspondence with a basis vector from $H_a$ $(H_b)$. Product states in $H_a\otimes H_b$ correspond to a product of linear forms from $C[\{a_j\}]\times C[\{b_j\}]$ which we refer to as a [*non-scalar*]{} multiplication. Thus, we obtain the following fact:
The minimum number of product states that contain the $\{|i\rangle_{AB}:1\le i\le m\}$, and hence the tensor rank of $|\phi\rangle$, is the same number of non-scalar multiplications $M_k=(\sum_{j=1}^{n_a}\alpha_{k,j}a_j)\times(\sum_{j=1}^{n_b}\beta_{k,j}b_j)$ needed to calculate the $\{f_i:1\le i\le m\}$.
We now use the technique outlined above to study the tensor rank of certain tripartite states.
(a’): For $\sum_{i=1}^N|i\rangle_A|i\rangle_B|i\rangle_C$, the support of $\rho_{AB}$ is spanned by $N$ product states. Thus by Prop. 1 and Lemma 2, a necessary condition for the given transformation is $rk(|\phi\rangle)\leq N$. Now suppose that $|\phi\rangle=\sum_{i=1}^k|a_i\rangle|b_i\rangle|c_i\rangle$ where $k\leq N$. Since $\{|i\rangle_A:1\le i\le N\}$ is an orthonormal set, we can define the linear operator $A$ by $A|i\rangle_A=\begin{cases}|a_i\rangle, &1\leq i\leq k\\0,&k<i\leq N\end{cases}$. Similarly, operators $B$ and $C$ can be constructed. As noted in [@DVC-2000], the existence of such operators is sufficient for an SLOCC protocol since $|\phi\rangle$ will be obtained when Alice performs the local measurement $\{\frac{A}{||A||},\sqrt{I_A-\frac{1}{||A||^2}A^\dagger A}\}$ and similarly for Bob and Charlie. Note that (unnormalized) $|GHZ\rangle^{\otimes n}$ can be expressed as $\sum_{i=1}^{2^n}|i\rangle_A|i\rangle_B|i\rangle_C$. (b’): One can verify by direct computation that $|W\rangle^{\otimes 2}$ expands as: $$\begin{aligned}
\label{W2}
\big(|11\rangle_A|00\rangle_B+|10\rangle_A|01\rangle_B+|01\rangle_A|10\rangle_B+|00\rangle_A|11\rangle_B\big)&|00\rangle_C\notag\\
+\big(|10\rangle_A|00\rangle_B+|00\rangle_A|10\rangle_B\big)&|01\rangle_C\notag\\
+\big(|01\rangle_A|00\rangle_B+|00\rangle_A|01\rangle_B\big)&|10\rangle_C\notag\\
+\big(|00\rangle_A|00\rangle_B\big)&|11\rangle_C.\end{aligned}$$ The structure of $|W\rangle^{\otimes 2}$ becomes more manageable when working with its corresponding bilinears $f_i$ since they can be succinctly expressed through the matrix multiplication $$\begin{pmatrix}
f_{00}\\
f_{01}\\
f_{10}\\
f_{11}\\
\end{pmatrix}
=
\begin{pmatrix}
a_{11}&a_{10}&a_{01}&a_{00}\\
a_{10}&0&a_{00}&0\\
a_{01}&a_{00}&0&0\\
a_{00}&0&0&0\\
\end{pmatrix}
\cdot
\begin{pmatrix}
b_{00}\\
b_{01}\\
b_{10}\\
b_{11}\\
\end{pmatrix}.$$ We make use of the following identity [@Fiduccia-1972-CCC], where a “$\cdot$” means a $0$ entry:
$$\begin{aligned}
\label{w2decomp}
\begin{pmatrix}
a_{11}&a_{10}&a_{01}&a_{00}\\
a_{10}&0&a_{00}&0\\
a_{01}&a_{00}&0&0\\
a_{00}&0&0&0\\
\end{pmatrix}
&=
\begin{pmatrix}
a_{10}&a_{10}&\cdot&\cdot\\
a_{10}&a_{10}&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot\\
\end{pmatrix}
+
\begin{pmatrix}
a_{01}&\cdot&a_{01}&\cdot\\
\cdot&\cdot&\cdot&\cdot\\
a_{01}&\cdot&a_{01}&\cdot\\
\cdot&\cdot&\cdot&\cdot\\
\end{pmatrix}
+
\begin{pmatrix}
a_{00}&\cdot&\cdot&a_{00}\\
\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot\\
a_{00}&\cdot&\cdot&a_{00}\\
\end{pmatrix}\notag\\
&+
\begin{pmatrix}
\cdot&\cdot&\cdot&\cdot\\
\cdot&a_{00}&a_{00}&\cdot\\
\cdot&a_{00}&a_{00}&\cdot\\
\cdot&\cdot&\cdot&\cdot\\
\end{pmatrix}
+
\begin{pmatrix}
a_{11}-a_{10}-a_{01}-a_{00}&\cdot&\cdot&\cdot\\
\cdot&-a_{10}-a_{00}&\cdot&\cdot\\
\cdot&\cdot&-a_{01}-a_{00}&\cdot\\
\cdot&\cdot&\cdot&-a_{00}
\end{pmatrix}.\end{aligned}$$
Note that rank one matrices require only one non-scalar multiplication: $\begin{pmatrix} a_i&a_i\\a_i&a_i\end{pmatrix}\cdot\begin{pmatrix} b_1\\b_2\end{pmatrix}=\begin{pmatrix} a_i(b_1+b_2)\\a_i(b_1+b_2)\end{pmatrix}$, while any $n\times n$ diagonal matrix requires $n$ multiplications: $\begin{pmatrix} \lambda_1&{}&{}&{}\\{}&\lambda_2&{}&{}\\{}&{}&\lambda_3&{}\\{}&{}&{}&\lambda_4\end{pmatrix}\cdot\begin{pmatrix}b_1\\b_2\\b_3\\b_4\end{pmatrix} =\begin{pmatrix}\lambda_1b_1\\\lambda_2b_2\\\lambda_3b_3\\\lambda_4b_4\end{pmatrix}$. Hence, a total of eight non-scalar multiplications is sufficient to compute each $f_i$. These multiplications correspond to product states that contain $supp(Tr_C(|W\rangle\langle W|^{\otimes 2}))$. By Lemma 2 then, $rk(|W\rangle^{\otimes 2})\leq 8$. In fact, expansion gives the eight product states that contain $supp(Tr_C(|W\rangle\langle W|^{\otimes 2}))$ enabling us to rewrite Alice and Bob’s vector attached to $|i\rangle_C:i\in\{00,01,10,11\}$ in as a combination of these eight states. To our knowledge, this is the first observed non-additivity of the Schmidt measure for pure states. (c’): Up to a local unitary transformation on Alice’s part, the corresponding bilinear forms of $|\Phi^3\rangle$ match the set of polynomials obtained when multiplying two $2\times 2$ matrices: $$\label{mm}
\begin{pmatrix}
a_{00}&a_{01}\\
a_{10}&a_{11}\\
\end{pmatrix}
\begin{pmatrix}
b_{00}&b_{01}\\
b_{10}&b_{11}\\
\end{pmatrix}
=\begin{pmatrix}
f_{00}&f_{01}\\
f_{10}&f_{11}\\
\end{pmatrix}.$$ An algorithm for obtaining the $f_i$ using only seven multiplications was discovered by Strassen [@Strassen69] and later proven to be optimal by Winograd [@winograd-1971-4]. These seven non-scalar multiplications correspond to a minimum number of product states containing $supp(Tr_C(|\Phi^3\rangle\langle\Phi^3|))$ and so $rk(|\Phi^3\rangle)=7$. As an eight term expansion for $|W\rangle^{\otimes 2}$ can easily be obtained from expansion , it is straightforward to find a seven term expansion of $|\Phi^3\rangle$ from Strassen’s algorithm given in [@Strassen69]. Since the explicit expressions are not of primary interest here, we omit the calculations. (d’): By taking multiple tensor products of the matrices in , we see that for $n$ copies of $|\Phi^3\rangle$, the corresponding polynomials are represented by $2^n\times 2^n$ matrix multiplication. Hence, $rk({|\Phi^3\rangle}^{\otimes n})$ is the complexity of this operation.
In conclusion, we have found that no easy test exists for determining whether two general tripartite states are probabilistically convertible. The difficulty arises because any general solution involves a tripartite tensor rank computation. As a result, one must consider tripartite transformations on a case-by-case basis. In this letter we have done this for special states in which the tensor rank has already been studied or can be calculated with mild effort. Performing this analysis led to an improved GHZ state to W state SLOCC transformation rate as well as the first demonstration of obtaining EPR pairs from GHZ states at a rate greater than one with a nonzero probability.
The connection between tensor rank and entanglement transformation is perhaps most beautifully exemplified by the equivalence of matrix multiplication complexity and the optimization of stochastic EPR distillation from many copies of GHZ. This relationship opens many avenues of further research as the techniques of algebraic complexity theory might teach us more about the nature and limitations of SLOCC transformations. Conversely, constructing explicit SLOCC entanglement transformations using results in quantum information may be useful to obtain bounds for the multiplicative complexity of a particular set of bilinear forms.
This work was partially supported by the National Science Foundation of the United States under Awards 0347078 and 0622033. R. Duan was partially supported by the National Natural Science Foundation of China (Grant Nos. 60702080, 60736011, and 60621062), the FANEDD under Grant No. 200755, and the Hi-Tech Research and Development Program of China (863 project) (Grant No. 2006AA01Z102).
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abstract: 'In real life applications, certain images utilized are corrupted in which the image pixels are damaged or missing, which increases the complexity of computer vision tasks. In this paper, a deep learning architecture is proposed to deal with image completion and enhancement. Generative Adversarial Networks (GAN), has been turned out to be helpful in picture completion tasks. Therefore, in GANs, Wasserstein GAN architecture is used for image completion which creates the coarse patches to filling the missing region in the distorted picture, and the enhancement network will additionally refine the resultant pictures utilizing residual learning procedures and hence give better complete pictures for computer vision applications. Experimental outcomes show that the proposed approach improves the Peak Signal to Noise ratio and Structural Similarity Index values by 2.45% and 4% respectively, when compared to the recently reported data.'
author:
- Vaishnav Chandak
- Priyansh Saxena
- Manisha Pattanaik
- Gaurav Kaushal
title: Semantic Image Completion and Enhancement using Deep Learning
---
Introduction
============
Image completion and enhancement are two prime issues in the field of image processing and machine learning. In the past couple of years, the incredible developments of deep learning on different issues in low level as well as in high-level computer visions applications have been seen. The low level computer vision issues includes picture completion & its enhancement and are much foreseen to happen during image processing. Numerous image processing based techniques are proposed to take care of low-level vision issues. Many of them handle these issues separately [@dummy:3], [@dummy:6]; however, most of the time image completion and denoising issues occur simultaneously. Many real world computer vision tasks endure from missing and masked areas in images which leads to poor quality images and hence the complexity of corresponding computer vision tasks increases. These problems are difficult to deal as estimating the missing region in the image is not easy [@dummy:1].\
Image completion refers to filling missing regions in the image based on the available visual data. On the other hand, image enhancement attempts to eliminate unwanted noise and blur from the image along with sustaining most of the image details. In this paper, an efficient image completion and enhancement model is proposed, which intends to recover the corrupted and masked regions in images and then refining the image further to increase the quality of output image. The method is motivated by generative adversarial networks (GANs) [@dummy:8], [@dummy:11]. Wasserstein GAN architecture of the GANs is used for image completion which recovers the missing regions by filling the corrupted part in the damaged image, and the enhancement network will further refine the completed images using residual learning techniques and then provide a better quality image as output.
RELATED WORK
============
Yizhen Chen and Haifeng Hu have proposed an upgraded approach for semantic inpainting of images. The proposed method named progressive inpainting using generative models in which they first estimated corrupted image distribution and then moderately refined image details. However, the model still could not handle large missing regions in images [@dummy:3]. Jia-Bin Huang and Ahuja have proposed an advance-knowledge approach which used contextual information for image completion. However, in case the corrupted region is large or is irrelevant to visual data, or if the complexity of the image is high, the output of the method would be quite unsatisfactory [@dummy:6]. Connelly Barnes and Eli have proposed patch matching algorithm for image completion for nonparametric texture construction. The algorithm performed satisfactorily and was able to identify similar patches. However, it failed when the original image lacked adequate data to complete the missing regions [@dummy:1]. Yunjin Chen and Thomas Pock have proposed nonlinear response dispersion model, which consists of a feed forward network with a fixed number of gradient descent stages. Trainable nonlinear reaction-diffusion accomplished promising execution in any case; its display was prepared for a specific noise level. It was unfit to perform well on pictures with obscure noise levels. Additionally, it requires the output which is expected by the network during training [@dummy:2]. In 2011, Deng transformed the inpainting task to the graph-labeling task using graph Laplace method. However, this method required images samples of the image to be inpainted be included in the training data, which was not practical in real life applications [@dummy:4]. A viable face inpainting algorithm utilizing a generative model was proposed by Yijun Li, Sifei Liu, and Jimei Yang. From background inpainting task, face inpainting is a challenging task because it regularly needs to produce semantically newer pixels areas in the missing region parts like eyes and nose, which can vary from person to person. Even though the model had the capacity to produce semantically conceivable and outwardly satisfying content, it has a few constraints. The model still could not deal with some unaligned faces. also, it did not wholly misuse the spatial connections between nearby pixels [@dummy:8]. Ruijun and Yang proposed an improved generative translation model. The paper proposed a semantic image completion method using regional completions for painting completion. Using the generator and discriminator network, the missing region is generated, which should be consistent with the surrounding region. However, image completion work is restricted to only face data and needed to be improved to ensure that the entire painting work could be recovered [@dummy:9]. Deepak Pathak put forward Context Encoders(CE) which estimated missing areas in images based on its surroundings. However, during training it needed a mask on the corrupted regions of the image, that is a significant disadvantage of the approach, and also context encoders led to blurry and noisy results in the inpainted parts [@dummy:12]. Peyr proposed an adjustable low-dimensional manifold for images. It included inpainting task on synthetic as well as texture data. However, the employed work was quite away in giving solutions to real-world images of a face [@dummy:13]. Ren proposed a novel CNN architecture named Shepard Convolutional Neural Networks which efficiently equips conventional CNN with the ability to learn missing data. However, in case the corrupted region was large or was irrelevant to visual data, or if the complexity of the image is high, the output of the method would be quite unsatisfactory [@dummy:14]. In [@dummy:15], low-light enhancement model using convolutional neural network and Retinex theory was proposed. It showed an equivalence between multi-scale Retinex and feedforward convolutional neural network using Gaussian kernels. However, because of the limited receptive field in their model, very smooth regions such as clear sky are sometimes attacked by the halo effect. Jeremias sulam, formulated trainlets, to construct large adaptable atoms using various datasets of facial images using dictionary learning algorithm. Because of the computational constraints, this method was applied to tiny regions of the image and not on the entire image. As a result, this approach did not give satisfactory results on large regions in images [@dummy:17]. Raymond and Chen [@dummy:19] proposed another picture completion technique that can be utilized to fix any state of gaps. In any case, such training depends on the data used in training. In the meantime, the processing of surface and structure was not sufficiently impeccable. Kai Zhang and Yunjin Chen proposed a picture denoising approach in which they built feed-forward denoising convolutional neural systems using residual learning and batch normalization. However, this methodology was unfit to recover missing regions, and it just denoised the picture. Likewise, it was unfit to refine pictures with genuine complex commotion and other general picture restoration tasks [@dummy:20]. Zhao, Liu, and Hiang proposed deep neural networks to inpaint and de-noise the corrupted image. However, in this method, the handling of structure and texture was not satisfactory [@dummy:21].
THE PROPOSED MODEL
==================
The methodology could be separated into three different steps. In the first step, data-preprocessing on CelebA-hq dataset [@dummy:10] is done to run and test the developed model. In the second step, a Wasserstein GAN based model to complete the missing pixels in the image is developed. The image completion GAN gives a complete image with a blurry filled area. The generator of the GAN generates real looking images, but in the process of generation, the noise gets unavoidably added. So, in the third step, the output of the generator is passed through the enhancement network to make the filled area clear and to refine the completed image further.
DATA PREPROCESSING
------------------
The following data preprocessing steps were followed:
- The dataset is splited into 15000 training images and around 1000 testing images.
- Each face image in the dataset is resized to 64\* 64\* 3 pixels to train the Wasserstein GAN model.
- Masking- A binary mask is used with values 0 or 1. 0 corresponds to the corrupted region while 1 corresponds to the uncorrupted region in the image. This binary mask is applied to all images to make them corrupted which will serve as input of the training process.
- The enhancement network is trained using 2000 image pairs containing blurry images and its corresponding clean images.
WASSERSTEIN GAN
---------------
The concept of GANs was put forward by Goodfellow[@dummy:11] which trains two networks simultaneously: the generator network G to learn the distribution of training data and the critic network C which distinguishes the generated samples from the original samples as in Figure 1. The generative network is trained to generate patches to complete the missing regions in the image. Meanwhile, it is difficult for the critic network to classify the output of the generator from the original dataset sample. The GAN architecture used is Wasserstein GAN, which uses Wasserstein distance as to train the generator so that it can capture training data distribution and generate images similar to those in the training data.
WASSERSTEIN DISTANCE AS LOSS FUNCTION FOR THE TRAINING OF GENERATOR
-------------------------------------------------------------------
Wasserstein distance is a measure of the distance between two probability distributions. For the generated data distribution $p_{g}$ and the real data distribution $p_{r}$, it can be mathematically defined as the cost for the cheapest plan from $p_{g}$ to $p_{r}$. It is also called [**Critic loss**]{} or [**Wasserstein distance**]{}. The Wasserstein distance loss function L to train the generator can be mathematically represented as[@dummy:16]: $$L=\displaystyle\mathop{\mathbb{E}}_{\overset{\sim}{x}\sim P_{g}}[C(\overset{\sim}{x})]-\displaystyle\mathop{\mathbb{E}}_{x\sim P_{r}}[C(x)]$$ Here, the first term represents the expectation of the distribution generated by the generator, and the second term represents the expectation of the real training data distribution. By minimizing the difference between the two, the generator learns to generate samples having probability distribution similar to training data distribution. Now, to make the learning faster and make model convergence faster gradient penalty term is added to our loss function. So, the overall loss function L of Wasserstein GAN becomes: $$L=\displaystyle\mathop{\mathbb{E}}_{\overset{\sim}{x}\sim P_{g}}[C(\overset{\sim}{x})]-\displaystyle\mathop{\mathbb{E}}_{x\sim P_{r}}[C(x)]+gradient\,penalty$$ where, gradientpenalty will be given by $$gradient\,penalty=\lambda\displaystyle\mathop{\mathbb{E}}_{\hat{x}\sim P_{g}}[(||\bigtriangledown_{\hat{x}} C(\hat{x})||_{2}-1)^2]$$ here $\lambda$ is the gradient penalty coefficient.
IMAGE COMPLETION WITH WASSERSTEIN GAN
-------------------------------------
After training the generator to generate samples which look real, the next aim is to ensure that the missing region generated has a similar context to the non-missing region so that sensible looking completed images as output can be obtained. For this, the following is done:\
A binary mask with values 0 or 1 is used. 0 corresponds to the corrupted region while 1 corresponds to the uncorrupted region in the image. Let y represents the uncorrupted image. $M\odot y$ gives the uncorrupted part of the image. Let G($z^{'}$) be some image generated by the generator which suitably completes the missing region in the image. (1-M) $\odot$ G($z^{'}$) represents the completed region which when added to the uncorrupted region gives the reconstructed image [@dummy:18] as output:
$$x_{reconstructed} = M\odot y+(1-M)\odot G(z^{'})$$
To find $z^{'}$ that suitably completes the image following loss functions are defined [@dummy:18]:\
[**Contextual Loss:**]{} To ensure both generated and the input image have same context, ensure that the uncorrupted pixel in original image y are same as the pixels in the generated image G(z) at a particular location. For this, pixel wise difference between the uncorrupted part of the two images is taken and then this difference is minimized. $$L_{contextual}(z) = ||M\odot G(z) - M\odot y ||_1$$ where $||x||_1$ represents $l_1$ norm of some vector x.\
[**Perceptual Loss:**]{} It ensures that the output image looks real. For this, the following perceptual loss: $$L_{perceptual}(z) = log(1-C(G(z)))$$ [**Total loss**]{}: It is a sum of perceptual and contextual loss and is denoted by L(z): $$L(z) = L_{contextual}(z) + Q L_{perceptual}(z)$$ Q is a hyper-parameter and we minimize this loss function to ensure completed image is contextally similar to input image.
ENHANCEMENT NETWORK USING RESIDUAL LEARNING
-------------------------------------------
In enhancement network to refine completed images, the residual learning approach is used. The input to the network is blurry image y = x + v, here x is the clear image,v represents the blur added. The residual network is trained to grasp the mapping R(y)$ \approx $v , to get the clear image x as x = y- R(y). Mathematically, the average mean square error among the output residual image by the model and the actual residual images is used as error function for getting the parameter $ \Theta $ to train the enhancement network. $$L(\theta)=\frac{1}{2N}\sum_{i=1}^{N}||R(y_{i};\theta)-(y_{i}-x_{i})||^2$$ Here, L is the training error of the enhancement network and N are total training images.\
Enhancement network consists of following layers as shown in Figure 2: (i) Conv+ReLU: It creates feature maps, and ReLU adds the non-linearity. (ii) Conv+BN+ReLU: This layers contains added batch normalization between Conv and ReLU. (iii) Conv: It is used to get the output residual image.
RESULTS AND DISCUSSION
======================
The following plot was obtained by training the enhancement network on 2000 celeba-hq image pairs of clean and its corresponding blurr images.
In Figure 3, as the training proceeds, the average mean square error among the output residual image by the model and the actual residual images decreases. As a result, according to Equation (8), the training error decreases. Finally, around 200th epoch, the enhancement network is sufficiently trained ,which is evident as the training error becomes constant at a particular value, and there is no further decrease.\
The following Wasserstein distance plot was obtained while training Wasserstein GAN on around 15000 Celeba-hq images for 10000 epochs and batch size of 128.
In Figure 4, from Equation (2) it can be seen that in the initial stages of learning the expectation of the distribution generated by the generator is different from the expectation of the distribution of real data and hence the difference between the two is higher resulting in higher Wasserstein distance values. However, as the learning proceeds generator learns the distribution of the real data and then generates samples having a similar distribution with the real data, and hence the difference in their expectation decreases resulting in lower Wasserstein distance values. Now, around 10000 epochs the generator has sufficiently learned, and hence the Wasserstein distance values do not decrease further and becomes constant around a particular lower value.\
The following contextual, perceptual and total loss plots were obtained while training the Wasserstein GAN for image completion for 1250 epochs on 15000 Celeba-hq images.
In Figure 5, in the initial stage, the context in the uncorrupted region of the generated samples and the original samples are different, so from Equation (5), it can be seen that the resulting contextual loss is higher. As the training moves further using Adam’s optimizer (z) gets trained, and hence, there is a significant decrease in the contextual loss values. Around 1200th epoch, the context in the uncorrupted region of the generated samples and the original samples becomes quite familiar, and hence the contextual loss becomes constant around a particular value.
In Figure 6, initially the distribution of generated images and real images is different, so the critic is able to distinguish the generated samples from the real ones and hence the value of C(G(z)) is close to 0 and as a result 1-C(G(z)) becomes close to 1 as a result from Equation (6) the loss is higher. However, as learning proceeds, the generator generates real looking samples as a result C(G(z)) becomes close to 1 and 1-C(G(z)) becomes close to 0, resulting in lower perceptual loss values from Equation (6).
The perceptual loss values are quite lower compared to contextual loss values, and as a result from Equation (7), the total image completion loss is almost equal to contextual loss, and as a result Figure 7 which is total image completion loss plot is almost similar to contextual loss for image completion plot Figure 5.
The following two evaluation metrics to evaluate the quality of the output images by the model:
Peak Signal-to-Noise Ratio (PSNR):
----------------------------------
PSNR [@dummy:5] is measured in decibels (dB). The higher the PSNR, the better image has been completed to match the original image. $$MSE=\frac{1}{mn}\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}||f(i,j)-g(i,j)||^2$$ $$PSNR=20\log_{10}(\frac{MAX_{f}}{\sqrt{MSE}})$$ Here, f is the original image, g represents completed image through the model, m represents image pixel rows, n represents image pixel columns, i and j represents row and column index respectively. MAX$_{f}$ is a constant equal to 255.
[ Structural Similarity Index (SSIM):]{}
========================================
The Structural Similarity (SSIM) Index [@dummy:5] depends on computation of terms, namely the luminance, contrast and structural term. $$SSIM(x,y)=[C(x,y)]^\alpha\times[I(x,y)]^\beta\times[S(x,y)]^\gamma$$
where, C represents contrast, I represents luminance, S represents structural term, x represents original, y represents completed images. The parameters $\alpha > 0$, $\beta > 0 $, and $\gamma > 0$, are used to adjust the relative importance of the three components.
The following PSNR and SSIM values through the proposed approach:
**Methods** **CE[@dummy:12]** **PI [@dummy:3]** **This work**
-------------- ------------------- ------------------- ---------------
**PSNR(dB)** 22.85 21.45 23.41
: Comparison of PSNR values
**Methods** **CE[@dummy:12]** **PI [@dummy:3]** **This work**
------------- ------------------- ------------------- ---------------
**SSIM** 0.872 0.851 0.9074
: Comparison of SSIM values
It can be seen that the approach performs well CelebA-hq dataset compared to other proposed image completion techniques, which is evident from the above PSNR and SSIM values.\
Some of the results obtained through the proposed image completion approach using Wasserstein GAN are shown below in Figure 8(a-e):
Original Input Output\
\
\
\
\
\
CONCLUSION
==========
In this paper, the Wasserstein Generative Adversarial Network (WGAN) is first trained to generate the missing patches in the image and then passed the completed image given by the WGAN architecture is passed through an enhancement network to remove the blur and unwanted noise. By integrating image completion and enhancement task into a single process, the proposed approach provides better inpainting solutions by improving the Peak Signal to Noise ratio and Structural Similarity Index values by 2.45% and 4% respectively when compared to the recently used approaches. However, in this approach, overall training is highly depended on the data used for training. In future, work can be done towards optimizing the overall structure of the network and raise the network’s capability to grasp minute details of the image further and to improve the image completion model.
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abstract: 'We have carried out observations of CCH and its two $^{13}$C isotopologues, $^{13}$CCH and C$^{13}$CH, in the 84 – 88 GHz band toward two starless cores, L1521B and L134N (L183), using the Nobeyama 45 m radio telescope. We have detected C$^{13}$CH with a signal-to-noise (S/N) ratio of 4, whereas no line of $^{13}$CCH was detected in either the dark clouds. The column densities of the normal species were derived to be ($1.66 \pm 0.18$)$\times 10^{14}$ cm$^{-2}$ and ($7.3 \pm 0.9$)$\times 10^{13}$ cm$^{-2}$ ($1 \sigma$) in L1521B and L134N, respectively. The column density ratios of $N$(C$^{13}$CH)/$N$($^{13}$CCH) were calculated to be $>1.1$ and $>1.4$ in L1521B and L134N, respectively. The characteristic that $^{13}$CCH is less abundant than C$^{13}$CH is likely common for dark clouds. Moreover, we find that the $^{12}$C/$^{13}$C ratios of CCH are much higher than those of HC$_{3}$N in L1521B by more than a factor of 2, as well as in Taurus Molecular Cloud-1 (TMC-1). In L134N, the differences in the $^{12}$C/$^{13}$C ratios between CCH and HC$_{3}$N seem to be smaller than those in L1521B and TMC-1. We discuss the origins of the $^{13}$C isotopic fractionation of CCH and investigate possible routes that cause the significantly high $^{12}$C/$^{13}$C ratio of CCH especially in young dark clouds, with the help of chemical simulations. The high $^{12}$C/$^{13}$C ratios of CCH seem to be caused by reactions between hydrocarbons (e.g., CCH, C$_{2}$H$_{2}$, $l,c$-C$_{3}$H) and C$^{+}$.'
author:
- Kotomi Taniguchi
- Eric Herbst
- Hiroyuki Ozeki
- Masao Saito
title: 'Investigation of $^{13}$C Isotopic Fractionation of CCH in Two Starless Cores: L1521B and L134N'
---
Introduction {#sec:intro}
============
Exotic unsaturated carbon-chain molecules are one of the crucial constituents of approximately 200 molecules detected in the interstellar medium (ISM) and circumstellar shells. In fact, they account for around 40% of the interstellar molecules. Therefore, it is important for astrochemists to understand carbon-chain chemistry.
These carbon-chain species have long been associated with young starless cores such as Taurus Molecular Cloud-1 [TMC-1; @1992ApJ...392..551S; @2004PASJ...56...69K]. They are formed by gas-phase ion-molecule reactions and neutral-neutral reactions before carbon atoms are converted into CO molecules. Besides the classical carbon-chain chemistry, an ion-molecule chemistry occurring at somewhat higher temperatures, and starting from gaseous methane (CH$_{4}$), named warm carbon-chain chemistry [WCCC; @2013ChRv..113.8981S], was found to occur around low-mass Class 0/I protostars such as L1527. In particular, methane sublimated from dust grains reacts with ionic carbon (C$^{+}$) in the gas phase, which is a trigger of successive carbon-chain formation. It was recently found that formation of cyanopolyynes (HC$_{2n+1}$N, $n=1,2,3,...$) occurs in the warm dense gas around high-mass protostellar objects [@2018ApJ...854..133T; @2019ApJ...872..154T].
The formation and destruction mechanisms of carbon-chain molecules were investigated mainly by astrochemical simulations [e.g., @1992ApJ...392..551S]. In these early stages, it was unclear what specific reactions significantly contribute to the formation of carbon-chain species. Another method to investigate the main formation pathways of carbon-chain molecules consists of observations of the $^{13}$C isotopic fractionation [e.g., @1998AA...329.1156T].
The first hint of $^{13}$C isotopic fractionation for carbon-chain molecules was found in HC$_{5}$N toward TMC-1 using the Nobeyama 45 m radio telescope [@1990ApJ...361L..15T]. However, signal-to-noise ratios were not high enough to confirm the differences in abundances among its five $^{13}$C isotopologues and such studies were left for future work. The confirmation of the $^{13}$C isotopic fractionation was achieved for HC$_{3}$N in TMC-1 [@1998AA...329.1156T]. Other observations including fractionation studies targeting different carbon-chain molecules were carried out in TMC-1. These species include CCS [@2007ApJ...663.1174S], CCH [@2010AA...512A..31S], C$_{3}$S and C$_{4}$H [@2013JPCA..117.9831S], HC$_{5}$N [@2016ApJ...817..147T], and HC$_{7}$N [@2018MNRAS.474.5068B].
Based on the observations mentioned in the preceding paragraph, some possible main formation pathways of cyanopolyynes were investigated [@1998AA...329.1156T; @2016ApJ...817..147T; @2018MNRAS.474.5068B]. In the case of HC$_{3}$N, the abundances of H$^{13}$CCCN and HC$^{13}$CCN are similar to each other, and HCC$^{13}$CN is more abundant than the others. From the results, the reaction between C$_{2}$H$_{2}$ and CN was suggested as the main formation pathway of HC$_{3}$N [@1998AA...329.1156T]. On the other hand, there is no significant difference in abundance among the five $^{13}$C isotopologues of HC$_{5}$N. Reactions between hydrocarbon ions (C$_{5}$H$_{n}^{+}$, $n=3,4,5$) and nitrogen atoms followed by dissociative recombination reactions were found to be the most plausible route to explain the observed $^{13}$C isotopic fractionation of HC$_{5}$N [@2016ApJ...817..147T]. These proposed reactions were supported by the $^{14}$N/$^{15}$N ratios of HC$_{3}$N and HC$_{5}$N [@2017PASJ...69L...7T]. In the case of HC$_{7}$N, the fractionation results and proposed main formation mechanism are similar to those of HC$_{5}$N [@2018MNRAS.474.5068B].
The main formation mechanisms of HC$_{3}$N were investigated in other sources including the L1527 low-mass star-forming core and the G28.28–0.36 high-mass star-forming core [@2016ApJ...830..106T], as well as two starless cores [L1521B and L134N; @2017ApJ...846...46T]. Except for the case of L134N, the reaction between C$_{2}$H$_{2}$ and CN has been proposed as its main formation mechanism [@2016ApJ...830..106T; @2017ApJ...846...46T], while the reaction between CCH and HNC could explain the observed $^{13}$C isotopic fractionation in L134N [@2017ApJ...846...46T]. The proposed main formation pathway of HC$_{3}$N in the star-forming cores L1527 and G28.28–0.36 is consistent with model calculations for WCCC [@2008ApJ...681.1385H] and for hot cores [@2019arXiv190611296T]. The differences among starless cores are probably caused by their different ages; L134N is considered to be more evolved than L1521B and TMC-1 [@2017ApJ...846...46T].
Another interesting feature involving carbon isotopes was the observation that the $^{12}$C/$^{13}$C ratios of carbon-chain molecules are higher than the elemental ratio in the local interstellar medium [$60-70$; @2005ApJ...634.1126M], an effect known as the dilution of the $^{13}$C species[^1]. This dilution is considered to be caused at least in part by the low $^{13}$C$^{+}$ abundance which occurs via the reaction: $$\label{rea:co}
^{13}{\rm {C}}^{+} + {\rm {CO}} \rightarrow {\rm {C}}^{+} + ^{13}\!{\rm {CO}} + \Delta E \; (35 \; {\rm {K}}),$$ a reaction that is efficient especially in low-temperature conditions [@1984ApJ...277..581L]. The importance of this reaction stems from the fact that the initial step in the formation of carbon-chain molecules in dark clouds occurs via gas-phase ion-molecule reactions or neutral-neutral reactions with C$^{+}$ or C. Hence, the loss of the $^{13}$C$^{+}$ abundance leads to the high $^{12}$C/$^{13}$C ratios of carbon-chain molecules. However, the different degrees of the dilution of the $^{13}$C species among carbon-chain molecules found in TMC-1 cannot be explained only by reaction (\[rea:co\]) [@2016ApJ...817..147T].
In this paper, we report the observations of the $N=1-0$ transition lines of CCH and its two $^{13}$C isotopologues in L1521B ($d=140$ pc) and L134N ($d=110$ pc) using the Nobeyama 45 m radio telescope. We describe our observations in Section \[sec:obs\]. The results and derived parameters with the methods utilized are presented in Section \[sec:resana\]. The differential fractionation between the two $^{13}$C-containing CCH isotopologues in L1521B and L134N and possible mechanisms causing the heavy dilution of $^{13}$C-containing species especially of CCH in dark clouds are discussed with the help of a chemical simulation in Sections \[sec:dis1\] and \[sec:dis2\], respectively. Our conclusions are summarized in Section \[sec:con\].
Observations {#sec:obs}
============
The observations were carried out in 2019 January with the Nobeyama 45-m radio telescope (Proposal ID: CG181003, PI: Kotomi Taniguchi, 2018-2019 season). The $N=1-0$ transition lines of CCH and its two $^{13}$C isotopologues in the 84 – 88 GHz band were observed simultaneously with the T70 receiver. The beam size and main beam efficiency ($\eta_{\rm {mb}}$) were 19and 55%, respectively. The system temperatures were between 170 and 270 K depending on the weather conditions and elevation. We used the SAM45 FX-type digital correlator in the frequency setup whose bandwidth and frequency resolution were 125 MHz and 30.52 kHz, respectively. The frequency resolution corresponds to the velocity resolution of 0.1 km s$^{-1}$ at 86 GHz. We conducted the 2-channel binning in the final spectra, and thus the velocity resolution of the final spectra is 0.2 km s$^{-1}$.
The position-switching mode was employed. The observed positions were ($\alpha_{2000}$, $\delta_{2000}$) = (04$^{\rm h}$24$^{\rm m}$1267, +2636528) and (15$^{\rm h}$54$^{\rm m}$1272, -0249474) for L1521B and L134N, respectively. The off position for L1521B was set to be ($\Delta \alpha$, $\Delta \delta$) = (+4, +4) away from the on-source position, and that for L134N was set at $+3\arcmin$ away in the right ascension. The scan pattern was 20 s and 20 s for on-source and off-source positions, respectively. The chopper-wheel calibration method was adopted and hence the absolute calibration error was approximately 10%.
We checked the pointing accuracy by observations of the SiO ($J=1-0$) maser lines from NML Tau at ($\alpha_{2000}$, $\delta_{2000}$) = (03$^{\rm h}$53$^{\rm m}$2886, +1124224) and WX-Ser at ($\alpha_{2000}$, $\delta_{2000}$) = (15$^{\rm h}$27$^{\rm m}$4705, +1933518) during the observations of L1521B and L134N, respectively. The pointing observations were conducted using the H40 receiver every 1.5 hour. The pointing accuracy was within 3.
Results and Analyses {#sec:resana}
====================
Results
-------
[lcccccccc]{} & & & & & & & &\
CCH & $J=3/2-1/2, F= 1- 1$ & 87.284156 & 0.17 & 0.588 (9) & 0.423 (8) & 6.5 & 0.265 (6) & 4.7\
& $J=3/2-1/2, F= 2- 1$ & 87.316925 & 1.67 & 1.165 (12) & 0.459 (5) & 6.4 & 0.569 (9) & 4.7\
& $J=3/2-1/2, F= 1- 0$ & 87.328624 & 0.83 & 0.843 (10) & 0.4223 (6) & 6.5 & 0.379 (7) & 4.7\
& $J=1/2-1/2, F= 1- 1$ & 87.402004 & 0.83 &1.13 (2) & 0.405 (9) & 6.4 & 0.485 (14) & 5.6\
& $J=1/2-1/2, F= 0- 1$ & 87.407165 & 0.33 & 0.865 (15) & 0.408 (8) & 6.4 & 0.375 (10) & 5.6\
& $J=1/2-1/2, F= 1- 0$ & 87.446470 & 0.17 & 0.678 (12) & 0.405 (8) & 6.4 & 0.292 (8) & 5.6\
$^{13}$CCH & $J=3/2-1/2, F_{1}= 2- 1, F=5/2-3/2$ & 84.119329 & 2.00 & $<0.016$ & ... & ... & $<0.007$ & 3.2\
& $J=3/2-1/2, F_{1}= 2- 1, F=3/2-1/2$ & 84.124143 & 1.22 & $<0.016$ & ... & ... & $<0.007$ & 3.0\
& $J=3/2-1/2, F_{1}= 1- 0, F=1/2-1/2$ & 84.151352 & 0.66 & $<0.016$ & ... & ... & $<0.007$ & 2.9\
C$^{13}$CH & $J=3/2-1/2, F_{1}= 2- 1, F=5/2-3/2$ & 85.229326 & 2.00 & 0.026 (13) & 0.31 (19) & 6.5 & 0.0084 (7) & 3.8\
& $J=3/2-1/2, F_{1}= 2- 1, F=3/2-1/2$ & 85.232792 & 1.25 & 0.025 (7) & 0.53 (16) & 6.5 & 0.0139 (6) & 3.8\
& $J=3/2-1/2, F_{1}= 1- 0, F=1/2-1/2$ & 85.247708 & 0.65 & 0.016 (5) & 0.7 (3) & 6.4 & 0.0129 (6) & 3.5\
& $J=3/2-1/2, F_{1}= 1- 0, F=3/2-1/2$ & 85.256952 & 1.28 & $<0.02$ & ... & ... & $<0.009$ & 3.6\
[**[L134N (L183)]{}**]{} & & & & & & & &\
CCH & $J=3/2-1/2, F= 1- 1$ & 87.284156 & 0.17 & 0.358 (11) & 0.311 (12) & 2.6 & 0.118 (6) & 4.8\
& $J=3/2-1/2, F= 2- 1$ & 87.316925 & 1.67 & 0.853 (8) & 0.379 (4) & 2.5 & 0.344 (5) & 4.8\
& $J=3/2-1/2, F= 1- 0$ & 87.328624 & 0.83 & 0.637 (12) & 0.339 (8) & 2.5 & 0.229 (7) & 4.8\
& $J=1/2-1/2, F= 1- 1$ & 87.402004 & 0.83 & 0.777 (11) & 0.335 (5) & 2.5 & 0.277 (6) & 5.4\
& $J=1/2-1/2, F= 0- 1$ & 87.407165 & 0.33 & 0.538 (16) & 0.320 (12) & 2.5 & 0.183 (9) & 5.4\
& $J=1/2-1/2, F= 1- 0$ & 87.446470 & 0.17 & 0.415 (15) & 0.303 (14) & 2.5 & 0.134 (8) & 5.4\
$^{13}$CCH & $J=3/2-1/2, F_{1}= 2- 1, F=5/2-3/2$ & 84.119329 & 2.00 & $<0.016$ & ... & ... & $<0.006$ & 3.1\
& $J=3/2-1/2, F_{1}= 2- 1, F=3/2-1/2$ & 84.124143 & 1.22 & $<0.016$ & ... & ... & $<0.006$ & 3.1\
& $J=3/2-1/2, F_{1}= 1- 0, F=1/2-1/2$ & 84.151352 & 0.66 & $<0.016$ & ... & ... & $<0.006$ & 2.9\
C$^{13}$CH & $J=3/2-1/2, F_{1}= 2- 1, F=5/2-3/2$ & 85.229326 & 2.00 & 0.016 (6) & 0.7 (3) & 2.6 & 0.011 (6) & 3.4\
& $J=3/2-1/2, F_{1}= 2- 1, F=3/2-1/2$ & 85.232792 & 1.25 & 0.020 (6) & 0.42 (14) & 2.5 & 0.009 (4) & 3.3\
& $J=3/2-1/2, F_{1}= 1- 0, F=1/2-1/2$ & 85.247708 & 0.65 & $<0.02$ & ... & ... & $<0.007$ & 3.6\
& $J=3/2-1/2, F_{1}= 1- 0, F=3/2-1/2$ & 85.256952 & 1.28 & $<0.02$ & ... & ... & $<0.007$ & 3.7\
We conducted the data reduction using Java NEWSTAR, which is the software for data reduction and analyses of the Nobeyama data. The total on-source integration times were 18.75 hr and 23.5 hr for L1521B and L134N, respectively. We fitted the spectra with a Gaussian profile, and the obtained spectral line parameters are summarized in Table \[tab:t1\].
Figure \[fig:f1\] shows the spectra of the normal species of CCH in L1521B and L134N. The black vertical lines indicate the systemic velocities of each source, which are 6.5 km s$^{-1}$ and 2.5 km s$^{-1}$ in L1521B and L134N, respectively. The velocity components of all the transition lines are consistent with the systemic velocities of each source within their errors of 0.2 km s$^{-1}$. Figures \[fig:f2\] and \[fig:f3\] show the spectra of the isotopomers $^{13}$CCH (left panels) and C$^{13}$CH (right panels) in L1521B and L134N, respectively. In either the sources, no line of $^{13}$CCH was detected. Two strong transition lines ($J=3/2-1/2, F_{1}= 2- 1, F=5/2-3/2$ and $J=3/2-1/2, F_{1}= 2- 1, F=3/2-1/2$) of C$^{13}$CH were detected in L1521B and L134N with a signal-to-noise (S/N) ratio of 4. In addition, the weaker transition line ($J=3/2-1/2, F_{1}= 1- 0, F=1/2-1/2$) was tentatively detected with an S/N ratio of 3 in L1521B.
Analyses
--------
[lcc]{} $N$(CCH) \[cm$^{-2}$\] & ($1.66 \pm 0.18$)$\times 10^{14}$ & ($7.3 \pm 0.9$)$\times 10^{13}$\
$N$($^{13}$CCH) \[cm$^{-2}$\] & $< 6.2 \times 10^{11}$ & $< 5.3 \times 10^{11}$\
$N$(C$^{13}$CH) \[cm$^{-2}$\] & ($7 \pm 2$)$\times 10^{11}$ & ($7.2 \pm 1.4$)$\times 10^{11}$\
$N$(CCH)/$N$($^{13}$CCH) & $>271$ & $>142$\
$N$(CCH)/$N$(C$^{13}$CH) & $252^{+77}_{-48}$ & $ 101^{+24}_{-16}$\
$N$(H$_{2}$) \[cm$^{-2}$\] & $9.5 \times 10^{21}$ & $1.2 \times 10^{22}$\
$X$(CCH) & ($1.75 \pm 0.19$)$\times 10^{-8}$ & ($6.1 \pm 0.8$)$\times 10^{-9}$\
We derived the column densities and excitation temperatures of the normal species with the non-LTE code RADEX . The gas kinetic temperature is assumed to be 10 K, which is a typical value in dark clouds [@1998ApJ...503..717H]. The collision rate coefficients were taken from @2012MNRAS.421.1891S. We calculated the parameters using two H$_{2}$ densities ($n_{\rm {H_{2}}}$) in each source. The assumed H$_{2}$ densities are $1.0 \times 10^{5}$ cm$^{-3}$ [@1998ApJ...503..717H] and $5.0 \times 10^{4}$ cm$^{-3}$ [@2004ApJ...617..399H] in L1521B, and $1.0 \times 10^{5}$ cm$^{-3}$ [@1998ApJ...503..717H] and $2.1 \times 10^{4}$ cm$^{-3}$ [@2000ApJ...542..870D] in L134N.
We derived the column densities and excitation temperatures from the intensities of the two weakest hyperfine components by a least-squares method [@2010AA...512A..31S][^2]. The derived column densities and excitation temperatures of CCH are ($1.66 \pm 0.18$)$\times 10^{14}$ cm$^{-2}$ and $6.6 \pm 0.7$ K ($1\sigma$) for $n_{\rm {H_{2}}} = 1 \times 10^{5}$ cm$^{-3}$, and ($2.05 \pm 0.2$)$\times 10^{14}$ cm$^{-2}$ and $5.3 \pm 0.6$ K for $n_{\rm {H_{2}}} = 5 \times 10^{4}$ cm$^{-3}$ in L1521B. For L134N, the column densities and excitation temperatures of CCH were derived to be ($7.3 \pm 0.9$)$\times 10^{13}$ cm$^{-2}$ and $6.4 \pm 0.8$ K for $n_{\rm {H_{2}}} = 1 \times 10^{5}$ cm$^{-3}$, and ($1.36 \pm 0.17$)$\times 10^{14}$ cm$^{-2}$ and $4.1 \pm 0.5$ K for $n_{\rm {H_{2}}} = 2.1 \times 10^{4}$ cm$^{-3}$, respectively. Because an excitation temperature of $\sim 6.5$ K is consistent with typical values of carbon-chain molecules in dark clouds [@1992ApJ...392..551S] and the critical density of the $N=1-0$ transition of CCH is $1 \times 10^{5}$ cm$^{-3}$ , we employ the values obtained with $n_{\rm {H_{2}}} = 1 \times 10^{5}$ cm$^{-3}$ in the following sections. The excitation temperature of $\sim 6.5$ K is lower than the gas kinetic temperature, but such low excitation temperatures have been derived in prestellar cores .
We derived the column densities of the $^{13}$C isotopologues assuming the LTE condition using the following formulae [@2016ApJ...817..147T]: $$\label{tau}
\tau = - {\mathrm {ln}} \left[1- \frac{T_{\rm mb} }{J(T_{\rm {ex}}) - J(T_{\rm {bg}})} \right],$$ where $$\label{tem}
J(T) = \frac{h\nu}{k}\Bigl\{\exp\Bigl(\frac{h\nu}{kT}\Bigr) -1\Bigr\} ^{-1},$$ and $$\begin{aligned}
\label{col}
N = \tau \frac{3h\Delta v}{8\pi ^3 S}\sqrt{\frac{\pi}{4\mathrm {ln}2}}Q\frac{1}{\mu ^2}\frac{1}{J_{\rm {lower}}+1}\exp\Bigl(\frac{E_{\rm {lower}}}{kT_{\rm {ex}}}\Bigr) \times \Bigl\{1-\exp\Bigl(-\frac{h\nu }{kT_{\rm {ex}}}\Bigr)\Bigr\} ^{-1}.\end{aligned}$$ In equation (\[tau\]), $T_{\rm mb}$ is the peak intensity (Table \[tab:t1\]) and $\tau$ is the optical depth. $T_{\rm {ex}}$ and $T_{\rm {bg}}$ are the excitation temperature and the cosmic microwave background temperature (2.73 K), respectively. We assumed that the excitation temperatures of the $^{13}$C isotopologues of CCH are equal to those of the normal species. We then used the excitation temperatures of $6.6 \pm 0.7$ K and $6.4 \pm 0.8$ K in L1521B and L134N, respectively. [*J*]{}([*T*]{}) in equation (\[tem\]) is the effective temperature equivalent to that in the Rayleigh-Jeans law. In equation (\[col\]), [*N*]{} denotes the column density, $\Delta v$ the line width (FWHM), $S$ the line strength, $Q$ the rotational partition function, $\mu$ the permanent electric dipole moment, and $E_{\rm {lower}}$ the energy of the lower rotational energy level. The permanent electric dipole moment is 0.769 Debye for both the $^{13}$C isotopologues [@1995CPL...244...45W]. Taking into account the evaluation of the Gaussian fitting, we derived the column densities of C$^{13}$CH from the line of $J=3/2-1/2, F_{1}= 2- 1, F=5/2-3/2$ in L1521B and the line of $J=3/2-1/2, F_{1}= 2- 1, F=3/2-1/2$ in L134N. In the case of $^{13}$CCH, we derived the upper limits of column density from the $3\sigma$ upper limits of the peak intensities. We used the average line widths of the normal species: 0.42 km s$^{-1}$ and 0.35 km s$^{-1}$ in L1521B and L134N, respectively. We summarize the column densities derived in each source in Table \[tab:t2\].
Table \[tab:t2\] summarizes the H$_{2}$ column density, $N$(H$_{2}$), at the observed positions. We obtained the $N$(H$_{2}$) value in L1521B from the fits file of the column density map[^3]. The map was made using the Herschel data . We derived the H$_{2}$ column density in L134N from the archival data of the 1.2 mm dust continuum emission obtained by the MAMBO bolometer array installed on the IRAM 30 m telescope[^4], using the following formula : $$\label{H2}
N({\rm{H}}_{2}) = 6.69 \times 10^{20} \times F_{\nu},$$ where $F_{\nu}$ is the flux intensity in unit of mJy beam$^{-1}$. The flux intensity is 17.9 mJy beam$^{-1}$ at the observed position, and the derived H$_{2}$ column density is $1.2 \times 10^{22}$ cm$^{-2}$ using equation (\[H2\]). The fractional abundances of CCH, $X$(CCH)$=N$(CCH)/$N$(H$_{2}$), are calculated at ($1.75 \pm 0.19$)$\times 10^{-8}$ and ($6.1 \pm 0.8$)$\times 10^{-9}$ in L1521B and L134N, respectively.
Discussion {#sed:dis}
==========
$^{13}$C Isotopic Fractionation of CCH in L1521B and L134N {#sec:dis1}
----------------------------------------------------------
In this section, we compare the column densities between the two $^{13}$C isotopologues of CCH, namely the fractionation between the two $^{13}$C isotopomers, in the observed two starless cores. Because we could not detect $^{13}$CCH with an S/N ratio above 3 in the two observed starless cores, we derived lower limits of the $N$(C$^{13}$CH)/$N$($^{13}$CCH) ratio, which are $>1.1$ and $>1.4$ in L1521B and L134N, respectively[^5]. In both sources, the $^{13}$CCH isotopomer is less abundant than the C$^{13}$CH isotopomer. This result is the same as found for TMC-1 and L1527 by @2010AA...512A..31S, where the C$^{13}$CH/$^{13}$CCH abundance ratios were derived to be $1.6 \pm 0.4$ and $1.6 \pm 0.1$ ($3 \sigma$) respectively. Based on these results, it may be common that $^{13}$CCH tends to be less abundant in starless cores.
Two possible mechanisms causing the $^{13}$C isotopomer fractionation in CCH have been proposed: fractionation via the formation pathway [@2010AA...512A..31S] and via an isotopomer-exchange reaction [@2011ApJ...731...38F]. @2010AA...512A..31S discussed the formation pathways of CCH that could cause its $^{13}$C isotopic fractionation. They considered the following three reactions: $$\label{rea:r1}
{\rm {C}}_{2}{\rm {H}}_{2}^{+} + {\rm {e}}^{-} \rightarrow {\rm {C}}_{2}{\rm {H}} + {\rm {H}},$$ $$\label{rea:r2}
{\rm {C}}_{2}{\rm {H}}_{3}^{+} + {\rm {e}}^{-} \rightarrow {\rm {C}}_{2}{\rm {H}} + {\rm {H}}_{2},$$ and $$\label{rea:r3}
{\rm {C}}{\rm {H}}_{2} + {\rm {C}} \rightarrow {\rm {C}}_{2}{\rm {H}} + {\rm {H}}.$$ Among the above three reactions, only reaction (\[rea:r3\]) is able to cause the $^{13}$C isotopic fractionation in CCH, because the two carbon atoms are not clearly equivalent. Hence, @2010AA...512A..31S deduced that the observed differences in the abundances between $^{13}$CCH and C$^{13}$CH would reflect the significant contribution of reaction (\[rea:r3\]). However, the contributions of each reaction to the overall formation of CCH were not investigated in detail.
We chose to calculate the contributions of each formation pathway of CCH using the astrochemical code Nautilus [@2016MNRAS.459.3756R]. Our model calculation and the reaction network utilized are described in Appendix \[sec:a1\]. Figure \[fig:f4\] shows the results of the model calculation. The upper panel shows the CCH abundance with respect to total hydrogen as a function of time, as well as horizontal lines for the observed abundances in L1521B and L134N. Given the standard level of agreement between calculated and observed abundances in dark clouds, it can be argued that the modeled abundance shows substantial agreement with the observed abundances in both sources over significant periods of time. In addition, estimates of dark cloud ages based on the agreement between observed and calculated abundances for large numbers of molecular species indicate a much tighter constraint on ages centered on the so-called “early time” of $\approx 10^{5}$ yr .
The lower panel of Figure \[fig:f4\] shows the contribution of each major reaction to the rate of formation of CCH. We exclude reactions which have fractions below 10%. Before $10^{3}$ yr, reaction (\[rea:r1\]) is the major formation pathway of CCH. The C$_{2}$H$_{2}^{+}$ ion has two equivalent carbon atoms, so this reaction cannot explain the differences in abundances between the two $^{13}$C isotopologues of CCH. The following reaction has the second highest contribution in this time range: $$\label{rea:r4}
{\rm {C}}_{3}{\rm {H}}^{+} + {\rm {e}}^{-} \rightarrow {\rm {C}}_{2}{\rm {H}} + {\rm {C}}.$$ The three carbon atoms in C$_{3}$H$^{+}$ are not equivalent, and this can explain the $^{13}$C isotopic fractionation, unless scrambling of the carbon atoms occurs efficiently during the electron recombination reaction. However, this reaction contributes a significantly smaller amount to the formation of CCH than reaction (\[rea:r1\]), and we cannot conclude that reaction (\[rea:r4\]) significantly contributes to the observable $^{13}$C isotopic fractionation of CCH. In addition, although unlikely in such a small system, a scrambling of carbon atoms may occur during ion-molecule reaction that produces C$_{3}$H$^{+}$ [@2016ApJ...817..147T]. In that case, we would not recognize clear differences in abundances among the $^{13}$C isotopologues. The contribution of reaction (\[rea:r4\]) to the $^{13}$C isotopic fractionation of CCH is still unclear.
After $10^{3}$ yr, reaction (\[rea:r3\]) has the largest contribution to the formation of CCH. As first explained by @2010AA...512A..31S, this reaction can result in $^{13}$C fractionation because the carbon atoms are not identical. Another way of looking at the problem is that unless the insertion of a carbon atom into the C-H bond occurs at the same rate as its addition to the C of CH$_{2}$ and the C atoms can scramble, fractionation will occur. The dominant contribution of reaction (\[rea:r3\]) extends from somewhat greater than 10$^{3}$ yr to almost 10$^{5}$ yr, which is a much longer period than the time range when reaction (\[rea:r1\]) is dominant. Moreover, its range of dominance coincides more closely with the age range determined in multi-molecule fits to abundances in dark clouds . This reaction should therefore contribute to the differences in abundances between $^{13}$CCH and C$^{13}$CH, as mentioned before.
Now consider the case of the isotopomer-exchange reaction: $${\rm ^{13}C^{12}CH + H \rightleftharpoons ^{12}C^{13}CH + H }+ \Delta E (8.1 K),$$ which is exothermic as written from left-to-right so that the net effect is to increase the abundance of the $^{12}$C$^{13}$CH isotopomer at the expense of $^{13}$C$^{12}$CH isotopomer. @2011ApJ...731...38F used $1\times 10^{-10}$ cm$^{3}$ s$^{-1}$ as the rate coefficient of the forward reaction of the isotopomer-exchange reaction in their calculation. They also applied 8.1 K to the zero-point energy difference of $^{13}$CCH and C$^{13}$CH. The efficiency of this process depends upon a number of factors including whether the exothermicity of 8.1 K is sufficient to cause the difference in observed abundance, whether the abundance of atomic hydrogen is large enough, and whether or not a barrier to reaction exists. The differences in abundances between $^{13}$CCH and C$^{13}$CH can be seen even at the early stages [Figure 3 in @2011ApJ...731...38F]. This isotopomer-exchange reaction appears to be able to explain at least partially the $^{13}$C isotopic fractionation in both L1521B and L134N.
In summary, we found that C$^{13}$CH is more abundant than $^{13}$CCH in L1521B and L134N from our observations. This tendency agrees with the previous observations in TMC-1 and L1527 [@2010AA...512A..31S]. The higher abundance of C$^{13}$CH compared with $^{13}$CCH may indeed be common for dark clouds. Both reaction (\[rea:r3\]) and the isotopomer-exchange reaction likely contribute to the $^{13}$C isotopic fractionation of CCH both in L1521B and L134N.
The Dilution of the $^{13}$C Species in Dark Clouds {#sec:dis2}
---------------------------------------------------
### Comparisons of the $^{12}$C/$^{13}$C Ratios between CCH and HC$_{3}$N among Dark Clouds {#sec:dis2_1}
[lccc]{} $^{13}$CCH & $>271$ & $>142$ & $>250$\
C$^{13}$CH & $252^{+77}_{-48}$ & $101^{+24}_{-16}$ & $>170$\
H$^{13}$CCCN & $117 \pm 16$ & $61 \pm 9$ & $79 \pm 11$\
HC$^{13}$CCN & $115 \pm 16$ & $94 \pm 26$ & $75 \pm 10$\
HCC$^{13}$CN & $76 \pm 6$ & $46 \pm 9$ & $55 \pm 7$\
It is interesting to compare the $^{12}$C/$^{13}$C ratios of CCH and HC$_{3}$N among the three dark clouds L1521B, L134N, and TMC-1. Table \[tab:frac\] summarizes the $^{12}$C/$^{13}$C ratios of CCH and HC$_{3}$N in the three dark clouds and Figure \[fig:f5\] shows the comparisons. In the local interstellar medium, the elemental $^{12}$C/$^{13}$C ratio has been determined to be $60-70$ [e.g., @2005ApJ...634.1126M], which we indicate as the yellow range in Figure \[fig:f5\].
From Figure \[fig:f5\], we find that the dilution of the $^{13}$C species in carbon-chain molecules holds for the three observed dark clouds. In addition, the $^{12}$C/$^{13}$C ratios of CCH tend to be higher than those of HC$_{3}$N in all of the dark clouds. It had already been suggested that the degrees of the dilution are different among carbon-chain species in TMC-1 [@2016ApJ...817..147T]. We now can confirm the different degrees of the dilution among the carbon-chain species in the other dark clouds from our observations.
As mentioned in Section \[sec:intro\], the reaction between CCH and HNC was proposed as the main formation pathway of HC$_{3}$N in L134N [@2017ApJ...846...46T]. In this reaction, the carbon atom in HNC attacks the carbon atom with an unpaired electron in CCH forming HC$_{3}$N via HCCCNH [@1997ApJ...489..113F]. In that case, we can distinguish all the carbon atoms and trace them during the reaction scheme. We would expect that HC$_{3}$N/HC$^{13}$CCN is equal to CCH/$^{13}$CCH and HC$_{3}$N/H$^{13}$CCCN is equal to CCH/C$^{13}$CH. We indicated for the former pair (HC$^{13}$CCN/HC$_{3}$N and $^{13}$CCH/CCH) with red symbols and for the latter pair with blue symbols in Figure \[fig:f5\]. The latter pair agree within their $2\sigma$ error bars (CCH/C$^{13}$CH $=69-149$ and HC$_{3}$N/H$^{13}$CCCN $=43-79$) taking the $2\sigma$ errors into consideration. The former pair may marginally lie within the $2\sigma$ error bars (CCH/$^{13}$CCH $>142$ and HC$_{3}$N/HC$^{13}$CCN $=42-146$), taking the $2\sigma$ error into consideration, but we cannot strongly confirm it due to the non-detection of $^{13}$CCH. We need data with higher sensitivity in order to reach this conclusion.
In L1521B and TMC-1, the $^{12}$C/$^{13}$C ratios of CCH are higher than those of HC$_{3}$N by more than a factor of 2. The reaction between C$_{2}$H$_{2}$ and CN was proposed as the main formation pathway of HC$_{3}$N in both the dark clouds [@1998AA...329.1156T; @2017ApJ...846...46T], and the carbon atom which is next to the nitrogen atom in HC$_{3}$N should originate from CN, while the other two carbon atoms in HC$_{3}$N should come from C$_{2}$H$_{2}$ [@1997ApJ...489..113F]. Taking these points into consideration, the $^{12}$C/$^{13}$C ratios of H$^{13}$CCCN and HC$^{13}$CCN could reflect those of C$_{2}$H$_{2}$. The C$_{2}$H$_{2}$ molecule is mainly formed by the electron recombination reaction of C$_{2}$H$_{4}^{+}$, which is formed by the reaction between C$_{2}$H$_{2}^{+}$ and H$_{2}$ (Figure \[fig:f6\]). The C$_{2}$H$_{2}^{+}$ ion also forms CCH via its electron recombination reaction. Hence, the $^{12}$C/$^{13}$C ratio of C$_{2}$H$_{2}$ is expected to be similar to those of CCH, if the $^{12}$C/$^{13}$C ratios are determined during their bottom-up formation from C$^{+}$ and/or C. However, the observational results show discrepancies in the $^{12}$C/$^{13}$C ratio between CCH and C$_{2}$H$_{2}$. This suggests that the different degrees of the dilution of the $^{13}$C species are not induced during the carbon-chain growth from C$^{+}$ and/or C, but during reactions occurring after their production. In the following subsection, we discuss possible routes which could cause the significantly high $^{12}$C/$^{13}$C ratios of CCH, especially at a young dark cloud stage.
### Possible Routes to Produce the Significantly High $^{12}$C/$^{13}$C Ratios in CCH {#sec:dis2_2}
From our model calculation, we uncovered efficient formation and destruction pathways of small hydrocarbons as shown in Figure \[fig:f6\]. These pathways were found as we searched for possible routes which cause the high $^{12}$C/$^{13}$C ratio in CCH.
An important cycle, which seems to be efficient in increasing the $^{12}$C/$^{13}$C ratio in CCH in stages as early as $t < 10^{3}$ yr, is highlighted as a red triangle in Figure \[fig:f6\]. In this cycle, more than 80% of CCH is destroyed by the reaction with C$^{+}$ to form C$_{3}^{+}$ before $10^{3}$ yr. CCH is reformed by the reaction of C$_{3}^{+}$ with H$_{2}$ followed by the dissociative recombination of C$_{3}$H$^{+}$. If $^{13}$C$^{+}$ is diluted due to reaction (\[rea:co\]), the $^{12}$C/$^{13}$C ratios of CCH will become higher during the cycle because this cycle involves $^{13}$C$^{+}$. Another possible route to produce the high $^{12}$C/$^{13}$C ratio in CCH involves the following reaction: $${\rm {C}}_{2}{\rm {H}}_{2} + {\rm {C}}^{+} \rightarrow {\rm {C}}_{3}{\rm {H}}^{+} + {\rm {H}}.$$ The C$_{3}$H$^{+}$ product once again reacts dissociatively with electrons to form CCH + C. If the $^{12}$C/$^{13}$C ratio of C$_{3}$H$^{+}$ is high, the ratio of CCH also should be high. The following ion-neutral bimolecular reaction could transfer the high $^{12}$C/$^{13}$C ratio to C$_{3}$H$^{+}$: $$c,l-{\rm {C}}_{3}{\rm {H}} + {\rm {C}}^{+} \rightarrow {\rm {C}}_{3}{\rm {H}}^{+} + {\rm {C}}.$$ The high $^{12}$C/$^{13}$C ratio of C$_{3}$H$^{+}$ may increase the $^{12}$C/$^{13}$C ratio of $l,c$-C$_{3}$H$_{3}^{+}$. However, most electron recombination reactions of $l,c$-C$_{3}$H$_{3}^{+}$ lead to the formation of $l,c$-C$_{3}$H$_{2}$ and $l,c$-C$_{3}$H. In that case, the high $^{12}$C/$^{13}$C ratio of $l,c$-C$_{3}$H$_{3}^{+}$ will not significantly affect the ratio of C$_{2}$H$_{2}$. Therefore, the high $^{12}$C/$^{13}$C ratio of C$_{3}$H$^{+}$ will not significantly affect the ratio of C$_{2}$H$_{2}$, but produce the high ratio of CCH.
@2011ApJ...731...38F plotted the temporal variation of the $^{12}$C$^{+}$/$^{13}$C$^{+}$ ratio. In fact, this ratio takes extremely high value of $\geq450$ around $t \simeq 10^{3}$ yr with the assumed density of $5 \times 10^{4}$ cm$^{-3}$ [Figure 1 in @2011ApJ...731...38F]. Therefore, there is a possibility that the above reactions cause the significantly high $^{12}$C/$^{13}$C ratios in CCH at an early time.
After $t = 10^{3}$ yr, the abundance of ionic carbon (C$^{+}$) rapidly decreases (see the middle panel of Figure \[fig:f4\]), and the reactions with C$^{+}$ are suppressed. In addition, the $^{12}$C$^{+}$/$^{13}$C$^{+}$ ratio becomes lower in the later stage [$^{12}$C$^{+}$/$^{13}$C$^{+}$$\approx 200$ at $2 \times 10^{3} < t < 3 \times 10^{4}$ yr; @2011ApJ...731...38F]. Hence, the $^{12}$C/$^{13}$C ratios of CCH would not increase due to the above reactions. These model results may support the lower $^{12}$C/$^{13}$C ratios of CCH in L134N compared to younger clouds of L1521B and TMC-1[@2004ApJ...617..399H; @2000ApJ...542..870D; @2017ApJ...846...46T].
@2011ApJ...731...38F computed and displayed the temporal variation of the $^{12}$C/$^{13}$C ratio of CCH. The modeled ratio takes its peak value of $\sim220$ at a time around $3 \times 10^{2}$ yr and quickly decreases to 100 – 60. The predicted peak value is still lower than the observed value in L1521B, and their prediction is not consistent with the observational results quantitatively. However, the observed results that the $^{12}$C/$^{13}$C ratios of CCH in the chemically evolved dark cloud (L134N) are lower compared to the chemically young dark clouds, L1521B and TMC-1, are qualitatively consistent with the simulation [@2011ApJ...731...38F]. As they pointed out, @2011ApJ...731...38F may lack some mechanisms which cause the dilution of the $^{13}$C species. For example, @2011ApJ...731...38F did not take the selective photodissociation into consideration, but it will increase the $^{12}$C/$^{13}$C ratios. If the selective photodissociation of CCH occurs, the $^{13}$C isotopologues are destroyed in denser regions where the normal species can survive. CCH seems to be optically thick because the optical thickness of the weakest hyperfine components are around 0.2 in L1521B.
Conclusions {#sec:con}
===========
We have carried out observations of the $N=1-0$ transition lines of CCH and its two $^{13}$C isotopologues toward two starless cores, L1521B and L134N, using the Nobeyama 45 m radio telescope. The isotopologue of C$^{13}$CH is detected with an S/N ratio of 4, while the other isotopologue, $^{13}$CCH, could not be detected with an S/N ratio above 3. The $N$(C$^{13}$CH)/$N$($^{13}$CCH) ratios are derived to be $>1.1$ and $>1.4$ in L1521B and L134N, respectively. The characteristic that C$^{13}$CH is more abundant than $^{13}$CCH seems to be common for cold dark clouds. Such a difference in abundances between the two $^{13}$C isotopologues, namely the $^{13}$C isotopic fractionation, of CCH could be caused during its formation pathway and by the isotopomer-exchange reaction after the molecule is formed.
The derived $^{12}$C/$^{13}$C ratios of CCH in L1521B and L134N are higher than the elemental ratio in the local interstellar medium. We compared the $^{12}$C/$^{13}$C ratios of CCH and HC$_{3}$N among the three dark clouds. The $^{12}$C/$^{13}$C ratios of CCH are higher than those of HC$_{3}$N by more than a factor of 2 in L1521B and TMC-1, while the differences in the $^{12}$C/$^{13}$C ratios between CCH and HC$_{3}$N seem to be smaller in L134N. We discussed possible routes to produce the significantly high $^{12}$C/$^{13}$C ratios only in CCH based on the chemical network simulation. We found a possible cycle which occurs efficiently in the early stage of dark clouds. The previous study shows that the $^{12}$C$^{+}$/$^{13}$C$^{+}$ ratio becomes extremely high above 450 in the early time of dark cloud [@2011ApJ...731...38F]. Taking the predicted $^{12}$C$^{+}$/$^{13}$C$^{+}$ into account, the $^{12}$C/$^{13}$C ratio of CCH will also become high in the cycle because this cycle involves C$^{+}$. Besides, the reactions of “C$_{2}$H$_{2}$ + C$^{+}$" and “$l,c$-C$_{3}$H + C$^{+}$" can contribute to the high $^{12}$C/$^{13}$C ratios in CCH.
We are deeply grateful to the staff of the Nobeyama Radio Observatory. The Nobeyama Radio Observatory is a branch of the National Astronomical Observatory of Japan (NAOJ), National Institutes of Natural Science (NINS). K. T. would like to thank the University of Virginia for providing the funds for her postdoctoral fellowship in the Virginia Initiative on Cosmic Origins (VICO) research program. E. H. would like to thank the National Science Foundation for support of his program in astrochemistry.
Model Calculation {#sec:a1}
=================
We calculated the abundance of CCH and the contributions of each formation/destruction pathway using the astrochemical code Nautilus [@2016MNRAS.459.3756R]. The initial elemental abundances with respect to total hydrogen are taken from [@2017ApJ...850..105A] as summarized in Table \[tab:ie\]. Initially, all of hydrogen is the form in H$_{2}$. The initial form of hydrogen does not affect our discussion. We ran the model calculation including 7646 gas-phase reactions and 498 gas-phase species, mainly taken from the Kinetic Database for Astrochemistry (KIDA)[^6]. There are 5323 grain-surface reactions and 431 grain-surface species including suprathermal species [@2018PCCP...20.5359S]. The surface reactions come mainly from @2013ApJ...765...60G, with additional data taken from @2018ApJ...852...70B and @2018ApJ...857...89H. The self shielding effects of H$_{2}$ , CO , and N$_{2}$ are included.
The assumed density, gas temperature, visual extinction ($A_{\rm {v}}$), and cosmic-ray ionization rate ($\zeta$) are $2 \times 10^{4}$ cm$^{-3}$, 10 K, 10 mag, and $1.3 \times 10^{-17}$ s$^{-1}$, respectively. We assume that the dust temperature is equal to the gas temperature. These values are considered to be the typical values for dark clouds .
[cc]{} H$_{2}$ & 0.5\
He & 0.09\
C$^{+}$ & $7.3 \times 10^{-5}$\
N & $2.14 \times 10^{-5}$\
O & $1.76 \times 10^{-4}$\
F & $1.8 \times 10^{-8}$\
Si$^{+}$ & $8 \times 10^{-9}$\
S$^{+}$ & $8 \times 10^{-8}$\
Fe$^{+}$ & $3 \times 10^{-9}$\
Na$^{+}$ & $2 \times 10^{-9}$\
Mg$^{+}$ & $7 \times 10^{-9}$\
Cl$^{+}$ & $1 \times 10^{-7}$\
P$^{+}$ & $2 \times 10^{-10}$\
Excitation temperature and optical depth of each hyperfine component {#sec:a2}
====================================================================
Table \[tab:radex\] summarizes the excitation temperatures and optical depths of CCH of each hyperfine component derived by the RADEX.
[lcccccccccc]{} $J=3/2-1/2, F= 1- 1$ & 0.17 & 6.8 & 0.16 & 5.4 & 0.26 & & 6.6 & 0.10 & 4.1 & 0.33\
$J=3/2-1/2, F= 2- 1$ & 1.67 & 6.5 & 0.38 & 5.1 & 0.73 & & 6.4 & 0.27 & 4.0 & 1.30\
$J=3/2-1/2, F= 1- 0$ & 0.83 & 6.3 & 0.27 & 4.9 & 0.52 & & 6.3 & 0.20 & 3.8 & 0.95\
$J=1/2-1/2, F= 1- 1$ & 0.83 & 6.2 & 0.40 & 5.0 & 0.75 & & 6.1 &0.27 & 3.9 & 1.21\
$J=1/2-1/2, F= 0- 1$ & 0.33 & 6.3 & 0.29 & 5.0 & 0.51 & & 6.2 & 0.17 & 3.9 & 0.70\
$J=1/2-1/2, F= 1- 0$ & 0.17 & 6.5 & 0.20 & 5.3 & 0.33 & & 6.3 & 0.13 & 4.1 & 0.41\
Acharyya, K., & Herbst, E. 2017, , 850, 105 Bergantini, A., G[ó]{}bi, S., Abplanalp, M. J., & Kaiser, R. I. 2018, , 852, 70 Burkhardt, A. M., Herbst, E., Kalenskii, S. V., et al. 2018, , 474, 5068 Dickens, J. E., Irvine, W. M., Snell, R. L., et al. 2000, , 542, 870 Fukuzawa, K., & Osamura, Y. 1997, , 489, 113 Furuya, K., Aikawa, Y., Sakai, N., & Yamamoto, S. 2011, , 731, 38 Garrod, R. T. 2013, , 765, 60 Hassel, G. E., Herbst, E., & Garrod, R. T. 2008, , 681, 1385 Hirota, T., Maezawa, H., & Yamamoto, S. 2004, , 617, 399 Hirota, T., Yamamoto, S., Mikami, H., & Ohishi, M. 1998, , 503, 717 Hudson, R. L., & Moore, M. H. 2018, , 857, 89 Kaifu, N., Ohishi, M., Kawaguchi, K., et al. 2004, , 56, 69 Kauffmann, J., Bertoldi, F., Bourke, T. L., Evans, N. J., II, & Lee, C. W. 2008, , 487, 993 Kauffmann, J., Goldsmith, P. F., Melnick, G., et al. 2017, , 605, L5 Langer, W. D., Graedel, T. E., Frerking, M. A., & Armentrout, P. B. 1984, , 277, 581 Lee, H.-H., Herbst, E., Pineau des Forets, G., Roueff, E., & Le Bourlot, J. 1996, , 311, 690 Li, X., Heays, A. N., Visser, R., et al. 2013, , 555, A14 McElroy, D., Walsh, C., Markwick, A. J., et al. 2013, , 550, A36 Milam, S. N., Savage, C., Brewster, M. A., Ziurys, L. M., & Wyckoff, S. 2005, , 634, 1126 M[ü]{}ller, H. S. P., Schl[ö]{}der, F., Stutzki, J., & Winnewisser, G. 2005, Journal of Molecular Structure, 742, 215 Padovani, M., Walmsley, C. M., Tafalla, M., et al. 2009, , 505, 1199 Palmeirim, P., Andr[é]{}, P., Kirk, J., et al. 2013, , 550, A38 Ruaud, M., Wakelam, V., & Hersant, F. 2016, , 459, 3756 Sakai, N., Ikeda, M., Morita, M., et al. 2007, , 663, 1174 Sakai, N., Takano, S., Sakai, T., et al. 2013, Journal of Physical Chemistry A, 117, 9831 Sakai, N., Saruwatari, O., Sakai, T., Takano, S., & Yamamoto, S. 2010, , 512, A31 Sakai, N., & Yamamoto, S. 2013, Chemical Reviews, 113, 8981 Shingledecker, C. N., & Herbst, E. 2018, Physical Chemistry Chemical Physics (Incorporating Faraday Transactions), 20, 5359 Spielfiedel, A., Feautrier, N., Najar, F., et al. 2012, , 421, 1891 Suzuki, H., Yamamoto, S., Ohishi, M., et al. 1992, , 392, 551 Takano, S., Masuda, A., Hirahara, Y., et al. 1998, , 329, 1156 Takano, S., Suzuki, H., Ohishi, M., et al. 1990, , 361, L15 Taniguchi, K., Herbst, E., Caselli, P., et al. 2019a, arXiv e-prints, arXiv:1906.11296 Taniguchi, K., Ozeki, H., & Saito, M. 2017, , 846, 46 Taniguchi, K., Ozeki, H., Saito, M., et al. 2016a, , 817, 147 Taniguchi, K., & Saito, M. 2017, , 69, L7 Taniguchi, K., Saito, M., & Ozeki, H. 2016b, , 830, 106 Taniguchi, K., Saito, M., Sridharan, T. K., & Minamidani, T. 2018, , 854, 133 Taniguchi, K., Saito, M., Sridharan, T. K., & Minamidani, T. 2019b, , 872, 154 van der Tak, F. F. S., Black, J. H., Sch[ö]{}ier, F. L., Jansen, D. J., & van Dishoeck, E. F. 2007, , 468, 627 Visser, R., van Dishoeck, E. F., & Black, J. H. 2009, , 503, 323 Wakelam, V., Herbst, ER., & Selsis, F. 2006, , 451, 551 Woon, D. E. 1995, Chemical Physics Letters, 244, 45
[^1]: In this paper, we define the dilution of the $^{13}$C species as the $^{12}$C/$^{13}$C ratios higher than 70, which is the mean $^{12}$C/$^{13}$C ratio in the local interstellar medium.
[^2]: The derived excitation temperatures and optical depths of each hyperfine component are summarized in Table \[tab:radex\] in Appendix \[sec:a2\].
[^3]: Taken from <http://www.herschel.fr/cea/gouldbelt/en/Phocea/Vie_des_labos/Ast/ast_visu.php?id_ast=66>
[^4]: Taken from <http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/487/993>
[^5]: These lower limits have the errors of 0.3, which are derived from the standard deviation of $N$(C$^{13}$CH).
[^6]: http://kida.obs.u-bordeaux1.fr
|
---
abstract: 'This study aims at advancing mathematical and computational techniques for reconstructing the interior structure of a small Solar System body via Computed Radar Tomography (CRT). We introduce a far-field model for full-wave CRT and validate it numerically for an orbiting distance of 5 km using a synthetic 3D target asteroid and sparse limited-angle data. As a potential future application of the proposed method, we consider the Deep Interior Scanning CUbeSat (DISCUS) concept in which the goal is to localize macroporosities inside a rubble pile near-Earth asteroid with two small spacecraft carrying a bistatic radar.'
author:
- 'M. Takala, P. Bambach, J. Deller, E. Vilenius, M. Wittig, H. Lentz, H. M. Braun, M. Kaasalainen, S. Pursiainen [^1] [^2] [^3] [^4] [^5] [^6]'
bibliography:
- 'pursiainen.bib'
- 'references.bib'
title: 'Far-Field Inversion for the Deep Interior Scanning CubeSat'
---
Small Solar System Bodies, Near-Earth Asteroids, Far-Field Measurements, Inverse Imaging, Computed Radar Tomography.
Introduction
============
The aim of this study is to advance Computed Radar Tomography (CRT) [@persico2014; @devaney2012] for reconstructing the deep interior structure of a small solar system body (SSSB) [@pursiainen2016; @su2016; @herique2016; @kofman2015; @asphaug2003]. The first such attempt, the Comet Nucleus Sounding Experiment by Radio-wave Transmission (CONSERT), was made as a part of the ESA’s [*Rosetta*]{} mission to comet 67P/Churyumov-Gerasimenko. In CONSERT, a tomographic radar signal was transmitted between the orbiter and the [*Philae*]{} lander [@kofman2007; @kofman2015]. At the moment, several space organizations aim to rendezvous SSSBs. In 2018, the Osiris-REx by NASA [@berry2013; @lauretta2012] and the Hayabusa-2 [@kawaguchi2008; @tsuda2013] by Japan Aerospace Exploration Agency (JAXA) will arrive at the asteroids 101955 Bennu and 162173 Ryugu (1999 JU3), respectively. Future CRT experiments have recently been planned for the ESA’s proposed Asteroid Impact Mission (AIM) to asteroid 65803 Didymos (1996 GT) [@herique2016; @michel2016]. This paper introduces and validates a three-dimensional far-field extension for the full-wave CRT model presented in [@pursiainen2016]. We use realistic parameter values, orbiting distance and target scaling in order to support the design of the future planetary missions. In spaceborne CRT, the ability to simulate and invert far-field data is essential, since inserting a spacecraft into a stable SSSB orbit is a major challenge due to the low escape velocity of the SSSB. As the potential future application of the present inversion approach we consider the Deep Interior Scanning CUbesat (DISCUS) concept [@deller2017; @bambach2017] in which two small spacecraft carrying a bistatic (dual-antenna) radar [@willis2008] record penetrating radar data at about a few kilometers distance to a 260–600 m (Itokawa-size [@abe_mass_2006]) rubble pile near-Earth asteroid (NEA). Using two CubeSats for a bistatic radar measurement was first proposed for the AIM mission [@herique2016].
Rubble-pile asteroids are celestial bodies composed of aggregates bound together by gravitation and weak cohesion [@bottke2002; @michel2015; @richardson2005; @polishook2016; @sanchez2014; @deller2015; @deller2016]. Therefore, they are likely to contain macroporosity, e.g., internal voids or cracks. From the observed rotation period cut-off limit of about $>$$2.2\,\text{hours}$ for asteroids with diameters greater than 200–300 m, it has been concluded that these bodies are not monolithic, but in the vast majority rubble pile asteroids [@Pravec2000]. The estimated density for the rubble piles suggests that macroporosity exists [@carry_density_2012] but the actual proof is still missing. The radar onboard DISCUS would use a center frequency which is advantageous regarding both void detection [@binzel2005; @daniels2004] and also the measurement noise due to the Sun [@burke2009; @stone2000; @kraus1967]. As an independent mission, DISCUS would demonstrate a new and affordable mission concept to gain knowledge of the inside of NEAs.
In the numerical experiments, we investigated a three-dimensional asteroid model containing deep interior anomalies and a surface dust layer. The inversion accuracy and reliability were explored for several different noise levels and also for sparse limited-angle data, i.e., a situation in which the trajectory of the spacecraft does not allow measurements from all directions. Full-wave CRT was applied in order to maximize the imaging quality and to allow the sparsity of the measurements which is vital for the many limitations of the space missions, such as low data transfer rates and comparably short instrument lifetime [@agrawal2014; @doody2010]. Volumetric full-wave CRT in a realistic geometry is a computationally challenging imaging technique which requires 3D wave simulation and inversion of a large system of equations. In order to achieve a sufficiently short computation time, a state-of-the-art cluster of graphics processing units (GPUs) was applied in the forward simulation. The waveform data were linearized and inverted using a high-end dual-processor workstation computer.
The results obtained suggest that the proposed mathematical model can be applied to invert full-wave far-field data for realistic asteroid sizes and shapes. Furthermore, based on the results, it seems that the existing background noise in the solar system and the expected level of modeling errors allow detection of macroporosities from the planned orbiting distance. The bistatic and stepped-frequency measurement techniques present in the DISCUS concept were found to be vital for improving the signal-to-noise ratio and for reducing the effects of the measurement noise and modeling errors. A comparison between full- and limited-angle CRT results suggests that the interior structure of the targeted NEA with a typical spin orientation can be reconstructed without requiring an orbital plane that includes shadow phases of the spacecraft and without the need to cover the entire surface of the SSSB.
This paper is organized as follows. Section \[mm\] briefly describes the DISCUS radar, numerical experiments and the full-wave CRT model for the far-field measurements. Sections \[r\] and \[d\] include the results and the discussion.
Materials and Methods {#mm}
=====================
DISCUS radar
------------
The DISCUS concept comprises two identical CubeSats. Both carry an identical radar instrument capable of a 10 W signal transmission. The CubeSats are equipped with a half-wavelength dipole antenna with a center frequency $f_c$ between 20 and 50 MHz and a bandwidth $B$ of at least 2 MHz. The bistatic measurement approach is used, since the scattering waves can be headed away from the transmitter spacecraft (Figure \[bistatic\_measurement\]). That is, one of the CubeSats both transmits and receives the signal, and the other one serves as an additional receiver.
The stepped-frequency measurement technique is applied [@iizuka1984; @gill2001; @paulose1994]. That is, the signal is a pulse sequence of narrow frequency lines $\psi_1, \psi_2, \ldots, \psi_N$ which allow one to approximate a given function $f$ via the sum $$f = \sum_{\ell = 1}^{N} c_\ell \psi_\ell. \label{f_formula}$$ If $\varphi_\ell$ is the received signal corresponding to frequency line $\psi_\ell$, then the data $g$ resulting from a transmission $f$ is approximately given by $g = \sum_{\ell = 1}^{N} c_\ell \varphi_\ell$ with $c_1, c_2, \ldots, c_N$ following from (\[f\_formula\]). Consequently, the data $g$ for any transmission $f$ within the given frequency range can be approximated, if the function pairs $\psi_\ell$ and $\varphi_\ell$ for $\ell = 1, 2, \ldots, N$ are given, i.e., if the stepped-frequency measurement data are available. Recording the frequency lines separately is advantageous with regard to the measurement accuracy, since the signal-to-noise ratio between the received power and the noise power is inversely proportional to the bandwidth of a single line.
The CubeSats follow their target asteroid in a polygonal plane which will be nearly perpendicular to the ecliptic of the solar system. The signals will be transmitted and captured via the two half-wavelength dipole antennas which will be pointed towards the Sun during the measurement to suppress the effect of the solar radiation (Figure \[bistatic\_measurement\]). In order to optimize the radar performance for the target NEA’s interior structure, the spacecraft will be equipped with a camera so that an accurate optical surface model can be created.
Numerical Modeling
------------------
In this study, we validate the radar concept of DISCUS numerically using the publicly available surface model of the asteroid 1998 KY26 scaled to 550 m diameter. The center frequency and bandwidth of the radar measurement are assumed to be $f_c = 20$ MHz (wavelength $\lambda = 15$ m in vacuum) and $B \approx 2.4$ MHz, respectively. The essential model parameters and their values can be found in Table \[scaling\]. In the numerical simulation, we use a scalable unitless parameter presentation. The unitless values can be scaled to SI-units as shown in Table \[scaling\_2\].
The signal power $P_{RX}$ (dB) received by the radar is estimated via the equation $$P_{RX} = P_{TX} + G_{TX} + S + G_{RX},$$ where $P_{TX} = 0$ dB is the power transmitted, $G_{TX} = G_{RX} = 2.15$ dBi (1.64) follows from the gain of the half-wavelength dipole antenna and $S$ is the signal power at the receiver location obtained through an isotropic radiator model, i.e., with an isotropic source and effective antenna aperture $A_{eff} = \lambda^2/(4 \pi)$.
In the stepped frequency measurement, the bandwidth of a single frequency line is given approximately by $B_\ell = 1 / T_\ell$, where $T_\ell$ is the pulse duration. We assume that the data is collected at a 5 km distance to the target asteroid. The duration of a single line is set to be $T_\ell = 32$ $\mu$s, i.e., 96 % of the signal travel-time, in order to minimize the corresponding bandwidth (here $B_\ell = 31$ kHz) and, thereby, also the relative amplitude of the measurement noise.
Item Unitless SI-units Relative to $\lambda$
--------------------------------------------- ----------------------- ------------- -----------------------
Asteroid diameter 0.262 550 m 36.7
Geometry scaling factor $s$ 1 2100 m 140
Conductivity $\sigma$ 24 3E-5 S/m
Attenuation rate 52.5 dB/(unit length) 25 dB/km
Orbiting distance 2.4 5 km 330
Void diameter 0.029–0.048 60–100 m 4–6.7
Dust layer thickness 0.019 40 m 2.7
Void $\varepsilon_r$ 1 1
Dust $\varepsilon_r$ 3 3
Body $\varepsilon_r$ 4 4
Final pulse duration after processing $T_0$ 0.12 0.84 $\mu$s
Radar bandwidth $B \approx 2/T_0$ $16.7$ 1/(unit time) $2.4$ MHz
Center frequency $f_c$ 140 20 MHz
Center wavelength in vacuum $\lambda$ 7.1e-2 15 m 1
Final data duration after processing $T$ 0.7 5.0 $\mu$s
Frequency line duration $T_\ell$ 4.6E-3 32 $\mu$s
Frequency line bandwidth $B_\ell$ 217 1/(unit time) 31 kHz
Item Unitless SI-units
------------------------------------------ ----------------- ------------------------------------------------
Dielectric permittitivity $\varepsilon_r$ $\varepsilon_r$
Electrical conductivity $\sigma$ $(\mu_0 / \varepsilon_0)^{-1/2} s^{-1} \sigma$
Position $\vec{x}$ $s {\vec x}$
Time $t$ $(\varepsilon_0 \mu_0)^{1/2} s t$
Frequency $t^{-1}$ $(\varepsilon_0 \mu_0)^{-1/2} s^{-1} t^{-1}$
Velocity (${\mathsf c} = \varepsilon_r$) ${\mathsf c}$ $(\varepsilon_0 \mu_0 )^{-1/2} {\mathsf c}$
: Formulas for scaling between the unitless and SI-unit expressions. In these, the permittivity and magnetic permeability of the vacuum are given by $\varepsilon_0 = 8.85 \cdot 10^{-12}$ F/m, and $\mu_0 = 4 \pi \cdot 10^{-7}$ b/m, respectively, and $s$ (meters) denotes the spatial scaling factor.\[scaling\_2\]
### Permittivity and Conductivity
The present inverse problem of the CRT is to reconstruct the target asteroid’s internal dielectric permittivity distribution $\varepsilon_r$ which contains, in this study, a 40 m surface layer (e.g. dust or sand) with $\varepsilon_r = 3$ and three 60–100 m interior voids with $\varepsilon_r = 1$ (vacuum), respectively. Otherwise, $\varepsilon_r$ is assumed to be 4, which is typical, e.g., for Kaolinite and Dunite [@herique2002].
The motivation for using this synthetic and relatively simple permittivity distribution was to enable validating the current far-field inversion approach both for the surface and deep interior structures with an exact measure for the inversion accuracy. The size of the details is mainly determined by the estimated tomography resolution. According to the recent studies, macroporosity structures of the present size and permittivity range can exist in rubble pile asteroids [@carry_density_2012; @michel2015].
The electrical conductivity distribution inside the asteroid was assumed to be an unknown nuisance parameter, i.e., not of primary interest. It was set to be $\sigma = 5 \varepsilon_r$ matching roughly to 3E-5 S/m and a loss rate of around 25 dB/km which is typical, e.g., for porous basalt (pyroxene) [@olhoeft1981; @kofman2012]. Outside the asteroid the $\sigma$ was assumed to be zero.
### Measurements
The data are simulated for 128 measurement points (Figure \[measurement\_points\]) evenly distributed on an origin centric sphere. The angle $\omega$ between the asteroid spin and the normal of the orbiting plane determines the coverage of the measurements. We investigate the following three different cases (A) $\omega = 90^\circ$, (B) $\omega = 30^\circ$ and (C) $\omega = 10^\circ$. In (A), a full-angle dataset can be recorded, meaning that the spacecraft will form a dense network of points enclosing the whole asteroid. The limited-angle cases (B) and (C), include an aperture of $90^\circ - \omega$ around the spin axis.
The following two measurement approaches were compared:
1. [*Monostatic measurement*]{}. A single spacecraft both transmits and receives the signal. The measurement is made each time at the point of the transmission. The total number of measurement positions is 128, 64, and 24 for (A), (B) and (C), respectively.
2. [*Bistatic measurement*]{}. Two spacecraft are used; one transmits the signal and both record the backscattered wave (Figure \[measurement\_points\]). Each measurement is made simultaneously at two different positions. The additional measurement is made at the point closest to 25 degrees apart from the transmission location with respect to the asteroid’s center of mass. The total number of transmission/measurement points is 128/256, 64/128 and 24/48 for (A), (B) and (C), respectively. In each case, the total number of unique measurement positions in the point set is the same as in the respective monostatic case.
![[**Left:**]{} A bistatic measurement approach is used, since the scattering waves can be headed away from the transmitter spacecraft. One spacecraft (red) is utilized to both transmit and receiver the signal. The other one (blue) serves as an additional receiver. [**Right:**]{} The spacecraft are assumed to orbit their target asteroid in a plane which will be nearly perpendicular to the ecliptic of the solar system. Each spacecraft is equipped with a single half wavelength dipole antenna (yellow) which will be pointed towards the Sun during the measurement in order to suppress the effect of the solar radiation. That is, the Sun is assumed to be located in the direction of the normal of the orbiting plane. \[bistatic\_measurement\] ](far_field_satellite_2.png){height="3.5cm"}
![[**Left:**]{} A bistatic measurement approach is used, since the scattering waves can be headed away from the transmitter spacecraft. One spacecraft (red) is utilized to both transmit and receiver the signal. The other one (blue) serves as an additional receiver. [**Right:**]{} The spacecraft are assumed to orbit their target asteroid in a plane which will be nearly perpendicular to the ecliptic of the solar system. Each spacecraft is equipped with a single half wavelength dipole antenna (yellow) which will be pointed towards the Sun during the measurement in order to suppress the effect of the solar radiation. That is, the Sun is assumed to be located in the direction of the normal of the orbiting plane. \[bistatic\_measurement\] ](asteroid_satellites.png){height="3.7cm"}
![ A schematic (non-scaled) illustration showing the sparse full-angle configuration (A) of 128 measurement points (left) and two sparse limited-angle configurations (B) and (C) of 64 and 24 points (right), respectively. The angle $\omega$ between the orbiting plane normal and the asteroid spin is 90, 30 and 15 degrees. The limited-angle configurations include an aperture around the z-axis. \[measurement\_points\]](points_90.png "fig:"){width="3cm"}\
(A) : $\omega = 90^\circ$
![ A schematic (non-scaled) illustration showing the sparse full-angle configuration (A) of 128 measurement points (left) and two sparse limited-angle configurations (B) and (C) of 64 and 24 points (right), respectively. The angle $\omega$ between the orbiting plane normal and the asteroid spin is 90, 30 and 15 degrees. The limited-angle configurations include an aperture around the z-axis. \[measurement\_points\]](points_30.png "fig:"){width="3cm"}\
(B) : $\omega = 30^\circ$\
0.2cm ![ A schematic (non-scaled) illustration showing the sparse full-angle configuration (A) of 128 measurement points (left) and two sparse limited-angle configurations (B) and (C) of 64 and 24 points (right), respectively. The angle $\omega$ between the orbiting plane normal and the asteroid spin is 90, 30 and 15 degrees. The limited-angle configurations include an aperture around the z-axis. \[measurement\_points\]](points_10.png "fig:"){width="3cm"}\
(C) : $\omega = 10^\circ$
Far-Field Forward Model for CRT
-------------------------------
The forward problem of CRT is to predict the voltage of the antenna given the computation geometry and the unknown parameters. In this study, we model a sparse set of (simulated) measurements in the spatio-temporal domain $[0, T] \times \Omega$ applying the wave propagation model presented in Appendix.
### Weak form
The signal transmission at point $\vec{p}$ is modeled as a point source of the form $$\label{point_source} \frac{\partial \vec{f}}{\partial t}(t, \vec{x})\, = \frac{\partial \vec{h}}{\partial t}(t) \, \delta(\vec{x} - \vec{p}) \quad \hbox{with} \quad h(0) = \frac{\partial \vec{h}}{\partial t}(0) = 0,$$ where $\vec{h}$ denotes the dependence of $\vec{f}$ on time and $\delta(\vec{x} - \vec{p})$ is the Dirac’s delta function satisfying $\int q(\vec{x})\delta (\vec{x} - \vec{p}) \, d V = q(\vec{p})$ for any sufficiently regular function $q$. Physically $\vec{f}$ can be interpreted as the current density of the antenna (Appendix).
The $i$-th component $E = E_i$ electric field $\vec{E} = E_i$ evoked by $\vec{f}$ satisfies the following weak form: [ $$\begin{aligned}
\label{weak_form_1} a ( E, {\bf g} ; {\bf w}; \Omega) & = & 0, \label{weak_form_2a}\\
b ( E, {\bf g}; v; \Omega) & = & - \langle f, v ;\Omega \rangle \label{weak_form_2}\end{aligned}$$]{} where ${\bf g} = (\vec{g}^{\,(1)}, \vec{g}^{\, (2)}, \vec{g}^{\, (3)})$ with $\vec{g^{\,(j)}} = \int_0^t \nabla E_i (\tau, \vec{x}) \, d \tau$ for $j = 1, 2, 3$ and $\langle f, v ;\Omega \rangle = \int_\Omega f v \, \hbox{d} V$. The bilinear forms $a$ and $b$ correspond to the right-hand sides of the equations (\[wf1\]) and (\[wf2\]), respectively, and $v \in H^1(\Omega)$, ${\bf w} = (\vec{w}^{(1)}, \vec{w}^{(2)}, \vec{w}^{(3})$ with $\vec{w}^{(i)} \in [L_2(\Omega)]^3$ are test functions. Under regular enough initial conditions this weak form [@pursiainen2016] has a unique solution $E : [0,T] \to H^1(\Omega)$ [@evans1998].
For modeling the far-field, we assume that the domain $\Omega$ consists of two sub-domains: an outer part $\Omega_1$ and an enclosed ball $\Omega_2$ containing the target asteroid. The spherical inner boundary is denoted with $\mathcal{S} = \Omega_1 \cap \Omega_2$ (Figure \[far\_field\_calculation\]). The spacecraft position $\vec{p}$ is assumed to be located outside $\Omega$.
### Incident and Scattered Field
The total field $E$ is expressed as the sum of the incident and scattered field, i.e., $E = E_{\mathtt I} + E_{\mathtt S}$, where the incident field $E_{\mathtt I}$ emanates from the source and vanishes in the interior of $\Omega_2$, that is, in $\Omega_{2} \setminus \mathcal{S}$. The scattered field $E_{\mathtt S}$ is the total field in $\Omega_{2}$, and its restriction to the surface $\mathcal{S}$ is utilized in calculating the measured far-field. The incident field $E_{\mathtt I}$ satisfies the following weak form in $\Omega_{1}$: [ $$\begin{aligned}
\label{weak_form_3} a ( E_{\mathtt I}, {\bf g}_{\mathtt I}; {\bf w}; \Omega_1) & = & 0, \\
b ( E_{\mathtt I}, {\bf g}_{\mathtt I}; v; \Omega_1) & = & - \langle f, v ; \Omega_1 \rangle + \langle {\bf g}_{\mathtt I }, v ;\mathcal{S} \rangle , \label{weak_form_4}\end{aligned}$$]{} in which $$\label{surface_source}
\langle {\bf g}_{\mathtt{I}}, v ; \mathcal{S} \rangle = \int_\mathtt{S} (\vec{g}_{\mathtt{I}} \cdot \vec{n}) v \, \hbox{d} S$$ with $\vec{g}_\mathtt{I} = \int_0^t \nabla E_\mathtt{I} (\tau, \vec{x}) \, d \tau$. The inner product $\langle {\bf g}_{\mathtt{I}}, v ; \mathcal{S} \rangle$ corresponds to the nonzero surface term in the integral $$\int_{\Omega_i} (\nabla \cdot \vec{g}_{\mathtt{I}} ) \, v \, d V = (-1)^{i} \int_\mathtt{S} (\vec{g}_{\mathtt{I}}\cdot \vec{n}) v \, \hbox{d} S - \int_{\Omega_i} \vec{g}_{\mathtt{I}}\cdot \nabla v \, \hbox{d} V,$$ where $\vec{n}$ denotes the outward unit normal of $\mathcal{S}$ and $j = 1, 2$. For the scattered field $E_{\mathtt S}$, it holds $E_{\mathtt S} = E - E_{\mathtt I}$. Thus, its weak form in $\Omega_i$ for $i = 1, 2$ can be obtained by subtracting both sides of (\[weak\_form\_3\]), (\[weak\_form\_4\]) from (\[weak\_form\_1\]), (\[weak\_form\_2\]) written for $\Omega_1$ and $\Omega_2$, respectively, taking into account that $\langle f, v; \Omega_2 \rangle_2 = 0$ and that the incident field vanishes in $\Omega_{2} \setminus \mathcal{S}$. It follows that [ $$\begin{aligned}
\label{weak_form_5} a ( E_{\mathtt S}, {\bf g}_{\mathtt S}; {\bf w}; \Omega_i) & = & 0, \\
b ( E_{\mathtt S}, {\bf g}_{\mathtt S}; v; \Omega_i) & = & - \langle {\bf g}_{\mathtt I}, v ;\mathcal{S} \rangle \label{weak_form_6}\end{aligned}$$]{} for $i = 1 , 2$. Summing both sides of (\[weak\_form\_5\]), (\[weak\_form\_6\]) for $\Omega_1$ and $\Omega_2$ together leads to the following full domain weak form for the scattered field: [ $$\begin{aligned}
\label{weak_form_7} a ( E_{\mathtt S}, {\bf g}_{\mathtt S}; {\bf w}; \Omega) & = & 0, \\
b ( E_{\mathtt S}, {\bf g}_{\mathtt S}; v; \Omega) & = & - 2 \, \langle {\bf g}_{\mathtt I }, v ;\mathcal{S} \rangle . \label{weak_form_8}\end{aligned}$$]{} This formulation is otherwise similar to (\[weak\_form\_1\])–(\[weak\_form\_2\]), but instead of a single point, the source function $\bf{g}_{\mathtt I}$ is evaluated on the sphere $\mathcal{S}$.
### Incident Far-Field
In empty space ($\varepsilon_r = 1 $ and $\sigma = 0$), the incident field for a monopolar (isotropic) point source (\[point\_source\]) placed at $\vec{p}$ can be expressed as the following convolution: $$E_{\mathtt I} = - \mathcal{G} \ast_\tau \frac{\partial h}{\partial t}= - \frac{ 1 }{4 \pi |\vec{x} - \vec{p}|} \left[ \frac{\partial h}{\partial t}\right],
\label{incident_field}$$ where $h \ast_\tau k(t, \vec{x}) = \int_0^\infty h (t - \tau, \vec{x}) k(\tau, \vec{x}) \, d \tau$, $[ {\partial h}/{\partial t}] $ is a retarded signal evaluated at $t - |\vec{x} - \vec{p}|$ and $\mathcal{G} = {\delta(t - |\vec{x} - \vec{p}|)}/{(4 \pi |\vec{x} - \vec{p}|)}$ is a Green’s function with $\delta$ denoting the Dirac’s delta function defined with respect to time. That is, $h(t, \vec{x}) = \int_0^\infty h(t-\tau, \vec{x}) \delta(\tau) \, d \tau$ for any sufficiently regular $h$. It follows from (\[incident\_field\]) through a straightforward calculation that [ $$\begin{aligned}
\nabla E_{\mathtt I} & = & \frac{(\vec{x} - \vec{p})}{4 \pi |\vec{x} - \vec{p}|^3} \left[ \frac{\partial h}{\partial t}\right] + \frac{(\vec{x} - \vec{p}) }{4 \pi |\vec{x} - \vec{p}|^2} \left[ \frac{\partial^2 h}{\partial t^2}\right] \\
\vec{g}_{\mathtt I} & = & \int_0^t \! \nabla u_{\mathtt{I}} d \tau \! = \! \frac{\vec{x} - \vec{p} }{4 \pi |\vec{x} - \vec{p}|^3} [{h}] \! + \! \frac{\vec{x} - \vec{p}}{4 \pi |\vec{x} - \vec{p}|^2} \left[ \frac{\partial h}{\partial t}\right] \! .\end{aligned}$$]{}
The incident field $E_{\mathtt I}$ needs to be evaluated only on its first arrival at $\mathcal{S}$, that is, in the subset $\mathcal{S}^- = \{ \vec{x} \in \mathcal{S} \, | \, (\vec{x} - \vec{p}) \cdot \vec{n} < 0 \} $. In the remaining part $\mathcal{S}^+ = \mathcal{S} \setminus \mathcal{S}^-$, $E_{\mathtt I}$ is set to be zero. Hence, it follows that $\langle \vec{g}_{\mathtt{I}}, v ; \mathcal{S}^- \rangle = \langle \vec{g}_{\mathtt{I}}, v ; \mathcal{S} \rangle$ with $$\langle \vec{g}_{\mathtt{I}}, v ; \mathcal{S}^- \rangle \! = \int_{\mathcal{S}^{+}} \! \left( \frac{( \vec{x} \! - \! \vec{p} ) \cdot \vec{n}}{4 \pi |\vec{x} \! - \! \vec{p}|^3} [{h}] \! + \! \frac{(\vec{x} \! - \! \vec{p}) \cdot \vec{n}}{4 \pi |\vec{x} \! - \! \vec{p}|^2} \left[ \frac{\partial h}{\partial t}\right] \right) v \, \hbox{d} S .
\label{s_integral}$$ This can be verified by extending the surface $\mathcal{S}^-$ with the tangent cone of $\mathcal{S}$ which intersects $\vec{p}$, and further with a $\vec{p}$-centric sphere with a radius larger than $T$. For the resulting surface $\mathcal{C}$, it holds that $\langle \vec{g}_{\mathtt{I}}, v ; \mathcal{C} \rangle = \langle \vec{g}_{\mathtt{I}}, v ; \mathcal{S}^- \rangle$. Namely, the integrand in (\[s\_integral\]) is zero on the tangent cone and within the distance $> T$ from $\vec{p}$ the incident wave is zero for $t \in [0, T]$.
### Scattered Far-Field
![ The incident field needs to be evaluated only on its first arrival at $\mathcal{S}$ (dark gray circle), that is, on the subset $\mathcal{S}^- = \{ \vec{x} \in \mathcal{S} \, | \, (\vec{x} - \vec{p}) \cdot \vec{n} < 0 \} $ (dark blue). On the remaining part $\mathcal{S}^+ = \mathcal{S} \setminus \mathcal{S}^-$, it is set to be zero. Surface $\mathcal{C}$ (dark and light blue) results from extending the surface $\mathcal{S}^-$ with the tangent cone (light blue) of $\mathcal{S}$ which intersects $\vec{p}$, and further with a $\vec{p}$-centric sphere with a radius larger than $T$. It holds that $\langle \vec{g}_{\mathtt{I}}, v ; \mathcal{C} \rangle = \langle \vec{g}_{\mathtt{I}}, v ; \mathcal{S}^- \rangle$. Namely, the integrand in (\[s\_integral\]) is zero on the tangent cone and within the distance $> T$ from $\vec{p}$ the incident wave is zero for $t \in [0, T]$. \[far\_field\_calculation\]](far_field_satellite.png){width="3.9cm"}
Following from the Green’s integral identities [@jackson2007], the empty space ($\varepsilon_r = 1, \sigma = 0$) wave radiating out of $\mathcal{S}$ satisfies the following Kirchhoff integral equation, which we utilize to extrapolate the scattered far-field (total field) at $\vec{p}$: [ $$\begin{aligned}
\label{far_field_measurement_1}
E(\vec{p}) & = & \int _{\mathcal{S}} [E_{\mathcal{S}}] \frac{\partial \mathcal{G}}{\partial \vec{n}} \, \hbox{d} S - \int_\mathtt{S} \mathcal{G} \left[ \frac{\partial {E_{\mathcal{S}}}}{\partial \vec{n}}\right] \hbox{d} S \\ \label{far_field_measurement_2} & = & - \frac{1}{4 \pi} \int_\mathtt{S} \frac{(\vec{x} - \vec{p}) \cdot \vec{n}}{ |\vec{x} - \vec{p}|^2} \left( \frac{[E_{\mathcal{S}}]}{ |\vec{x} - \vec{p}|} + \left[ \frac{\partial {E}_{\mathcal{S}}}{\partial t}\right] \right) \hbox{d} S \nonumber \\ & & - \frac{1}{4 \pi} \int_\mathtt{S} \frac{1}{ |\vec{x} - \vec{p}|} \left[ \frac{\partial {E_{\mathcal{S}}}}{\partial \vec{n}}\right] \hbox{d} S. \label{far_field_measurement_3}\end{aligned}$$]{}
Linearized Forward Model, Signal and Noise
------------------------------------------
We use the following linearized forward model [@pursiainen2016] in which the point of linearization is a constant background permittivity $\varepsilon_r^{(\hbox{\scriptsize bg})} $: $${\bf y} = {\bf L} {\bf x} + {\bf y}^{(\hbox{\scriptsize bg})} + {\bf n}.$$ The vectors ${\bf y}$ and ${\bf y}^{(\hbox{\scriptsize bg})}$ contain the measured and simulated data for $\varepsilon_r$ and $\varepsilon_r^{(\hbox{\scriptsize bg})}$, respectively, ${\bf x}$ is the coordinate vector for $\varepsilon_r$, ${\bf L}$ denotes the Jacobian matrix resulting from the linearization, and ${\bf n}$ contains both the measurement and forward modeling errors.
A single Blackman-Harris window $h(t)$ [@harris1978; @nuttall1981] is used as the signal pulse. That is, [ $$\begin{aligned}
h(t) = 0.359 & - & 0.488 \cos \left (\frac{2 \pi t}{T_0} \right) \nonumber \\ & + & 0.141 \cos \left ( \frac{4 \pi t}{T_0} \right) - 0.012 \cos \left ( \frac{6 \pi t}{T_0} \right) \end{aligned}$$]{} for $t \in [0, T_0]$ and $h(t) = 0$, otherwise. We choose $T_0 = 0.12$ as the final (unitless) pulse duration obtained after combining the frequency lines. For each measurement point, the received signal ${\bf y}$ is recorded for unitless time values 0.1–0.7, that is, 0.7–5.0 $\mu$s. The corresponding sampling rate is 15 MHz.
The present noise estimates are based on the targeted antenna specifications of the DISCUS mission concept. We assume that the measurement errors contained by ${\bf n}$ will be mainly caused by the galactic background noise and the Sun. At $20$ MHz, the galactic noise can be estimated to be around 5E-20 W/($\hbox{m}^2$Hz). The radiation from the Sun at a distance of $1\,\text{AU}$ and 20 MHz is about 2E-19 W/($\hbox{m}^2$Hz) and 2E-23 W/($\hbox{m}^2$Hz) for its active and inactive (quiet) phase of sunspot activity, i.e., for surface temperatures 1E+6 and 1E+10 K, respectively [@barron1985; @kraus1967]. During the active phase, the Sun emits radio-frequency waves in time scales varying from seconds to hours. As the reference level for modeling inaccuracies, we use the observation that, in CONSERT [@kofman2015], the peaks related to unpredictable echoes stayed mainly $- 20$ dB below the main signal peak.
The total error ${\bf n}$ is assumed to be an independent Gaussian white noise term with standard deviation between -25 and 0 dB with respect to the maximal entry of the difference $|{\bf y} - {\bf y}^{(\hbox{\scriptsize bg})}|$ between the measured and simulated signal. A Gaussian noise model is used, since ${\bf n}$ is expected to include both unknown forward and measurement errors, and as the sum of different independent and random error sources approaches a Gaussian distribution by the central limit theorem [@rice2006]. The measurement errors due to the spectral radiation flux density $F$ \[W/($\hbox{m}^2$Hz)\] of the galactic background and the Sun are approximated using this relative scale. We estimate the relative standard deviation of the measurement noise with the formula $$\sigma_m = \sqrt{ \frac{ F A_{\hbox{\scriptsize eff}} B_{\ell} }{ P_{TX} }},$$ i.e., it is the square root of the ratio between the absorbed noise and transmitted signal amplitude. With $\sigma_m$, the noise will have the desired average power given by $F$. As the antenna aperture, the standard approximation for the half-wavelength dipole antenna $A_{\hbox{\scriptsize eff}} = G \lambda^2 / (4 \pi)$ is used. The antenna of each spacecraft is assumed to be parallel to the orbiting plane normal and pointed towards the Sun during the measurements (Figure \[bistatic\_measurement\]).
Inversion Process
-----------------
In this study, the process of finding an estimate $\varepsilon^\ast_r$ for the relative permittivity $\varepsilon_r$, the unknown of the inverse problem, consists of the following three stages presented in [@pursiainen2016]:
1. [*Forward Simulation*]{}. For each transmission and/or measurement position, the signal can be propagated using the leap-frog iteration. A parallel computing cluster can be used, as the iteration is an independent process between different source points. In this study, altogether sixteen Nvidia Tesla P100 GPUs belonging to the [*Narvi*]{} cluster of Tampere University of Technology were used. Running the wave simulation for a single source position took about 71 and 184 minutes in the case of the background and exact data, respectively. The matrix–vector products including the inverse of the mass matrix [@pursiainen2016] were evaluated using the preconditioned conjugate gradient method with the lumped diagonal preconditioner, i.e., a matrix with row sums on its diagonal. The system matrices required a total of about 4 GB memory of which the mass matrix took 0.62 GB. The total GPU memory consumption during the forward simulation was around 10 GB, since the transposed matrices were stored separately in the memory to speed up the matrix-vector products of the iteration.
2. [*Linearization*]{}. The forward model can be linearized in a coarse nested mesh [@pursiainen2016] covering the asteroid which allows achieving an invertible system size. To speed up the parsing of the Jacobian matrix, the Tikhonov regularized deconvolution routine applied can be formulated as a pixelwise parallel algorithm. In the linearization stage, we applied a coarse mesh of 80000 tetrahedra. Running a parallelized 32-thread version of the parsing routine took 340 and 720 seconds for the monostatic and bistatic full-angle data in a Lenovo P910 workstation equipped with two Intel Xeon E5 2697A v4 2.6 GHz 16-core processors and 128 GB of RAM. The size of the resulting Jacobian matrix took 4.7 GB and 9.4 GB memory space, respectively.
3. [*Reconstruction procedure*]{}. The relative permittivity distribution can be reconstructed via the total variation (TV) regularized iteration presented in [@pursiainen2016]. The reconstruction was found via one iteration step using the regularization parameter values $\alpha = 0.1$ and $\beta = 0.001$ [@pursiainen2016]. The first one of these controls the overall regularization level and the second one the weighting ratio between the TV and norm-based regularization (the larger the value the more weight on the norm). To invert the regularized system matrix, we applied the conjugate gradient method with the stopping criterion 1E-5 for the relative residual norm. Computing a single reconstruction in the P910 workstation required around 500 and 400 conjugate gradient steps for the monostatic and bistatic full-angle system. The computation time was 100 and 170 seconds, respectively.
### Inversion Accuracy {#inversion_accuracy}
The accuracy of the estimate $\varepsilon^\ast_r$ obtained is measured using the relative overlap and value error (ROE and RVE), that is, the percentages [$$\begin{aligned}
\hbox{ROE} & = & 100 \left( 1 - \frac{\hbox{Volume}( \mathbf{T} \cap \mathbf{V}) }{ \hbox{Volume}(\mathbf{S})} \right) \\ \hbox{RVE} & = & 100 \left( 1 - \left| \frac{\int_{\mathbf{T}} (\varepsilon^\ast_r - \varepsilon_r^{(\hbox{\scriptsize bg})}) \, \hbox{d} V}{ \int_{\mathbf{T}} (\varepsilon_r -\varepsilon_r^{(\hbox{\scriptsize bg})}) \, \hbox{d} V } \right| \right). \end{aligned}$$]{} where $\varepsilon_r$ denotes the actual permittivity distribution and $\varepsilon_r^{(\hbox{\scriptsize bg})}$ is the background (initial guess). The set $\mathbf{S}$ is a region of interest (ROI) including both the surface layer and the voids and $\mathbf{V} = \mathbf{S} \cap \mathbf{R}$ denotes the overlap between the ROI and the set $\mathbf{R}$ in which a given reconstruction is smaller than a limit such that $\hbox{Volume}(\mathbf{R}) = \hbox{Volume}(\mathbf{S})$. The target set $\mathbf{T}$ refers to (1) the full ROI, i.e., $\mathbf{T} = \mathbf{S}$, (2) its surface part or (3) voids.
Discretization
--------------
The spatial domain was discretized as presented in [@pursiainen2016] and Appendix, using an unstructured tetrahedral mesh with accurate interior surfaces for all the modeled structures including the asteroid and the sphere $\mathcal{S}$. The fields $\vec{E}$ and ${g}^{(i)}$, $i = 1,2,3$ were discretized using piecewise linear and (element-wise) constant finite element basis functions, respectively. The temporal interval $[0, T]$ was divided into regular subintervals. The leap-frog based finite element time-domain algorithm (FETD) [@li2012; @schneider2016; @carley2008] presented in [@pursiainen2016] was utilized to obtain the fields and their linearizations. The scattered far-field for the source (\[point\_source\]) was simulated as follows:
1. Calculate the incident far-field on the surface $\mathcal{S}^-$
2. Solve the weak form (\[weak\_form\_7\])–(\[weak\_form\_8\]).
3. Extrapolate the scattered far-field via the restriction of $E_\mathtt{S}$ to $\mathcal{S}$.
The surface integral terms of (\[surface\_source\]) and (\[far\_field\_measurement\_1\])–(\[far\_field\_measurement\_3\]) were evaluated over the triangulated surface of $\mathcal{S}$ using the one point (barycenter) quadrature rule. In order to prevent numerical noise due to non-smoothness of the incident field at the boundary between $\mathcal{S}^-$ and $\mathcal{S}^+$, the following smoothed approximation for $\langle \vec{g}_{\mathtt I}, \! v ; \mathcal{S} \rangle$ was applied: $$\langle \vec{g}_{\mathtt I}, \! v ; \mathcal{S} \rangle \! \approx \! \! \int_{\mathcal{S^-}} \! \! \! \! ( \vec{g}_{\mathtt I} \cdot \vec{n} ) v \, \hbox{d} S + \int_{\mathcal{S^+}} \! \! \! \! ( \vec{g}_{\mathtt I} \cdot \vec{n} ) v \exp \! \left( \frac{- \gamma (\vec{x} \! - \! \vec{p}) \! \cdot \! \vec{n}}{1 \! - \! (\vec{x} \! - \! \vec{p}) \! \cdot \! \vec{n}} \right) \! \hbox{d} S$$ with the parameter $\gamma=10$ determining the decay rate for the second term. The shortest distance between $\vec{p}$ and $\mathcal{S}$ was utilized as a time shift to cancel out the signal travel-time between $\vec{p}$ and $\mathcal{S}$ in the simulated data sequence in a systematic way.
### Domain
![The present test domain $\Omega = \Omega_1 \cup \Omega_2$ (left) is an origin centric cube. The interior of the sphere $\mathcal{S}$ centered at origin formed the subdomain $\Omega_2$ (right) the exterior of which $\Omega_1$ contained a split-field perfectly matched layer [@pursiainen2016; @schneider2016] to simulate open field scattering. As the test target, we utilized the surface model (12260 triangles) of asteroid 1998 KY26 which can be associated through scaling with a 550 m diameter asteroid (Table \[scaling\]). The volumetric finite element discretization used in the reconstruction procedure was created based on the multi-layer surface mesh model illustrated in the pictures. \[domain\_fig\]](3d_domain.png){width="3.5cm"}
![The present test domain $\Omega = \Omega_1 \cup \Omega_2$ (left) is an origin centric cube. The interior of the sphere $\mathcal{S}$ centered at origin formed the subdomain $\Omega_2$ (right) the exterior of which $\Omega_1$ contained a split-field perfectly matched layer [@pursiainen2016; @schneider2016] to simulate open field scattering. As the test target, we utilized the surface model (12260 triangles) of asteroid 1998 KY26 which can be associated through scaling with a 550 m diameter asteroid (Table \[scaling\]). The volumetric finite element discretization used in the reconstruction procedure was created based on the multi-layer surface mesh model illustrated in the pictures. \[domain\_fig\]](3d_object_1.png){width="3.5cm"}
The present test domain $\Omega = \Omega_1 \cup \Omega_2$, is an origin centric cube (Figure \[domain\_fig\]). The interior of the sphere $\mathcal{S}$ centered at origin formed the subdomain $\Omega_2$ the exterior of which $\Omega_1$ contained a split-field perfectly matched layer [@pursiainen2016; @schneider2016] to simulate open field scattering. As the target, we utilized the surface model of asteroid 1998 KY26 which can be associated through scaling with a rubble pile asteroid (Table \[scaling\]). Two different conforming tetrahedral finite element meshes were used for simulating background and exact data, i.e., ${\bf y}^{(\hbox{\scriptsize bg})}$ and ${\bf y}$, respectively. Each one consisted of a total of 14.9 M elements of which 5.3 M were contained by the asteroid. All the surfaces present in the model were modeled accurately as triangular finite element mesh boundaries. The interior structure was homogeneous in the case of the background. The exact model included the following inhomogeneities to be reconstructed: (i) a surface layer together and (ii) three deep interior anomalies. Two different meshes were applied in order to avoid overly good data fit, i.e., the [*inverse crime*]{} [@colton1998]. The wave was propagated from zero time to the (unitless) value $T = 0.7$ (5.0 $\mu$s) using the FETD method. As the time increment for the background and exact asteroid mesh we employed the (unitless) values $\Delta t =$ 1E-4 and $\Delta t =$ 3.7E-5, respectively.
[**Overall**]{} ROE and RVE for (A).\
0.2cm
![The [**overall**]{} relative overlap and value error (ROE and RVE) in percents (%) in the full ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. Each bar is based on a sample of 30 reconstructions obtained with independent realizations of the total noise vector. The narrow part visualizes the interval between the minimum and maximum value in the sample. The thick part is called the interquartile range (IQR) or spread, that is, the interval between the 25 % and 75 % quantile. The white line shows the median. \[results\_total\]](roe_tot_90.png "fig:"){height="3.1cm"}\
Total Noise (dB)
![The [**overall**]{} relative overlap and value error (ROE and RVE) in percents (%) in the full ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. Each bar is based on a sample of 30 reconstructions obtained with independent realizations of the total noise vector. The narrow part visualizes the interval between the minimum and maximum value in the sample. The thick part is called the interquartile range (IQR) or spread, that is, the interval between the 25 % and 75 % quantile. The white line shows the median. \[results\_total\]](rve_tot_90.png "fig:"){height="3.1cm"}\
Total Noise (dB)
\
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[**Overall**]{} ROE and RVE for (B).\
0.2cm
![The [**overall**]{} relative overlap and value error (ROE and RVE) in percents (%) in the full ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. Each bar is based on a sample of 30 reconstructions obtained with independent realizations of the total noise vector. The narrow part visualizes the interval between the minimum and maximum value in the sample. The thick part is called the interquartile range (IQR) or spread, that is, the interval between the 25 % and 75 % quantile. The white line shows the median. \[results\_total\]](roe_tot_30.png "fig:"){height="3.1cm"}\
Total Noise (dB)
![The [**overall**]{} relative overlap and value error (ROE and RVE) in percents (%) in the full ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. Each bar is based on a sample of 30 reconstructions obtained with independent realizations of the total noise vector. The narrow part visualizes the interval between the minimum and maximum value in the sample. The thick part is called the interquartile range (IQR) or spread, that is, the interval between the 25 % and 75 % quantile. The white line shows the median. \[results\_total\]](rve_tot_30.png "fig:"){height="3.1cm"}\
Total Noise (dB)
\
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[**Overall**]{} ROE and RVE for (C).\
0.2cm
![The [**overall**]{} relative overlap and value error (ROE and RVE) in percents (%) in the full ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. Each bar is based on a sample of 30 reconstructions obtained with independent realizations of the total noise vector. The narrow part visualizes the interval between the minimum and maximum value in the sample. The thick part is called the interquartile range (IQR) or spread, that is, the interval between the 25 % and 75 % quantile. The white line shows the median. \[results\_total\]](roe_tot_10.png "fig:"){height="3.1cm"}\
Total Noise (dB)
![The [**overall**]{} relative overlap and value error (ROE and RVE) in percents (%) in the full ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. Each bar is based on a sample of 30 reconstructions obtained with independent realizations of the total noise vector. The narrow part visualizes the interval between the minimum and maximum value in the sample. The thick part is called the interquartile range (IQR) or spread, that is, the interval between the 25 % and 75 % quantile. The white line shows the median. \[results\_total\]](rve_tot_10.png "fig:"){height="3.1cm"}\
Total Noise (dB)
[**Surface**]{} ROE and RVE for (A).\
0.2cm
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**surface part**]{} of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_surf\]](roe_surf_90.png "fig:"){height="3.1cm"}\
Total Noise (dB)
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**surface part**]{} of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_surf\]](rve_surf_90.png "fig:"){height="3.1cm"}\
Total Noise (dB)
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[**Surface**]{} ROE and RVE for (B).\
0.2cm
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**surface part**]{} of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_surf\]](roe_surf_30.png "fig:"){height="3.1cm"}\
Total Noise (dB)
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**surface part**]{} of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_surf\]](rve_surf_30.png "fig:"){height="3.1cm"}\
Total Noise (dB)
\
0.3cm
[**Surface**]{} ROE and RVE for (C).\
0.2cm
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**surface part**]{} of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_surf\]](roe_surf_10.png "fig:"){height="3.1cm"}\
Total Noise (dB)
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**surface part**]{} of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_surf\]](rve_surf_10.png "fig:"){height="3.1cm"}\
Total Noise (dB)
[**Deep**]{} ROE and RVE for (A).\
0.2cm
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**deep interior**]{} part of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_deep\]](roe_deep_90.png "fig:"){height="3.1cm"}\
Total Noise (dB)
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**deep interior**]{} part of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_deep\]](rve_deep_90.png "fig:"){height="3.1cm"}\
Total Noise (dB)
\
0.3cm
[**Deep**]{} ROE and RVE for (B).\
0.2cm
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**deep interior**]{} part of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_deep\]](roe_deep_30.png "fig:"){height="3.1cm"}\
Total Noise (dB)
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**deep interior**]{} part of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_deep\]](rve_deep_30.png "fig:"){height="3.1cm"}\
Total Noise (dB)
\
0.3cm
[**Deep**]{} ROE and RVE for (C).\
0.2cm
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**deep interior**]{} part of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_deep\]](roe_deep_10.png "fig:"){height="3.1cm"}\
Total Noise (dB)
![The relative overlap and value error (ROE and RVE) in percents (%) for the [**deep interior**]{} part of ROI $\mathbf{S}$. The results for the monostatic and bistatic data correspond to the light grey and dark blue box plot bars, respectively. \[results\_deep\]](rve_deep_10.png "fig:"){height="3.1cm"}\
Total Noise (dB)
Clipping plane: $z = 0$\
{width="3.0cm"}\
Set $\mathbf{S}$: Exact
{width="3.0cm"}\
Set $\mathbf{V}$: [**(i)**]{}
{width="3.0cm"}\
Set $\mathbf{V}$: [**(ii)**]{}
{width="3.0cm"}\
Set $\mathbf{V}$: [**(iii)**]{}
\
Clipping plane: $y = 0$\
{width="3.0cm"}\
Set $\mathbf{S}$: Exact
{width="3.0cm"}\
Set $\mathbf{V}$: [**(i)**]{}
{width="3.0cm"}\
Set $\mathbf{V}$: [**(ii)**]{}
{width="3.0cm"}\
Set $\mathbf{V}$: [**(iii)**]{}
\
Clipping plane: $x = 0$\
{width="3.0cm"}\
Set $\mathbf{S}$: Exact
{width="3.0cm"}\
Set $\mathbf{V}$: [**(i)**]{}
{width="3.0cm"}\
Set $\mathbf{V}$: [**(ii)**]{}
{width="3.0cm"}\
Set $\mathbf{V}$: [**(iii)**]{}
Rec. Conf. Measure Total (%) Surface (%) Deep (%)
--------------- -------- --------- ----------- ------------- ----------
[**(i)**]{} (A) ROE 33 34 21
RVE 22 5 48
[**(ii)**]{} (B) ROE 39 41 20
RVE 34 27 57
[**(iii)**]{} (C) ROE 45 45 46
RVE 62 55 85
: ROE and RVE for reconstructions [**(i)**]{}, [**(ii)**]{} and [**(iii)**]{} obtained with bistatic data, total noise of -15 dB and configurations (A), (B) and (C), respectively (Figure \[result\_comparison\]). The values have been calculated separately for the total ROI $\mathbf{S}$ and for its intersection with the surface layer and voids. \[ROE\_RVE\_table\]
Results {#r}
=======
The mathematical far-field model introduced in this paper was found to perform adequately for the test asteroid model with different levels of the total noise. The results have been included in Figures \[results\_total\]–\[result\_comparison\] and Tables \[ROE\_RVE\_table\] and \[noise\_table\].
Figures \[results\_total\], \[results\_surf\] and \[results\_deep\] illustrate the ROE and RVE as a function of the total noise for the full ROI $\mathbf{S}$, the surface layer and voids, respectively. At each investigated noise level, a sample of 30 different reconstructions, obtained with independent realizations of the total noise, has been visualized as a box plot bar. The overall error level can be observed to be elevated for the -5 and 0 dB total noise levels. In comparison between the monostatic and bistatic CRT, the latter was found to be more robust with respect to noise and limited-angle data.
Figure \[result\_comparison\] visualizes reconstructions [**(i)**]{}, [**(ii)**]{} and [**(iii)**]{} obtained with the total noise level -15 dB, bistatic data and configurations (A), (B) and (C), respectively. The errors of the limited-angle reconstructions [**(ii)**]{} and [**(iii)**]{}, are particularly concentrated around the z-axis, that is, around the aperture in the limited-angle measurement positions. In [**(iii)**]{}, they are also notably more spread than in [**(ii)**]{}. Table \[ROE\_RVE\_table\] includes the ROE and RVE for [**(i)**]{}, [**(ii)**]{} and [**(iii)**]{} calculated separately for the full ROI $\mathbf{S}$ and for its intersection with the surface layer and the deep part (voids).
Table \[noise\_table\] includes the relative signal amplitudes noise levels for bistatic measurement. Both galactic noise and the upper estimate for the Solar radiation (active Sun) were observed to stay at a tolerable level with median peak levels of -14 dB and -8 dB, respectively. The lower Solar radiation estimate (quiet Sun) was below the investigated noise range with the median -48 dB.
Source Min Max Median
--------------------------- ------------ ------ ------ -------- --
Relative signal amplitude -136 -120 -130
Relative noise $\sigma_m$ Galactic -24 -7 -14
Active Sun -18 -1 -8
Quiet Sun -58 -41 -48
: The minimum, maximum and median values (dB) of the relative signal amplitude measurement noise standard deviation $\sigma_m$ with respect to the signal peak intensity estimated for bistatic data in the set of the simulated signals. \[noise\_table\]
Discussion {#d}
==========
This paper introduced and validated a mathematical far-field model applicable in the Computed Radar Tomography (CRT) imaging [@persico2014; @devaney2012] of small solar system bodies (SSSBs). In particular, the Deep Interior Scanning CUbeSat (DISCUS) mission concept [@deller2017; @bambach2017] was examined as a potential application of this method. The numerical results obtained with the test asteroid model and the relative total noise range of -25–0 dB suggest that a sparse set of full-wave measurements can be inverted with the planned DISCUS mission specifications. A sufficient reconstruction accuracy was obtained with sparse full- and limited-angle data. Furthermore, the results suggest that a bistatic (dual spacecraft) measurement technique [@willis2008] with a fixed $25$ degrees angle between the transmitter and receiver can improve the reliability of the inversion as compared to the monostatic (single spacecraft) approach.
In this study, the simulations were conducted for orbit radius of $5$ km. The final orbit is determined by the spacecraft’s $\Delta v$ budget and orbit stability and is currently under investigation. Generally, the closer the approach the better is the signal-to-noise ratio of the measurements. We currently expect that a radius of about 5 km will be achievable by a CubeSat. For comparison, Rosetta orbited at around 10 km distance with its closest approach at about 4.5 km [@kofman2015] but was limited mostly by the environment of the active comet 67P/Churyumov-Gerasimenko. The present results suggest that a deeper descent is not needed for detecting a surface dust layer and deep interior voids. It is noteworthy that performing close observations of SSSBs is a recent tendency in the planetary research. For example, during Rosetta’s final descent its Osiris wide-angle camera took its final image at 20 m altitude [@barbieri2017; @clery2016]. Another example is the ongoing Osiris-REx mission [@berry2013; @lauretta2012] in which the goal is to achieve a 730 m orbiting distance and also to bring a sample of the regolith back to Earth. Moreover, advanced active control strategies for hovering in the vicinity of an SSSB have been developed [@broschart2005; @lee2016].
The present results suggest that an appropriate reconstruction quality can be achieved, if the standard deviation of the total noise is below -10 dB. Of the investigated sources of measurement errors [@barron1985; @kraus1967], the Sun’s radiation during its active phase seems to exceed this limit for the 5 km orbiting distance which will need to be taken into account in the mission design. A natural way to reduce the noise would be to point the radar antennas towards the Sun. Namely, a dipole antenna is practically insensitive to radiation propagating along its axis. The galactic background noise, which cannot be reduced, seems to remain on an acceptable level with a median of around -14 dB. Other potential noise sources not investigated in this study include for example Jupiter’s radiation the magnitude of which depends largely on the target asteroid’s position in the solar system. Furthermore, it seems obvious that achieving a feasibly low measurement noise with the 10 W transmitting power applied in this study will, in practice, require applying the stepped-frequency technique [@iizuka1984; @gill2001; @paulose1994] which allows dividing the total radar bandwidth and, thereby, also the noise, into narrow frequency lines.
The errors related to the forward modeling can be significant and require further research. In the CONSERT measurements, the unpredicted noise peaks were observed to stay mainly -20 dB below the actual signal peak, suggesting that also those errors remain tolerable. Akin to [@pursiainen2014], the echo reflecting from the surface opposite to the spacecraft was found to be noisy. Achieving the best possible reconstruction quality necessitated excluding this echo from the measurement data. That is, the recorded time interval had to be limited to 5.0 $\mu$s.
Based on the comparison between the full- and limited-angle tomography results, it seems that the interior structure can be reconstructed, if targeted NEA has a typical spin without the need to alter the orbiting plane of the spacecraft which greatly simplifies operations. Namely, the spin latitude is close to -90 degrees for a large majority of the small NEAs [@la2004], suggesting that a better measurement coverage than in the 30 degrees limited-angle test can be achieved.
The bistatic CRT was found to an essential way to improve the inversion reliability as compared to the monostatic approach. The advantage of the bistatic measurement was observed to be particularly emphasized for a high total noise level and sparse limited-angle observation both of which are potential scenarios for a space mission. Hence, we propose that maximizing the reliability of the data requires measurements between two spacecraft in addition to recording the backscattering data at the point of transmission.
In this study, the smallest (low-noise) ROE and RVE were in some cases obtained with the monostatic approach. This was obviously due to the larger polarization shift in the signal captured by the second spacecraft, following from the non-direct reflection. Consequently, spacecraft positioning can have a major effect on the signal quality. The present choice for the distance between the transmitter and receiver is based on our preliminary numerical tests using the test asteroid model. Further optimization can be done in the future.
The current computational implementation is scalable and allows using any asteroid geometry. In a three-dimensional spatial scaling, the system matrix size is roughly proportional to $(s_1 / s_0)^3$, where $s_0$ and $s_1$ denote the scaling factors of the original and scaled domain. With the current measurement setting, the system size would be approximately 64 GB, for the diameter of 1100 m, i.e., two times the current one. The 0.62 GB mass matrix of the current system would be of the size 10 GB for the scaled one. A system size of 64 GB is feasible regarding an implementation in a computation cluster. If the cluster nodes are equipped with a GPU with more than 10 GB memory, then the mass matrix can be inverted rapidly in the GPU which, due to its parallel computing capabilities, may be assumed to provide a faster solution for sparse matrix–vector multiplication than the central processing unit.
Based on the diameter distribution of the NEAs [@trilling2017; @mainzer2011], the range of potential target diameters for DISCUS extends to at least 1000 m above which the distribution decays. For a target diameter significantly larger than 1000 m, it might be reasonable to limit the imaging to some [*a priori*]{} estimated depth estimated, e.g., based on maximal observed signal penetration. If necessary, it is also possible to compress the memory consumption by replacing some of the matrices with matrix-free functions returning a given matrix–vector product. Also methodological development can be considered. For example, the discontinuous Galerkin time-domain method can be compared to the current finite element time-domain implementation [@angulo2014; @lu2005] and the leap-frog iteration can be replaced with the well-known Runge-Kutta algorithm.
The reconstruction of interior structures containing less contrast than the structure analyzed in this study might also prove more difficult, and the existence of multiple scattering surfaces in the interior might increase the signal to noise required for a good reconstruction. It is therefore planned to assess the reconstruction of more complex interior structures, as for example an interior filled with spherical monolithic fragments following a power law size distribution as motivated in [@deller2015; @deller2016].
An ongoing future work is to verify the current results for a carefully simulated orbit. Additionally, the effect of the polarization and sparse limited-angle data on the reconstruction quality will be studied further. An asteroid flyby can be considered as an alternative way to do tomographic measurements. Based on this study, achieving a sufficient signal-to-noise ratio might be challenging for a flyby, since flybys are usually made in the range of 1000 km. For example, the closest point of Rosetta’s flyby at the asteroids Lutetia and Steins was 3170 km and 800 km [@accomazzo2012; @keller2010], respectively. On the contrary, CRT seems to require an extremely close 5 km rendezvous. Therefore, to achieve a reasonable signal to noise level during a flyby mission, concepts of very close flyby configurations with low relative velocity have to be developed.
Acknowledgements {#acknowledgements .unnumbered}
================
MT, MK and SP were supported by the AoF Centre of Excellence in Inverse Problems. MT and SP were funded by the Academy of Finland Key Project number 305055. Special thanks to Juha Herrala and Kari Suomela for support in computing resources.
\[appendix\_1\]
The present wave equation and its weak form can be derived from the (unitless) Maxwell’s equations [ $$\begin{aligned}
\nabla \cdot \varepsilon_r \vec{E} & = & 0 \label{mx1} \\
\nabla \cdot \vec{B} & = & 0 \label{mx2} \\
\nabla \times \vec{E} & = & - \frac{\partial \vec{B}}{\partial t } \label{mx3} \\
\nabla \times \vec{B} & = & \vec{J} + \varepsilon_r \frac{\partial \vec{E}}{\partial t} \label{mx4}\end{aligned}$$]{} in which $\vec{E}$ and $\vec{B}$ denote the electric and magnetic field, respectively, and $\vec{J} = \sigma \vec{E} + \vec{f}$ is the total current density with $\vec{f}$ denoting the current density of the antenna. The curl of the third equation (\[mx3\]) can be written as $$\nabla \times \nabla \times \vec{E} = \nabla (\nabla \cdot \vec{E}) - \nabla^2 \vec{E} = - \sigma \frac{\partial \vec{E}}{\partial t} - \varepsilon_r \frac{\partial^2 \vec{E}}{\partial t^2} - \frac{\partial \vec{f}}{\partial t}. \label{mx_spec}$$ Expressing the electric field and the position vector in the component-wise form, i.e., $\vec{E} = (E_1, E_2, E_3)$ and $\vec{x}= (x_1, x_2, x_3)$, respectively, one obtains the following general form of the present wave propagation model: $$\varepsilon_r \frac{\partial^2 {E}_i }{\partial t^2} + \sigma \frac{\partial E_i}{\partial t} - \sum_{j = 1}^3 \frac{\partial^2 E_i}{\partial x_j^2} + \frac{\partial}{\partial x_i} \sum_{j=1}^3 \frac{\partial E_j}{\partial x_j} = - \frac{\partial f_i}{\partial t}. \label{wave1}$$ The first-order formulation of this equation is given by [$$\begin{aligned}
\label{first-order_system} \label{first_order_form}
\frac{\partial {g}^{\,{(i)}}_{j}}{\partial t} - \frac{\partial E_i}{\partial x_j} & = & 0, \\
\varepsilon_r \frac{\partial {E}_i}{\partial t} + \sigma {{E}_i} - \sum_{j = 1}^3 \frac{\partial {g}^{\,(i)}_j}{\partial x_j} + \frac{\partial}{\partial x_i} \sum_{j=1}^3{g}^{\,(j)}_{j} & =& - f_i,
\end{aligned}$$ ]{} in which the first equation holds for $i = 1,2,3$ and the second one for $i,j = 1, 2,3$. The entries of the vectors $\vec{g}^{\, (1)}, \vec{g}^{\, (2)}$ and $\vec{g}^{\, (3)}$ are given by ${g}^{\, (i)}_{j} = \int_0^t \frac{\partial E_i}{\partial x_j}(\tau, \vec{x}) \, d \tau$ for $j = 1, 2, 3$. Multiplying the first and the second equation of (\[first-order\_system\]) by the test functions $\vec{w}^{(i)} \in [L_2(\Omega)]^3$ and $v_i \in H^1(\Omega)$, respectively, and integrating by parts yields the system [ $$\begin{aligned}
\label{wf1} 0 & = &
\frac{\partial }{\partial t} \sum_{j=1}^3 \int_{\Omega} \!\! {\vec g}^{\,(i)}_j {w}^{\,(i)}_j \, \hbox{d} V \! \! \nonumber \\ & & - \sum_{j=1}^3 \int_{\Omega} \!\! {w}^{\,(i)}_j \frac{\partial E_i}{\partial x_j} \, \hbox{d} V, \\ \label{wf2}
- \int_\Omega f_i v_i \, \hbox{d} V & = & \frac{\partial}{\partial t} \! \int_{\Omega} \! \! \varepsilon_r \, E_i \, v_i \, \hbox{d} V \!\! + \!\! \int_{\Omega} \! \! \sigma \, E_i \, v_i \, \hbox{d} V \!\! \nonumber\\ & & + \sum_{j=1}^3 \int_{\Omega} \!\! {g}^{\,(i)}_j \frac{ \partial{v_i}}{\partial x_j} \, \hbox{d} V \nonumber \\
& & - \!\! \sum_{j=1}^3 \int_{\Omega} \!\! {g}^{\,(j)}_{j} \frac{ \partial{v_i}}{\partial x_i} \, \hbox{d} V. \end{aligned}$$]{} This system can be discretized via the approach presented in the two-dimensional study [@pursiainen2016]. Using the notation of [@pursiainen2016], the last right-hand side term affecting the polarization of the wave, absent in the 2D case, is of the form $- {{\bf B}^{(i)}}^T \sum_{j = 1}^3 {\bf q}_k^{(k)}$, where ${\bf q}_j^{(i)}$ denotes the coordinate vector of ${g}_i^{(j)}$.
Received xxxx 20xx; revised xxxx 20xx.
[^1]: M. Takala (corresponding author) , M. Kaasalainen and S. Pursiainen are with the Laboratory of Mathematics, Tampere University of Technology, P.O. Box 692, 33101 Tampere, Finland.
[^2]: M. Takala is with Laboratory of Pervasive Computing, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland.
[^3]: P. Bambach, J. Deller and E. Vilenius are with Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany.
[^4]: M. Wittig is with MEW-Aerospace, Hameln, Germany.
[^5]: H. Lentz and H. M. Braun are with RST Radar Systemtechnik AG, Ebenaustrasse 8, 9413 Oberegg, Switzerland.
[^6]: Contact sampsa.pursiainen@tut.fi for further questions about this work.
|
---
abstract: 'We present axisymmetric, orbit superposition models for 12 galaxies using data taken with the [*Hubble Space Telescope (HST)*]{} and ground-based observatories. In each galaxy, we detect a central black hole (BH) and measure its mass to accuracies ranging from 10% to 70%. We demonstrate that in most cases the BH detection requires [*both*]{} the [*HST*]{} and ground-based data. Using the ground-based data alone does provide an unbiased measure of the BH mass (provided they are fit with fully general models), but at a greatly reduced significance. The most significant correlation with host galaxy properties is the relation between the BH mass and the velocity dispersion of the host galaxy; we find no other equally strong correlation, and no second parameter that improves the quality of the mass-dispersion relation. We are also able to measure the stellar orbital properties from these general models. The most massive galaxies are strongly biased to tangential orbits near the BH, consistent with binary BH models, while lower-mass galaxies have a range of anisotropies, consistent with an adiabatic growth of the BH.'
author:
- 'Karl Gebhardt, Douglas Richstone, Scott Tremaine, Tod R. Lauer, Ralf Bender, Gary Bower, Alan Dressler, S.M. Faber, Alexei V. Filippenko, Richard Green, Carl Grillmair, Luis C. Ho, John Kormendy, John Magorrian, and Jason Pinkney'
title: Axisymmetric Dynamical Models of the Central Regions of Galaxies
---
Introduction
============
Most nearby galaxies contain massive compact dark objects at their centers. The number density and masses of these objects are consistent with the hypothesis that they are dead quasars: massive black holes that grew mainly by gas accretion and were once visible as quasars or other active galactic nuclei from radiation emitted during the accretion process (see Kormendy & Richstone 1995 for a review).
We have obtained [*Hubble Space Telescope (HST)*]{} spectra of the centers of 12 nearby galaxies, using first the square aperture of the Faint Object Spectrograph (FOS) and later the long-slit on the Space Telescope Imaging Spectrograph (STIS). Additional ground-based spectra have been obtained at the MDM Observatory. Pinkney [[et al.]{}]{}(2002a) describe the data collected by our group for the 10 galaxies observed with STIS, and we present the data for the two galaxies observed with FOS in the Appendix of this paper. Section 2 discusses how we incorporate the data into the dynamical models.
An overall discussion of the dynamical modeling methods is given in Gebhardt [[et al.]{}]{} (2000a) and Richstone [[et al.]{}]{} (2002). The models are axisymmetric and based on superposition of individual stellar orbits. Section 3 provides the details of the models for these galaxies. Five other galaxies have stellar-dynamical data and models of comparable quality. Three of these are from the Leiden group: M32 (van der Marel [[et al.]{}]{} 1998; Verolme [[et al.]{}]{} 2002), NGC 4342 (Cretton & van den Bosch 1999), and IC 1459 (Cappellari [[et al.]{}]{} 2002). The remaining two are NGC 3379 (Gebhardt [[et al.]{}]{} 2000a) and NGC 1023 (Bower [[et al.]{}]{} 2001). Results from these five additional galaxies are included in the analysis in Section 4.
We use orbit-based models rather than parameterized models of the distribution function because parameterization can lead to biased black hole (BH) mass estimates. Parameterized models can even imply the presence of a BH when none exists. Orbit-based models do not suffer from this bias. However, we do make various assumptions whose consequences must be examined (Section 5). In particular, we model galaxies as axisymmetric. Triaxial, and worse yet, asymmetric galaxies, may be poorly represented by axisymmetric models. However, these effects are likely to be random and therefore it is reasonable to expect that the assumption of axisymmetry will not cause an overall bias in the BH mass.
In addition to measuring the BH mass ($M_{BH}$) and stellar mass-to-light ratio ([[*M/L*]{}]{}, assumed to be independent of position), our models constrain the orbital structure in the galaxy. It appears from this study and those of Verolme [[et al.]{}]{} (2002) and Cappellari [[et al.]{}]{}(2002) that the distribution function in axisymmetric galaxies depends on all three integrals of motion, not just the energy and angular momentum.
Preliminary BH masses for these galaxies have been reported by Gebhardt [[et al.]{}]{} (2000b); these masses are based on a coarser grid of models (explained in Section 4) and thus have larger uncertainties than those presented here. However, the best-fit values for the BH masses are nearly the same in the two studies.
Most distances in this paper have been measured with the surface-brightness fluctuation method (SBF, Tonry [[et al.]{}]{} 2000); for those galaxies without an SBF distance we assume the distance in an unperturbed Hubble flow and $H_0 = 80$ km s$^{-1}$ Mpc$^{-1}$.
Data
====
The data consist of images and spectra from ground-based and [*HST*]{} observations. The high spatial resolution of [*HST*]{} is essential to measure the mass of the central BH. The ground-based data are essential to constrain the stellar orbital distribution and mass-to-light ratio. Since we are using two-dimensional galaxy models, we must have data along various position angles to constrain adequately the orbital structure.
Imaging
-------
Most of the sample galaxies were imaged with WFPC2 during [*HST*]{} Cycles 4 and 5; the exception is NGC 4697, which was observed with WFPC1 (see Lauer [[et al.]{}]{} 1995). In general, each galaxy was observed in both the F555W ($V$) and F814W ($I$) filters. The typical total integration time in each filter was $\sim1200$ s, but subdivided into shorter exposures to allow for the identification of cosmic-ray events. The exposure levels at the centers of all galaxies exceeded $10^4$ photons pixel$^{-1}$, and were often nearly an order of magnitude higher. After the sub-exposures were compared to detect and eliminate cosmic-ray events, they were combined and then deconvolved using Lucy (1974) and Richardson (1972) deconvolution. The point-spread functions (PSFs) were provided by standard-star observations obtained during the routine photometric monitoring of WFPC2. Typically, 40 iterations of Lucy-Richardson deconvolution were used. Lauer [[et al.]{}]{} (1998) demonstrate that this procedure allows accurate recovery of the intrinsic galaxy brightness distribution for all but the central pixel. Brightness profiles were then measured from the deconvolved images using the high-resolution Fourier isophote-fitting program of Lauer (1985). The present work uses the $V$-band profiles, given their intrinsically higher spatial resolution. The [*HST*]{} imaging provides adequate coverage out to around 10; beyond that we rely on ground-based imaging to complete the radial coverage. Ground-based imaging comes primarily from Peletier [[et al.]{}]{} (1990).
Figure 1 presents the luminosity density profiles of the galaxies, which are input to the dynamical models. We determine the luminosity density distribution from the surface-brightness distribution by assuming that the luminosity density is axisymmetric, and constant on similar spheroids. (In one galaxy, NGC 4473, we have included a stellar disk in addition to the spheroidal luminosity distribution.) We use the non-parametric techniques described by Gebhardt [[et al.]{}]{}(1996), which involve smoothing the surface brightness and then inverting the Abel integral equation that relates surface brightness and luminosity density. We note that without this or some other restrictive assumption on the shape of the equidensity surfaces, the deprojection is not unique, except for edge-on galaxies (Gerhard & Binney 1996, Kochanek & Rybicki 1996). In particular, Magorrian & Ballantyne (2001) show that deprojection uncertainties, and in particular face-on disks, can significantly increase the uncertainties in the measured orbital distribution. We do not attempt a complete treatment of deprojection uncertainties in this paper, but do discuss possible consequences and biases in Section 4.9.
HST Kinematics
--------------
Pinkney [[et al.]{}]{} (2002a) present the spectra and kinematics from the [*HST*]{} STIS observations. Most of the galaxies in our sample have spectra taken with STIS, except for NGC 3377 and NGC 5845, which were observed with a single FOS aperture. The kinematic results for these two galaxies are presented in the Appendix. We use the line-of-sight velocity distributions (LOSVDs) in the modeling (i.e., we fit to the binned LOSVD, not its moments). For most of the galaxies, we use 13 equally-spaced velocity bins to represent the LOSVD. The width of the velocity bins is generally around 40% of the galaxy’s velocity dispersion. The uncertainty in the signal in each velocity bin is determined using Monte Carlo simulations. We reproduce a sample of the velocity profiles in Figure 2 (for NGC 4564).
The LOSVD can be biased by several systematic effects, including the choice of stellar template, continuum shape, spectral range used in the fit, and amount of smoothing. These are discussed by Pinkney [[et al.]{}]{} (2002a). In general, the most significant bias is probably template mismatch. However, most of our data are observed in the Ca II near-triplet region (8500 Å), and in this region the LOSVD is not very sensitive to template variations.
There is scattered light in STIS that is about 0.2% of the incoming light. The scattering occurs after the light has passed through the grating and so is not due to the PSF of [*HST*]{}. We measure this light using the spectral lamp images where we can use the high signal-to-noise ratio (S/N) to study the wings of the profile. There is a broad component that has a standard deviation equal to 25 pixels, presumably due to scattering in the STIS optics. We have run extensive tests to determine whether this scattered light affects our results. It is possible that a bright nucleus can scatter light into neighboring pixels which would not be reflected in the assumed PSF. We simulate this effect in both NGC 3377 (a power-law galaxy) and M87 (a core galaxy). We simulate the two-dimensional image by inputing kinematic profiles consistent with those measured in both galaxies, and then convolving those kinematics with both the narrow and broad components of the PSF. We then extract and fit the profiles ignoring the broad component, and compare with the input values. There was essentially no effect in M87, as expected since its kinematic profile does not vary strongly with radius. For NGC 3377, the change in the second moment was negligible; however, the velocity profile showed a 5% reduction in peak amplitude and the dispersion profile showed a 5% increase near the center. Since the second moment was hardly changed, this broad component has no effect on the measured BH mass and is not included in subsequent analysis.
The STIS spectral resolution with the G750M grating is around 55 [$\rm {km}~\rm s^{-1}$]{}(FWHM), with 37 [$\rm {km}~\rm s^{-1}$]{} per binned pixel (we bin 2x1 pixels on the chip for most of our data). Since we always use a template star convolved with the LOSVD to match the galaxy spectrum, we do not have to worry as much about the detailed shape of the spectral PSF (as opposed to understanding the spatial PSF) because the velocity dispersions of our galaxies are much larger than the spectral FWHM. The main concern is whether we are illuminating the slit with the templates in the same way that we illuminate with the galaxy. For galaxies with point-like nuclei, this is not a concern, but for those with shallow light profiles we must consider the effect. The concern is whether the velocity variation across the slit adds to the dispersion measured in the galaxy, which would not be true for a point source. We can calculate the effect using the results of Bower [[et al.]{}]{} (2001). For the 0.1 slit, the velocity variation from slit edge to edge is around 20 [$\rm {km}~\rm s^{-1}$]{}, and for the 0.2 slit, it is 40 [$\rm {km}~\rm s^{-1}$]{}. Given the FWHM of the spectral lines, 55 [$\rm {km}~\rm s^{-1}$]{}, the velocity variation provides a 7–25% increase in the required instrumental spectral FWHM. However, for the galaxies where this effect is the largest (i.e., the core galaxies), the galaxy dispersion is the greatest and therefore the broadening in the galaxy is insignificantly affected by the exact instrumental profile. For example, in the case of NGC 3608, the central dispersion is 300 [$\rm {km}~\rm s^{-1}$]{}; with a change in the instrumental dispersion from 20 [$\rm {km}~\rm s^{-1}$]{} to 25 [$\rm {km}~\rm s^{-1}$]{} (25% higher), the inferred galaxy dispersion differs by only 0.1%. Given the insignificant difference, we apply no correction for illumination effects. Bower [[et al.]{}]{} (2001) find a similar result for NGC 1023.
Ground-Based Kinematics
-----------------------
Nearly all of the ground-based data come from the MDM Observatory (Pinkney [[et al.]{}]{} 2002a). Briefly, most of the spectra were taken around the Ca II triplet (near 8500 Å) and the rest were taken near the Mg $b$ region (5100 Å). The instrumental resolution varied slightly from run to run, but was generally around 40 [$\rm {km}~\rm s^{-1}$]{}, which is more than adequate given the dispersions of the galaxies studied. The spatial resolution varied from 0.5 to 1.5. We included the appropriate spatial PSF for each of the ground-based spectra in the modeling.
In an axisymmetric system, the velocity profile at a radius on one side of the galaxy will be identical to a profile that is flipped about zero velocity on the other side of the galaxy at the same radius. There are three options that we can use to include this symmetry in the models. First, we can fit the same, but appropriately flipped, observed velocity profile on the two spectra from opposite sides of the galaxy during the extraction. In this way, we only include one profile at a given radius. Second, we can independently fit velocity profiles from opposite sides of the galaxy, and then average these two velocity profiles (after flipping one of them) to provide one profile for that radius. Third, we can include the two independently fit velocity profiles directly into the models. Each of these has their own advantages and disadvantages. For example, if there is a bad spot on the detector or a star on one side of the galaxy, then the most reliable measure would be to use the third option (since one can then exclude the affected region).
We have tried all three methods and find little differences between the results. We choose to use the first option since, in that case, the S/N used for the extraction of the velocity profile is increased by $\sqrt2~$ compared to the other cases, and this serves to alleviate potential biases. This increase arises because we use two spectra to measure one velocity profile, as opposed to measuring two independent velocity profiles. The uncertainty (and hence, the S/N) in the resultant velocity profile is the same regardless of the method used to estimate it, but our reason for using the first option is motivated by alleviating potential biases in the extraction of the velocity profile. For low S/N data, there are often biases in the velocity profile (mainly due to the need to use more smoothing as the signal is lowered), and we decrease these biases by forcing axisymmetry during the spectral extraction. The alternative of using individual profiles from both sides of the galaxy is not optimal.
Dynamical Models
================
Richstone [[et al.]{}]{} (2002) provide a complete account of the construction of the dynamical models, including analytic tests. Here, we provide a basic summary and include the details that are specific to these galaxy models. Other groups discuss the use of and tests for similar orbit-based models (van der Marel [[et al.]{}]{} 1998, Cretton & van den Bosch 1999, Cretton [[et al.]{}]{} 1999, 2000, Cappellari [[et al.]{}]{} 2002, Verolme & de Zeeuw 2002, and Verolme [[et al.]{}]{} 2002). These studies used models similar to each other; the models presented here and in Gebhardt [[et al.]{}]{} (2000a) differ from those above in small but important ways. As discussed in Richstone [[et al.]{}]{} our models use a maximum likelihood approach to find the orbital weights as opposed to using a regularization method, and ours also use the full LOSVD as opposed to using parameterized moments. There are positives and negatives associated with the different approaches and a full comparison can only be studied when the different models are applied to identical datasets.
The dynamical models are constructed as follows: we first determine the luminosity density from the surface brightness profile. Although we have constructed models with a variety of inclinations, we generally assume that the galaxy is edge-on, for reasons given in Section 4. In this case, the deprojection is unique. To determine the potential we assume a stellar mass-to-light ratio and a BH mass. In this potential we run a representative set of orbits (typically 7000) that cover phase space adequately. We then find the non-negative set of weights for those orbits that provides the best match to the available data (in the sense of minimum $\chi^2$). In order to have a smooth phase space distribution, we use a maximum entropy method as described below. We repeat this analysis for different BH masses and different mass-to-light ratios to find the overall best fit.
We measure the velocity moments of our models on a two-dimensional grid in radius and angle relative to the symmetry axis of the galaxy. We generally use 20 radial and 5 angular grid elements. The parameters of this grid (spacing and extent) are designed to maximize the S/N in both the kinematics and the photometry. The angular bins have centers at latitudes 5.8, 17.6, 30.2, 45.0, and 71.6, where the angle is defined from the major to the minor axis; we use the same binning scheme whether we are in projected or internal space. We have run tests in which we both double and halve the number of bins and we find insignificant differences. Since STIS provides kinematic information along a slit, we need to specify how to extract the data along that slit to optimize the S/N to measure the BH mass. Pinkney [[et al.]{}]{} (2002a) describe the 20 radial extraction windows that we use. We define our radial binning scheme in the models with the same configuration as that used in the data extraction.
We specify the galaxy potential and the forces on a grid that is five times finer than the grids used in the data comparisons, in order to assure accurate orbit integration. If we have $N$ radial bins, labeled $i=1,\ldots,N$, our goal is to have at least one orbit with apocenter and pericenter in every possible pair of bins $(i,j)$ in order to cover phase space well; this requires $N(N-1)/2$ orbits, times two to include stars with the opposite sign of rotation. This leads to 380 orbits; however, we must also cover the angular dependence and to do this we include 20 additional angular bins. Thus, the total number of orbits is around 7000. We track the velocity information by storing the LOSVD for each orbit in each grid element. For each galaxy we use 13 velocity bins, spanning the maximum and minimum velocities generated for the whole orbit distribution. It is important to include all velocity information, particularly in the LOSVD wings where the effects from the BH are the strongest. Our final models consist of 7000 (orbits) x 20 (radial) x 5 (angular) x 13 (velocity) elements. For each galaxy, we generally try about 10 different BH masses, 10 or more values of [[*M/L*]{}]{}, and sometimes a few different inclinations. We have also run models where we have both doubled and halved the number of orbits. In either case, we find no difference in the best fit to the data.
In addition to the projected quantities, we track the internal properties including the velocity moments and luminosity density. For the dynamics, we only track the zeroth, first, and second moments of the velocity profile. The internal moments are presented in Section 4.7 below.
It is important to include the effects of the PSF of [*HST*]{} in the dynamical models. We use the same PSF as measured by Bower [[et al.]{}]{}(2001), which has FWHM = 0.08 along the slit at 8500 Å. At 8500 Å, the first diffraction peak is visible and is included in the PSF model. This profile comes from a highly sampled PSF using a cut in the spatial direction for a star at various columns on the chip. Since there is a 6-row shift of the star across the STIS chip, the PSF is sampled differently in each column, thereby producing a well-sampled profile. Unfortunately, this procedure only produces a PSF in one dimension (along the slit), and we have no measurement of the PSF in the spectral direction. As discussed by Bower [[et al.]{}]{} (2001), the PSF is expected to be circularly symmetric, so we assume that the PSF across the slit is the same as the PSF measured along the slit. We run all orbit libraries with no PSF included and then convolve with the appropriate PSF before we fit to the kinematic data. In this way, we can include a different PSF for each kinematic observation, if necessary.
We have ground-based kinematic data along 2–4 position angles for each galaxy, covering over half of the radial bins. Thus, we typically have 20 positions on the sky, each with 13 LOSVD bins, for a total of 260 data bins. However, as explained by Gebhardt [[et al.]{}]{} (2000a) and discussed further in §4.1, the number of degrees of freedom is difficult to estimate. The main problem is that the smoothing in the LOSVD estimation introduces covariance between velocity bins. For a typical galaxy, every two velocity bins are correlated. This factor of two is determined through simulations where we vary the smoothing parameter in the velocity profile estimate and measure the effect on $\chi^2$ (see Gebhardt [[et al.]{}]{} 2000a). Thus, the number of degrees of freedom is reduced by a factor of roughly two compared to the total observed parameters.
The orbit weights are chosen so that the luminosity density in every spatial bin matches the observations to better than 1%. Typically, the match is better than 0.1%. We regard matching the luminosity density in each bin as a set of constraints, rather than a set of data points. Thus, the photometric data do not contribute to the total number of degrees of freedom. We make this choice for two reasons: first, the uncertainties in the photometry are much smaller than those in the kinematics; second, including photometric uncertainties would require compiling a far larger set of orbit libraries (one for each tested photometric profile).
To ensure that the phase-space distribution function is smooth, we maximize the entropy as in Richstone & Tremaine (1984). We do this by defining a function $f\equiv \chi^2-\alpha S$, where $\chi^2$ is the sum of squared residuals to the data, $S$ is the entropy, and $\alpha$ is a parameter describing the relative weights of entropy and residuals in the fit. Our goal is to minimize $f$. We start with a large value of $\alpha$ and then gradually reduce it until further improvement in $\chi^2$ is no longer possible. At first, the entropy determines the orbital weights but, at the end of the minimization, the entropy has no influence on the quality of the fit. The entropy constraint does affect those regions where we do not have kinematic data, but we never use results from those regions. Solving for the 7000 orbital weights with 200–500 observations is the most computationally expensive part of the analysis. We have tried a variety of initial conditions for the orbital weights and entropy forms, and find that neither the minimum value of $\chi^2$, nor the BH mass and stellar mass-to-light ratio, nor the orbital structure is sensitive to these choices.
We need to determine the uncertainties in the BH mass and the stellar [[*M/L*]{}]{}. These are correlated, of course, and we generally use two-dimensional $\chi^2$ distributions to determine the uncertainties. The uncertainties in the parameters are determined from the change in $\chi^2$ as we vary one of the variables; in this case, the 68% confidence band is reached when $\chi^2$ increases above its minimum value by 1. This parameter estimation is different from hypothesis testing: a tested hypothesis is consistent with the data if $\chi^2$ per degree of freedom is approximately unity, while the allowed range of a parameter is determined by the change in $\chi^2$ from its minimum value. For example, since the BH has no effect on the kinematics at large radii, we could always ensure that a galaxy is consistent with the hypothesis that there is no BH by adding more and more kinematic data at large radii.
Thus, we advocate that one must use $\Delta\chi^2$ in order to determine the uncertainties in the parameters. This conclusion was also discussed in both van der Marel [[et al.]{}]{} (1998) and Cretton [[et al.]{}]{}(2000). The difficulty for the parameter estimation is that we need to measure the uncertainties in the kinematics accurately. The uncertainties on the kinematics are difficult to quantify; problems due to template mismatch and continuum estimation, for example, can have a significant effect on the results. We have tried to take this into account during the Monte Carlo simulations that we use to generate the errors. A more natural approach would be to use a Bayesian analysis, but given the large number of unknown variables (the 7000 orbital weights), this is impractical.
Figure 2 shows the data/model comparison for three LOSVDs in NGC 4564. For this galaxy, we actually have 33 velocity profiles but only show three here. Thus, there is significantly more data that has gone into the models. For NGC 4564, the signature for the BH comes from the central few bins in each of the three position angles. However, it is only by examining the full dataset can one understand the global fit for any particular model. The change in $\chi^2$ is given in the bottom plot. The difference between the best-fit BH and zero BH model in just these three bins is equal to 8. Using the full dataset the difference is 53, implying an extremely high significance against the zero BH model.
Results
=======
The three main properties which we obtain from the models are the BH mass, mass-to-light ratio, and the orbital structure. The BH mass and mass-to-light ratio are fitted by choosing a grid of parameters for them and then examining their $\chi^2$ distribution. The orbital structure, however, results from finding the orbital weights for each specified potential (i.e., BH mass and mass-to-light ratio) that provides the minimum $\chi^2$. Each BH mass/mass-to-light ratio pair produces a best fit orbital structure, but the overall best model is that which has the one global minimum. In this section, we discuss results for each parameter, and consider possible biases and additional uncertainties.
BH mass and mass-to-light ratio
-------------------------------
Figure 3 presents $\chi^2$ as a function of BH mass and [[*M/L*]{}]{}. The contours are drawn using a two-dimensional smoothing spline (Wahba 1980). As in Wahba (1980), Generalized Cross-Validation determines the smoothing value; however, the modeled values are relatively smooth and little smoothing is necessary. We plot only those points near the $\chi^2$ minimum; we have tried many more models that lie outside the limits shown in the plot but only highlight the center to show the contour shape. Models that lie outside these limits are excluded at much greater than 99% confidence. Each approximately vertical sequence represents models with the same ratio of BH mass to galaxy mass (or [[*M/L*]{}]{}), all of which can use the same orbit library except for a trivial rescaling of the velocities.
Table 1 presents the properties of the galaxies in this sample. The columns are galaxy name (col. 1), galaxy type (col. 2), absolute $B$-band bulge magnitude (col. 3), BH mass (col. 4) and uncertainty $\sigma_e$ (col. 5), which is defined in Section 4.6, distance in Mpc (col. 6), mass-to-light ratio and the band (col. 7), central slope of luminosity density (col. 8), shape of the velocity dispersion tensor in the central model bin (col. 9), shape of the velocity dispersion tensor at a quarter of the bulge half-light radius (col. 10), and half-light radius of the bulge (col. 11). Bulge magnitudes come from Kormendy & Gebhardt (2001). The half-light radii come from Faber [[et al.]{}]{} (1989) and Baggett [[et al.]{}]{} (2000).
In all but two of the galaxies, there is little covariance between BH mass and [[*M/L*]{}]{}. The reason is that we are probing those regions where the BH mass dominates the potential with multiple resolution elements. Since the stars contribute a small fraction of the total mass in this region, varying their mass-to-light ratios has little effect on the enclosed mass. The two cases in which there is some covariance, NGC 3377 and NGC 5845, have high-resolution kinematic data from only a single FOS aperture. Thus, they have poorer spatial sampling inside of the region where the BH dominates the potential.
Figure 4 shows $\chi^2$ as a function of BH mass for NGC 4564. This plot has been marginalized over [[*M/L*]{}]{}. For this galaxy, we have spectra at 33 spatial positions. With 13 velocity bins each, we then have 429 kinematic measurements. The velocity profiles have a smoothing width of about two bins, and thus the number of degrees of freedom is about 210. For NGC 4564, we ran a large number of models in order to inspect the shape of the $\chi^2$ distribution and its asymptotic shape at small mass. Near the minimum of the $\chi^2$, there is noise at the level of $\Delta\chi^2 \approx 0.5$.
Figure 5 shows the $\chi^2$ distributions for the whole sample of 12 galaxies. These plots have been marginalized over [[*M/L*]{}]{} so $\chi^2$ is a function of only one variable, $M_{BH}$. In these plots, $\Delta\chi^2=1$ corresponds to $1\sigma$ uncertainty or 68%. Thus the detection of a BH — or, strictly speaking, of a massive dark object — is very significant in most of these galaxies. The least significant detection is NGC 2778 where the difference in $\chi^2$ is less than ten between the best-fit BH mass and zero BH mass.
We note the difference in the convention used here for the contour levels compared to other orbit-based studies. We report uncertainties that are based on one degree of freedom (i.e., marginalizing over the other parameters) and are at the 68% level ($1\sigma$). Other studies have used different values. Van der Marel [[et al.]{}]{} (1998) report $3\sigma$ uncertainties based on two degrees of freedom (BH mass and mass-to-light ratio) corresponding to $\Delta\chi^2=11.8$. Cretton & van den Bosch (1999) report $1\sigma$ uncertainties with two degrees of freedom corresponding to $\Delta\chi^2=2.3$. Cappellari [[et al.]{}]{}(2002) and Verolme [[et al.]{}]{} (2002) report $3\sigma$ uncertainties with three degrees of freedom (including inclination) corresponding to $\Delta\chi^2=14.2$. Our intention is to use the BH masses reported here in galaxy parameter studies, and so we desire a BH mass uncertainty that has been marginalized over all other parameters. Furthermore, convention suggests that $1\sigma$ uncertainties are the most useful for parameter studies. Thus, we use $1\sigma$, 1 degree of freedom uncertainties. We also convert the uncertainties from the orbit-based studies above to our convention of $\Delta\chi^2=1.0$. The three galaxies for which we converted the uncertainties are M32 (Verolme [[et al.]{}]{} 2002), NGC 4342 (Cretton & van den Bosch 1999), and IC1459 (Cappellari [[et al.]{}]{} 2002). These are reported in Table 3. In order to do this properly requires sampling the dynamical models finely enough to see $\Delta\chi^2=1.0$ variations. Since the above studies were not concerned with the uncertainties at this level, we must use an approximation. We can use the $\chi^2$ contours from our sample to approximate the change in BH mass uncertainty relative to change in $\chi^2$. This is not ideal but does serve as a first approximation. Thus, $\Delta\chi^2$ changing from 14.3 to 1.0 implies an average change in the BH mass uncertainty of a factor of 4 (which we use for M32 and IC1459). Going from $\Delta\chi^2$ of 2.3 to 1.0 implies an uncertainty change of a factor of 1.6 (which we use for NGC 4342).
Most of the galaxies in our sample have significant flattenings. Since it appears that the distribution of intrinsic flattenings peaks at axis ratio 0.7 (Alam & Ryden 2002), most of our galaxies should not be far from edge-on. Except for NGC 4473, all of the models presented in Figure 5 assume edge-on configuration. We have run a few inclined models for NGC 3608 and NGC 5845. In both cases, the the BH mass is within the uncertainty given for the edge-on model. For the more face-on configurations, the BH mass increased by 30% for NGC 3608, and decreased by 20% for NGC 5845. Gebhardt [[et al.]{}]{} (2000a) found that some inclined models for NGC 3379 had BH masses larger by a factor of two compared to the edge-on case (but still within the uncertainty). However, the data used for these studies have limited spatial kinematic coverage. The two-dimensional kinematic dataset used for M32 (Verolme [[et al.]{}]{} 2002) provides the optimal way to study inclination effects. They find that the more face-on case gives a 30% decrease in the BH mass measured from the edge-on model. Since we do not have adequate angular kinematic data to constrain the inclination, we rely on the above studies and the few cases that we have run to determine the inclination effect. On average, it appears that inclination will cause a 30% random change in the BH mass. Our BH mass uncertainties range from 10% to 70%, with the most flattened, nearly edge-on, galaxies tending to have the smallest uncertainties. For those galaxies where inclination might be a concern, their uncertainties are larger than 30%. Thus we do not include any additional uncertainty that might be caused from using the incorrect inclination. The uncertainties given in Table 1 include those as measured from the edge-on models alone (or from the one inclined case for NGC 4473). Inclined models do, however, affect the [[*M/L*]{}]{}; as the galaxy approaches a more face-on configuration, it becomes more intrinsically flat in order to give the same projected flattening. Since it must maintain a similar projected dispersion, then the smaller column depth for the more flattened galaxy requires a higher [[*M/L*]{}]{}, which is what is seen in the models.
The uncertainty in the BH mass determination clearly depends on both the spatial resolution and the S/N of the data. The spatial resolution can be parametrized relative to the radius of the sphere of influence, $GM_{BH}/\sigma^2$. For our galaxies, these radii range from 0.02 in NGC 2778 to 0.75 in NGC 4649. The size of the central bin used in the modelling is 0.05. Thus, for NGC 2778, the sphere of influence is more than a factor of two below our resolution limit. Because of this small radius, we have tried a variety of different datasets applied to the NGC 2778 models and find similar results. The zero BH mass model for NGC 2778 is ruled out at only the 95% confidence limit — our least confident detection. For all other galaxies, the sphere of influence is larger or equal to our resolution limit.
The uncertainties in the BH mass come from the shape of the one dimensional $\chi^2$ contours. It is important to check whether this estimate of the uncertainties properly reflects the true uncertainties. We can check this to a limited extent through Monte Carlo simulations of the kinematics. We note that this study will only determine the uncertainties within our assumptions; we discuss effects from relaxing our assumptions in Section 4.9. We use the same Monte Carlo realizations that were used for the spectra (as described in Pinkney [[et al.]{}]{}). For each realization of the set of LOSVDs we find the BH mass that provides the minimum $\chi^2$. With 100 realizations, we then determine the 68% confidence limits from the Monte Carlo and compare that to the same limits as determined from the shape of the $\chi^2$ contour. The uncertainties as measured from both techniques are in excellent agreement. Assuming that the Monte Carlo simulations should provide the most accurate uncertainties, we find no reason to question the uncertainties as measured from the $\chi^2$ shape, due to their concordance. We have run this experiment only on NGC 3608 but believe these results to be general.
Gebhardt [[et al.]{}]{} (2000b) presented preliminary BH masses based on this analysis. Most of the preliminary masses are the same as those presented here, except for changes due to the change in assumed distance ($M_{BH} \propto \hbox{distance}$). The few other differences arise because we now use a higher resolution grid of models, so the minimum of $\chi^2$ is located more accurately. The changes in both the best-fit mass and the uncertainties are generally less than $0.5\sigma$. The most extreme change is in NGC 5845 since the lower limit for that galaxy was defined using a very poorly sampled grid.
Individual Galaxies
-------------------
Pinkney [[et al.]{}]{} (2002a) provide observational notes for the ten galaxies in our sample that were observed with STIS. Here we report any additional details of the dynamical models for our sample of twelve galaxies. In addition, we include notes for five other galaxies taken from the literature that have similar models and are used in the analysis in Sections 4.5 and 4.7.
[*NGC 821:*]{} There are 312 velocity constraints, coming from 24 spatial positions each with 13 LOSVD bins. Given that 2–3 adjacent bins are correlated from the velocity profile smoothing, the number of degrees of freedom is about 100, so the minimum $\chi^2$ of 128 (Table 2) indicates a good fit.
[*NGC 2778:*]{} There are additional ground-based data from Fisher [[et al.]{}]{} (1995) along the major axis. NGC 2778 is important because its BH mass is low relative to the $M_{BH}/\sigma$ relation. We have modelled NGC 2778 using three ground-based datasets: the STIS data plus our ground-based data, the STIS data plus the Fisher [[et al.]{}]{} data, and the STIS data plus both ground-based datasets. All three best-fit BH masses are consistent at the 68% confidence level. The data from Fisher [[et al.]{}]{} have higher S/N and so the results we present are based on the STIS data plus the Fisher [[et al.]{}]{} data.
[*NGC 3377:*]{} The black hole mass in NGC 3377 was first measured in Kormendy [[et al.]{}]{} (1998) using only ground-based data; the mass that we find here is within their uncertainties. We use ground-based data from Kormendy [[et al.]{}]{} (1998) along the major and minor axes. Kormendy [[et al.]{}]{}present only the first two moments of the velocity distribution. Since our models require data on the full LOSVD, we convert these moments into a Gaussian velocity profile. The uncertainties are generated through a Monte Carlo procedure; we generate 1000 velocity profiles consistent with the means and uncertainties of the moments. The uncertainty at each velocity bin is given from the 68% range about the mean in the simulations. From [*HST*]{}, we have two FOS observations, which we present in the Appendix. Since we only use the first two moments to generate the LOSVD and since galaxies can have significantly non-Gaussion LOSVDs, we have checked whether including additional moments affects the results. We have included a variety of H3 and H4 components for the ground-based data, using values that are consistent with those from other galaxies. We find little difference in the BH mass as reported in Table 1. The main reason for this is that the HST data shows a dramatic increase in the central dispersion and in the rotation relative to the ground-based data. Thus, the BH mass is determined mainly from the radial change in the kinematics and not from the higher order moments of the LOSVD.
[*NGC 3384:*]{} NGC 3384 is one of the two galaxies that show a smaller velocity dispersion in the [*HST*]{} data than in the ground-based data. The reason for this drop is that the STIS kinematics are coming mainly from a cold edge-on disk. The dynamical models are free to include as many circular, or nearly circular, orbits as necessary, and so they easily match the kinematic profile. NGC 3384 is one of the more significant BH detections.
[*NGC 4473:*]{} NGC 4473 shows a flattening in the central isophotes and also a significant decrease in the central dispersion. Both of these indicate the presence of a central disk (see Pinkney [[et al.]{}]{}2002a). Central stellar disks are seen in many elliptical galaxies (Jaffe [[et al.]{}]{} 1994, Lauer [[et al.]{}]{} 2002). In order to provide the best representation for the dynamical models, we include a central disk. The parameters of the disk are measured from the [*HST*]{} images. We use a spheroidal representation for the bulge component and model the residual with a zero-thickness disk with an exponent of 0.5. The parameters for the exponential disk are $4.9\times10^7{L_\odot}/{\rm arcsec}^2$ for the central surface bightness and 1.0 for the scale length. The best-fit inclination is 72 which we also assume for the galaxy. The mass of the disk inside of 1 is 20% higher than the bulge mass in that region. Thus, it does have a noticeable effect on the kinematics. The models have no problem matching the high rotation and low dispersion of the disk.
[*NGC 4649:*]{} NGC 4649 is the largest galaxy in our sample and has the lowest surface brightness. We spent 22 [*HST*]{} orbits exposing on this galaxy. The central dispersion, 550 [$\rm {km}~\rm s^{-1}$]{}, is the highest ever observed. Despite the large dispersion and low surface brightness, both of which strongly affect the S/N, the uncertainty in the BH mass is only 30%.
[*NGC 4697:*]{} There is a gas disk in the center of this galaxy, and the gas kinematics for this galaxy are measured by Pinkney [[et al.]{}]{}(2002b). NGC 4697 has the most significant BH detection. The difference in $\chi^2$ between the zero BH mass model and the best fit model is 155.
[*NGC 7457:*]{} There is a central point nucleus in NGC 7457. When measuring the surface brightness profile, we first subtract a point source from the center. Thus, for the stellar luminosity density, we assume that the point source is coming from nonthermal emission and does not contribute to the stellar density. If the point source is a nuclear star cluster instead of weak nuclear activity, then we will bias our BH mass since we would have then ignored some of the stellar mass. The total light in the point source is substantial, $V \approx
18.1$ mag. This amount of light translates into $1 \times
10^7~{L_\odot}$. Given the BH mass that we measure of $3.5 \times
10^6~{M_\odot}$, assuming that the point source is stellar is inconsistent with the STIS kinematics. The other effect that it may have is in the kinematics since the radius at which the point source is contributing light may be much smaller than the STIS pixels. Thus, the smaller radius would imply a smaller BH mass for the same dispersion measure. We do not have a good way to estimate this effect since it would depend strongly on the actual size of the assumed point source, but we can get some feel by comparing results from models using only ground-based observations. For those data, we measure a BH mass similar to that when including the STIS data, suggesting that the point source does not have a dramatic effect on the kinematics.
There are two main observations that suggest that the point source is nonthermal. First, the STIS kinematics show a significant decrease in the equivalent widths of the Ca II triplet lines. The drop is around 40% suggesting nearly equal contribution from stellar and continuum sources. This drop is also seen in the ground-based data which had a spatial FWHM of $\sim 1$. Second, the point source is unresolved at [*HST*]{} resolution. At 13.2 Mpc, the implied scale for the source is less than 2 pc. Given the luminosity of the source, this radial scale implies an extremely dense structure, denser than any known stellar cluster. These two facts lead us to conclude that the source is nonthermal and must be excluded from the dynamical analysis. The most likely explanation is that the point source is a weak active galactic nucleus. Ravindranath [[et al.]{}]{} (2001) find nuclear sources in 40% of galaxies that they observed with [*HST*]{}, and they conclude that most of these are likely weak AGN. Ho [[et al.]{}]{} (1995) see no obvious nuclear emission from NGC 7457, and we conclude that it is most likely a weak BL Lac object. The luminosity density for NGC 7457 in Figure 1 excludes the central point source.
Below are notes for the other galaxies with orbit superposition models taken from the literature:
[*M32:*]{} Verolme [[et al.]{}]{} (2002) have used both STIS spectroscopy and high S/N ground-based two-dimensional spectra to provide one of the best measured BH masses using orbit-based models.
[*NGC 1023:*]{} The results for NGC 1023 are given by Bower [[et al.]{}]{} (2001) and will not be repeated here. The only difference is the assumed distance which changes both the BH mass ($M_{BH} \propto$ distance) and the mass-to-light ratio ([[*M/L*]{}]{} $\propto$ 1/distance).
[*NGC 3379:*]{} The data and orbit superposition models are presented by Gebhardt [[et al.]{}]{} (2000a). NGC 3379 has only a single FOS pointing using the 0.21 aperture.
[*NGC 4342:*]{} Cretton & van den Bosch (1999) use seven FOS aperture pointings and ground-based data along several position angles. The FOS aperture had 0.26 diameter.
[*IC 1459:*]{} Cappellari [[et al.]{}]{} (2002) use both STIS spectra and extensive two-dimensional ground-based spectral coverage. This galaxy is very important since it also has a measurement of the BH mass from gas kinematics (Verdoes-Kleijn [[et al.]{}]{} 2000). The stellar kinematic measurement is almost a factor of six higher than the gas measurement. This discrepancy is far larger than the typical error from our sample measured from stellar kinematics. However, important uncertainties attach to gas measurements, such as the orientation of the innermost gas disk and the assumption that the gas is in perfectly circular orbits. The large residual here suggests that these uncertainties perhaps deserve more attention than they have received to date.
-80pt
-30pt
Quality of the Fit
------------------
Figure 6 presents the root-mean-square (rms) line-of-sight velocity as a function of radius for both the data and best-fit model. \[Strictly, we show $(V^2+\sigma^2)^{1/2}$, where $V$ and $\sigma$ are the mean velocity and dispersion of the Gaussian that appear in the Gauss-Hermite expansion of the LOSVD.\] We stress that the model uses the velocity profiles in the fitting and not the second moments directly. Thus, there are more parameters that control the quality of the fit than those shown in Figure 6. In addition, some galaxies have several position angles and we show only one in Figure 6.
For each galaxy, the solid red line and the dashed blue line come from the same model; the only difference is that they use a different PSF. For example, the central ground-based measurements for NGC 3377, NGC 3608, NGC 4564, and NGC 7457 are all significantly different from the central STIS measurement. This is due to smaller PSF of [*HST*]{} which is taken into account in the model. There are two galaxies, NGC 821 and NGC 4564, which show a spike in the second moment at the outer STIS radius. This spike is an artifact since we are using only $V$ and $\sigma$ from the Gauss-Hermite fits, as opposed to including the higher-order moments, H3 and H4, in the estimate of the second moment.
In order to judge the quality of the fit, we have to compare $\chi^2$ to the numbers of degrees of freedom (ndof). The ndof is difficult to measure mainly because there is a smoothing parameter in the estimation of the velocity profiles; this effect typically decreases the ndof by a factor of two. An additional difficulty in calculating the ndof arises since we often include the outer regions of the velocity profiles where they have no light in the models. Sometimes these regions extend to velocities that are either outside of those measured in the velocity profiles or are very uncertain there. In the regions that are beyond the velocities in which the LOSVDs probe, we use the uncertainty at the last measured velocity. Since we have no velocity profile there, we also set the observed LOSVDs in those bins to zero. Thus, the result is to add zero to the overall $\chi^2$, yet increase the ndof. The problem can most easily be seen in a galaxy that has a significant dispersion gradient with radius. For the modeling, we use a fixed velocity interval and bins for the LOSVD. In such a galaxy with a large dispersion gradient, the outer edges of the velocity profile in the center of the galaxy will contain some light, while those regions at large radii will not. This effect can be dramatic in some galaxies, causing about half of the velocity bins to have zero light for the large-radii LOSVD. Thus, there is a further reduction in the ndof that one needs to apply in order to judge the quality of the fit. Column 6 in Table 2 reports the total numbers of fitted parameters for each of the galaxies. Comparing these numbers to the total minimum $\chi^2$ (Col. 7, Table 2) shows that the $\chi^2$ values are about 2–3 times lower than the ndof, implying reduced $\chi^2$ values near 0.4. However, this low reduced $\chi^2$ is in good agreement with the reduction expected from the two effects above.
The Need for [*HST*]{}
----------------------
The high-resolution spectral data presented here represent over 100 orbits of [*HST*]{} time. It is illuminating to determine the importance of these observations compared to ground-based data. For each galaxy, we have re-computed the best-fit models using [*only*]{} the ground-based spectra (we still use both ground-based and [*HST*]{} photometry). Figure 7 plots the $\chi^2$ as a function of BH mass for both sets of data (the [*HST*]{}+ground and ground only). In every case, inclusion of the [*HST*]{} data makes a substantial improvement in the significance of the BH detection (see also Table 2).
The two galaxies with the strongest BH detection based on the ground-based data are NGC 4649 and NGC 4697. Of the twelve galaxies in the sample, these two have the largest angular sphere of influence, 0.75 and 0.4 respectively. The ground-based data come from MDM where the seeing is typically 1. Thus, it is not too surprising that we can detect the BH in these galaxies without [*HST*]{} data. However, when the [*HST*]{} data are included, in both of these galaxies the significance is greatly increased.
We can also check whether the BH masses estimated from the two sets of data are the same. Figure 8 plots this comparison. All of the masses estimated from the two sets of data are consistent at the 1$\sigma$ level (i.e. all of the error bars in Figure 8 overlap the straight line). There is no evidence that masses based on ground-based data alone are systematically high; if anything, the use of ground-based alone appears to slightly [*underestimate*]{} the BH mass. A striking feature of Figure 8 is that even when the 1$\sigma$ uncertainty in the BH mass from ground-based data includes zero, the best-fit mass from these data is very similar to the best-fit mass from the full dataset.
Magorrian [[et al.]{}]{} (1998) presented masses based on ground-based data and two-integral axisymmetric models. Subsequent analysis shows that some of the BH masses were overestimated by up to a factor of three. Merritt & Ferrarese (2001) argue that this bias is due to use of the ground-based data. From the results presented here, it appears that the problem does not lie in using ground-based data, but more likely in the model assumptions. For the eight galaxies common to the present paper and Magorrian [[et al.]{}]{}, we find that using our higher-resolution HST+MDM kinematics has little effect on the BH masses found by the two-integral models. Therefore the error in the BH masses of Magorrian [[et al.]{}]{} is due to their assumption of isotropy. In particular, the axisymmetric models in this paper exhibit some radial anisotropy in the velocity-dispersion tensor at mid-range radii; as Magorrian [[et al.]{}]{} point out, radial anisotropy will cause the simpler isotropic models used in that paper to overestimate the masses. For the 12 galaxies in Magorrian [[et al.]{}]{} (1998) that have non-zero BH mass estimates and are also in the Tremaine [[et al.]{}]{} (2002) sample, the mean overestimate in $\log M$ is 0.22 dex.
We have also investigated whether reliable BH masses can be obtained from [*HST*]{} spectral data alone, excluding the ground-based spectra. We ran the models on the one galaxy that should have produced the strongest BH detection based on [*HST*]{} alone. NGC 3608 shows a dispersion increase by a factor of two just in the [*HST*]{} data, from 0–1. For these data, we find [*no*]{} significant detection for a BH, suggesting that the ground-based data are necessary to measure one. The reason is that the stellar [[*M/L*]{}]{} is unconstrained by the [*HST*]{} data alone. The [[*M/L*]{}]{} implied for NGC 3608 from the [*HST*]{} data is about a factor of two higher than that found when using all of the data together. This increase in the [[*M/L*]{}]{} causes the significance of the BH detection to disappear in the [*HST*]{} data alone.
Black Hole Correlations with Galaxy Properties
----------------------------------------------
We are now in a position to compare BH masses with other host galaxy properties, to look for underlying relationships that may inform us about the formation process of the BH. In Figure 9, we plot ten galaxy properties, including the BH mass, against one another. For example, in the first plot on the left on the top row, we plot the BH mass along the abscissa and the bulge luminosity along the ordinate. In addition to the galaxies with BH masses measured in this paper, we have added other galaxies with reliable mass estimates, for a total sample of 31 galaxies. Tremaine [[et al.]{}]{} (2002) report some of the properties of the galaxies not included here. In Figure 9, the number of galaxies in each panel changes depending on whether that particular value exists for all 31 galaxies. However, we do differentiate between those galaxies studied with orbit-based models (filled symbols) and those with other models (open symbols).
The galaxy properties that we report are bulge luminosity, BH mass, effective dispersion (discussed below), radial to tangential dispersion at the galaxy center (discussed below), mass-to-light ratio in the $V$ band, bulge half-light radius $R_e$, central luminosity density slope ($d\log\nu/d\log r$, where $\nu$ is the luminosity density), radial to tangential dispersion on the major axis at a quarter of the half-light radius, bulge stellar mass, and BH mass offset from the $M_{BH}/\sigma$ correlation. The bulge luminosities are taken from Kormendy & Gebhardt (2001) and the calculations are given by Kormendy [[et al.]{}]{} (2002). The bulge half-light radii come from Faber [[et al.]{}]{} (1989) and Baggett [[et al.]{}]{} (2000). The BH mass offset is calculated using the relation in Tremaine [[et al.]{}]{} (2002). For the mass-to-light ratios, we use only those galaxies that have a measured value in the $V$ band. To measure the total mass, we use the bulge total $B$-band light, convert to $V$ using $B-V$ from RC3 (de Vaucouleurs [[et al.]{}]{} 1991), and multiply by the $V$-band mass-to-light ratio.
There are five galaxies that have axisymmetric orbit-based models from previous studies. We include the internal velocity structure of these five galaxies in Figure 9 and subsequent analysis. Table 3 reports their internal velocity moments and the reference.
Besides the obvious and expected correlation between total mass and total light, the most significant correlation is the $M_{BH}/\sigma$ relation reported by Gebhardt [[et al.]{}]{} (2000b) and Ferrarese & Merritt (2000). Tremaine [[et al.]{}]{} (2002) discuss the differences in the measured slope of this relation. Most other significant correlations between various galaxy properties and the BH mass can be regarded as a result of well-known correlations of other galaxy properties with dispersion. There is a significant correlation between the shape of the dispersion tensor at $R_e/4$ and the BH mass. Larger BHs tend to live in galaxies that have more radial motion. This trend may be a clue to the formation process of the BH but, also, could represent a secondary correlation between galaxy anisotropy and galaxy dispersion. There is a suggestion that larger BH mass offsets occur in galaxies that have less radial energy near the center, possibly signifying additional evolutionary effects. However, we need more data to decide on the significance, since it is only at the 20% level in the anisotropy. A full treatment of the correlations should include a proper principal component analysis (PCA); however, given the uncertainties and small sample, we are not in a position to explore PCA, especially since the $M_{BH}/\sigma$ relation provides such small scatter already.
Effective Dispersion
--------------------
We use the effective dispersion, $\sigma_e$, as a representation of the galaxy velocity dispersion; $\sigma_e$ is the second moment of the velocity profile integrated from $-R_e$ to $+R_e$ along the major axis with a slit width of 1. The idea is to represent the galaxy by one dispersion estimate. There are many ways to do this; for example, Jørgensen [[et al.]{}]{} (1996), Faber [[et al.]{}]{} (1989), and the Sloan Survey (Bernardi [[et al.]{}]{} 2002) use the second moment inside a circular aperture of radius $R_e$/8. Tremaine [[et al.]{}]{} (2002) discuss the effect that the BH can have on either the effective dispersion or the dispersion inside $R_e$/8. In most cases, the BH has little effect, $<$3%, but in some galaxies the effect can be as large as 30%.
The effective dispersions are given in Table 1. In all cases the S/N is very high, over 100. The corresponding statistical uncertainty in $\sigma_e$ ranges from 1–3%. However, at this level, systematic uncertainties dominate, particularly continuum estimation and template mismatch. We have investigated both of these effects by varying the continuum level and using different templates. We find that at any S/N, using an appropriate range of systematic variables, the overall uncertainty is no better than 5%. Therefore, we adopt 5% accuracy for the effective dispersion measurements. We discuss below how this choice affects the main results.
Velocity Dispersion Tensor
--------------------------
We show the shape of the velocity dispersion tensor in Figure 10. We define the tangential dispersion as $\sigma_t =
[(\sigma_\theta^2+\sigma_\phi^2)/2]^{1/2}$, so that for an isotropic distribution the radial and tangential dispersions are equal. Note that $\sigma_\phi$ includes both random and ordered motion (i.e., it represents the second moment of the azimuthal velocity relative to the systemic velocity, not relative to the mean rotation speed). The most obvious trend in Fig 10 is that the tangential motion tends to become more important towards the center (discussed below) in all galaxies except for NGC 4697.
-180pt -100pt -100pt
We use two methods to measure the uncertainties on these quantities. We have run Monte Carlo simulations for two galaxies, and for the remaining we use the simple alternative of using the smoothness in both the radial and angular profiles to estimate the uncertainties. For the Monte Carlo simulations we use the same realizations that are used to generate the LOSVD uncertainties (Pinkney [[et al.]{}]{}) and to generate the BH mass uncertainties. For each set of LOSVD realizations, we find the best-fit model and examine the orbital structure. We have run these simulations on two galaxies: the core galaxy NGC 3608 and the power-law galaxy NGC 4564. Figure 11 plots the results. For the 100 realizations we estimate the 68% confidence bands by choosing the 16% and 84% values in the sorted internal moments at each radii. Figure 11 shows that the drop in the radial motion near the center is statistically significant for both galaxies. At the radii outside of our last measured kinematic measurements, the uncertainties become very large, as expected. We note that the radial profiles are not very smooth. The level of non-smoothness is consistent with the measured uncertainties. This noise is likely a result of using the same orbit library with each new LOSVD realization. Thus, the remaining noise is due to using a limited number of stellar orbits. Ideally, we should include a random sampling of the photometry, and hence the stellar potential in the Monte Carlo simulations. By doing so, we would average over noise from a particular set of orbits. However, this is computationally prohibitive and we rely on the present simulations to provide the uncertainties.
For the other galaxies, we use deviations from smoothness as an estimate of the internal orbital structure uncertainties. An expectation is that the radial and angular gradient of the internal moments may be smooth, albeit details due to recent merger and accretion history may cause small scale variations. Thus, deviations from a smooth profile may be indicative of the measurement uncertainty. The three galaxies with the smallest number of kinematic measurements—NGC 2778, NGC 3377 and NGC 7457—show the largest radial and angular variations, suggesting that these are due to increased uncertainties from not having as much kinematic constraints. By inspection of variations seen Fig. 10 and Fig. 11, we estimate that the uncertainties on ratio of the internal moments is around 0.1 to 0.2 for most galaxies. This uncertainty is also consistent with the angular variations. The models are free to have very different dispersion ratios at different position angles and radii. The fact that the ratios are similar at different angles suggests that the measurements are robust for these models. The core galaxies (denoted with an asterisk) show a larger decrease in the ratio towards the center which we discuss next.
Although the models produce the internal moments everywhere in the galaxies (i.e., Fig. 10), Figure 12 shows them only along the major axis and at two radii: the central bin in the models and an average of the three bins nearest $R_e/4$. The BH dominates the potential in the central bin.
Figure 12 shows that galaxies with shallow central density profiles have orbits with strong tangential bias near their centers. At larger radii, the orbits tend to be isotropic or slightly radial. There is a concern that this change in dispersion ratio may simply reflect our assumption that the mass-to-light ratio is independent of radius. If a dark halo were present, so that the mass-to-light ratio increased outwards, a galaxy with isotropic orbits will appear to become tangentially anisotropic at large radii. However, it is unlikely that the dark halo makes a significant contribution to the potential within $R_e/4$, and in any case the sign of this trend (increasing tangential anisotropy with radius) is opposite to the one we observe. Figure 12 only includes results from radii that are small enough to be unaffected by the presence of a dark halo. In any event, the most likely bias is that, by not including a dark halo, we will overestimate the amount of tangential anisotropy at large radii. At small radii, the dark halo assumption will have no effect. Therefore, we are confident of the gradient seen in Figure 12.
The bottom panel in Figure 12 plots the change in the radial-to-tangential motion between the center and $R_e/4$ as a function of central slope. This plot reiterates that the gradient in shape of the dispersion tensor is larger for core galaxies than for power-law galaxies.
The shape of the velocity dispersion tensor depends on the galaxy formation process. Tangentially biased orbits at small radii can occur through the destruction or ejection of stars on high-eccentricity orbits that pass near the BH. There are now 17 galaxies for which we have measurements of orbital anisotropies; most come from the dynamical models presented here, but similar results are found using other orbit superposition studies (M32: van der Marel [[et al.]{}]{} 1998; Verolme [[et al.]{}]{} 2002; NGC 4342: Cretton & van den Bosch 1999; and IC 1459: Cappellari [[et al.]{}]{} 2002). Spherical theoretical models predict a range of both central cusps and anisotropies. Three models that have been studied are adiabatic BH growth (Quinlan [[et al.]{}]{} 1995), fall-in of a single BH (Nakano & Makino 1999), and BH binary models (Quinlan & Hernquist 1997). The adiabatic models grow the black hole by slow accretion of material (gas or stars). The BH fall-in models start with a galaxy without a black hole and then a BH is placed a large radii where is falls in due to dynamical friction. The BH binary model assumes the galaxy has an existing black hole at the center and then a second black hole falls in due to dynamical friction, and they subsequently form a binary BH. The models predict a different value of the central anisotropy: single BH infall models have $1>\sigma_r/\sigma_t > 0.87$, adiabatic models have $\sigma_r/\sigma_t
\approx 0.87$, and BH binary models have $\sigma_r/\sigma_t \approx
0.7$. The increased tangential anisotropy for the binary models is due to the orbital motion of the binary in the galaxy core, which causes it to affect more stars on radial orbits.
The models that agree best with both observed anisotropies and density slopes depend on the type of galaxy, whether it is a core or power-law. Examining the solid points in Figure 12, one notices that those galaxies with shallow central densities have the largest tangential motion. The central $\sigma_r/\sigma_t$ for core galaxies is around 0.4 (highly tangentially biased), while that for the power-law galaxies range from 0.45 to 1.05 with an average of 0.8. Thus, it appears that the core galaxies are more consistent with the BH binary models, and the power-law galaxies are more consistent with adiabatic growth. These conclusions are similar to those of Faber [[et al.]{}]{} (1997) and Ravindranath [[et al.]{}]{} (2002). Unfortunately, these comparison models are limited by their simplistic initial conditions (i.e., isotropic velocity dispersion tensor, spherical potential). Fortunately, the large number of researchers working in this area now (e.g., Milosavljevic & Merritt 2001; Holley-Bockelmann [[et al.]{}]{} 2002; Sellwood 2001) will provide improved theoretical comparisons.
The Need for Three Integrals
----------------------------
It is important to know whether the distribution function of these galaxies depends on three integrals of motion or only on the two classical integrals (energy and $z$-component of angular momentum). In two-integral models the dispersion in the $R$ and $z$ directions must be equal (where $R$, $\phi$, $z$ are the usual cylindrical coordinates). Figure 13 plots this ratio for the 12 galaxies. There are no obvious radial trends. For seven galaxies, at small radii the radial dispersion is higher than in the $\theta$ direction. For two, the $\theta$ motion significantly dominates there. At other radii, the results show a variety of trends with some galaxies having rising ratios while others having falling ones. The [*average*]{} ratio along the major axis is close to unity (i.e., a two-integral model), but the radial run demonstrates that the best fit model is inconsistent with having only two integrals of the motion. This result is similar to that found in other orbit-based models (Verolme [[et al.]{}]{} 2002; Cappellari [[et al.]{}]{} 2002).
We discussed that most of the galaxies (10 of the 12) have substantial tangential motion near their centers, but not whether this is due to the $\theta$ or $\phi$ motion. By comparison of Fig 13 to Fig 10, we notice that at most of the radii, the curves are similar which implies that the $\theta$ and $\phi$ dispersion are similar. However, Figure 13 shows a significant increase in the contribution from the $\theta$ direction near their centers for only two of the galaxies, whereas from Fig 10, we see that most galaxies show a dramatic increase in the tangential motion near their centers. Thus, the dominant component in nearly all of the galaxies near the center is in the $\phi$ direction. At the center, the $\phi$ dispersion generally has similar contributions from random and ordered motion. The theoretical models discussed in §4.7 do not provide the difference between the $\theta$ and $\phi$ dispersions, but these could potentially be important constraints.
Possible Concerns
-----------------
Our models are limited to axisymmetry. Triaxial and non-symmetric structures may be common attributes of galaxies. Of the 12 galaxies in this sample, at least four—NGC 3377, NGC 3608, NGC 4473, and NGC 7457—show signs of non-axisymmetric structure in the kinematics. An incorrect assumption of axisymmetry could bias our results. For example, if a bar is observed down its long axis, the radial streaming motions along the bar may increase the projected velocity dispersion. This measured increase may mimic that expected with a central mass concentration (Gerhard 1988). We have not investigated the effects of triaxiality on BH mass determinations in detail, but believe that these effects average to zero when the system is viewed from many different orientations; thus, triaxiality may contribute to the scatter in our mass determinations but should not produce a systematic bias. Furthermore, the scatter due to not considering triaxiality may be a function of galaxy size, since large core galaxies possibly are more triaxial than the smaller power-law galaxies. Clearly, triaxial models should be used to quantify these effects. Any bias caused by using an inappropriate stellar distribution could be more dramatic if we only had data at larger radii where the stellar potential needs to be included. However, as seen in Figure 8, the ground-based data alone do a fairly good job at measuring the BH mass compared to when including the [*HST*]{}. Thus, at least at the level of our uncertainties, the BH mass is unaffected by using the large radial data, suggesting that if the galaxies are not axisymmetric then either the non-axisymmetry is unimportant for the modeling or it is constant with radius. The best way to test biases with the axisymmetry assumption is to either model the same galaxies with triaxial codes or to run the axisymmetric code on an analytic triaxial galaxy.
We have assumed that the surfaces of constant luminosity density in all of our galaxies are similar spheroids (with the exception of NGC 4473, where we add a disk component). This assumption is consistent with the observation that the ellipticity of the surface-brightness distribution is similar at all radii, but other density distributions are also consistent with this observation. The question is whether the assumption of spheroidal equidensity surfaces could bias our BH mass determinations. Some guidance comes from the analysis of Magorrian & Ballantyne (2001), who study the influence of embedded stellar disks. In this case, they find that face-on disks in round galaxies in projection may bias a spherical model toward having radial anisotropy. This effect is primarily seen at large radii and is unlikely to bias the BH mass since we are measuring the kinematics so close to the BH. But, this effect may be important for the orbital structure that we measure in these galaxies. Again, at small radii, the influence of a disk is likely to be small since we do not see strong signatures of one (except in NGC 4473), however it would be difficult to measure a disk at larger radii. Thus, there is a concern that we may be biased by this effect in some of the galaxies at larger radii. Fortunately, only four of our galaxies are rounder than E3, so this is unlikely to alter the overall conclusions. There are many other possibilities other than embedded stellar disks that can lead to non-uniqueness in the deprojection. The best way to understand their effects is to run models with different deprojections. For four of the galaxies, we have tried a variety of inclinations and find insignificant changes to both the BH mass and orbital structure.
We have not included a dark halo in this analysis. It appears that in most elliptical galaxies the dark halo becomes important at about the half-light radius (Kronawitter [[et al.]{}]{} 2000; Rix [[et al.]{}]{} 1997). Even though we have data and model results at these large radii, and they are plotted in Figure 10, we do not use the model results from these radii because they may be seriously comprised by the exclusion of a dark halo. At radii less than $R_e/4$ the stars and central BH dominate the potential. For Figure 12, we choose $R_e$/4; at this radius the stellar potential dominates. The BH mass is determined almost exclusively by the small-radii data; thus, we are confident that exclusion of a dark halo is unimportant for the BH mass estimate. Gebhardt [[et al.]{}]{} (2000a) include various dark-halo profiles and find little difference in the results inside of $R_e$/2. The next step in the data analysis is to run models in order to measure both the BH mass and dark halo properties.
We have assumed that the mass-to-light ratio is constant with radius. As we discuss above, the exclusion of a dark halo is unlikely to affect either the black hole mass or the orbital structure, however, variation at small radii can have an effect. For example, a dramatic increase in the stellar mass-to-light ratio in the central regions can decrease the measured BH mass if not accounted for. We have not done a detailed spectral analysis to determine the stellar makeup but we can use the color gradients to provide some constraints. For the 12 galaxies, the [*largest*]{} mass-to-light variation from 10 to the center is $V-I=0.1$; the average is around 0.04. Models of Worthey (1994) suggest that a $V-I=0.1$ imply an mass-to-light change of about 20%. We do not include that small variation here but note that Gebhardt [[et al.]{}]{} (2000a) use an even larger variation and find no change in the measured BH mass. Cappellari [[et al.]{}]{} (2002) find a similar result for IC1459. Thus, we conclude that inclusion of a small mass-to-light variation at small radius will have insignificant effect on the BH mass.
Conclusions
===========
The twelve galaxies in this paper all have significant BH detections, with a typical statistical significance in the masses of around 30%. The average significance of detection is well above 99% and the least significant detection (NGC 2778) has about 90% confidence. Thus, for this sample, every galaxy has a BH. In fact, only one nearby galaxy with high-resolution spectral data lacks any significant BH detection: the pure disk galaxy M33 (Gebhardt [[et al.]{}]{}2001). The most obvious difference between M33 and the galaxies with significant BH detection is that the latter have a bulge component.
For a few of these galaxies, ground-based spectra alone yield reasonably precise BH masses. The masses based on ground-based data alone are generally remarkably close to the masses based on ground-based and [*HST*]{} data; there is no evidence that masses based on ground-based data alone are systematically high. The most important aspect of using ground-based data is assure that the models are fit using full generality (i.e., without assumptions about the orbital structure).
The most significant correlation with BH mass is with the velocity dispersion. The present intrinsic scatter is around 0.23 dex in BH mass (Tremaine [[et al.]{}]{} 2002). It will be extremely illuminating to include more galaxies at both extremes, the low mass and high mass ends. The next most significant correlation is with the radial-to-tangential velocity dispersion at $R_e$/4. We do not know whether this is simply a secondary correlation due to that with the velocity dispersion, or if it represents an evolutionary pattern due to the growth of the BH. Detailed theoretical and N-body models are required to understand this. The BH mass also significantly correlates with both galaxy bulge luminosity and bulge mass, but neither of these is as strong as with dispersion.
The uncertainties in the BH masses reported here are only statistical. We have not attempted to include uncertainties from the assumptions in our models or systematic errors in our analysis outlined in §4.9. We believe that the increase in the uncertainties is likely to be small, but additional tests are required in order to substantiate this. We can use the $M_{BH}/\sigma$ correlation as an approximate constraint on the uncertainties. If there is an underlying physical mechanism that causes a [*perfect*]{} correlation between $M_{BH}$ and $\sigma$, then any scatter seen in the correlation must be measurement error. Since the current scatter is comparable to the measurement error, we probably have a reasonable estimate of our uncertainties; any additional uncertainties caused by our assumptions should be smaller than 0.23 dex in BH mass. However, this argument applies only to random errors. If, for example, galaxies deviate from our assumptions systematically, then the $M_{BH}/\sigma$ correlation may still have small scatter but incorrect BH masses. The only way to test this is to include a larger sample with general dynamical models that cover a wide variety of input configurations.
The orbit-based models provide a look into the internal orbital structure of an axisymmetric system. Based on the small sample of galaxies shown here and the limited theoretical comparisons, we are already able to place some constraints on the possible evolutionary history of the galaxy. The results in this paper suggest that core galaxies have tangentially biased orbits near their centers, while power-law galaxies show a range of tangential relative to radial motion. As suggested by Faber [[et al.]{}]{} (1997) and Ravindranath [[et al.]{}]{}(2002), it appears that the core galaxies are consistent with the BH/binary models, and the power-law galaxies are more consistent with adiabatic growth. This conclusion comes from analysis of the stellar surface brightness profiles, and now a similar conclusion comes from the stellar kinematics. Significant improvement in our understanding of the orbital structure will come from datasets with two-dimensional kinematics. De Zeeuw [[et al.]{}]{} (2002) and Bacon [[et al.]{}]{} (2002) present examples of datasets that can be exploited for this analysis. However, in order to make progress in this area we must understand possible systematic biases that can arise from various assumptions (e.g., dark halo, different deprojections, lack of axial symmetry, etc.)
K.G. is grateful for many interesting discussions with Tim de Zeeuw, Ellen Verolme, and Christos Siopis. We thank the referee, Tim de Zeeuw, for excellent comments which improved the manuscript. This work was supported by [*HST*]{} grants to the Nukers (GO–02600, GO–6099, and GO–7388), and by NASA grant G5–8232. A.V.F. acknowledges NASA grant NAG5–3556, and the Guggenheim Foundation for a Fellowship.
Alam, S. M. K., & Ryden, B. S. 2002, astro-ph/0201435 Bacon, R., [[et al.]{}]{} 2002, astro-ph/0204129 Baggett, W. E., Baggett, S. M., & Anderson, K. S. J. 1998 , 116, 1626 Beers, T. C., Flynn, K., & Gebhardt, K. 1990, , 100, 32 Bernardi, M., [[et al.]{}]{} 2002, , submitted (astro-ph/0110344) Bower, G., [[et al.]{}]{} 2001, , 550, 75 Cappellari, M., [[et al.]{}]{} 2002,, 578 (astro-ph/0202155) Cretton, N., de Zeeuw, P. T., van der Marel, R. P., & Rix, H.-W. 1999, , 124, 383 Cretton, N., Rix, H.-W., & de Zeeuw, P.T. 2000, , 536, 319 Cretton, N., & van den Bosch, F. 1999, , 514, 704 de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Buta, R. J., Paturel, G., & Fouquè, P. 1991, Third Reference Catalogue of Bright Galaxies (Springer, New York) (RC3) de Zeeuw, P. T., [[et al.]{}]{} 2002, , 329, 513 Faber, S., Wegner, G., Burstein, D., Davies, R., Dressler, A., Lynden-Bell, D., & Terlevich, R. 1989, , 69, 763 Faber, S. M., [[et al.]{}]{} 1997, , 114, 1771 Ferrarese, L., & Merritt, D. 2000, , 539, L9 Fisher, D., Illingworth, G., & Franx, M. 1995, , 438, 539 Gebhardt, K., [[et al.]{}]{} 1996, , 112, 105 Gebhardt, K., [[et al.]{}]{} 2000a, , 119, 1157 Gebhardt, K., [[et al.]{}]{} 2000b, , 539, L13 Gebhardt, K., [[et al.]{}]{} 2001, , 122, 2469 Gerhard, O. 1988, , 232, 13 Gerhard, O., & Binney, J. 1996, , 279, 993 Ho, L. C., Filippenko, A. V., & Sargent, W. L. W. 1995, , 98, 477 Holley-Bockelmann, K., Mihos, J. C., Sigurdsson, S., Hernquist, L., & Norman, C. 2002, , 567, 817 Jaffe, W., Ford, H., O’Connell, R., van den Bosch, F., & Ferrarese, L. 1994, , 108, 1567 Jørgensen, I., Franx, M., & Kjærgaard, P. 1996, , 280, 167 Kochanek, C., & Rybicki, G. 1996 ,, 280, 1257 Kormendy, J., Bender, R., Evans, A., & Richstone, D. 1998, , 115, 1823 Kormendy, J., & Gebhardt, K. 2001, in The 20th Texas Symposium on Relativistic Astrophysics, eds. H. Martel & J. C. Wheeler (New York: AIP), in press (astro-ph/0105230) Kormendy, J., & Richstone, D. 1995, , 33, 581 Kormendy, J., [[et al.]{}]{} 1996, , 459, L57 Kronawitter, A., Saglia, R. P., Gerhard, O., & Bender, R. 2000, , 144, 53 Lauer, T. R. 1985, , 57, 473 Lauer, T. R., Ajhar, E. A., Byun, Y.-I., Dressler, A., Faber, S. M., Grillmair, C., Kormendy, J., Richstone, D., & Tremaine, S. 1995, , 110, 2622 Lauer, T. R., Faber, S. M., Ajhar, E. A., Grillmair, C. J., & Scowen, P. A. 1998, , 116, 2263 Lauer, T. R., [[et al.]{}]{} 2002, , submitted Lucy, L. B. 1974, , 79, 745 Magorrian, J., [[et al.]{}]{} 1998, , 115, 2285 Magorrian, J., & Ballantyne, D. 2001, , 322, 702 Merritt, D., & Ferrarese, L. 2001, , 320, L30 Milosavljevic, M., & Merritt, D. 2001, , 563, 34 Nakano, T., & Makino, J. 1999, , 510, 155 Peletier, R.F., Davies, R.L., Illingworth, G.D., & Davies, L.E. 1990, , 100, 1091 Pinkney, J. [[et al.]{}]{} 2002a, submitted Pinkney, J. [[et al.]{}]{} 2002b, in preparation Quinlan, G., & Hernquist, L. 1997, NewA, 2, 533 Quinlan, G., Hernquist, L., & Sigurdsson, S. 1995, , 440, 554 Ravindranath, S., Ho, L.C., Peng, C., Filippenko, A.V., & Sargent, W. 2001, , 122, 653 Ravindranath, S., Ho, L.C., & Filippenko, A.V. 2002, , 566, 801 Richardson, W. H. 1972, J. Opt. Soc. A., 62, 52 Richstone, D. O., & Tremaine, S. 1984, , 286, 27 Richstone, D., [[et al.]{}]{} 2002, AJ, in preparation Rix, H.-W., de Zeeuw, P.T., Cretton, N., van der Marel, R.P., & Carollo, M. 1997, , 488, 702 Sellwood, J. A. 2001, astro-ph/0107353 Tonry, J., Blakeslee, J., Ajhar, E., & Dressler, A. 2000 , 530, 625 Tremaine, S., [[et al.]{}]{} 2002, , 574, 740 van der Marel, R. P., Cretton, N., de Zeeuw, P. T., & Rix, H.-W. 1998, , 493, 613 Verdoes Kleijn, G., van der Marel, R. P., Carollo, C. M., & de Zeeuw, P. T. 2000, , 120, 1221 Verolme, E., & de Zeeuw 2002, , 331, 959 Verolme, E., [[et al.]{}]{} 2002, , 335, 517 Wahba, G. 1990, Spline Models for Observational Data (Philadelphia: SIAM) Worthey, G. 1994, , 95, 107
FOS and Ground-Based Data for NGC 3377 and NGC 5845
===================================================
The ground-based velocities and dispersions for NGC 3377 come from Kormendy [[et al.]{}]{} (1998) and will not be repeated here. Since our models use the LOSVDs, we convert from the first two moments to the velocity profile using Monte Carlo simulations. Each velocity profile realization is a Gaussian with the mean chosen from a random draw from the measured mean using its uncertainty, and the sigma chosen from a random draw from the measured dispersion using its uncertainty. This procedure does not take into account the H3 and H4 components that are likely to be non-zero. However, the model results depend very little on the higher order moments, since it is mainly the radial run of the first two moments that determine the BH mass. The height of the LOSVD at a given velocity is then the mean of the simulations and the uncertainty is given by the 68% confidence bands of the simulations.
For NGC 5845, the data were taken with the MDM telescope. The observational setup and reductions are similar to those outlined in Pinkney [[et al.]{}]{} (2002a). We used the Ca II triplet region around 8500 Å. The plate scale is 0.59 per pixel. The wavelength scale is 1.44 Å per pixel, with an instrumental resolution of 0.75 Å or 26 [$\rm {km}~\rm s^{-1}$]{}. We observed along three position angles for NGC 5845: 0, 22, and 90 (defined from the major axis to the minor axis), with total exposure times of 3 hours for each position angle (9 hours in total). In Table 4, we report the first four velocity moments of the LOSVD for the three position angles. They are plotted in Figure A14.
For the FOS data, the reduction procedure is similar to that in Gebhardt [[et al.]{}]{} (2000a). Both galaxies were observed using the 0.21 square aperture. The wavelength range, 4566–6815 Å, includes the Mg I [*b*]{} lines near 5175 Å. The spectral dispersion is 1.09 Å pixel$^{-1}$. The instrumental velocity dispersion is $\sigma_{\rm instr} = $FWHM/2.35$ = 1.76 \pm 0.03$ pixels = $101\pm
2$ [$\rm {km}~\rm s^{-1}$]{} (internal error). This width is intrinsic to the instrument and is not strongly affected by how the aperture is illuminated (Keyes [[et al.]{}]{} 1995). We therefore make no aperture illumination corrections to the measured velocity dispersion. Flat fielding and correction for geomagnetically induced motions (GIM) were done as in Kormendy [[et al.]{}]{} (1996). The flat-field image uses the same aperture. Each galaxy exposure is divided into multiple subintegrations during the visits. There are four individual exposures for NGC 5845 with a total integration time of 2.43 hours. For NGC 3377, the central pointing had a single exposure of 0.66 hours, and two flanking exposures on both sides of the galaxy of 1.02 and 0.90 hours (each split into two subintegrations).
The most important step is to determine where the slit was actually placed. For NGC 3379 (Gebhardt [[et al.]{}]{} 2002a), this was a critical issue since the aperture was not placed exactly in the center of the galaxy. For both NGC 3377 and NGC 5845, the aperture placement is much more secure since both galaxies have central cusps. Since we take a setup image of the galaxy before the spectral observations, we know where the aperture was placed. For both galaxies, the center of the galaxy is at the center of the FOS aperture to better than 0.05. For NGC 3377, we have two additional apertures placed 0.2 away from the center along the major axis on opposite sides. We have checked their placement using the setup images and confirm that they were both placed at the requested position to within 0.05. Since the flanking spectra were placed at very similar radii on opposite sides of the galaxy, we have fit the same velocity profile, but appropriately flipped, to both spectra. This fit is the same as that done for all of the other data used in the models. Table 4, therefore, only reports the moments for one profile fitted to both spectra.
We have three template stars taken with the same FOS aperture, and all three provide similar results for the kinematics. Table 4 includes the Gauss-Hermite moments for the single pointing for NGC 5845 and the two pointings for NGC 3377.
Extraction of the LOSVD for the three [*HST*]{} spectra and the ground-based spectra use the procedure described by Gebhardt [[et al.]{}]{}(2000a) and Pinkney [[et al.]{}]{} (2002a). We use the full LOSVD in the models, but we report only its first four moments in Table 4.
[lcccccccccc]{} N821 & E4 & $-$20.41 & $3.7\times 10^7~(2.9,6.1)$ & 209 & 24.1 & 7.6,$V$ & –1.4 & 0.37 & 0.88 & 5.32\
N2778 & E2 & $-$18.59 & $1.4\times 10^7~(0.5,2.2)$ & 175 & 22.9 & 8.0,$V$ & –1.9 & 0.56 & 0.81 & 1.82\
N3377 & E5 & $-$19.05 & $1.0\times 10^8~(0.9,1.9)$ & 145 & 11.2 & 2.9,$V$ & –1.5 & 0.74 & 1.01 & 1.82\
N3384 & S0 & $-$18.99 & $1.6\times 10^7~(1.4,1.7)$ & 143 & 11.6 & 2.5,$V$ & –1.9 & 0.44 & 0.89 & 0.73\
N3608$^*$ & E2 & $-$19.86 & $1.9\times 10^8~(1.3,2.9)$ & 182 & 22.9 & 3.7,$V$ & –1.0 & 0.52 & 1.00 & 3.85\
N4291$^*$ & E2 & $-$19.63 & $3.1\times 10^8~(0.8,3.9)$ & 242 & 26.2 & 5.5,$V$ & –0.6 & 0.42 & 0.98 & 1.85\
N4473$^*$ & E5 & $-$19.89 & $1.1\times 10^8~(0.3,1.5)$ & 190 & 15.7 & 6.0,$V$ & –0.3 & 0.33 & 0.79 & 1.84\
N4564 & E3 & $-$18.92 & $5.6\times 10^7~(4.8,5.9)$ & 162 & 15.0 & 2.0,$I$ & –1.9 & 0.62 & 0.89 & 1.54\
N4649$^*$ & E1 & $-$21.30 & $2.0\times 10^9~(1.4,2.4)$ & 385 & 16.8 & 8.5,$V$ & –1.2 & 0.51 & 1.04 & 5.95\
N4697 & E4 & $-$20.24 & $1.7\times 10^8~(1.6,1.9)$ & 177 & 11.7 & 4.7,$V$ & –1.7 & 0.97 & 0.92 & 4.25\
N5845 & E3 & $-$18.72 & $2.4\times 10^8~(1.0,2.8)$ & 234 & 25.9 & 5.5,$V$ & –1.4 & 0.79 & 1.07 & 0.51\
N7457 & S0 & $-$17.69 & $3.5\times 10^6~(2.1,4.6)$ & 67 & 13.2 & 3.2,$V$ & –1.9 & 0.62 & 0.72 & 0.90\
[lllclrr]{} N821 & $3.7\times10^7~(2.9,6.1)$ & $3.0\times10^7~(0.0,8.0)$ & 2 (1,2) & 24 & 312 & 128\
N2778 & $1.4\times10^7~(0.5,2.2)$ & $0.0\times10^7~(0.0,6.0)$ & 1 (1,1) & 12 & 93 & 29\
N3377 & $1.0\times10^8~(0.9,1.9)$ & $1.2\times10^8~(0.5,2.0)$ & 2 (1,2) & 15 & 52 & 8\
N3384 & $1.6\times10^7~(1.4,1.7)$ & $1.4\times10^7~(1.1,3.0)$ & 2 (1,2) & 24 & 312 & 131\
N3608 & $1.9\times10^8~(1.3,2.9)$ & $1.4\times10^8~(0.7,3.0)$ & 2 (1,2) & 24 & 312 & 92\
N4291 & $3.1\times10^8~(0.8,3.9)$ & $2.0\times10^8~(0.0,5.0)$ & 3 (1,3) & 27 & 351 & 182\
N4473 & $1.1\times10^8~(0.3,1.5)$ & $2.0\times10^7~(1.0,9.9)$ & 2 (1,2) & 24 & 312 & 64\
N4564 & $5.6\times10^7~(4.8,5.9)$ & $1.0\times10^7~(0.6,5.5)$ & 3 (1,3) & 33 & 429 & 187\
N4649 & $2.0\times10^9~(1.4,2.4)$ & $1.5\times10^9~(0.7,2.5)$ & 3 (1,3) & 35 & 455 & 128\
N4697 & $1.7\times10^8~(1.6,1.9)$ & $2.5\times10^8~(1.6,3.1)$ & 3 (1,3) & 30 & 390 & 192\
N5845 & $2.4\times10^8~(1.0,2.8)$ & $3.0\times10^8~(0.4,4.5)$ & 3 (1,3) & 25 & 325 & 205\
N7457 & $3.5\times10^6~(2.1,4.6)$ & $3.1\times10^6~(0.0,9.9)$ & 3 (1,3) & 20 & 260 & 76\
[lcccccc]{} N221=M32 & $2.5\times10^6~(2.4,2.6)$ & –1.6 & 1.01 & 0.72 & 0.15 & 1\
N1023 & $4.4\times10^7~(3.9,4.9)$ & –1.8 & 0.57 & 1.04 & 1.98 & 2\
N3379$^*$ & $1.0\times10^8~(0.5,1.6)$ & –1.0 & 0.41 & 1.06 & 1.76 & 3\
N4342 & $3.0\times10^8~(2.4,4.1)$ & –1.7 & 1.03 & 1.00 & 0.47 & 4\
IC1459 & $2.5\times10^9~(2.4,2.6)$ & –1.4 & 0.81 & 1.12 & 4.48 & 5\
[rrrrrr]{} 0& 0.00 & 0.0$\pm$6.5 & 258.0$\pm$6.0 &–0.05$\pm$0.02 & 0.00$\pm$0.02\
0&–0.20 & 100.0$\pm$5.0 & 215.0$\pm$5.0 &–0.06$\pm$0.02 &–0.01$\pm$0.02\
0& 0.00 & 17.2$\pm$20.8 & 292.9$\pm$17.0 &–0.09$\pm$0.05 &–0.05$\pm$0.04\
0&–0.13 & 63.0$\pm$7.8 & 250.0$\pm$7.2 & 0.03$\pm$0.03 &–0.02$\pm$0.02\
0& 0.46 & –36.6$\pm$11.5 & 239.4$\pm$8.3 & 0.10$\pm$0.02 & 0.02$\pm$0.03\
0& 1.05 & –76.7$\pm$12.2 & 224.9$\pm$13.3 & 0.11$\pm$0.03 &–0.01$\pm$0.04\
0& 1.93 &–122.6$\pm$7.3 & 185.2$\pm$6.3 & 0.07$\pm$0.02 &–0.02$\pm$0.02\
0& 3.41 & –95.8$\pm$10.1 & 170.3$\pm$6.7 &–0.02$\pm$0.04 &–0.09$\pm$0.01\
0& 5.77 & –44.9$\pm$12.3 & 140.4$\pm$18.1 & 0.09$\pm$0.09 & 0.01$\pm$0.07\
0& 9.60 &–103.0$\pm$27.3 & 170.7$\pm$33.5 & 0.15$\pm$0.09 & 0.11$\pm$0.08\
0& 15.80 & –21.3$\pm$53.3 & 163.5$\pm$68.9 & 0.10$\pm$0.12 & 0.01$\pm$0.20\
0&–0.72 & 138.7$\pm$9.4 & 226.2$\pm$10.4 &–0.09$\pm$0.04 &–0.00$\pm$0.03\
0&–1.31 & 173.4$\pm$9.8 & 191.2$\pm$12.9 &–0.14$\pm$0.04 & 0.03$\pm$0.03\
0&–1.90 & 172.5$\pm$9.1 & 194.5$\pm$11.1 &–0.03$\pm$0.04 &–0.04$\pm$0.03\
0&–2.79 & 143.1$\pm$7.9 & 175.7$\pm$7.6 & 0.01$\pm$0.03 &–0.05$\pm$0.03\
0&–4.26 & 92.6$\pm$6.2 & 138.4$\pm$10.3 &–0.05$\pm$0.03 &–0.05$\pm$0.02\
0&–6.62 & 142.9$\pm$17.1 & 161.6$\pm$27.0 &–0.07$\pm$0.07 & 0.00$\pm$0.07\
0&–10.16& 171.7$\pm$20.2 & 154.6$\pm$24.6 &–0.01$\pm$0.07 &–0.04$\pm$0.06\
22&–0.33 & 34.7$\pm$11.4 & 253.0$\pm$9.8 & 0.01$\pm$0.03 &–0.01$\pm$0.04\
22& 0.26 & –45.8$\pm$8.1 & 230.4$\pm$8.1 & 0.07$\pm$0.02 &–0.04$\pm$0.03\
22& 0.85 &–102.8$\pm$9.6 & 209.6$\pm$9.1 & 0.11$\pm$0.03 & 0.00$\pm$0.03\
22& 1.74 &–105.7$\pm$7.8 & 197.7$\pm$7.2 & 0.11$\pm$0.02 &–0.01$\pm$0.02\
22& 3.21 & –71.0$\pm$8.5 & 175.0$\pm$8.1 & 0.04$\pm$0.03 & 0.02$\pm$0.03\
22& 5.57 & –59.4$\pm$10.2 & 147.2$\pm$10.4 &–0.05$\pm$0.03 &–0.02$\pm$0.03\
22& 9.41 & –52.2$\pm$20.9 & 142.0$\pm$43.1 & 0.04$\pm$0.06 &–0.03$\pm$0.16\
22&–0.92 & 93.6$\pm$10.9 & 237.2$\pm$10.9 &–0.05$\pm$0.04 & 0.02$\pm$0.04\
22&–1.51 & 119.3$\pm$8.7 & 203.0$\pm$7.6 &–0.04$\pm$0.03 &–0.05$\pm$0.03\
22&–2.39 & 113.9$\pm$7.6 & 189.0$\pm$5.8 &–0.00$\pm$0.03 &–0.05$\pm$0.03\
22&–3.87 & 112.1$\pm$7.2 & 162.4$\pm$6.4 & 0.03$\pm$0.03 &–0.03$\pm$0.02\
22&–6.23 & 87.8$\pm$9.7 & 139.4$\pm$14.5 & 0.00$\pm$0.05 &–0.03$\pm$0.04\
22&–10.06& 139.1$\pm$29.7 & 160.7$\pm$31.7 &–0.18$\pm$0.12 & 0.11$\pm$0.15\
90& 0.07 & 42.6$\pm$11.7 & 280.1$\pm$11.4 & 0.05$\pm$0.04 & 0.02$\pm$0.04\
90& 0.66 & 49.8$\pm$11.7 & 246.6$\pm$8.6 &–0.03$\pm$0.04 &–0.07$\pm$0.03\
90& 1.25 & 65.5$\pm$8.7 & 218.8$\pm$6.6 &–0.01$\pm$0.02 &–0.01$\pm$0.03\
90& 2.13 & 31.5$\pm$8.3 & 173.5$\pm$7.5 &–0.07$\pm$0.03 &–0.05$\pm$0.02\
90& 3.61 & 42.0$\pm$6.1 & 159.5$\pm$8.6 & 0.00$\pm$0.02 &–0.05$\pm$0.01\
90& 5.67 & 54.3$\pm$14.6 & 133.6$\pm$13.2 &–0.00$\pm$0.04 &–0.04$\pm$0.04\
90& 9.21 & 21.9$\pm$52.0 & 160.0$\pm$75.3 & 0.01$\pm$0.25 & 0.01$\pm$0.25\
90&–0.52 & 38.5$\pm$12.0 & 236.5$\pm$11.0 & 0.05$\pm$0.04 &–0.01$\pm$0.03\
90&–1.11 & 20.6$\pm$10.9 & 223.8$\pm$6.9 & 0.06$\pm$0.04 &–0.04$\pm$0.03\
90&–1.70 & 29.2$\pm$11.0 & 197.3$\pm$10.4 & 0.04$\pm$0.03 &–0.05$\pm$0.03\
90&–2.59 & 20.2$\pm$12.3 & 183.9$\pm$13.6 & 0.06$\pm$0.06 &–0.02$\pm$0.02\
90&–4.06 & 27.1$\pm$10.3 & 155.4$\pm$8.7 & 0.01$\pm$0.03 &–0.03$\pm$0.02\
90&–6.42 & 35.9$\pm$19.3 & 167.6$\pm$18.3 & 0.10$\pm$0.07 &–0.01$\pm$0.05\
90&–10.26& 78.5$\pm$53.7 & 195.8$\pm$93.1 & 0.07$\pm$0.16 &–0.03$\pm$0.27\
|
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abstract: 'We provide an economical description of mass and flavor based on strong interactions and some dynamical assumptions. We include a discussion of CP violation in the quark sector and its relation to neutrino masses.'
author:
- |
B. Holdom [^1]\
\
M5S1A7, CANADA
title: Mass and flavor from strong interactions
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(0,0)(0,0) (310,205)[UTPT-98-07]{} (310,190)[hep-ph/9804312]{}
Introduction
============
Over the years there have two basic approaches to finding the theory of flavor and mass.[^2] One approach is to consider only those theories we fully understand. That is, one constructs theories which either are perturbative, or are based on some rigorous results of strong interactions, such as those emerging in the study of strongly interacting supersymmetric theories. But so far at least, this approach has led to models of mass and flavor which appear overly complicated, with numerous new interactions and/or matter multiplets. The other approach is to consider only those theories with an economical structure. But so far at least, this approach has forced the model builder to make dynamical assumptions about the behavior of strong interactions. In other words, it is not known that the models being proposed actually work as advertised.
It is not surprising that the first approach has proven much more popular over the years; it is preferable to know what one is talking about! On the other hand, the theory of hadronic interactions, QCD, has more in common with the second approach. QCD has a simple and economical structure, and yet it is often difficult to extract physical results. QCD is the theory of the hadronic mass spectrum, but we have yet to see this spectrum fully emerge from a theoretical calculation. Even the concept of confinement is still closer to a dynamical assumption than a rigorously derived result of the theory. But none of this shakes our acceptance of QCD, since there have been other ways to get a handle on QCD which have allowed for experimental checks and confirmation of the theory.
Our experience with QCD thus suggests that it is not necessarily fatal for a theory to rely on plausible dynamical assumptions, as long as the structure of the theory is rigid enough to lead to testable consequences. The correct theory of flavor and mass may be such as to *not* allow for a calculation of the fermion mass spectrum with current tools. But even without being able to provide precise numbers, the following are examples of what we may hope to glean from the correct theory.
- patterns in new flavor dependent effects
- patterns in CP violation
- patterns in neutrino masses
- predictions for the lightest of the new particles
Another common theme in the search for the theory of mass and flavor is to first deal with the question of electroweak symmetry breaking. There are two widely reported approaches to that question, supersymmetry and technicolor, which both provide attractive answers to that single question. These are then taken as the two possible starting points in the search for the theory of mass and flavor. But as indicated in Fig. 1, many obstacles must be overcome in each case before one can approach a comprehensive theory of mass and flavor. In both cases, after the various hurdles are passed, the resulting proposed theories are looking quite complicated and convoluted.
Here we shall consider the possibility that the key to electroweak symmetry breaking is neither supersymmetry or conventional technicolor. We will hope to identify an alternative which leads more simply and naturally to a theory of flavor. The price we will pay is to have electroweak symmetry breaking associated with some aspect of strong interactions which is less familiar to us, i.e. associated with a dynamical assumption.
Our basic picture is as follows.
- A new strong interaction breaks close to a TeV, unlike technicolor which remains unbroken.
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- Associated with this symmetry breaking are the dynamically generated masses for a fourth family of quarks and leptons, which in turn is responsible for electroweak symmetry breaking.
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- The new strong interaction is a remnant flavor interaction, and it only acts on the third and fourth families.
- At a higher “flavor scale”, say 100–1000 TeV, the remnant flavor interaction merges with the full flavor interaction, which involves all quarks and leptons.
The full flavor interaction is some strong, chiral gauge interactions which partially breaks itself. The important point is that this symmetry breaking does not include the breakdown of ${\mathit{SU}(2)_{L}}\times {\mathit{U}(1)_{Y}}$, and thus the known quarks and leptons receive no mass at the flavor scale. The exception are the right-handed neutrinos, which can serve as bilinear order parameters for the flavor breaking. The theory above the flavor scale may also be left-right symmetric, in which case the right-handed neutrino condensates also serve to break ${\mathit{SU}(2)_{L}}\times {\mathit{SU}(2)_{R}}\times
{\mathit{U}(1) _{B - L}}$ down to ${\mathit{SU}(2)_{L}}\times
{\mathit{U}(1)_{Y}}$.
The basic two-scale structure of the model is shown in Fig. 2. The physics at the flavor scale shows up on lower scales through 4-fermion operators and other nonrenormalizable operators. These effects, combined with the mass generation at a TeV, feed down masses from the fourth family to the ligher families. The following are some key ingredients for understanding the origin of a complicated fermion mass spectrum.
- There are a wide variety of possible 4-fermion operators, due to strong-coupled flavor physics, and different operators can contribute to different elements of the mass matrices.
- Operators have various transformation properties under the remnant flavor interaction, and in particular, operators have various numbers of fermions coupling to this interaction.
<!-- -->
- Since the remnant flavor interaction is strong, we can expect that anomalous scaling gives large relative enhancement of operators. This will be one of the sources of quark and lepton mass hierarchies.
One problem which has plagued the technicolor approach has been the difficulty in understanding the origin of the large isospin breaking inherent in the top mass, in a way compatible with the electroweak correction parameter $\delta \rho $. In our approach, isospin breaking originates at the flavor scale, for example through a dynamical breakdown of ${\mathit{SU}(2)_{R}}$. The remnant flavor interaction remaining down to a TeV is isospin preserving and it is this interaction which is responsible for electroweak symmetry breaking. This in itself produces no contribution to $\delta \rho $. Isospin breaking is communicated to the TeV scale via 4-fermion operators, and an operator in particular which must be present is the $t$-mass operator $\overline{\mathit{t'}}\mathit{t'}
\overline{t}t$, where primes denote fourth family members. (It may be that the corresponding $b$-mass operator, $\overline{\mathit{b'}}\mathit{b'}
\overline{b}b$, is generated as a weak radiative ${\mathit{SU}(2)_{R}}$ correction to the $t$-mass operator.) It turns out that the contribution of the $\overline{\mathit{t'}}\mathit{t'}
\overline{t}t$ operator to $\delta \rho $ is suppressed by $({m_{t}}/{m_{\mathit{t'}}})^{4}$ where ${m_{\mathit{t'}}}\approx
1\mathrm{\ TeV}$, and thus the $t$-mass does not directly imply a significant problem for $\delta \rho $. Indeed we are relying on how small the $t$ mass is small relative to the fundamental TeV scale, which differs from the usual emphasis on how large the $t$ mass is. We shall find a dynamical reason as to why the $t$-mass operator is the largest isospin violating operator, due to the anomalous scaling mentioned above.
A minimal model
===============
We will now specify the model in more detail [@a]. The main object here is to show how a complicated fermion mass spectrum can arise from a simple underlying structure. It is sufficient for us to present the minimal model, since we do not have adequate understanding of the strong dynamics to judge which variation of the model will produce the assumed symmetry breaking pattern.[^3] We will consider a 4 family model where the flavor gauge symmetry is ${\mathit{U}(2)_{V}}$.[^4] Two pairs of families transform as $(2, +)$ and $(\overline{2}, -)$ under ${\mathit{SU}(2)_{V}}{\times}{\mathit{U}(1) _{V}}$; we label these two pairs of families as $[Q, L]$ and $[\underline{Q}, \underline{L}]$. The basic structure of the model, including the right-handed neutrino masses which are assumed to occur at the flavor scale and the resulting breakdown of ${\mathit{U}(2)_{V}}$ to ${\mathit{U}(1)_{X}}$, are depicted in Fig. 3. Notice that ${\mathit{U}(1)_{X}}$ couples only to the two heavy families, and that the fermion basis depicted in the figure is not the mass eigenstate basis.
The main dynamical assumption we make is in the form of the fourth family masses. These masses must be generated by the strong, and broken, ${\mathit{U}(1)_{X}}$ interaction along with possible 4-fermion interactions.[^5]
- $\mathit{t'}$ and $\mathit{b'}$ quark masses: ${\overline{\underline{Q}}_{\mathit{L1}}}{Q_{\mathit{R1}}}$
- $\tau '$ mass: ${\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}$
- ${\nu _{\tau '}}$ mass: ${\underline{N}_{\mathit{L1}}^2}$
Now consider the 4-fermion operators which feed these masses to the lighter quark and leptons. We find that interesting results follow from the following subset of operators. Other operators may also contribute, but we assume that this subset dominates. The unique characteristics of these operators can of course provide a dynamical reason for their dominance.
- They have the chiral structure ${\overline{\psi }_{L}}{\psi
_{R}}{\overline{\psi }_{L}}{\psi _{R}}$, and hence must be generated dynamically by the strong flavor interactions.
<!-- -->
- They preserve ${\mathit{SU}(2)_{L}}{\times}{\mathit{U
}(1)_{Y}}$ but display maximal ${\mathit{SU}(2)_{R}}$ breaking.
- They preserve the strong ${\mathit{SU}(2)_{V}}$. They may thus be composed of the ${\mathit{SU}(2)_{V}}$ singlet bilinears: ${\overline{Q}_{\mathit{Li}}}{Q_{\mathit{Ri}}}$, ${\overline{Q}_{\mathit{Li}}}{\underline{Q}_{\mathit{Rj}}}{\varepsilon
_{\mathit{ij}}}$, etc.
Some of these operators break ${\mathit{U}(1)_{X}}$, and we will assume that this generates an $X$ mass of order a TeV, or somewhat higher.
Quark masses
============
We now briefly describe the various operators which are responsible for quark and lepton masses. We first discuss the quark sector. The following operators feed mass down from $\mathit{t'}$ and $\mathit{b'}$ (and in the last case from $t$): $$\left[ {\begin{array}{cc}
{\overline{U}_{\mathit{L1}}}{\mathit{D}_{\mathit{R1}}}{\overline{\underline{D}}_{\mathit{L1}}}{\underline{U}_{\mathit{R1
}}} & {\cal{B}} \\
{\overline{\mathit{D}}_{\mathit{L1}}}{U_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{L1}}}{\underline{D}_{\mathit{R1
}}} & {\tilde{\cal{B}}} \\
{\overline{U}_{\mathit{L1}}}{\mathit{D}_{\mathit{R1}}}{\overline{\underline{D}}_{\mathit{L1}}}{U_{\mathit{R2}}}
& {\cal{C}} \\
{\overline{\mathit{D}}_{\mathit{L1}}}{U_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{L1}}}{\mathit{D}_{\mathit{R2}}
} & {\tilde{\cal{C}}} \\
{\overline{\underline{U}}_{\mathit{L2}}}{\mathit{D}_{\mathit{R1}
}}{\overline{\underline{D}}_{\mathit{L1}}}{\underline{U}_{\mathit{R1}}}
& {\cal{D}} \\
{\overline{\underline{D}}_{\mathit{L2}}}{U_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{L1}}}{\underline{D}_{\mathit{R1
}}} & {\tilde{\cal{D}}} \\
{\overline{\underline{Q}}_{\mathit{Li}}}{U_{\mathit{Rj}}}{\varepsilon
_{\mathit{ij}}}{\overline{\underline{Q}}_{\mathit{Lk
}}}{\mathit{D}_{\mathit{Rl}}}{\varepsilon _{\mathit{kl}}} & {\cal{E}}
\\ {\overline{Q}_{\mathit{Li}}}{\underline{U}_{\mathit{Rj}}}{\varepsilon
_{\mathit{ij}}}{\overline{Q}_{\mathit{Lk}}}{\underline{D}_{\mathit{Rl}}}{\varepsilon
_{\mathit{kl}}} & {\cal{F}}
\end{array}}
\right]$$ while the following operators feed mass down from $\tau '$: $$\left[ {\begin{array}{cc}
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{U}_{\mathit{L1}}}{U_{\mathit{R1}}}
& {{\cal{G}}_{1}} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{U}_{\mathit{L2}}}{U_{\mathit{R2}}}
& {{\cal{G}}_{2}} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{L1}}}{\underline{U}_{\mathit{R1}}}
& {{\cal{H}}_{1}} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{L2}}}{\underline{U}_{\mathit{R2}}}
& {{\cal{H}}_{2}} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{U}_{\mathit{Li}}}{\underline{U}_{\mathit{Rj}}}
{\varepsilon _{\mathit{ij}}} & {\cal{I}} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{Li}}}{U_{\mathit{Rj}}}
{\varepsilon _{\mathit{ij}}} & {\cal{J}}
\end{array}}
\right]$$ The main point is that each operator contributes to a different mass element. $${M_{u}}= \left[ {\begin{array}{cccc} 0 &
{{\cal{G}}_{2}} & {\cal{I}} & 0 \\ {{\cal{H}}_{2}} & {\cal{E}} &
{\cal{D}} & {\cal{J}}
\\ {\cal{I}} & {\cal{C}} & {\cal{B}} & {{\cal{G}}_{1}}
\\ 0 & {\cal{J}} & {{\cal{H}}_{1}} & {\cal{A}}
\end{array}}
\right]$$ $${M_{d}}= \left[ {\begin{array}{cccc} {\cal{F}} & 0 & 0
& 0 \\ 0 & {\cal{E}} & {\tilde{\cal{D}}} & 0 \\ 0 & {\tilde{\cal{C}}} &
{\tilde{\cal{B}}} & 0 \\ 0 & 0 & 0 & {\cal{A}}
\end{array}}
\right]$$ Note that essentially all of the CKM mixing arises in the up sector, and that the mass matrices are not symmetric. Various mass hierarchies arise for the following reasons.
- Various operators experience different power-law scaling enhancements from the strong ${\mathit{U}(1)_{X}}$. Basically, operators containing heavy fermions in both Lorentz and ${\mathit{U}(1)_{X}}$ singlet combinations are expected to be enhanced the most. The ${\cal{B}}$ operator, which is the $t$-mass operator, is expected to be the largest. $$\begin{aligned}
&&{\cal{B}}>{\cal{C}},
{\cal{D}}>{\cal{E}}\\&&{{\cal{G}}_{1}}, {{\cal{H}}_{1}}>{\cal{I}},
{\cal{J}}>{{\cal{G}}_{2}}, {{\cal{H}}_{2}}\end{aligned}$$
- There are different heavy masses, ${m_{\mathit{t',b'}}}>{m_{\tau ^{'
}}}>{m_{t}}$, being fed down. $$\begin{aligned}
&&{\cal{E}}>{\cal{F}}\\&&{\cal{B}}>{{\cal{G}}_{1}},
{{\cal{H}}_{1}}\\&&{\cal{C}}, {\cal{D}}>{\cal{I}},
{\cal{J}}\end{aligned}$$
- ${\tilde{\cal{B}}}$, ${\tilde{\cal{C}}}$ and ${\tilde{\cal{D}}}$ can arise from weak radiative corrections (from ${\mathit{SU}(2)_{R}}$). $${\cal{B}}, {\cal{C}}, {\cal{D}}>{\tilde{\cal{B}}},
{\tilde{\cal{C}}}, {\tilde{\cal{D}}}$$
- Operators are affected differently by the axial interaction mentioned in footnote 3. $${\cal{G}}>{\cal{H}}$$
We get one approximate relation due to the similarity of the ${\cal{E}}$ and ${\cal{F}}$ operators. $$\frac {{m_{d}}}{{m_{s}}}\approx \frac
{{m_{t}}}{{m_{\mathit{t'}}}}$$
Lepton masses
=============
We now turn to the charged lepton mass matrices where we find that the mixed quark-lepton operators play a crucial role. The following operators feed mass down from $\mathit{t'}$, $$\left[ {\begin{array}{cc}
{\overline{E}_{\mathit{L1}}}{U_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}
& {{\cal{B}}_{{\ell}}} \\
{\overline{E}_{\mathit{L1}}}{U_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{L1}}}{E_{\mathit{R2}}}
& {{\cal{C}}_{{\ell}}} \\
{\overline{\underline{E}}_{\mathit{L2}}}{U_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1
}}} & {{\cal{D}}_{{\ell}}} \\
{\overline{\underline{E}}_{\mathit{L2}}}{U_{\mathit{R1}}}{\overline{\underline{U}}_{\mathit{L1}}}{E_{\mathit{R2}}}
& {{\cal{E}}_{{\ell}}}
\end{array}}
\right]$$ while the following operators feed mass down from $t$. $$\left[ {\begin{array}{cc}
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{U}_{\mathit{R1}}}{\overline{U}_{\mathit{L1}}}{E_{\mathit{R1}}}
& {{\cal{F}}_{{\ell}}} \\
{\overline{E}_{\mathit{L2}}}{\underline{U}_{\mathit{R1}}}{\overline{U}_{\mathit{L1}}}{E_{\mathit{R1}}}
& {{\cal{G}}_{{\ell}}} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{U}_{\mathit{R1}}}{\overline{U}_{\mathit{L1}}}{\underline{E}_{\mathit{R2}}}
& {{\cal{H}}_{{\ell}}} \\
{\overline{E}_{\mathit{L2}}}{\underline{U}_{\mathit{R1}}}{\overline{U}_{\mathit{L1}}}{\underline{E}_{\mathit{R2}}}
& {{\cal{I}}_{{\ell}}}
\end{array}}
\right]$$ The following operators are the only ones we mention which are generated by ${\mathit{SU}(2)_{V}}$ exchange, and they feed mass down from the $\tau '$ and $\tau $. $$\left[ {\begin{array}{cc}
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{E}}_{\mathit{R2}}}{\underline{E}_{\mathit{L2}}}
& {{\cal{J}}_{{\ell}}} \\
{\overline{E}_{\mathit{L1}}}{E_{\mathit{R1}}}{\overline{E}_{\mathit{R2}}}{E_{\mathit{L2}}}
& {{\cal{K}}_{{\ell}}}
\end{array}}
\right]$$ Here is the resulting matrix. $${M_{{\ell}}}= \left[ {\begin{array}{cccc}
{{\cal{K}}_{{\ell}}} & {{\cal{I}}_{{\ell}}} & {{\cal{G}}_{{\ell}}} & 0 \\
{{\cal{E}}_{{\ell}}} & {{\cal{J}}_{{\ell}}} & 0 & {{\cal{D}}_{{\ell}}} \\
{{\cal{C}}_{{\ell}}} & 0 & 0 & {{\cal{B}} _{{\ell}}} \\ 0 &
{{\cal{H}}_{{\ell}}} & {{\cal{F}}_{{\ell}}} & {{\cal{A}}_{{\ell}}}
\end{array}}
\right]$$ The $\mu $ mass is reasonable, $${m_{\mu }}\approx \frac
{(1\mathrm{TeV})^{3}}{(100\mathrm{TeV}
)^{2}}{\label{mu}}$$ and there is a relation due to the similarity of the ${{\cal{J}}_{{\ell}}}$ and ${{\cal{K}}_{{\ell}}}$ operators: $$\frac {{m_{e}}}{{m_{\mu }}}\approx \frac {{m_{\tau
}}}{{m_{\tau '}}}$$
We now turn to neutrinos. We have already mentioned that the RH neutrinos have mass at the flavor scale, and that the 4th LH neutrino has a dynamical mass in the 100 GeV to 1 TeV range. The remaining 3 LH neutrino masses can only come from 6-fermion operators. For example, the operator $${\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{N}_{\mathit{L2}}}{\overline{N}_{\mathit{L2}}}$$ is generated by two ${\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{N}_{\mathit{L2}}}{N_{\mathit{R2}}}$ operators after integrating out the large ${N_{\mathit{R2}}}$ mass. The result is that the $\tau '$ mass feeds down to produce a small ${N_{\mathit{L2}}}$ (i.e. ${\nu _{e}}$) mass.[^6] This is very similar to the standard see-saw mechanism involving scalar fields, except that the dimensions of the operators involved here are much larger. This allows the right-handed neutrino mass scale to be at the relatively low flavor scale we are discussing.
The whole set of 4-fermion operators which can contribute in this way to neutrino masses are: $$\left[ {\begin{array}{cc}
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{N}_{\mathit{L2}}}{N_{\mathit{R2}}}
& {{\cal{B}}_{\nu }} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{N}}_{\mathit{L2}}}{\underline{N}_{\mathit{R2}}}
& {{\cal{C}}_{\nu }} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{N}_{\mathit{L1}}}{\underline{N}_{\mathit{R2}}}
& {{\cal{D}}_{\nu }} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{N}_{\mathit{L1}}}{N_{\mathit{R1}}}
& {{\cal{E}}_{\nu }} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{N}}_{\mathit{L2}}}{N_{\mathit{R1}}}
& {{\cal{F}}_{\nu }} \\
{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{N}_{\mathit{L2}}}{\underline{N}_{\mathit{R1}}}
& {{\cal{G}}_{\nu }}
\end{array}}
\right]$$ After labelling the heavy right-handed neutrino masses as $$\left[ {\begin{array}{cc} {N_{\mathit{R2}}^2} & {m_{1}} \\
{\underline{N}_{\mathit{R2}}^2} & {m_{2}} \\
{N_{\mathit{R2}}}{\underline{N}_{\mathit{R2}}} & {m_{3}} \\
{N_{\mathit{R1}}}{\underline{N}_{\mathit{R1}}} & {m_{4}}
\end{array}}
\right]$$ we find for the light neutrino mass matrix the following. $$\left[ {\begin{array}{ccc} {\displaystyle \frac {{{\cal{B}}_{\nu
}^2}}{{m_{1}}}} & {\displaystyle \frac {{{\cal{B}}_{\nu
}}{{\cal{C}}_{\nu } }}{{m_{3}}}} + {\displaystyle \frac {{{\cal{F}}_{\nu
}}{{\cal{G}}_{\nu }}}{{m_{4}}}} & {\displaystyle \frac
{{{\cal{B}}_{\nu }}{{\cal{D}}_{\nu }}}{{m_{3}}}} + {\displaystyle
\frac {{{\cal{E}}_{\nu }}{{\cal{G}}_{\nu } }}{{m_{4}}}} \\[2ex]
{\displaystyle \frac {{{\cal{B}}_{\nu }}{{\cal{C}}_{\nu } }}{{m_{3}}}}
+ {\displaystyle \frac {{{\cal{F}}_{\nu }}{{\cal{G}}_{\nu }}}{{m_{4}}}}
& {\displaystyle \frac {{{\cal{C}}_{\nu }^2}}{{m_{2}}}} & {\displaystyle
\frac {{{\cal{C}}_{\nu }}{{\cal{D}}_{\nu }}}{{m_{2}}}} \\[2ex]
{\displaystyle \frac {{{\cal{B}}_{\nu }}{{\cal{D}}_{\nu } }}{{m_{3}}}}
+ {\displaystyle \frac {{{\cal{E}}_{\nu }}{{\cal{G}}_{\nu }}}{{m_{4}}}}
& {\displaystyle \frac {{{\cal{C}}_{\nu }}{{\cal{D}}_{\nu
}}}{{m_{2}}}} & {\displaystyle \frac {{{\cal{D}}_{\nu
}^2}}{{m_{2}}}}
\end{array}}
\right]$$ We see that this matrix bears no resemblence to quark or charged lepton mass matrices. Large mixings are expected, with masses unrelated to the family hierarchy. CP violation is also expected, and it is to that topic which we now turn.
CP violation
============
We first note that the quantity most sensitive to possible CP violation in new 4-fermion effects is the $\varepsilon $ parameter in the $K$-$\overline{K}$ system. In other words we have a natural setting for a superweak model of CP violation. Superweak models are especially attractive in the context of the strong CP problem, since they allow for the quark mass matrix to be real, or very close to it, which would account for why the strong CP violating parameter $\overline{\theta }$ is close to vanishing. When we consider how this can arise in the present model, we find that CP violation in the quark sector may arise in a way similar to neutrino masses; that is, via 6-fermion operators only. This provides a natural suppression mechanism which can go a long way towards suppressing strong CP violation to acceptable levels.
In our picture we assume that above the flavor scale we have a CP invariant gauge theory of massless fermions. Our dynamical assumption is that lepton-number violation, ${\mathit{SU}(2)_{V}}$ breaking and CP violation all originate in the right-handed neutrino condensates (both bilinear and multilinear). We may then consider the operators which feed CP violation feed into the quark sector. It can be shown that they must violate lepton-number or ${\mathit{SU}(2)_{V}}$ or both. This in turn requires 6-fermion operators, of which the following are two examples. $$\begin{aligned}
&&{\overline{\mathit{D}}_{\mathit{L2}}}{\mathit{D}_{\mathit{R2}}
}{\overline{\mathit{D}}_{\mathit{L2}}}{\mathit{D}_{\mathit{R2
}}}{\overline{\underline{N}}_{\mathit{L1}}}{\overline{\underline{N}}_{\mathit{L1}}}\\&&{\overline{\underline{D}}_{\mathit{L2}}}{\underline{D}_{\mathit{R2}}}{\overline{\underline{D}}_{\mathit{L2}}}{\underline{D}_{\mathit{R2}}}{\overline{\underline{N}}_{\mathit{L1}}}{\overline{\underline{N}}_{\mathit{L1}}}\end{aligned}$$ From the mass matrices we have given it can be seen that in the presence of the heavy ${\nu _{\tau '}}$ mass, these generate the $\Delta S=2$ operators $({\overline{d}_{L}}{s_{R}})^{2}$ and $({\overline{s}_{L}}{d_{R}})^{2}$. If the 6-fermion operators have coefficients of order $1/(100\mathrm{\ TeV})^{5}$ and $\langle
{\underline{N}_{\mathit{L1}}^2}\rangle \approx ( 1\mathrm{\
TeV})^{3}$, then the coefficients of the $\Delta S=2$ operators are of the right size to give $\varepsilon $ in $K$–$\overline{K}$ mixing. $\varepsilon '$ on the other hand requires $d$–$s$ mass mixing, and thus is negligible.
The following operators $$\begin{aligned}
&&{\overline{\mathit{D}}_{\mathit{L2}}}{\underline{D}_{\mathit{R1
}}}{\overline{\mathit{D}}_{\mathit{L2}}}{\underline{D}_{\mathit{R1}}}{\overline{\underline{N}}_{\mathit{L1}}}{\overline{\underline{N}}_{\mathit{L1}}}\\&&{\overline{\mathit{D}}_{\mathit{L1}}}{\underline{D}_{\mathit{R2
}}}{\overline{\mathit{D}}_{\mathit{L1}}}{\underline{D}_{\mathit{R2}}}{\overline{\underline{N}}_{\mathit{L1}}}{\overline{\underline{N}}_{\mathit{L1}}}\end{aligned}$$ correspond to the $\Delta b=2$ operators $({\overline{d}_{L}}{b_{R}})^{2}$ and $({\overline{b}_{L}}{d_{R}})^{2}$. These should similar in size to the $\Delta S=2$ operators, in which case the CP violation in the $b$ sector is 3 or 4 orders of magnitude smaller than in the standard model. We have recovered a classic superweak model.
The following 6-fermion operators can also feed CP violation into the quark masses, and thus into $\overline{\theta }$. $$\begin{aligned}
&&{\overline{\mathit{D}}_{\mathit{L2}}}{\mathit{D}_{\mathit{R2}}
}{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{N}}_{\mathit{L1}}}{\overline{\underline{N}}_{\mathit{L1}}}\\&&{\overline{\underline{D}}_{\mathit{L2}}}{\underline{D}_{\mathit{R2}}}{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{N}}_{\mathit{L1}}}{\overline{\underline{N}}_{\mathit{L1}}}\\&&{\overline{\mathit{D}}_{\mathit{Li}}}{\underline{D}_{\mathit{Rj
}}}{\varepsilon
_{\mathit{ij}}}{\overline{\underline{E}}_{\mathit{L1}}}{\underline{E}_{\mathit{R1}}}{\overline{\underline{N}}_{\mathit{L1}}}{\overline{\underline{N}}_{\mathit{L1}}}\end{aligned}$$ Thus the CP violating parts of quark masses can be of similar magnitude to the neutrino masses. This by itself does not sufficiently suppress $\overline{\theta }$, but the detailed structure of the quark mass matrices can lead to further suppression.
Conclusion
==========
There can be many other effects of the new flavor physics, through nonrenormalizable effects and in particular through the effects of the $X$ boson. For example we can expect anomalous couplings of standard model gauge bosons to the third family. Flavor changing effects may surface in $B$–$\overline{B}$ and $\mathit{D}$–$\overline{\mathit{D}}$ mixing, with the result that the $B$ factories may uncover flavor changing effects rather than CP violation.
In conclusion, we have bypassed the usual approaches to electroweak symmetry breaking and proceeded straight to the flavor problem. We have suggested that there is a dynamically broken flavor gauge symmetry around 100 to 1000 TeV which generates a wide variety of multi-fermion operators. Close to a TeV the remnant flavor symmetry breaks, fourth family masses arise, and electroweak symmetry breaking occurs. We have explored the interplay between quark and lepton sectors in the generation of mass matrices. We have also seen how the suppression of CP violation in the quark sector is similar to the suppression of neutrino masses. One of the first signals of this picture could be the absence of CP violation at B factories.
Acknowledgement {#acknowledgement .unnumbered}
===============
This research was supported in part by the Natural Sciences and Engineering Research Council of Canada. I thank the organizers of this workshop for their support, and the KEK theory group where this report was prepared.
[99]{} B. Holdom, Phys. Rev. **D57** (1998) 357. B. Holdom and F. S. Roux, hep-ph/9804311. B. Holdom and T. Torma, in preparation.
[^1]: Talk given at the workshop on Fermion Mass and CP violation, Hiroshima, Japan, March 1998.
[^2]: Theories which simply parametrize fermion masses, such as the standard model, do not qualify as theories of flavor and mass.
[^3]: But see [@b]. Less minimal versions will likely have a new sector of strongly interacting fermions which play little role in quark and lepton mass generation. The most minimal model would seem to be preferred by the constraints on $S$ and $T$, but see [@c].
[^4]: The subscript reminds us that this is a vectorial interaction in a certain basis. An additional axial interaction at the flavor scale, which plays little role in our discussion, is needed to make the strong interaction chiral.
[^5]: The fact that the 4th family and not the 3rd family masses form must be due to a cross-channel coupling, which should be familiar to builders of multi-Higgs potentials. Note that the $\tau '$ mass forms in the ${\mathit{U}(1)_{X}}$-singlet channel, unlike the $q'$ masses, which may be due to flavor induced 4-fermion operators which distinguish quarks and leptons and which are enhanced by ${\mathit{U}(1)_{X}}$ anomalous scaling, e.g. $\overline{{\tau
_{L}}'}{\tau _{R}}'
\overline{{\tau _{R}}'}{\tau _{L}}'$.
[^6]: Using numbers similar to those in (\[mu\]), and accounting for the anomalous scaling enhancement built into those numbers, can yield neutrino masses in the eV range. We also note that in comparing these operators to those containing quark fields, the latter operators may be dynamically favored due to QCD effects.
|
---
abstract: 'Belonging to the group of B\[e\] stars, V921Scorpii is associated with a strong infrared excess and permitted and forbidden line emission, indicating the presence of low- and high-density circumstellar gas and dust. Many aspects of V921Sco and other B\[e\] stars still remain mysterious, including their evolutionary state and the physical conditions resulting in the class-defining characteristics. In this Letter, we employ VLTI/AMBER spectro-interferometry in order to reconstruct high-resolution ([$\lambda/2B=0.0013$]{}) model-independent interferometric images for three wavelength bands around 1.65, 2.0, and 2.3 $\mu$m. In our images, we discover a close ($25.0 \pm 0.8$ milliarcsecond, corresponding to $\sim 29 \pm 0.9$ AU at 1.15 kpc) companion around V921Sco. Between two epochs in 2008 and 2009, we measure orbital motion of $\sim 7^{\circ}$, implying an orbital period of $\sim 35$ years (for a circular orbit). Around the primary star, we detect a disk-like structure with indications for a radial temperature gradient. The polar axis of this AU-scale disk is aligned with the arcminute-scale bipolar nebula in which V921Sco is embedded. Using Magellan/IMACS imaging, we detect multi-layered arc-shaped sub-structure in the nebula, suggesting episodic outflow activity from the system with a period of $\sim 25$ years, roughly matching the estimated orbital period of the companion. Our study supports the hypothesis that the B\[e\] phenomenon is related to dynamical interaction in a close binary system.'
author:
- 'Stefan Kraus, Nuria Calvet, Lee Hartmann, Karl-Heinz Hofmann, Alexander Kreplin, John D. Monnier, and Gerd Weigelt'
title: |
On the nature of the Herbig B\[e\] star binary system V921Scorpii:\
Discovery of a close companion and relation to the large-scale bipolar nebula
---
Introduction
============
B\[e\] stars are intermediate-mass stars associated with substantial amounts of circumstellar gas and dust, as indicated by permitted and forbidden line emission, in particular of \[\] and \[\], and a strong infrared excess [@all76]. These class-defining characteristics have been observed in a wide range of evolutionary stages [@lam98], including pre-main-sequence stars (Herbig Ae/B\[e\]), post-main-sequence stars (supergiants, symbiotic stars, or compact planetary nebulae), and stars of unknown nature (unclassified B\[e\] stars). In the pre-main-sequence stage, about half of all intermediate-mass young stars show the [*B\[e\] phenomenon*]{} [@oud06], although significant differences in the strength of the forbidden line emission can be observed.
It has been proposed that the B\[e\] phenomenon might be related to the presence of a close binary system, where the line-emitting gas is ejected in recurring mass-loss events triggered by the companion [e.g. @she00; @mir07]. To test this hypothesis, it is important to search for close companions around B\[e\] stars. Given the kiloparsec distance of B\[e\] stars, this task requires high angular resolution, which we achieve in this Letter using infrared interferometry.
We observed the unclassified B\[e\] star V921Sco using the Very Large Telescope Interferometer (VLTI). The AU-scale environment around this B0Ve-type star has been investigated by two earlier interferometric studies, which determined the spatial extension of the Br$\gamma$-line emitting region [@kra08b] and the structure of the continuum-emitting disk [@kre12]. V921Sco is associated with an intriguing nebula, which shows reflection as well as absorption characteristics [@hut90] and which has been imaged both at visual [@hut90] and mid-infrared wavelengths [@boe09]. There is some debate on the distance [$\sim 1.15$ kpc, @bor07] and nature of V921Sco, with authors arguing both for an evolved [@hut90; @bor07] and young [@ben98; @hab03; @ack05; @ack06] evolutionary stage.
In this Letter, we report on Magellan wide-field imaging and VLTI aperture synthesis imaging observations (Sect. \[sec:observations\]), which reveal the presence of a close companion around V921Sco. The astrometry of the binary system and the orientation of the circumprimary disk will be derived using quantitative modeling (Sect. \[sec:continuum\]) and interpreted in Sect. \[sec:interp\]. A summary of our results will be presented in Sect. \[sec:conclusions\].
Observations {#sec:observations}
============
VLTI/AMBER spectro-interferometry {#sec:obsAMBER}
---------------------------------
Our observations on V921Sco were carried out using the AMBER instrument [@pet07], which allowed us to coherently combine the light from three of the VLTI 1.8m auxiliary telescopes. The observations were conducted using AMBER’s low spectral resolution mode, which covers the near-infrared $H$- and $K$-bands ($1.44$ to 2.50 $\mu$m) with a spectral resolution $R \sim 35$. The observations were carried out between 2008-04-03 and 2009-03-19 on four different array configurations (Tab. \[tab:obslog\]), providing a good $uv$-coverage with baseline lengths between 10 and 127m. For all observations, we used a detector integration time (DIT) of 100ms.
For a significant fraction of our data we find that the closure phases (CPs) vary significantly between subsequent exposures (i.e. on time scales of several minutes). As discussed in Sect. \[sec:continuum\], these rapid variations are very likely due to the presence of a companion star. Therefore, in contrast to the standard AMBER data reduction procedure, we decided not to average the individual exposures, but to fit the quantities derived from the individual data exposures separately.
Each observation on V921Sco was accompanied by observations on interferometric calibrator stars of known intrinsic diameter, allowing us to monitor the instrumental and atmospheric transfer function. Both for the science and calibrator star observations, we extract raw visibilities and CPs using the amdlib (V3.0) data reduction software [@tat07b; @che09]. In order to associate the CP sign with the on-sky orientation, we use a reference data set[^1] on the binary star $\theta^1$Orionis C [@kra09a].
[ccccc]{} 2008-05-21 & 09:20 & 10 & A0-D0-H0 & HD159941\
2008-05-22 & 04:18 & 9 & A0-D0-H0 & HD159941\
2008-05-22 & 05:25 & 9 & A0-D0-H0 & HD159941\
2008-05-22 & 06:47 & 9 & A0-D0-H0 & HD159941\
2008-05-24 & 06:29 & 11 & A0-D0-H0 & HD159941\
2008-05-24 & 09:24 & 9 & A0-D0-H0 & HD159941\
2008-09-21 & 01:54 & 5 & A0-K0-G1 & HD159941\
2008-04-28 & 08:32 & 5 & D0-H0-G1 & HD163197\
2008-04-28 & 09:18 & 5 & D0-H0-G1 & HD163197\
2008-05-26 & 07:28 & 5 & D0-H0-G1 & HD159941\
2008-07-04 & 02:59 & 5 & D0-H0-G1 & HD159941\
2008-07-04 & 04:02 & 5 & D0-H0-G1 & HD159941\
2008-07-04 & 04:46 & 5 & D0-H0-G1 & HD159941\
2008-07-05 & 01:15 & 5 & D0-H0-G1 & HD159941\
2008-07-05 & 06:24 & 5 & D0-H0-G1 & HD159941\
2008-04-03 & 05:52 & 8 & E0-G0-H0 & HD152040\
2008-04-05 & 07:16 & 5 & E0-G0-H0 & HD159941\
2008-04-05 & 08:08 & 12 & E0-G0-H0 & HD159941\
2008-06-04 & 07:03 & 5 & E0-G0-H0 & HD159941\
2008-06-04 & 08:54 & 5 & E0-G0-H0 & HD159941\
2008-06-07 & 08:32 & 5 & E0-G0-H0 & HD159941\
2009-02-18 & 09:05 & 5 & D0-H0-G1 & HD159941\
2009-02-19 & 08:58 & 5 & D0-H0-G1 & HD159941\
2009-03-19 & 08:54 & 15 & D0-H0-G1 & HD159941
Magellan/IMACS+FIRE wide-field imaging {#sec:obsIMACS}
--------------------------------------
In order to investigate the large-scale environment around V921Sco, we employed the IMACS instrument on the Magellan/Baade 6.5m telescope. The images were recorded on 2011-03-13 under exceptional atmospheric conditions (seeing FWHM $\sim$0.4) and cover a field of [$15.4$]{} with a pixel size of 0.11/pixel. We employed the Bessell B, V, and R filters and two narrowband filters centered around the and H$\alpha$ line (filters “676circular” and “Halpha656”) using DITs of 240s, 240s, 180s, 300s, and 360s, respectively. The images were bias-subtracted and flat-fielded using standard IRAF data reduction routines and are shown in Fig. \[fig:IMACS\][*A-F*]{}.
Using the acquisition camera of the FIRE spectrograph [@sim08] at Magellan/Clay we also recorded on 2011-03-12 a $J$-band image of V921Sco (394$\times$394 pixels with a pixel size 0.147/pixel), which is shown in Fig. \[fig:IMACS\][*H*]{}.
Results: Continuum geometry {#sec:continuum}
===========================
Aperture-synthesis imaging {#sec:imaging}
--------------------------
![ Aperture-synthesis images constructed without [*(left column)*]{} and with maximum entropy regularization [*(right column)*]{} from our VLTI/AMBER data for the $H$ [*(top)*]{}, $K_1$ [*(2nd from top)*]{}, and $K_2$ [*(3rd from top)*]{} wavelength bins. The contours decrease from peak intensity by factors of $\sqrt{2}$. In the bottom row, the three wavelength bins are merged in a color composite (red: $K_2$, green: $K_1$, blue: $H$). All images were convolved to the formal resolution $\lambda/2B$, which is 1.3 mas ($H$), 1.6 mas ($K_1$), and 1.8 mas ($K_2$). []{data-label="fig:imaging"}](fig2.eps)
In order to derive the basic source structure of V921Sco, we reconstructed model-independent aperture-synthesis images from our AMBER data. Since the object structure might change with wavelength, we subdivide our data set in three wavelength bins, which we denote with $H$ ($1.4 \leq \lambda < 1.9~\mu$m), $K_{1}$ ($1.9 \leq \lambda < 2.15~\mu$m), and $K_{2}$ ($2.15 \leq \lambda < 2.5~\mu$m). For each wavelength bin, we reconstruct a separate image (Fig. \[fig:imaging\], [*top*]{}) and then combined the independent images in a color composite (Fig. \[fig:imaging\], [*bottom*]{}).
For the image reconstruction, we employed the building block mapping algorithm [@hof93], which was already used in several of our earlier long-baseline interferometric imaging projects [@kra07; @kra09a; @kra10]. The presented images were obtained without (Fig. \[fig:imaging\], [*left*]{}) and with (Fig. \[fig:imaging\], [*right*]{}) regularisation function. Image reconstruction with regularisation means the minimization of the cost function $$J[o_k({\bf x})] := Q[o_k({\bf x})] + \mu\cdot H[o_k({\bf x})],$$ where $Q[o_k({\bf x})]$ describes the $\chi^2$ function of the measured bispectrum data $O^{(3)}({\bf f_u},{\bf f_v})$ and the bispectrum of the actual iterated image $o_k({\bf x})$. $H[o_k({\bf x})]$ is a regularisation term, and $\mu$ is a weighting factor called the Lagrange multiplicator. For our reconstructions we used the maximum entropy regularisation function $$H[o_k({\bf x})] := \int \left[o_k({\bf x})\cdot \log\left({\frac{o_k({\bf x})}{p({\bf x})}}\right) - o_k({\bf x}) + p({\bf x})\right] d{\bf x}.$$ As prior function $p({\bf x})$ we used a smooth version of a building block reconstruction obtained without a regularisation function ($\mu = 0$). The start image was a circumsymmetric Gaussian with a size obtained by fitting the measured visibilities. Reconstructions for different Lagrange multiplicators ($\mu = 10^{-6}$ to $10^{-3}$) and for different reconstruction windows (radii: 60mas to 100mas) were obtained. The presented images are the best fit reconstructions, i.e. those reconstructions with minimum reduced $\chi^2$ values of the squared visibilities and CPs.
Each of the reconstructed images clearly reveals a close companion, which is located at a separation of $\sim 25$ mas north of the primary star[^2] ($\Theta \sim 353{\deg}$) and which we denote in the following V921Sco B. In our images, V921Sco B appears point-like, while the primary star is clearly associated with extended emission that appears elongated along position angle (PA) $\phi \sim 145{\deg}$. In the images, it is also evident, that the contributions of V921Sco B to the total flux decrease with wavelength.
Model fitting {#sec:modeling}
-------------
In order to better characterize the geometry and physical conditions of the disk-like structure in our image and to derive the relative astrometry of V921Sco A-B, we fitted geometric models to our interferometric data.
In all models, the primary star is included as a point source, where the photospheric emission at a given wavelength is given by the flux ratio of a B0 ($T_{\rm eff}=14,000$ K, $g=4.04$) Kurucz model atmosphere [@kur70] to the measured total SED flux $F_{\rm tot}$, with contributions $F_{\rm A}/F_{\rm tot}$ ranging from 0.27 (at 1.5 $\mu$m), 0.13 (at 2.0 $\mu$m) to 0.07 (at 2.5 $\mu$m), where $F_{\rm A}$ denotes the photospheric flux contribution of the primary star. We adopt $A_V=4.8\pm0.2$, a stellar radius of $R_{\star} = 17.3\pm0.6~R_{\sun}$, and a distance of $d=1150\pm150$ pc [@bor07]. Given that the spectral type of the primary was determined in earlier studies without knowledge of the companion star, we note that these spectral type estimates are potentially biased, which could affect the photospheric flux of the primary star in our models. However, the absolute level and wavelength-dependence of the photospheric emission in the near-infrared wavelength regime depends only weakly on the precise spectral type and will therefore not significantly affect our modeling results.
The companion star is described by five free parameters, namely the PA ($\Theta$) and separation ($\rho$) measured from the primary star, the angular extent of the circumstellar material (parameterized as Gaussians with FWHM $\theta_{B}$), and two parameters ($\frac{F_{\rm B}}{F_{\rm tot}}(\lambda_{\rm ref})$, $s$) to describe the photospheric flux contributions of V921Sco B to the total flux. The flux contributions might change with wavelength due to differences in the stellar effective temperatures, local extinction effects, or circumstellar emission. Given the still relatively short wavelength coverage, we approximate these effects with the linear relation $\frac{F_{\rm B}}{F_{\rm tot}}(\lambda) = \frac{F_{\rm B}}{F_{\rm tot}}(\lambda_{\rm ref}) + s \cdot (\lambda - \lambda_{\rm ref})$, where we choose $\lambda_{\rm ref}:=2~\mu$m as arbitrary reference wavelength.
Free parameters of the circumprimary disk are the PA ($\phi$), angular extend along the major axis (Gaussian FWHM $\theta_{A}$), and inclination angle ($i$, measured from the polar axis). Due to temperature gradients in the circumstellar material, it is possible that the source brightness distribution might change with wavelength. To test this hypothesis, we performed our geometric fits both to the complete data set and to the aforementioned data subsets ($H$, $K_1$, $K_2$ wavelength bins).
In order to find the best fit, we employ a Levenberg-Marquardt least square fitting procedure and minimize the likelihood-estimator $\chi_{r}^2 = \chi_{r,V}^2 + \chi_{r,\Phi}^2$, where $\chi_{r,V}^2$ and $\chi_{r,\Phi}^2$ are the reduced least square between the measured and model visibilities and CPs, respectively (eqs. 1 and 2 in @kra09b). For our best-fit solution (Tab. \[tab:modelfitting\]), the uncertainties on the individual parameters have been estimated using the bootstrapping technique. The fit provides a slightly better representation of the CP than the visibility-data, as indicated by the $\chi_{r,V}^2$ and $\chi_{r,\Phi}^2$-values. Likely, this indicates that the circumprimary disk geometry is not well represented by a Gaussian-brightness distribution and we will consider more physical models in an upcoming study (Kraus et al., in prep.).
Given that the position of the companion might change notably between 2008 and 2009, we first fitted only the 2008 data in order to characterize the properties of the stellar components and the circumstellar material.
[cccccccccccccc]{} 2008 & $H$ & 353.0 & 25.0 & $\leq 0.5$ & 0.074 & 0 && 6.12 & 51.1 & 143.2 & 2.46 & 1.40 & 1.73\
2008 & $K_1$ & 353.4 & 25.0 & $\leq 0.5$ & 0.070 & 0 && 6.85 & 48.6 & 143.5 & 2.50 & 2.14 & 2.25\
2008 & $K_2$ & 353.1 & 25.0 & $\leq 0.5$ & 0.064 & 0 && 7.55 & 48.5 & 149.0 & 3.91 & 1.97 & 3.27\
2008 & all & $353.8$ & $25.0$ & $\leq 0.2$ & $0.054$ & $-0.0056$ && $7.5$ & $50.3$ & $147.8$ & 4.30 & 3.09 & 4.88\
& & $\pm1.6$ & $\pm0.8$ & & $\pm0.018$ & $\pm0.019$ && $\pm0.2$ & $\pm1.9$ & $\pm4.3$ & & &\
2009 & all & $347.3$ & $25.5$ & $\leq 0.2$ & $0.054$ & $-0.0056$ && $7.5$ & 50.3 & 147.8 & 4.17 & 1.78 & 3.39\
& & $\pm1.0$ & $\pm1.2$ & & & && & & & & &
Orbital motion measurement {#sec:orbitalmotion}
--------------------------
Following the detailed characterization of the V921Sco system for epoch 2008 (Sect. \[sec:modeling\]), we then investigated whether we find evidence for orbital motion between 2008 and 2009. Compared to 2008, a significantly smaller number of AMBER observations was recorded in 2009 (Tab. \[tab:obslog\]), which lead us to fix the geometry of the circumprimary disk using the best-fit parameters from epoch 2008 and treat only the two astrometric parameters ($\Theta$, $\rho$) and the flux ratio ($F_{\rm B}/F_{\rm tot}$) as free fitting parameters.
The resulting best-fit parameters can be found in Tab. \[tab:modelfitting\] and indicate that the position of the secondary has moved significantly in PA ($\sim 7$) during the covered $\sim 8$ months, while only marginal changes in separation ($\rho \lesssim 0.5$ mas) or in the flux ratio could be detected over this time period.
Interpretation {#sec:interp}
==============
Characterization of the detected companion star {#sec:interpcompanion}
-----------------------------------------------
Assuming a face-on circular orbit, we can estimate from the detected signs of orbital motion (Tab. \[tab:modelfitting\]) the period of the orbit to $P \sim 35$ yrs. Of course, for a precise mass determination, long-term follow-up observations will be necessary in order to derive the full dynamical orbit.
Both in our images and our model fits, the companion appears spatially unresolved (Gaussian FWHM $<0.3$ mas, corresponding to $<0.35$ AU at 1.15 kpc), which suggests that V921Sco B is not associated with circumstellar material. Based on this result, we can compute the flux ratio of V921Sco A and B for the $H$-band ($(F_{\rm B}/F_{\rm A})_{H}=0.83\pm0.15$) and $K$-band ($(F_{\rm B}/F_{\rm A})_{K}=1.18\pm0.12$). These values suggest that V921Sco B is of cooler temperature and later spectral type than the primary V921Sco A, although the current uncertainties are still too large for a quantitative spectral classification.
Relation with large-scale structures {#sec:interplargescale}
------------------------------------
Our IMACS narrowband and broadband images (Fig. \[fig:IMACS\]) reveal a remarkably complex environment around V921Sco. The overall shape of the nebula is bipolar, where the south-western part appears much fainter and less extended, maybe indicating that this part is facing away from the observer and therefore suffers from a larger amount of obscuration from material in the ambient cloud. Obscuration from fore-ground material might also be responsible for the dark filaments which appear in the south-eastern part of the nebula (shaded area in Fig. \[fig:IMACS\][*G*]{}).
Remarkably, the symmetry axis of the bipolar nebula ($50\pm5^{\circ}$) appears well aligned with the polar axis ($57.8\pm4.3^{\circ}$) of the AU-scale disk detected with interferometry, suggesting that the nebula might have been shaped through outflow-activity from V921Sco. From our kinematical modeling (Kraus et al., in prep.), we conclude that the disk is notably inclined, with the north-eastern disk axis facing towards Earth. Therefore, based on the measured disk inclination and orientation, one would expect the north-eastern lobe of the bipolar nebula to appear brighter, as is observed. This scenario provides a natural explanation for the large number of arc- and cone-shaped structures which can be seen in our IMACS images (Fig. \[fig:IMACS\]) and that appear centered on V921Sco. We have identified the most significant of these arc- and cone-shaped features in our deepest image ($R$-band, Fig. \[fig:IMACS\][*E-F*]{}). After confirming the features in images taken with other filters, we marked them in Fig. \[fig:IMACS\][*G*]{}.
The distribution of the individual arc-fragments, in particular in the north-eastern lobe, suggests that we might observe up to five layers of ejecta, which have been created during episodic events of extreme mass-loss. In order to obtain a rough estimate for the period of the mass-loss events, we measured the typical separation between the different layers ([4-5]{}) and apply a projection factor in order to correct for the system inclination angle of $50.3\pm1.9^{\circ}$ (Sect. \[tab:modelfitting\]). Assuming the maximum expansion velocity of 1400 kms$^{-1}$ measured by @bor09 and a distance of 1.15 kpc, we find that a mass-loss period of $\sim 25$ yrs would be required, which seems consistent with the period of the companion, as estimated for a circular orbit (Sect. \[sec:interpcompanion\]). Compared to the bow-shock structures observed around other high-mass YSOs [e.g. @kra10], the arcs around V921Sco have a clumpy and sometimes truncated structure, possibly indicating a lower degree of collimation in the V921Sco outflow.
In a field of about one arcminute, we detect in our IMACS and FIRE images at least 26 stellar sources. @hab03 obtained spectra for a subset of these sources (see identification in Fig. \[fig:IMACS\][*I*]{}) and classified them as embedded low- and intermediate-mass YSOs. Increasing the number density in the cluster, our observations strengthen the argument that V921Sco is likely in a young evolutionary stage and, as many massive YSOs, embedded in a star forming region of low- and intermediate-mass YSOs.
Binary interaction and the B\[e\] phenomenon {#sec:interpbinaryinteraction}
--------------------------------------------
Our detection of a companion around V921Sco is interesting in the context of earlier suggestions that the B\[e\] phenomenon might be related to dynamical interaction in a close binary system, either as result of a recent stellar merger [e.g. @pod06] or through material which might have been ejected during phases of binary interaction [e.g. @she00; @mir07; @kra10b]. Unfortunately, only very few B\[e\]-stars have been studied so far with sufficient angular resolution to make definite statements about multiplicity. One of the best-studied B\[e\] systems is HD87643, where @mil09 detected a close companion around this supergiant star. Both the projected separation ($\sim 50$ AU) and orbital period ($\sim 50$ yrs) of this system shows similarities with the characteristics which we deduce for V921Sco ($\sim 29$ AU, $P \sim 35$ yrs). Also, both systems are embedded in an extended nebulosity with shell-like sub-structure, indicating episodic mass-loss. Therefore, it is plausible that the mass-loss in both systems might be triggered by dynamical interaction of the companions with the circumprimary disks. The B\[e\]-star-characteristic permitted and forbidden line emission might then originate in low density material which is periodically stripped away from the circumprimary disk. This scenario works independently of the evolutionary status and physics of the circumprimary disks (accretion or excretion disks) and might explain the diversity of stellar systems (Herbig B\[e\], supergiant, symbiotic star, compact planetary nebulae), in which the B\[e\] phenomenon is observed.
Follow-up interferometric studies on V921Sco and HD87643 might confirm this scenario for instance by measuring the orbital eccentricity and by imaging the expected star-disk interaction effects during periastron passage.
Conclusions {#sec:conclusions}
===========
We have investigated the milliarcsecond-scale environment around the B\[e\] star V921Sco and summarize our findings as follows:
- In our model-independent interferometric imaging, we discover a close ($\sim 25.0$ mas) companion around the B-type star and detect signs of orbital motion ($\Delta\Theta \sim 7$) between the 2008 and 2009 observations. The newly discovered companion V921Sco B is apparently not associated with a circumstellar disk and of later spectral type than the primary, as indicated by the measured $H-K$ color.
- Around the primary star, our images show a spatially extended disk-like structure, seen under an intermediate inclination angle of $50.3\pm1.9$.
- As about half of all B\[e\] stars [@mar08], V921Sco is associated with an extended nebula. In the case of V921Sco, the nebula has an intriguing bipolar morphology, where the symmetry axis ($50\pm5^{\circ}$) coincides with the polar axis of the AU-scale disk resolved by our interferometric observations ($57.8\pm4.3^{\circ}$). In our narrowband- and broadband-filter images (Fig. \[fig:IMACS\]), we detect complex, partially arc-shaped sub-structures, which might have been shaped by episodic mass-loss events, possibly triggered by the periastron passage of the newly-discovered close companion.
- Our findings add new support to the hypothesis that the B\[e\] phenomenon might be a consequence of interaction effects in close multiple systems, where the forbidden line-emitting material is ejected during close companion encounters and then episodically repeated during the periastron passage.
We thank Wen-Hsin Hsu for sharing some Magellan/IMACS observing time. This work was done under contract with the California Institute of Technology (Caltech), funded by NASA through the Sagan Fellowship Program (S.K. is a Sagan Fellow).
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[^1]: The reference data set can be accessed on the website http://www.skraus.eu/files/amber.htm
[^2]: In the Letter, we define the primary star (=V921 Sco A) as the star, which is associated with the majority of the near-infrared emitting circumstellar material.
|
---
abstract: |
Deep neural networks (DNNs) have been employed for designing wireless systems in many aspects, say transceiver design, resource optimization, and information prediction. Existing works either use the fully-connected DNN or the DNNs with particular architectures developed in other domains. While generating labels for supervised learning and gathering training samples are time-consuming or cost-prohibitive, how to develop DNNs with wireless priors for reducing training complexity remains open. In this paper, we show that two kinds of permutation invariant properties widely existed in wireless tasks can be harnessed to reduce the number of model parameters and hence the sample and computational complexity for training. We find special architecture of DNNs whose input-output relationships satisfy the properties, called *permutation invariant DNN (PINN)*, and augment the data with the properties. By learning the impact of the scale of a wireless system, the size of the constructed PINNs can flexibly adapt to the input data dimension. We take predictive resource allocation and interference coordination as examples to show how the PINNs can be employed for learning the optimal policy with unsupervised and supervised learning. Simulations results demonstrate a dramatic gain of the proposed PINNs in terms of reducing training complexity.
*Deep neural networks, a priori knowledge, permutation invariance, training complexity*
author:
- '[^1]\'
bibliography:
- 'IEEEabrv.bib'
- 'GJ1.bib'
title: 'Constructing Deep Neural Networks with *a Priori* Knowledge of Wireless Tasks'
---
Introduction
============
Deep learning has been considered as one of the key enabling techniques in beyond fifth generation (5G) and sixth generation (6G) cellular networks. Recently, deep neural networks (DNNs) have been employed to design wireless networks in various aspects, ranging from signal detection and channel estimation [@ye2017power; @samuel2019learning], interference management [@sun2017learning], resource allocation [@YW19; @VTC18GJ; @Guo2018Exploiting; @sun2019pimrc; @liu2020optimizing], coordinated beamforming [@alkhateeb2018deep], traffic load prediction [@wang2017spatiotemporal], and uplink/downlink channel calibration [@huang2019deep], *etc*, thanks to their powerful ability to learn complex input-output relation [@Hornik1989UnivApprox]. For the tasks of transmission scheme or resource allocation, the output is a transceiver or allocated resource (e.g., beamforming vector or transmit power), the input is the environment parameter (e.g., channel gain), and the relation is a concerned policy (e.g., power allocation). For the tasks of information prediction, the relation is a predictor, which depends on the temporal correlation between historical and future samples of a time series (e.g., traffic load at a base station).
Existing research efforts focus on investigating what tasks in wireless communications can apply deep learning by considering the fully-connected (FC)-DNN [@ye2017power; @alkhateeb2018deep; @huang2019deep; @sun2017learning; @sun2019pimrc], and how deep learning is used for wireless tasks by integrating the DNNs developed in other domains such as computer vision and natural language processing [@wang2017spatiotemporal; @VTC18GJ; @YW19]. By finding the similarity between the tasks in different domains, various deep learning techniques have been employed to solve wireless problems. For example, convolutional neural network (CNN) is applied for wireless tasks where the data exhibit spatial correlation, and recurrent neural network (RNN) is applied for information prediction using the data with temporal correlation. Most previous works consider supervised learning. Noticing the fact that generating labels is time-consuming or expensive, unsupervised learning frameworks were proposed for learning to optimize wireless systems recently [@YW19; @liu2020optimizing]. Nonetheless, the number of samples required for training in unsupervised manner may still be very high. This impedes the practice use of DNNs in wireless networks where data gathering is cost-prohibitive. Although the computational complexity of off-line training is less of a concern in static scenarios, wireless systems often need to operate in highly dynamic environments, where the channels, number of users, and available resources, *etc.*, are time-varying. Whenever the environment parameters change, the model parameters and even the size of a DNN need to be updated (e.g., the DNN in [@alkhateeb2018deep] needs to be trained periodically in the timescale of minutes). Therefore, training DNNs efficiently is critical for wireless applications.
To circumvent the “curse of dimensionality” that leads to the unaffordable sample and computational complexity for training, AI society has designed DNN architectures by harnessing general-purposed priors, such that each architecture is applicable for a large class of tasks. One successful example is CNN specialized for vision tasks. By exploiting the knowledge that local groups of pixels in images are often highly correlated, sparse connectivity is introduced in the form of convolution kernels. Furthermore, by exploiting the knowledge that local statistics of images are invariant to positions, parameter sharing is introduced among convolution kernels in each layer [@lecun2015deep; @bengio2009learning]. Another example is RNN. Considering the temporal correlation feature of time series, adjacent time steps are connected with weights, and parameter sharing is introduced among time steps such that the weights between hidden layers are identical [@lecun2015deep]. In this way of using *a priori* knowledge to design the architecture of DNNs, the number of model parameters and hence the training complexity can be reduced. To reduce the training complexity, wireless society promotes model-and-data-driven methodology to combine the well-established communication domain knowledge with deep learning most recently [@HJW2019; @ZRD2019]. For instance, the models can be leveraged to generate labeled samples for supervised learning [@sun2017learning; @ZRD2019], derive gradients to guide the searching directions for stochastic gradient descent/ascent [@sun2019pimrc], embed the modules with accurate models into DNN-based systems, and first use traditional model-based solutions to initialize and then apply DNNs to refine [@HJW2019; @ZRD2019]. Despite that the basic idea is general and useful, mathematical models are problem specific, and hence the solutions with model-based DNNs have to be developed on a case by case basis. Nonetheless, the two branches of research that are respectively priori-based and model-driven, are complementary rather than mutual exclusion. Given the great potential of deep learning in beyond 5G/6G cellular networks, it is natural to raise the following question: are there any general priors in wireless tasks? If yes, how to design DNN architecture by incorporating the priors?
Each task corresponds to a specific relation (i.e., a function). In many wireless tasks, the relation between the concerned solutions and the relevant parameters satisfies a common property: permutation invariance. For example, if the channel gains of multiple users permute, then the resources allocated to the users permute accordingly. This is because the resource allocated to a user depends on its own channel but not on the permutation of other users’ channels [@sun2017learning; @alkhateeb2018deep; @VTC18GJ; @YW19]. While the property seems obvious, the way to exploit the knowledge is not straightforward. In this paper, we strive to demonstrate how to reduce training complexity by harnessing such *general knowledge*. We consider two kinds of permutation invariance properties, which widely exist in wireless tasks. For the tasks satisfying each kind of property, we find a DNN with special architecture to represent the relation between the solution and the concerned parameters, referred to as *permutation invariant DNN (PINN)*, where majority of the model parameters are identical. Different from CNN and RNN that exploit the characteristic of data, which is the input of the DNN, PINN exploits the characteristic of tasks, which decide the input-output relation. The architecture of PINN offers the flexibility in applying to different input data dimension. By jointly trained with a small size DNN that captures the impact of the input dimension, the constructed DNNs can adapt to wireless systems with different scales (e.g., with time-varying number of users). Except the DNN architecture, we show that the property can also be used to generate labels for supervised learning. Simulation results show that much fewer samples and much lower computational complexity are required for training the constructed PINNs to achieve a given performance, and the majority of labels can be generated with the permutation invariance property. The proposed PINNs can be applied for a broad range of wireless tasks, including but not limited to the tasks in [@sun2017learning; @wang2017spatiotemporal; @alkhateeb2018deep; @VTC18GJ; @YW19; @huang2019deep; @sun2019pimrc; @samuel2019learning].
The major contributions are summarized as follows.
- We find the sufficient and necessary conditions for tasks to satisfy two kinds of permutation invariant properties. For each kind of tasks, we construct a DNN architecture whose input-output relationship satisfies the permutation invariance property. The constructed PINNs are applicable to both unsupervised and supervised learning.
- We show how the PINNs can adapt to different input data dimension by introducing a factor to characterize the impact of the scale of a wireless system. In training phase, the complexities can be reduced by training DNNs with small size. In operation phase, the trained DNN can be adaptive to the input with time-varying dimension.
- We take predictive resource allocation and interference coordination as examples to illustrate how the PINNs can be applied to unsupervisely and supervisely learn the two kinds of permutation invariant functions, respectively. Simulation results demonstrate that the constructed PINNs can reduce the sample and computational complexities remarkably compared to the non-structural FC-DNN with same performance.
*Notations*: ${\mathbb E}\{\cdot\}$ denotes mathematical expectation, $\|\cdot\|$ denotes two-norm, $\|\cdot\|_1$ denotes the summation of the absolute values of all the elements in a vector or matrix, and $(\cdot)^{\sf T}$ denotes transpose, ${\bm 1}$ denotes a column vector with all elements being $1$, ${\bm 0}$ denotes a column vector or a matrix with all elements being $0$.
The rest of the paper is organized as follows. In section \[sec: param share\], we introduce two permutation invariance properties and construct two PINNs, and illustrate how the PINNs can adapt to the input dimension. In section \[sec: case study I PRA\] and \[sec: case study II IC\], we present two case studies. In section \[sec: simulation results\], we show that the PINNs can reduce training complexity, and illustrate that the properties can also be used for dataset augmentation. In section \[sec: conclusion\], we provide the concluding remarks.
DNN for Tasks with Permutation Invariance {#sec: param share}
=========================================
In this section, we first introduce two kinds of relationships (mathematically, two kinds of functions) with permutation invariant property, which are widely existed in wireless communication tasks. For each relationship, we demonstrate how to construct a parameter sharing DNN satisfying the property. Then, we show how to make the constructed DNN adaptive to the scale of wireless networks.
Definition and Example Tasks {#sec: definition of PI}
----------------------------
For many wireless tasks such as resource allocation and transceiver design, the optimized policy that yields the solution (represented as column vector $\bf y$ without the loss of generality) from environment parameters (represented as vector $\bf x$ or matrix ${\bf X}$) can be expressed as a function ${\bf y}=f(\bf x)$ or ${\bf y}=f(\bf X)$. Both ${\bf y}$ and ${\bf x}$ are composed of $K$ blocks, i.e., ${\bf y}=[{\bf y}_1^{\sf T}, \cdots, {\bf y}_K^{\sf T}]^{\sf T}$, ${\bf x}=[{\bf x}_1^{\sf T}, \cdots, {\bf x}_K^{\sf T}]^{\sf T}$, and ${\bf X}$ is composed of $K^2$ blocks, i.e., $$\label{X}
{\bf X} = \left[
\begin{tabular}{cccc}
${\bf x}_{11}$ & $\cdots$ & ${\bf x}_{1K}$ \\
$\vdots$ & $\ddots$ & $\vdots$ \\
${\bf x}_{K1}$ & $\cdots$ & ${\bf x}_{KK}$
\end{tabular}
\right],$$ where the block ${\bf y}_k$ and ${\bf x}_k$ can either be a scalar or a column vector, $k=1,\cdots,K$, and the block ${\bf x}_{mn}$ can be a scalar, vector or matrix, $m,n=1,\cdots,K$.
A property is widely existed in the optimized policies $f(\cdot)$ for wireless problems: one-dimensional (1D) permutation invariance of ${\bf y}=f(\bf x)$ and two-dimensional (2D) permutation invariance of ${\bf y}=f(\bf X)$. Before the formal definition, we first introduce two examples.
[**Ex 1**]{}: One example is the task of power allocation to $K$ users by a base station (BS), as shown in Fig. \[fig:fig-wlchnl-1d\]. In the figure, $K=2$, each user and the BS are with a single-antenna, $\bm \Lambda$ is a permutation matrix to be defined soon. Then, a block in ${\bf x}$, say ${\bf x}_k=\gamma_k$, is the scalar channel of the $k$th user, a block in ${\bf y}$, say ${\bf y}_k=p_k$, is transmit power allocated to the user, and ${\bf y}=f(\bf x)$ is the power allocation policy. If the users are permutated, then the allocated powers will be permutated correspondingly. Such a policy is 1D permutation invariant to ${\bf x}$.
![Illustration of 1D permutation invariance, power allocation to $K$ users, $K=2$.[]{data-label="fig:fig-wlchnl-1d"}](fig-wlchnl-1D){width="0.6\linewidth"}
[**Ex 2**]{}: Another example is the task of interference coordination among $K$ transmitters by optimizing transceivers, as shown in Fig. \[fig:fig-wlchnl\]. Here, a block in ${\bf X}$, say ${\bf x}_{mn}={\bm\gamma}_{mn}\in {\mathbb C}^{N_{\sf tx}\times 1}$, is the channel vector between the $m$th transmitter (Tx) and the $n$th receiver (Rx), a block in ${\bf y}$, say ${\bf y}_k={\bf p}_k\in{\mathbb R}^{N_{\sf tx}\times 1}$, is the beamforming vector for the $k$th user, $m,n,k=1,\cdots,K$, $N_{\sf tx}$ is the number of transmit antennas, and ${\bf y}=f(\bf X)$ is the interference coordination policy. If the Tx-Rx pairs are permutated, then the beamforming vectors are correspondingly permutated. Such a policy is 2D permutation invariant to ${\bf X}$.
![Illustration of 2D permutation invariance, interference coordination among $K$ Tx-Rx pairs, $K=2$.[]{data-label="fig:fig-wlchnl"}](fig-wlchnl-2D){width="0.65\linewidth"}
To define permutation invariance, we consider a column transformation matrix $\bm \Lambda$, which operates on blocks instead of the elements in each block. In other words, the permutation matrix $\bm \Lambda$ only changes the order of blocks (e.g., ${\bf x}_k$, ${\bf y}_k$ or ${\bf x}_{mn}$) but do not change the order of elements within each block (e.g., the $N_{\sf tx}$ elements in vector ${\bm \gamma}_{mn}$). An example of $\bm \Lambda$ for $K=3$ is, $${\bm \Lambda} = \left[
\begin{tabular}{ccc}
${\bf I}$ & ${\bf 0}$ & ${\bf 0}$ \\
${\bf 0}$ & ${\bf 0}$ & ${\bf I}$ \\
${\bf 0}$ & ${\bf I}$ & ${\bf 0}$
\end{tabular}
\right], \notag$$ where ${\bf I}$ and ${\bf 0}$ are respectively the identity matrix and square matrix with all zeros.
\[def: 1\] For arbitrary permutation to ${\bf x}$, i.e., ${\bm \Lambda}^{\sf T}{\bf x}=[{\bf x}_{N_1}^{\sf T},\cdots,{\bf x}_{N_K}^{\sf T}]^{\sf T}$ where $N_1,\cdots,N_K$ is arbitrary permutation of $1,\cdots,K$, if ${\bm \Lambda}^{\sf T}{\bf y}=f({\bm \Lambda}^{\sf T}{\bf x})=[{\bf y}_{N_1}^{\sf T},\cdots,{\bf y}_{N_K}^{\sf T}]^{\sf T}$, then $f({\bf x})$ is 1D permutation invariant to ${\bf x}$.
In the following, we provide the sufficient and necessary condition for a function ${\bf y}=f(\bf x)$ to be 1D permutation invariant.
\[pp: 1\] The function ${\bf y}=f({\bf x})$ is 1D permutation invariant to ${\bf x}$ if and only if, $$\label{eq: perm inva}
{\bf y}_k = \eta\Big(\psi({\bf x}_k), {\cal F}_{n=1,n\neq k}^K \phi({\bf x}_n)\Big), k=1,\cdots,K,$$ where $\eta(\cdot), \psi(\cdot)$ and $\phi(\cdot)$ are arbitrary functions, and ${\cal F}$ is arbitrary operation satisfying the commutative law.
See Appendix \[appendix: A\].
The operations satisfying the commutative law include summation, product, maximization and minimization, *etc*. To help understand this condition, consider a more specific class of functions ${\bf y}=f(\bf x)$ satisfying , where the $k$th block in ${\bf y}$ can be expressed as $$\label{eq: perm inva-ex}
{\bf y}_k = \eta\Big(\psi({\bf x}_k), \sum_{n=1,n\neq k}^K \phi({\bf x}_n)\Big), k=1,\cdots,K.$$ Such class of functions ${\bf y}=f({\bf x})$ are 1D permutation invariant to ${\bf x}$. This is because for any permutation of ${\bf x}$, $\tilde{\bf x}=[{\bf x}_{N_1}^{\sf T},\cdots, {\bf x}_{N_K}^{\sf T}]^{\sf T}$, the solution corresponding to $\tilde{\bf x}$ is $\tilde{\bf y}=[\tilde{\bf y}_1^{\sf T},\cdots,\tilde{\bf y}_K^{\sf T}]^{\sf T}=[{\bf y}_{N_1}^{\sf T},\cdots,{\bf y}_{N_K}^{\sf T}]^{\sf T}$, where the $k$th block of $\tilde{\bf y}$ is $\tilde{\bf y}_k = \eta(\psi({\bf x}_{N_k}),\sum_{n=1,n\neq N_k}^K \phi({\bf x}_n))={\bf y}_{N_k}$.
For [**Ex 1**]{}, the optimal power allocation can be expressed as (though may not be explicitly), where $\psi({\bf x}_k)$ reflects the impact of the $k$th user’s channel on its own power allocation, and $\sum_{n=1,n\neq k}^K \phi({\bf x}_n)$ reflects the impact of other users’ channels on the power allocation to the $k$th user. From or we can observe that: (i) the impact of the block ${\bf x}_k$ and the impact of other blocks ${\bf x}_n,n\neq k$ on ${\bf y}_k$ are different, and (ii) the impact of every single block ${\bf x}_n, n\neq k$ on ${\bf y}_k$ does not need to be differentiated.
This suggests that for a DNN to learn the permutation invariant functions, it should and only need to compose of two types of weights to respectively reflect the two kinds of impact.
\[def: 2\] For arbitrary permutation to the columns and rows of ${\bf X}$, i.e., ${\bm \Lambda}^{\sf T}{\bf X}{\bm \Lambda}$, if ${\bm \Lambda}^{\sf T}{\bf y}=f({\bm \Lambda}^{\sf T}{\bf X}{\bm \Lambda})$, then $f({\bf X})$ is 2D permutation invariant to ${\bf X}$.
Using the similar method as in Appendix \[appendix: A\], we can prove the following sufficient and necessary condition for a function ${\bf y}=f(\bf X)$ to be 2D permutation invariant.
\[pp: 3\] The function ${\bf y}=f({\bf X})$ is permutation invariant to ${\bf X}$ if and only if, $$\label{eq: perm inva2}
{\bf y}_k = \eta\Big(\psi({\bf x}_{kk}), {\cal F}_{n=1,n\neq k}^K \phi({\bf x}_{kn}), {\cal G}_{n=1,n\neq k}^K \xi({\bf x}_{nk}), {\cal H}_{m,n=1,m,n\neq k}^K \zeta({\bf x}_{mn})\Big), k=1,\cdots,K,$$ where $\eta(\cdot), \zeta(\cdot), \psi(\cdot), \xi(\cdot)$ and $\phi(\cdot)$ are arbitrary functions, and ${\cal F}, {\cal G}, {\cal H}$ are arbitrary operations satisfying the commutative law.
Similarly, from we can observe that: (i) the impact of ${\bf x}_{kk}$, $\{{\bf x}_{kn},n\neq k\}$, $\{{\bf x}_{nk},n\neq k\}$, $\{{\bf x}_{mn},m,n\neq k\}$ on ${\bf y}_k$ are different, (ii) the impact of every single block ${\bf x}_{kn},n\neq k$ (also ${\bf x}_{nk},n\neq k$ and ${\bf x}_{mn},m,n\neq k$) on ${\bf y}_k$ does not need to be differentiated.
For [**Ex 2**]{}, the optimized solution for a Tx-Rx pair (say ${\bf p}_1$ for the first pair) depends on the channels of four links: (i) the channel between Tx1 and Rx1 ${\bm\gamma}_{11}$, (ii) the channels between Tx1 and other receivers ${\bm\gamma}_{1k},k=1,\cdots,K, k\neq 1$, (iii) the channels between other transmitters and Rx1 ${\bm\gamma}_{k1},k=1,\cdots,K, k\neq 1$, and (iv) the channels between all the other transmitters and receivers ${\bm\gamma}_{mn},m,n=1,\cdots,K, m,n\neq 1$. Their impacts on ${\bf y}_k$ are reflected respectively by the four terms within the outer bracket of . When $N_{\sf tx}=1$, both ${\bm\gamma}_{mn}$ and ${\bf p}_k$ become scalers, while ${\bf y}=f({\bf X})$ is still 2D permutation invariant to ${\bf X}$.
DNN Architectures for the Tasks with One- and Two-dimensional Permutation Invariance {#sec: 1d-perm inva}
------------------------------------------------------------------------------------
When we design DNNs for wireless tasks such as resource allocation, the essential goal of a DNN is to learn a function ${\bf y}=f({\bf x}, {\bf W})$ or ${\bf y}=f({\bf X}, {\bf W})$ , where ${\bf x}$ or ${\bf X}$ and ${\bf y}$ are respectively the input and output of the DNN, and ${\bf W}$ is the model parameters that need to be trained.
In what follows, we demonstrate how to construct the architecture of the DNN for the tasks whose policies have the property of 1D or 2D permutation invariance.
### One-dimensional Permutation Invariance
To begin with, consider the FC-DNN, which has no particular architecture and hence can approximate arbitrary function. The input-output relation of a FC-DNN consisting of $L$ layers can be expressed as, $$\label{eq: FC func}
{\bf y}\!=\!f({\bf x}, {\bf W})\!\triangleq\! g^{[L]}\left({\bf W}^{[L-1,L]} g^{[L-1]}\Big(\cdots g^{[2]}({\bf W}^{[1,2]}{\bf x}+{\bf b}^{[2]})\cdots\Big)+{\bf b}^{[L]}\right),$$ where ${\bf W} = \{\{{\bf W}^{[l-1,l]}\}_{l=2}^L, \{{\bf b}^{[l]}\}_{l=2}^L\}$ represents the model parameters.
When $f({\bf x}, {\bf W})$ is 1D permutation invariant to ${\bf x}$, we can reduce the number of model parameters by introducing parameter sharing among the blocks into the FC-DNN. Inspired by the observation from or , we can construct a DNN with a special architecture to learn a 1D permutation invariant function. Denote the output of the $l$th hidden layer as ${\bf h}^{[l]}$. Then, the relation between ${\bf h}^{[l]}$ and ${\bf h}^{[l-1]}$ is, $${\bf h}^{[l]} = g^{[l]}({\bf W}^{[l-1,l]}{\bf h}^{[l-1]}+{\bf b}^{[l]}),$$ with the weight matrix between the $(l-1)$th layer and the $l$th layer as, $$\label{weight mat0}
{\bf W}^{[l-1,l]} = \left[
\begin{tabular}{cccc}
${\bf U}^{[l-1,l]}$ & ${\bf V}^{[l-1,l]}$ & $\cdots$ & ${\bf V}^{[l-1,l]}$ \\
${\bf V}^{[l-1,l]}$ & ${\bf U}^{[l-1,l]}$ & $\cdots$ & ${\bf V}^{[l-1,l]}$ \\
$\vdots$ & $\vdots$ & $\ddots$ & $\vdots$ \\
${\bf V}^{[l-1,l]}$ & ${\bf V}^{[l-1,l]}$ & $\cdots$ & ${\bf U}^{[l-1,l]}$
\end{tabular}
\right],$$ where ${\bf U}^{[l-1,l]}$ and ${\bf V}^{[l-1,l]}$ are sub-matrices with the numbers of rows and columns respectively equal to the numbers of elements in ${\bf h}^{[l]}_k$ and ${\bf h}^{[l-1]}_k$, and ${\bf h}^{[l]}_k$ and ${\bf h}^{[l-1]}_k$ are respectively the $k$th block in the output of the $l$th and $(l-1)$th hidden layers, $k=1,\cdots,K, l=2,\cdots,L$. ${\bf b}^{[l]}=[({\bf a}^{[l]})^{\sf T},\cdots,({\bf a}^{[l]})^{\sf T}]^{\sf T}$, ${\bf a}^{[l]}$ is sub-vector with number of elements equal to that of ${\bf h}^{[l]}_k$. $g^{[l]}(\cdot)$ is the element-wise activation function of the $l$th layer.
When $l=1, {\bf h}^{[l]}={\bf x}$, and when $l=L, {\bf h}^{[l]}={\bf y}$.
\[pp: 2\] When the weight matrices ${\bf W}^{[l-1,l]},l=2,\cdots,L$ are with the structure in , ${\bf y}=f({\bf x}, {\bf W})$ in is 1D permutation invariant to ${\bf x}$.
For notational simplicity, we omit the bias vector in this proof. With the weight matrices in , the output of the 2nd hidden layer is ${\bf h}^{[2]} = g^{[2]}({\bf W}^{[1,2]}{\bf x})$, and the output of the $l$th hidden layer $(2<l<L)$ can be written as, $$\begin{aligned}
\label{B1}
&{\bf h}^{[l]}& \notag\\ &=& g^{[l]}({\bf W}^{[l-1,l]}{\bf h}^{[l-1]}) \notag\\
&=&\Big[g^{[l]}\big({\bf U}^{[l-1,l]}{\bf h}^{[l-1]}_1+{\bf V}^{[l-1,l]}\sum_{k=2}^K {\bf h}^{[l-1]}_k\big),\cdots,\notag g^{[l]}\big({\bf U}^{[l-1,l]}{\bf h}^{[l-1]}_K+{\bf V}^{[l-1,l]}\sum_{k=1}^{K-1} {\bf h}^{[l-1]}_k\big)\Big],
\end{aligned}$$ where ${\bf h}^{[l]}=[({\bf h}^{[l]}_1)^{\sf T},\cdots,({\bf h}^{[l]}_K)^{\sf T}]^{\sf T}$. The $k$th block of ${\bf h}^{[l]}$ can be expressed as $$\label{eq: hlk}
{\bf h}^{[l]}_k=g^{[l]}\left({\bf U}^{[l-1,l]}{\bf h}^{[l-1]}_k\!+\!{\bf V}^{[l-1,l]}\sum_{n=1,n\neq k}^K {\bf h}^{[l-1]}_n\right).$$
We can see that the relation between ${\bf h}^{[l]}_k$ and ${\bf h}^{[l-1]}_k$ has the same form as in . Then, according to Proposition 1, ${\bf h}^{[l]}=g^{[l]}({\bf W}^{[l-1,l]}{\bf h}^{[l-1]})$ is 1D permutation invariant to ${\bf h}^{[l-1]}$.
Since the output of every hidden layer is permutation invariant to the output of its previous layer, and ${\bf y} =g^{[L]}({\bf W}^{[L-1,L]}{\bf h}^{[L-1]})$ is also permutation invariant to ${\bf h}^{[L-1]}$, $f({\bf x}, {\bf W})$ in is 1D permutation invariant to ${\bf x}$.
To help understand how a function with 1D permutation invariance property is constructed by the DNN in Proposition 3, consider a neural network with no hidden layers, and omit the superscript $[l-1,l]$ and the bias for easy understanding. Then, the $k$th output of the neural network can be expressed as $\textstyle {\bf y}_k = g({\bf U}{\bf x}_k + {\sum_{n=1,n\neq k}^K \bf V} {\bf x}_n)$. By comparing with , we can see that $\eta(\cdot)$ is constructed as the activation function $g(\cdot)$, $\psi(\cdot)$ and $\phi(\cdot)$ are respectively constructed as linear functions as $\psi({\bf x}_k)={\bf U}{\bf x}_k$ and $\phi({\bf x}_k)={\bf V}{\bf x}_k$, and the operation $\cal F$ is $\sum_{n=1,n\neq k}$.
In , all the diagonal sub-matrices of ${\bf W}^{[l-1,l]}$ are ${\bf U}^{[l-1,l]}$, which are the model parameters to learn the impact of ${\bf x}_k$ on ${\bf y}_k$. All the other sub-matrices are ${\bf V}^{[l-1,l]}$, which are the model parameters to learn the impact of ${\bf x}_n, n \neq k$ on ${\bf y}_k$. Since only two (rather than $K^2$ as in FC-DNN) sub-matrices need to be trained in each layer, the training complexity of the DNN can be reduced. We refer this DNN with 1D permutation invariance property as “**PINN-1D**”, which shares parameters among blocks in each layer as shown in Fig. \[fig:dnn\].
![Architecture of PINN-1D. The connections with the same color are with same sub-weight matrices (i.e., ${\bf U}^{[l-1,l]}$ and ${\bf V}^{[l-1,l]}$). The neurons within the dashed box belong to a block. ${\bf h}^{[l]}_k$ denotes the $k$th block in the output of the $l$th hidden layer.](DNN){width=".5\linewidth"}
\[fig:dnn\]
.
### Two-dimensional Permutation Invariance {#sec: 2d-perm inva}
When $f({\bf X}, {\bf W})$ is 2D permutation invariant to ${\bf X}$, we can also reduce the number of model parameters by sharing parameters among blocks, as inspired by the observation from . The constructed “**PINN-2D**” is shown in Fig. \[fig:dnn-2d\].
![Architecture of PINN-2D. Each circle represents a block instead of a neuron.[]{data-label="fig:dnn-2d"}](DNN-2D){width=".95\linewidth"}
Different from “**PINN-1D**”, the output of each layer is a matrix instead of a vector. Denote the output of the $l$th hidden layer as ${\bf H}^{[l]}$. To learn a 2D permutation invariant function, the relation between ${\bf H}^{[l]}$ and ${\bf H}^{[l-1]}$ is constructed as, $$\label{eq: 2DPS-hidden}
{\bf H}^{[l]} = g^{[l]}\Big({\bf P}^{[l-1,l]}{\bf H}^{[l-1]} ({\bf Q}^{[l-1,l]})^{\sf T}\Big),$$ with the weight matrices between the $(l-1)$th layer and the $l$th layer as, $$\label{weight mat1}
{\bf P}^{[l-1,l]} \!=\!\! \left[\!\!
\begin{tabular}{cccc}
${\bf A}^{[l-1,l]}$ & ${\bf B}^{[l-1,l]}$ & $\cdots$ & ${\bf B}^{[l-1,l]}$ \\
${\bf B}^{[l-1,l]}$ & ${\bf A}^{[l-1,l]}$ & $\cdots$ & ${\bf B}^{[l-1,l]}$ \\
$\vdots$ & $\vdots$ & $\ddots$ & $\vdots$ \\
${\bf B}^{[l-1,l]}$ & ${\bf B}^{[l-1,l]}$ & $\cdots$ & ${\bf A}^{[l-1,l]}$
\end{tabular}
\!\!\right]\!\!,
{\bf Q}^{[l-1,l]} \!=\!\! \left[\!\!
\begin{tabular}{cccc}
${\bf C}^{[l-1,l]}$ & ${\bf D}^{[l-1,l]}$ & $\cdots$ & ${\bf D}^{[l-1,l]}$ \\
${\bf D}^{[l-1,l]}$ & ${\bf C}^{[l-1,l]}$ & $\cdots$ & ${\bf D}^{[l-1,l]}$ \\
$\vdots$ & $\vdots$ & $\ddots$ & $\vdots$ \\
${\bf D}^{[l-1,l]}$ & ${\bf D}^{[l-1,l]}$ & $\cdots$ & ${\bf C}^{[l-1,l]}$
\end{tabular}
\!\!\right]\!\!,$$ where ${\bf A}^{[l-1,l]}$ and ${\bf B}^{[l-1,l]}$ are sub-matrices with the number of rows and columns respectively equal to the number of rows in ${\bf h}^{[l]}_{mn}$ and in ${\bf h}^{[l-1]}_{mn}$, ${\bf C}^{[l-1,l]}$ and ${\bf D}^{[l-1,l]}$ are sub-matrices with the number of rows and columns respectively equal to the number of columns in ${\bf h}^{[l]}_{mn}$ and in ${\bf h}^{[l-1]}_{mn}$, ${\bf h}^{[l]}_{mn}$ is the block in the $m$th row of the $n$th column of ${\bf H}^{[l]}$, $g^{[l]}$ is the element-wise activation function of the $l$th layer, $m,n=1,\cdots,K, l=2,\cdots,L$.
We can see from that both ${\bf P}^{[l-1,l]}$ and ${\bf Q}^{[l-1,l]}$ consist of two sub-matrices, where one of them is on the diagonal position and the other one is on the off-diagonal position.
Since the output of the DNN is a vector while the output of the last hidden layer ${\bf H}^{[L]}$ is a matrix, to satisfy permutation invariance we let ${\bf y}={\cal E}({\bf H}^{[L]})$ in the last layer, where ${\cal E}(\cdot)$ can be arbitrary operation satisfying ${\bf \Lambda}^{\sf T}{\bf y}={\cal E}({\bf \Lambda}^{\sf T}{\bf H}^{[L]}{\bf \Lambda})$. As an illustration, we set ${\bf y}$ as the diagonal elements of ${\bf H}^{[L]}$, i.e., ${\bf y}_k={\bf h}^{[L]}_{kk}, k=1,\cdots,K$. Then, the input-output relation of the constructed PINN-2D can be expressed as, $$\label{eq: PINN func}
{\bf y}=f({\bf X}, {\bf W})\triangleq {\sf diag}\left(g^{[L]}\Big({\bf P}^{[L-1,L]} g^{[L-1]}\big(\cdots g^{[2]}({\bf P}^{[1,2]}{\bf X}({\bf Q}^{[1,2]})^{\sf T}\cdots\big)({\bf Q}^{[L-1,L]})^{\sf T}\Big)\right),$$ where ${\bf W}=\{{\bf A}^{[l-1,l]}, {\bf B}^{[l-1,l]}, {\bf C}^{[l-1,l]}, {\bf D}^{[l-1,l]}\}_{l=2}^L$, and ${\sf diag}(\cdot)$ denotes the operation of concatenating diagonal blocks of a matrix into a vector.
When the weight matrices ${\bf P}^{[l-1,l]}$ and ${\bf Q}^{[l-1,l]}$ are with the structure in , ${\bf y}=f({\bf X}, {\bf W})$ in is 2D permutation invariant to ${\bf X}$.
With ${\bf P}^{[l-1,l]}$ and ${\bf Q}^{[l-1,l]}$ in , for arbitrary column transformation ${\bm \Lambda}$, it is easy to prove that ${\bf P}^{[l-1,l]}{\bm \Lambda}^{\sf T}={\bm \Lambda}^{\sf T} {\bf P}^{[l-1,l]}$ and ${\bf Q}^{[l-1,l]}{\bm \Lambda}^{\sf T}={\bm \Lambda}^{\sf T} {\bf Q}^{[l-1,l]}, l=2,\cdots,L$. Since $g^{[l]},l=2,\cdots,L$ are element-wise activation functions, from we have $$\begin{aligned}
{\bm \Lambda}^{\sf T}{\bf H}^{[l]}{\bm \Lambda}
&=& {\bm \Lambda}^{\sf T}g^{[l]}\Big({\bf P}^{[l-1,l]}{\bf H}^{[l-1]} ({\bf Q}^{[l-1,l]})^{\sf T}\Big){\bm \Lambda}
= g^{[l]}\Big({\bm \Lambda}^{\sf T}{\bf P}^{[l-1,l]}{\bf H}^{[l-1]} ({\bf Q}^{[l-1,l]})^{\sf T}{\bm \Lambda}\Big) \notag\\
&=& g^{[l]}\Big({\bf P}^{[l-1,l]}{\bm \Lambda}^{\sf T}{\bf H}^{[l-1]}{\bm \Lambda} ({\bf Q}^{[l-1,l]})^{\sf T}\Big). \notag
\end{aligned}$$ Further considering that it is easy to prove that ${\bm \Lambda}^{\sf T}{\bf y} = {\sf diag}({\bm \Lambda}^{\sf T}{\bf H}^{[L]}{\bm \Lambda})$, from we have ${\bm \Lambda}^{\sf T}{\bf y}=f({\bm \Lambda}^{\sf T}{\bf X}{\bm \Lambda}, {\bf W})$. Then, according to Definition \[def: 2\] we know that ${\bf y}=f({\bf X}, {\bf W})$ is 2D permutation invariant to ${\bf X}$.
To show how a function with 2D permutation invariance property is constructed by such a DNN, consider a PINN-2D with one hidden layer and omit the subscript $[l-1, l]$ and $[l]$ for notational simplicity. Then, relation between ${\bf y}_k$ and ${\bf X}$ can be obtained from and as, $$\label{eq: 2DPS-xy}
{\bf y}_k=g\left({\bf A}{\bf x}_{kk}{\bf C}^{\sf T}
+\sum_{n=1,n\neq k}^K {\bf A}{\bf x}_{kn}{\bf D}^{\sf T}
+\sum_{n=1,n\neq k}^K {\bf B}{\bf x}_{nk}{\bf C}^{\sf T}
+\sum_{m,n=1,m,n\neq k}^K {\bf B}{\bf x}_{mn}{\bf D}^{\sf T}\right),$$ which has the same form as , and the sub-matrices ${\bf A}, {\bf B}, {\bf C}, {\bf D}$ are used to learn the impact of ${\bf x}_{kk}, \{{\bf x}_{kn}, n\neq k\}, \{{\bf x}_{nk}, n\neq k\}, \{{\bf x}_{mn}, m,n\neq k\}$ on ${\bf y}_k$. By comparing and , we can see that $\eta(\cdot)$ is constructed as the activation function $g(\cdot)$, the functions $\psi(\cdot)$, $\phi(\cdot)$, $\xi(\cdot)$ and $\zeta(\cdot)$ are respectively constructed as bi-linear functions as $\psi({\bf x}_{kk})={\bf A}{\bf x}_{kk}{\bf C}^{\sf T}$, $\phi({\bf x}_{kn})={\bf A}{\bf x}_{kn}{\bf D}^{\sf T}$, $ \xi({\bf x}_{nk})= {\bf B}{\bf x}_{nk}{\bf C}^{\sf T}$, and $\zeta({\bf x}_{mn})= {\bf B}{\bf x}_{mn}{\bf D}^{\sf T}$. The operations ${\cal F}$ and ${\cal G}$ are $\sum_{n=1,n\neq k}^K$, and the operation ${\cal H}$ is $\sum_{m,n=1,m,n\neq k}^K$.
In the sequel, and refer both PINN-1D and PINN-2D as “**PINN**” when we do not need to differentiate them.
Network Size Adaptation {#sec: net size adapt}
-----------------------
The PINN is organized in blocks, i.e., the numbers of blocks in the input, output and hidden layers depend on $K$. In practice, the value of $K$, e.g., the number of users in a cell, is time-varying. In the following, we take PINN-1D as an example to illustrate how to make PINN adaptive to $K$, while PINN-2D can be designed in the same way.
In each layer (say the $l$th layer) of PINN-1D, the matrix ${\bf W}^{[l-1,l]}$ with $K^2$ blocks is composed of two sub-matrices ${\bf U}^{[l-1,l]}$ and ${\bf V}^{[l-1,l]}$, where each block corresponds to one of the sub-matrices. Therefore, the size of ${\bf W}^{[l-1,l]}$ can be flexibly controlled by adding or removing sub-matrices to adapt to different values of $K$. It is shown from that the impact of other blocks in the $(l-1)$th hidden layer on ${\bf h}^{[l]}_k$ grows with the value of $K$. When $K$ is large, the impact of ${\bf h}^{[l-1]}_k$ on ${\bf h}^{[l]}_k$ (i.e., the first term in ) diminishes. To avoid this, we multiply the sub-matrix ${\bf U}^{[l-1,l]}$ with a factor $\beta_K$ that is learned by a FC-DNN (denoted as FC-DNN-$\beta_K$) with the input as $K$, as shown in Fig. \[fig:dnn-dimgen\]. Then, the $k$th block of the output of the $l$th hidden layer becomes, $$\label{eq: dimgen}
{\bf h}^{[l]}_k=g^{[l]}\left(\beta_K{\bf U}^{[l-1,l]}{\bf h}^{[l-1]}_k+{\bf V}^{[l-1,l]}\sum_{n=1,n\neq k}^K {\bf h}^{[l-1]}_n\right).$$ In this way, the DNN can adaptive to $K$. We call the DNN in Fig. \[fig:dnn-dimgen\] as “**PINN-1D-Adp-$K$**”.
![Illustration of the PINN-1D architecture that can adapt to $K$, referred to as PINN-1D-Adp-$K$.[]{data-label="fig:dnn-dimgen"}](DNN-dimgen){width="0.7\linewidth"}
The model parameters of PINN-1D and FC-DNN-$\beta_K$ are jointly trained. Specifically, since $\beta_K$ is learned by inputting $K$, the relation between $\beta_K$ and $K$ can be written as $\beta_K=f_{\beta}(K, {\bf W}_{\beta})$, where ${\bf W}_{\beta}$ is the model parameters in FC-DNN-$\beta_K$. Then, the relation between output $\bf y$ and input ${\bf x}$ can be written as ${\bf y}=f({\bf x}, {\bf W}, f_{\beta}(K, {\bf W}_{\beta}))$, where ${\bf W} = \{\{{\bf U}^{[l-1,l]}\}_{l=2}^L, \{{\bf V}^{[l]}\}_{l=2}^L\}$ is the model parameters in PINN-1D. By minimizing a cost function $
{\cal L}({\bf y}) = {\cal L}\Big(f\big({\bf x}, {\bf W}, f_{\beta}(K, {\bf W}_{\beta})\big)\Big)$ with back propagation algorithm [@rumelhart1986learning], ${\bf W}$ and ${\bf W}_{\beta}$ can be optimized.
Both the training phase and operation phase can benefit from the architecture of PINN-1D-Adp-$K$. A PINN with small size can be first trained using the samples generated in the scenarios with small values of $K$. The training complexity can be reduced because only a small size DNN needs to be trained. Thanks to the FC-DNN-$\beta_K$, the trained PINN-1D-Adp-$K$ can operate in realistic scenarios where $K$ (say the number of users) changes over time.
In the following, we take predictive resource allocation and interference coordination as two examples to illustrate how to apply the PINN-1D and PINN-2D. Since the PINNs are applicable to different manners of supervision on training, we consider unsupervised learning for predictive resource allocation policy and consider supervised learning for interference coordination.
Case Study I: Predictive Resource Allocation {#sec: case study I PRA}
============================================
In this section, we demonstrate how the optimal predictive resource allocation (PRA) policy is learned by PINN-1D. Since generating labels from numerically obtained solutions is with prohibitive complexity for learning the PRA policy, we consider unsupervised learning.
Problem Statement
-----------------
### System Model
Consider a cellular network with $N_{\rm b}$ cells, where each BS is equipped with $N_{\sf tx}$ antennas and connected to a central processor (CP). The BSs may serve both real-time traffic and non-real-time (NRT) traffic. Since real-time service is with higher priority, NRT traffic is served with residual resources of the network after the quality of real-time service is guaranteed.
We optimize the PRA policy for mobile stations (MSs) requesting NRT service, say requesting for a file. Suppose that $K$ MSs in the network initiate requests at the beginning of a prediction window, and the $k$th MS (denoted as MS$_k$) requests a file with $B_k$ bits.
Time is discretized into frames each with duration $\Delta$, and each frame includes $T_s$ time slots each with duration of unit time. The durations are defined according to the channel variation, i.e., the coherence time of large scale fading (i.e., path-loss and shadowing) and small scale fading due to user mobility. The prediction window contains $T_f$ frames.
Assume that an MS is only associated to the BS with the highest average channel gain (i.e., large scale channel gain) in each frame. To avoid multi-user interference, we consider time division multiple access as an illustration, i.e., each BS serves only one MS with all residual bandwidth and transmit power after serving real-time traffic in each time slot, and serves multiple MSs in the same cell in different time slots. Then, maximal ratio transmission is the optimal beamforming. Assume that the residual transmit power is proportional to the residual bandwidth [@YTCOM2016], then the achievable rate of MS$_k$ in the $t$th time slot of the $j$th frame can be expressed as $R^{j,t}_k = W^{j,t}\log_2\Big(1+\frac{\alpha^j_k \|{\bm \gamma}^{j,t}_k\|^2}{\sigma_0^2} P_{\max}\Big)$, where $W^{j,t}$ and $P_{\max}$ are respectively the residual bandwidth and the maximal transmit power in the $t$th time slot of the $j$th frame, $\sigma_0^2$ is the noise power, ${\bm \gamma}_{k}^{j,t} \in \mathbb{C}^{N_{\sf tx} \times 1}$ is the small scale channel vector with ${\mathbb E}\{{\bm \gamma}_{k}^{j,t}\}=N_{\sf tx}$, $\alpha^j_k$ is the large scale channel gain. When $N_{\sf tx}$ and $T_s$ are large, it is easy to show that the time-average rate in the $j$th frame of MS$_k$ can be accurately approximated as, $$\begin{aligned}
\label{R}
R^j_k \triangleq \frac{1}{T_s}\sum_{t=1}^{T_s} R^{j,t}_k = \frac{1}{T_s}\sum_{t=1}^{T_s} W^{j,t}\log_2\Big(1+\frac{\alpha^j_k \|{\bm \gamma}^{j,t}_k\|^2}{\sigma_0^2}P_{\max}\Big)\approx W^j\log_2\Big(1+\frac{\alpha^j_k N_{\sf tx}}{\sigma_0^2}P_{\max}\Big),\end{aligned}$$ where $W^j=\frac{1}{T_s}\sum_{t=1}^{T_s}W^{j,t}$ is the time-average residual bandwidth in the $j$th frame. The time-average rates of each MS in the frames of the prediction window can either be predicted directly [@NB2018] or indirectly by first predicting the trajectory of each MS [@LSTMtrjactory17] and the real-time traffic load of each BS [@wang2017spatiotemporal] and then translating to average channel gains and residual bandwidth [@Guo2018Exploiting].
### Optimizing Predictive Resource Allocation Plan
We aim to optimize a resource allocation plan that minimizes the total transmission time required to ensure the quality of service (QoS) of each MS. The plan for MS$_k$ is denoted as ${\bf s}_k=[s^1_k,\cdots,s^{T_f}_k]^{\sf T}$, where $s^j_k$ is the fraction of time slots assigned to the MS in the $j$th frame.
The objective function can be expressed as $\sum_{k=1}^K\sum_{j=1}^{T_f} s^j_k$. To guarantee the QoS, the requested file should be completely downloaded to the MS before an expected deadline. For simplicity, we let the duration between the time instant when an MS initiates a request and the transmission deadline equals the duration of the prediction window. Then, the QoS constraint can be expressed as $\sum_{j=1}^{T_f} s_k^{j} R_k^j/B_k\Delta=1$.
Denote $r^j_k \triangleq R_k^j/B_k\Delta$ and ${\bf r}_k=[r^1_k,\cdots,r^{T_f}_k]^{\sf T}$, which is called *average rate* in the sequel. Then, the optimization problem can be formulated as,
\[P1-vector\] $$\begin{aligned}
{\bf P1}:\min_{\bf S} &~~~ \|{\bf S}\|_1 \\
{\rm s.t.} &~~~{\bf S}^{\sf T}\cdot{\bf R}\star{\bf I}={\bf I},\label{P1-c-vec}\\
&~~~{\bf S}\cdot{\bf M}_i^{\sf T}\star{\bf I} \preceq {\bf I}, i=1,\cdots,N_{\rm b},\label{P1-d-vec}\\
&~~~{\bf S}\succeq {\bf 0},
\end{aligned}$$
where ${\bf S} = [{\bf s}_1,\cdots, {\bf s}_K], {\bf R} = [{\bf r}_1,\cdots, {\bf r}_K]$, $({\bf M}_i)_{jk}=1$ or 0 if MS$_k$ associates or not associates to the $i$th BS in the $j$th frame, $(\cdot)_{jk}$ stands for the element in the $j$th row and $k$th column of a matrix. is the QoS constraint, and is the resource constraint that ensures the total time allocated in each frame of each BS not exceeding one frame duration. In and , “$\cdot$” denotes matrix multiplication, and “$\star$” denotes element wise multiplication, ${\bf A}\preceq{\bf B}$ and ${\bf A}\succeq{\bf B}$ mean that each element in ${\bf A}$ is not larger or smaller than each element in ${\bf B}$, respectively.
After the plan for each MS is made by solving **P1** at the start of the prediction window, a transmission progress can be computed according to the plan as well as the predicted average rates, which determines how much data should be transmitted to each MS in each frame. Then, each BS schedules the MSs in its cell in each time slot, see details in [@YTCOM2016].
Unsupervised Learning for Resource Allocation Plan
--------------------------------------------------
**P1** is a convex optimization problem, which can be solved by interior-point method. However, the computational complexity scales with ${\cal O}(KT_f)^{3.5}$, which is prohibitive. To reduce on-line computational complexity, we can train a DNN to learn the optimal resource allocation plan. To avoid the computational complexity in generating labels, we train the DNN with unsupervised learning. To this end, we transform **P1** into a functional optimization problem as suggested in [@sun2019pimrc]. In particular, the relation between the optimal solution of **P1** and the known parameters (denoted as ${\bf S}({\bm \theta})$) can be found from the following problem as proved in [@sun2019pimrc],
\[P2-vector\] $$\begin{aligned}
{\bf P2}:\min_{\bf S({\bm \theta})} &~~~ {\mathbb E}_{\bm \theta}\{\|{\bf S}({\bm \theta})\|_1\} \\
{\rm s.t.} &~~~{\bf S({\bm \theta})}^{\sf T}\cdot{\bf R}\star{\bf I}={\bf I},\label{P2-c-vec}\\
&~~~{\bf S({\bm \theta})}\cdot{\bf M}_i^{\sf T}\star{\bf I} \preceq {\bf I}, i=1,\cdots,N_{\rm b},\label{P2-d-vec}\\
&~~~{\bf S({\bm \theta})}\succeq {\bf 0} \label{P2-e-vec},
\end{aligned}$$
where ${\bm \theta} = \{{\bf R}, {\bf M}_1, \cdots, {\bf M}_{N_{\rm b}}\}$ are the known parameters.
Problem **P2** is convex, hence it is equivalent to its Lagrangian dual problem [@convexopt],
\[P3\] $$\begin{aligned}
{\bf P3}:\notag\\\max_{\bm \lambda(\bm \theta)}\min_{\bf S({\bm \theta})} &{\cal L}\triangleq
{\mathbb E}_{\bm \theta}\Bigg\{\|{\bf S}({\bm \theta})\|_1\!
+\!{\bm\mu}^{\sf T}({\bm \theta})\big({\bf S({\bm \theta})}^{\sf T}\cdot{\bf R}\star{\bf I}-{\bf I}\big)\!\cdot\!{\bm 1}
+\sum_{i=1}^{N_{\rm b}}{\bm\nu}_i^{\sf T}({\bm \theta})\big({\bf S({\bm \theta})}\cdot{\bf M}_i^{\sf T}\star{\bf I}- {\bf I}\big)\!\cdot\!{\bm 1}\notag\\
&\hspace{10.5cm}-{\bm\Upsilon}({\bm \theta})\star{\bf S}({\bm \theta})\Bigg\}\label{P3-o} \\
{\rm s.t.} &~~~{\bm\Upsilon}({\bm \theta})\succeq {\bm 0},{\bm\nu}_i(\bm \theta)\succeq {\bm 0}, \forall i\in\{1,\cdots,N_{\rm b}\},\label{P3-c-vec}
\end{aligned}$$
where ${\cal L}$ is the Lagrangian function, ${\bm\lambda}(\bm \theta)=\{{\bm\mu}(\bm \theta), {\bm\nu}_1(\bm \theta),\cdots, {\bm\nu}_{N_{\rm b}}(\bm \theta),{\bm\Upsilon}({\bm \theta})\}$ is the set of Lagrangian multipliers. Considering the universal approximation theorem [@Hornik1989UnivApprox], ${\bf S}({\bm \theta})$ and ${\bm\lambda}(\bm \theta)$ can be approximated with DNN [@sun2019pimrc].
### Design of the DNN {#sec: DNN design}
The input of a DNN to learn ${\bf S}({\bm \theta})$ can be designed straightforwardly as ${\bm \theta} = \{{\bf R}, {\bf M}_1, \cdots, {\bf M}_{N_{\rm b}}\}$, which is of high dimension. To reduce the input size, consider the fact that to satisfy constraint , we can learn the resource allocated by each BS with a neural network (called DNN-$s$), because the resource conflictions only exist among the MSs associated to the same BS. In this way, the input only contains the known parameters of a single BS instead of all the BSs in the network.
The input of DNN-$s$ is ${\bf x}_i={\rm vec}({\bf R} \star {\bf M}_i)=[({\bf x}_{1,i})^{\sf T}, \cdots, ({\bf x}_{K,i})^{\sf T}]^{\sf T}$, where ${\rm vec}(\cdot)$ denotes the operation of concatenating the columns of a matrix into a vector, ${\bf x}_{k,i} = [x_{k,i}^1, \cdots, x_{k, i}^{T_f}]^{\sf T}$, $x_{k,i}^j=r_k^j$ is the average rate of MS$_k$ if it is served by the $i$th BS in the $j$th frame, and $x_{k,i}^j=0$ otherwise. The output of DNN-$s$ is the resource allocation plan of all the MSs when they are served by the $i$th BS, which is normalized by the total resources allocated to each MS to meet the constraint in , i.e., $
\textstyle\hat{s}^j_k\!=\!\frac{\hat{s}^{j'}_{k} r^j_k}{\sum_{\tau=1}^{T_f}\hat{s}^{\tau'}_{k}r^{\tau}_k} \Big/r^j_k\!=\! \frac{\hat{s}^{j'}_{k}}{\sum_{\tau=1}^{T_f}\hat{s}^{\tau'}_{k}r^{\tau}_k},k\!=\!1,\!\cdots\!,K, j\!=\!1,\!\cdots\!,T_f$, where $\hat{s}^{j'}_{k}$ and $\hat{s}^j_k$ are respectively the output of DNN-$s$ before and after normalization. We use the commonly used `Softplus` (i.e., $y=g(x)\triangleq\log(1+\exp(x))$) as the activation function of the hidden layers and output layer to ensure the learned plan being equal or larger than $0$.
Since DNN-$s$ is used to learn ${\bf S}({\bm \theta})$ that is permutation invariant to ${\bf x}_i$, we can apply PINN-1D-Adp-$K$ whose input-output relation is $f_s({\bf x}_i, {\bf W}_s)$, where ${\bf W}_s$ denotes the model parameters in DNN$-s$. Both the input and output sizes of DNN-$s$ are $KT_f$, which may change since the number of MSs may vary over time.
To learn the Lagrange multipliers, we design a FC-DNN called DNN-$\lambda$, whose input-output relation is $f_{\nu}(\tilde{\bf x}_i, {\bf W}_{\nu})$. Since the constraint in is already satisfied due to the normalization operation in the output of DNN-$s$ and the constraint is already satisfied due to the `Softplus` operation in the output layer of DNN-$s$, we do not need to learn multiplier ${\bm \mu}$ and ${\bm \Upsilon}$ in and hence we only learn multiplier ${\bm \nu}_i$. Since ${\bm \nu}_i$ is used to satisfy constraint , which depends on ${\bf x}_i$, the input of DNN-$\lambda$ contains ${\bf x}_i$. Since the vector ${\bf x}_i$ is composed of the average rates of $K$ MSs, its dimension may vary with $K$. Since DNN-$\lambda$ is a FC-DNN whose architecture cannot change with $K$, we consider the maximal number of MSs $K_{\max}$ such that $K\leq K_{\max}$. Then, the input of DNN-$\lambda$ is $\tilde{\bf x}_i=[({\bf x}_{1,i})^{\sf T}, \cdots, ({\bf x}_{K_{\max},i})^{\sf T}]^{\sf T}$. When $K<K_{\max}$, ${\bf x}_{K_{\max},i}={\bf 0}$ for $\forall k>K$. The activation functions in hidden layers and output layer are `Softplus` to ensure the Lagrange multipliers being equal or larger than $0$, hence can be satisfied.
### Training Phase
DNN-$s$ and DNN-$\lambda$ are trained in multiple epochs, where in each epoch ${\bf W}_s$ and ${\bf W}_{\nu}$ are consecutively updated using the gradients of a cost function with respective to ${\bf W}_s$ and ${\bf W}_{\nu}$ via back-propagation. The cost function is the empirical form of , where ${\bf S}({\bm \theta})$ and ${\bm \lambda}({\bm \theta})$ are replaced by $f_s({\bf x}_i, {\bf W}_s)$ and $f_{\nu}({\bf x}_i, {\bf W}_{\nu})$. In particular, we replace ${\mathbb E}_{\bm\theta}\{\cdot\}$ in the cost function with empirical mean, because the probability density function of ${\bm\theta}$ is unknown. We omit the second and third term in because the constraint and can be ensured by the normalization and `Softplus` operation in the output of DNN-$s$, respectively. Moreover, we add the cost function with an augmented Lagrangian term [@hestenes1969multiplier] to make the learned policy to satisfy the constraints in **P2**. The cost function is expressed as, $$\begin{aligned}
\hat{\cal L}({\bf W}_s, {\bf W}_{\nu}) = \frac{1}{N} \sum_{n=1}^N \sum_{i=1}^{N_{\rm b}}\Bigg( \left\|{\bm f}_{s,i}^{(n)}\right\|_1 &+& ({\bm f}_{\nu,i}^{(n)})^{\sf T}\big( [{\bm f}_{s,i}^{(n)}]_{T_f\times {K_{\max}}} \cdot {\bf M}_i^{\sf T} \star {\bf I} - {\bf I}\big)\cdot{\bm 1}\notag\\
&+& \underbrace{\frac{\rho}{2}\left\|\big( [{\bm f}_{s,i}^{(n)}]_{T_f\times {K_{\max}}} \cdot {\bf M}_i^{\sf T} \star {\bf I} - {\bf I}\big)^+\cdot{\bm 1}\right\|^2}_{(a)}\Bigg),\notag \nonumber\end{aligned}$$ where $N$ is the number of training samples, ${\bm f}_{s,i}^{(n)}\triangleq f_s({\bf x}_i^{(n)}, {\bf W}_s)$, ${\bm f}_{\nu,i}^{(n)}\triangleq f_{\nu}(\tilde{\bf x}_i^{(n)}, {\bf W}_{\nu})$, ${\bf x}^{(n)}_i$ and $\tilde{\bf x}^{(n)}_i$ denote the $n$th sample of DNN-$s$ and DNN-$\lambda$, respectively, $[{\bf a}]_{m\times n}$ is the operation to represent vector ${\bf a}$ as a matrix with $m$ rows and $n$ columns, $(a)$ is the augmented Lagrangian term, which is a quadratic punishment for not satisfying the constraints. $(x)^+=x$ when $x\geq 0$ and $(x)^+=0$ otherwise, $\rho$ is a parameter to control the punishment. It is proved in [@hestenes1969multiplier] that the optimality can be achieved as long as $\rho$ is larger than a given value. Hence we can regard $\rho$ as a hyper-parameter.
In DNN-$s$, ${\bf W}_s$ is trained to minimize $\hat{\cal L}({\bf W}_s, {\bf W}_{\nu})$. In DNN-$\lambda$, ${\bf W}_{\nu}$ is trained to maximize $\hat{\cal L}({\bf W}_s, {\bf W}_{\nu})$. The learning rate is adaptively updated with Adam algorithm [@Kingma2014Adam].
### Operation Phase
For illustration, assume that $\bf R$ and ${\bf M}_i, i=1,\cdots,N_{\rm b}$ are known at the beginning of the prediction window. Then, by sequentially inputting the trained DNN-$s$ with ${\bf x}_i={\rm vec}({\bf R} \star {\bf M}_i), i=1,\cdots,N_{\rm b}$, DNN-$s$ can sequentially output the resource allocation plans for all MSs served by the $1,\cdots,N_{\rm b}$th BS.
Case Study II: Interference Coordination {#sec: case study II IC}
========================================
In this section, we demonstrate how an interference coordination policy considered in [@sun2017learning] is learned by PINN-2D. For a fair comparison, we consider supervised learning as in [@sun2017learning]. Consider a wireless interference network with $K$ single-antenna transmitters and $K$ single-antenna receivers, as shown in Fig. \[fig:fig-wlchnl\]. To coordinate interference among links, the power at each transmitter is controlled to maximize the sum-rate as follows,
\[P: max sum-rate\] $$\begin{aligned}
\max_{p_1,\cdots, p_K} ~~& \sum_{k=1}^K \log\left(1+\frac{|\gamma_{kk}|^2p_k}{\sum_{n=1,n\neq k}^K|\gamma_{nk}|^2 + \sigma_0^2}\right) \label{P: msr-1} \\
{\rm s.t.} ~~& 0\leq p_k \leq P_{\max}, \forall k=1,\cdots,K, \label{P: msr-2}
\end{aligned}$$
where $\gamma_{mn}\in {\mathbb C}$ is the channel between the $m$th transmitter and the $n$th receiver, $m,n=1,\cdots,K$, $P_{\max}$ is the maximal transmit power of each transmitter, and $\sigma_0^2$ is the noise power.
Problem is NP-hard, which can solved numerically by a weighted-minimum-mean-squared-error (WMMSE) algorithm[@sun2017learning].
We use PINN-2D-Adp-$K$ to learn the power control policy. The input of the DNN is the channel matrix, i.e., $$\label{eq: Xinput}
{\bf X} = \left[
\begin{tabular}{ccc}
$|\gamma_{11}|$ & $\cdots$ & $|\gamma_{1K}|$ \\
$\vdots$ & $\ddots$ & $\vdots$ \\
$|\gamma_{K1}|$ & $\cdots$ & $|\gamma_{KK}|$,
\end{tabular}
\right],$$ and the output is the transmit power normalized by the maximal transmit power, i.e., ${\bf y}=[p_1,\cdots,p_K]^{\sf T}/P_{\max}$. Then, the constraint becomes ${\bf 0}\preceq {\bf y}\preceq {\bf 1}$. The expected output of the DNN (i.e., the label) is the solution obtained by WMMSE algorithm that is also normalized by $P_{\max}$), i.e., ${\bf y}^*=[p_{1}^*,\cdots,p_{K}^*]^{\sf T}/P_{\max}$. The activation function of the hidden layers is the commonly used [Softplus]{} and the activation function of the output layer is [Sigmoid]{} (i.e., $y=1/(1+e^{-x})$) such that ${\bf 0}\preceq {\bf y}\preceq {\bf 1}$, hence constraint can be guaranteed. We add batch normalization in the output layer to avoid gradient vanishing [@ioffe2015batch].
The model parameters ${\bf W}=\{{\bf A}^{[l-1,l]}, {\bf B}^{[l-1,l]}, {\bf C}^{[l-1,l]}, {\bf D}^{[l-1,l]}\}_{l=2}^L$ are trained to minimize the empirical mean square errors between the outputs of the DNN and the expected outputs over $N$ training samples. Each sample is composed of a randomly generated channel matrix as in and the corresponding solution obtained from the WMMSE algorithm.
Simulation Results {#sec: simulation results}
==================
In this section, we evaluate the performance of the proposed solutions. We consider the two tasks in previous case studies, which are respectively 1D- and 2D-permutation invariant.
All simulations are implemented on a computer with one 14-core Intel i9-9940X CPU, one Nvidia RTX 2080Ti GPU, and 64 GB memory. The optimal solution of PRA is implemented in Matlab R2018a with the build-in interior-point algorithm, and the WMMSE algorithm is implemented in Python 3.6.4 with the open-source code of [@sun2017learning] from Github (available: <https://github.com/Haoran-S/SPAWC2017>). The training of the DNNs is implemented in Python 3.6.4 with TensorFlow 1.14.0.
![Simulation setup for predictive resource allocation to $K$ NRT users, $K$ randomly changes from 1 to $K_{\max}=40$.[]{data-label="fig:setup"}](traffic){width="0.8\linewidth"}
Predictive Resource Allocation {#sec: simu-PRA}
------------------------------
### Simulation Setups
Consider a cellular network with cell radius $R_{\rm b}=250$ m, where four BSs each equipped with $N_{\sf tx}=8$ antennas are located along a straight line. For each BS, $P_{\max}$ is 40 W, $W_{\max}=20$ MHz and the cell-edge SNR is set as 5 dB, where the intercell interference is implicitly reflected. The path loss model is $36.8 + 36.7\log_{10}(d)$, where $d$ is the distance between the BS and MS in meter. The MSs move along three roads of straight lines with minimum distance from the BSs as $50$ m, $100$ m and $150$ m, respectively. At the beginning of the prediction window, $K$ MSs at different locations in the roads initiate requests, where each MS requests a file with size of $B_k=6$ Mbytes (MB). Each frame is with duration of $\Delta=1$ second, and each time slot is with duration $10$ ms, i.e., each frame contains $T_s=100$ time slots.
To characterize the different resource usage status of the BSs by serving the real-time traffic, we consider two types of BSs: busy BS with average residual bandwidth in the prediction window $\overline{W}= 5$ MHz and idle BS with $\overline{W}= 10$ MHz, which are alternately located along the line as idle, busy, idle, busy, as shown in Fig. \[fig:setup\]. The results are obtained from 100 Monte Carlo trials. In each trial, $K$ is randomly selected from 1 to $K_{\max}=40$, the MSs initiate requests randomly at a location along the trajectory, and travel with speed uniformly distributed in $(10, 25)$ m/s and directions uniformly selected from 0 or +180 degree. The small-scale channel in each time slot changes independently according to Rayleigh fading, and the residual bandwidth at each BS in each time slot varies according to Gaussian distribution with mean value $\overline{W}$ and standard derivation $0.2\overline{W}$. The setup is used in the sequel unless otherwise specified.
Each sample for unsupervised training or for testing is generated as follows. For the $K$ MSs, the indicator of whether a MS is served by the $i$th BS, ${\bf M}_i$, can be obtained. The average channel gains of the MSs are computed with the path loss model. With the simulated residual bandwidth in each BS, the average rates of $K$ users within the prediction window, ${\bf R}$, can be computed with . Then, a sample can be obtained as ${\bf x}_i={\rm vec}({\bf R} \star {\bf M}_i),i=1,\cdots,N_{\rm b}$.
As demonstrated previously, the architecture of PINN can be flexibly controlled to adapt to different values of $K$, with which the training complexity can be further reduced. In order to show the complexity reduction respectively brought by the network size adaptation and by the parameter sharing among blocks, we train three different kinds of DNN-$s$ as follows, each of them is trained together with a DNN-$\lambda$.
- *PINN-1D-Adp-$K$:* This DNN-$s$ is with the architecture in Fig. \[fig:dnn-dimgen\]. The training samples are generated in the scenarios with different number of MSs, where the majority of the samples are generated when $K$ is randomly selected from $1 \sim 10$ and the rest of 1000 samples are generated when $K=K_{\max}=40$.
- *PINN-1D:* This DNN-$s$ is with the architecture in Fig. \[fig:dnn\]. The training samples are generated by a simulated system with $K=K_{\max}$ users.
- *FC-DNN:* This DNN-$s$ is the FC-DNN without parameter sharing, which is with the same number of layers and the same number of neurons with PINN-1D. The training samples are also generated in the scenario where $K=K_{\max}$.
The fine-tuned hyper-parameters for these DNNs when $T_f=60$ seconds and $K_{\max}=40$ are summarized in Table \[table\]. When $T_f$ changes, the hyper-parameters should be tuned again to achieve the best performance. The training set contains 10,000 samples and the test set contains 100 samples, where the testing samples are generated in the scenario where $K=K_{\max}=40$.
[c|c|c|c|c|c]{} &\
& &&&\
& PINN-1D & FC-DNN-$\beta_K$ & & & \
-----------------------
Number of input nodes
-----------------------
: Hyper-parameters for the DNNs When $T_f=60$ Seconds and $K_{\max}=40$.[]{data-label="table"}
& $KT_f$ & 1 &$K_{\max}T_f = 2400$&$K_{\max}T_f = 2400$& $K_{\max}T_f = 2400$\
-------------------------
Number of hidden layers
-------------------------
: Hyper-parameters for the DNNs When $T_f=60$ Seconds and $K_{\max}=40$.[]{data-label="table"}
& 2 & 1 &2&2& 2\
------------------------
Number of hidden nodes
------------------------
: Hyper-parameters for the DNNs When $T_f=60$ Seconds and $K_{\max}=40$.[]{data-label="table"}
&50$K$, 50$K$&10&2,000, 2,000&2000, 2000& 200, 100\
------------------------
Number of output nodes
------------------------
: Hyper-parameters for the DNNs When $T_f=60$ Seconds and $K_{\max}=40$.[]{data-label="table"}
& $KT_f$ & 1 &$K_{\max}T_f=2400$&$K_{\max}T_f=2400$& $T_f = 60$\
-----------------------
Initial learning rate
-----------------------
: Hyper-parameters for the DNNs When $T_f=60$ Seconds and $K_{\max}=40$.[]{data-label="table"}
&\
--------------------
Learning algorithm
--------------------
: Hyper-parameters for the DNNs When $T_f=60$ Seconds and $K_{\max}=40$.[]{data-label="table"}
&\
----------------------------
Back propagation algorithm
----------------------------
: Hyper-parameters for the DNNs When $T_f=60$ Seconds and $K_{\max}=40$.[]{data-label="table"}
&\
### Number of Model Parameters
In PINN-1D, the weight matrix ${\bf W}^{[l-1,l]}$ contains two sub-matrices ${\bf U}^{[l-1,l]}$ and ${\bf V}^{[l-1,l]}$, each of which contains $N^{[l-1,l]}$ model parameters. Hence, the total number of model parameters is $2\sum_{l=2}^LN^{[l-1,l]}$.
In PINN-1D-Adp-$K$, the number of model parameters is $2\sum_{l=2}^LN^{[l-1,l]}+\sum_{l=2}^LN_{\beta}^{[l-1,l]}$, where the first and second term respectively correspond to the model parameters in PINN-1D and FC-DNN-$\beta_K$, $N_{\beta}^{[l-1,l]}$ is the number of parameters in the weights between the $(l-1)$th and $l$th layer of FC-DNN-$\beta_K$. For the PINN-1D with hyper-parameters in Table \[table\], the input contains $K_{\max}=40$ blocks and each block contains $T_f=60$ elements, the first hidden layer also contains $K_{\max}=40$ blocks and each block contains $2,000/K_{\max}=50$ elements. Then, $N^{[1,2]}=60\times 50=3,000$. Similarly, $N^{[2,3]}=50\times 50=2,500$, and $N^{[3,4]}=50\times 60=3,000$. Hence, there are $2\times(3,000+2,500+3,000)=17,000$ model parameters in PINN-1D. For FC-DNN-$\beta_K$ with hyper-parameters in Table \[table\], $N_{\beta}^{[1,2]}=1\times 10=10$ and $N_{\beta}^{[2,3]}=10\times 1=10$, hence there are $\sum_{l=2}^LN_{\beta}^{[l-1,l]}=20$ model parameters, which is with much smaller size than PINN-1D.
In the FC-DNN with the same number of hidden layers and the same number of neurons in each hidden layer as PINN-1D, the number of parameters in ${\bf W}^{[l-1,l]}$ is $K_{\max}^2\sum_{l=2}^LN^{[l-1,l]}$, which is $K_{\max}^2/2$ as large as PINN-1D. For the FC-DNN with hyper-parameters in Table \[table\], the number of model parameters is $40^2\times(3,000+2,500+3,000)=13,600,000$, which increases by $K_{\max}^2/2=800$ times over PINN-1D.
### Sample and Computational Complexity {#sec: PRA:complexities}
*Sample complexity* is defined as the minimal number of training samples for a DNN to achieve an expected performance, and *computational complexity* is measured by the running time consumed by training the DNNs.
In Fig. \[fig: PRA-cplxty\], we provide the sample and computational complexities of all the DNNs when the objective in **P1** on the test set can achieve less than 20% performance loss from the optimal value (i.e., the total allocated time resource for all MSs), which is obtained by solving **P1** with interior-point method. In Fig. \[fig: PRA-cplxty\] (b), when $T_f=60$ s, the computational complexity of training FC-DNN is 600 s, which is out of the range of $y$-axis.
We can see that the training complexities of PINN-1D and PINN-1D-Adp-$K$ are much lower than “FC-DNN”, because the PINNs can converge faster thanks to the reduced model parameters by parameter sharing. The computational complexity of PINN-1D-Adp-$K$ is lower than PINN-1D due to the less number of neurons in each layer during the training phase. The computational complexity reduction of PINN-1D and PINN-1D-Adp-$K$ from “FC-DNN” grows with $T_f$. When $T_f=5$ s, the computational complexity of PINN-1D-Adp-$K$ is 67% less than “FC-DNN”, while when $T_f=60$ s, the complexity is reduced by 94%. The sample complexities of the two PINNs are comparable, since their numbers of model parameters are comparable.
It is noteworthy that although DNN-$\lambda$ is not with parameter sharing, the training complexity of PINNs is still much lower by only applying parameter sharing to DNN-$s$. This is because the fine-tuned DNN-$\lambda$ has much less hidden and output nodes, as shown in Table \[table\].
### Performance of PRA Learned with DNNs
To evaluate the dimensional generalization ability of PINN-1D-Adp-$K$, we compare the total transmission time required for downloading the files averaged over all MSs with the following methods.
- **Proposed-1**: The resource allocation plan is obtained by the well-trained PINN-1D-Adp-$K$ with unsupervised learning. The training set contains 16000 samples, which are generated in scenarios where the numbers of MSs $K$ change randomly from 1 to 10.
- **Proposed-2**: The only difference from “Proposed-1” lies in the training set, where we add 2000 training samples generated from the scenario with $K=K_{\max}=40$ in addition to 14000 samples generated with $K \in [1,10]$.
- **Supervised**: The resource allocation plan is obtained by the PINN-1D trained in the supervised manner, where the labels in the training samples are generated by solving **P1** with interior-point method.
- **Optimal**: The resource allocation plan is obtained by solving **P1** with interior-point method.
- **Baseline**: This is a non-predictive method [@su2015user], where each BS serves the MS with the earliest deadline in each time slot. If several MSs have the same deadline, then the MS with most bits to be transmitted is served firstly.
In Fig. \[fig:figtimearr\], we provide the average total transmission time required for downloading a file. We can see that “Proposed-1” performs closely to the optimal method when $K$ is less than 20, but the performance loss is larger when $K$ is large. Nonetheless, by adding some training samples generated with large value of $K$ to learn $\beta_K=f_{\beta}(K, {\bf W}_{\beta})$, “Proposed-2” performs closely to the optimal method, while the training complexity keeps small as shown in previous results. Besides, the proposed methods with unsupervised DNN outperforms the method with supervised DNN. This is because the resource allocation plan learned from labels cannot satisfy the constraints in problem **P1**, which leads to resource confliction among users. Moreover, all the PRA methods outperform the non-predictive baseline dramatically.
{width="0.45\linewidth"}
\[fig:figtimearr\]
Interference Coordination {#sec: simu-IC}
-------------------------
### Simulation Setups {#sec:simu:IC:setup}
Consider a wireless network with $K$ transmitters and $K$ receivers each equipped with a single antenna, where $K \leq K_{\max}$. A power control policy is obtained either by the WMMSE algorithm or by a trained DNN, as discussed in section \[sec: case study II IC\].
When training the DNN with supervision, the samples $\{\bf X, {\bf y}^*\}$ are generated via Monte Carlo trials. In each trial, the channel matrix $\bf X$ in is firstly generated with Rayleigh distribution, and then the label is obtained as ${\bf y}^*=[p_{1}^*,\cdots,p_{K}^*]^{\sf T}/P_{\max}$ by solving problem with WMMSE algorithm. The test set contains 1,000 samples.
We compare the sample and training complexities of three different DNNs, i.e., PINN-2D-Adp-$K$, PINN-2D and FC-DNN. When training “PINN-2D-Adp-$K$”, 80% training samples are generated in the scenario when $K$ is small[^2] and 20% samples are generated in the scenario when $K=K_{\max}$. The hyper-parameters of the three DNNs are as follows. When $K_{\max}\neq30$, the hyper-parameters need to be fine-tuned again to achieve the best performance.
[c|c|c|c|c]{} &\
& &&\
& PINN-2D & FC-DNN-$\beta_K$ & & \
-----------------------
Number of input nodes
-----------------------
: Hyper-parameters for the DNNs When $K_{\max}=30$.[]{data-label="table-wmmse"}
& $K^2$ & 1 &$K_{\max}^2 = 900$&$K_{\max}^2 = 900$\
-------------------------
Number of hidden layers
-------------------------
: Hyper-parameters for the DNNs When $K_{\max}=30$.[]{data-label="table-wmmse"}
& 2 & 1 & 2 & 3\
------------------------
Number of hidden nodes
------------------------
: Hyper-parameters for the DNNs When $K_{\max}=30$.[]{data-label="table-wmmse"}
& $9K^2$ & 10 & $90\times 90$ & $400, 300, 200$\
------------------------
Number of output nodes
------------------------
: Hyper-parameters for the DNNs When $K_{\max}=30$.[]{data-label="table-wmmse"}
& $K$ & 1 &$K_{\max}=30$&$K_{\max}=30$\
-----------------------
Initial learning rate
-----------------------
: Hyper-parameters for the DNNs When $K_{\max}=30$.[]{data-label="table-wmmse"}
& &\
--------------------
Learning algorithm
--------------------
: Hyper-parameters for the DNNs When $K_{\max}=30$.[]{data-label="table-wmmse"}
&\
----------------------------
Back propagation algorithm
----------------------------
: Hyper-parameters for the DNNs When $K_{\max}=30$.[]{data-label="table-wmmse"}
&\
### Number of Model Parameters {#sec: 1D-No.Params}
Since there are two weight matrices between the $(l-1)$th and the $l$th layer, each weight matrix contains two sub-matrices, and each sub-matrix contains $N^{[l-1,l]}$ weights, the number of model parameters in PINN-2D is $4\sum_{l=2}^L N^{[l-1,l]}$.
The number of parameters in PINN-2D-Adp-$K$ is $4\sum_{l=2}^LN^{[l-1,l]}+\sum_{l=2}^LN_{\beta}^{[l-1,l]}$, where the first and second term respectively correspond to the parameters in PINN-2D and FC-DNN-$\beta_K$.
For PINN-2D with hyper-parameters in Table \[table-wmmse\], the input contains $K_{\max}^2=900$ blocks and each block ${\bf x}_{mn}$ is a scalar, the second layer also contains $900$ blocks and each block ${\bf h}^{[2]}_{mn}$ is a $(90/K_{\max})\times (90/K_{\max})=3\times 3$ matrix. Recall the number of rows and columns of the sub-matrices defined in , there are $N^{[1,2]}=1\times 3=3$ model parameters in each sub-matrix. Similarly, $N^{[2,3]}=3\times 3=9$ and $N^{[3,4]}=3\times 1=3$. Hence, there are in total $4\sum_{l=2}^L N^{[l-1,l]}=60$ model parameters in PINN-2D. The number of model parameters in FC-DNN-$\beta_K$ is 20, hence there are $60+20=80$ model parameters in PINN-2D-Adp-$K$.
The FC-DNN with hyper-parameters in Table \[table-wmmse\] contains $900\times 400 +400\times 300 + 300\times 200 + 200\times 30=546,000$ model parameters. Hence, PINN-2D and PINN-2D-Adp-$K$ can reduce the model parameters by $546,000/60=9,100$ and $546,000/80=6,825$ times with respect to FC-DNN, respectively.
### Sample and Computational Complexity {#sample-and-computational-complexity}
In the following, we compare the training complexities for the PINNs to achieve an expected performance on the test set, which is set as the best performance that all the DNNs can achieve. When $K_{\max}=10, 20, 30$, the performance is 90%, 85%, 80% of the sum-rate that the WMMSE algorithm can achieve, respectively.
In Fig. \[fig: wmmse-cplxty\], we show the training complexity of the DNNs when $K_{\max}$ differs. As expected, both the complexities of training PINN-2D and PINN-2D-Adp-$K$ are much lower than “FC-DNN”, and the complexity reductions grow with $K_{\max}$. When $K_{\max}=30$, the sample and computational complexities of training PINN-2D is respectively reduced by 99% and 80% from “FC-DNN”, and the sample and computational complexities of training PINN-2D-Adp-$K$ is respectively reduced by 99% and 97%. Although the sample complexity of PINN-2D and PINN-2D-Adp-$K$ are almost the same, the complexity in generating labels for PINN-2D-Adp-$K$ is lower than PINN-2D. This is because most samples for training PINN-2D-Adp-$K$ are generated with $K<K_{\max}$, while all the samples for training PINN-2D are generated with $K=K_{\max}$.
### Permutation Invariance for Dataset Augmentation
Generating labels is time-consuming, especially when $K_{\max}$ is large. This is because more samples are required for training (shown in Fig. \[fig: wmmse-cplxty\] (a)), meanwhile generating each label costs more time to solve problem . In what follows, we show that the time consumed for generating labels can be reduced by dataset augmentation, i.e., generating more labels based on already obtained labels.
Specifically, by leveraging the permutation invariant relationship between ${\bf y}^*$ and ${\bf X}$, we know that for arbitrary permutation to ${\bf X}$, i.e., ${\bm \Lambda}^{\sf T}{\bf X}{\bm \Lambda}$, ${\bm \Lambda}^{\sf T}{\bf y}^*$ is the corresponding optimal solution. This suggests that we can generate a new sample $\{{\bm \Lambda}^{\sf T}{\bf X}{\bm \Lambda}, {\bm \Lambda}^{\sf T}{\bf y}^*\}$ based on an existed sample $\{{\bf X}, {\bf y}^*\}$. In this way of dataset augmentation, we can first generate a small number of training samples as in the setups in section \[sec:simu:IC:setup\], and then augment the dataset for more samples. For example, the possible permutations when $K_{\max}=30$ is $K_{\max}! \approx 2.65\times 10^{32}$, hence we can generate $2.65\times 10^{32}$ samples based on only a single sample!
In Table \[time-dataAug\], we compare the time consumption for generating training set with and without using dataset augmentation, when the trained “FC-DNN”[^3] can achieve the same sum-rate on the training set. The legend “generated samples” means the samples generated as in section \[sec:simu:IC:setup\], and “augmented samples” means the samples augmented with the permutation invariance.
[c|c|c|c|c]{} & &\
&
-------------------------------
Number of generated samples +
Number of augmented samples
-------------------------------
: Time consumed for generating samples, which is dominated by generating labels[]{data-label="time-dataAug"}
& Time consumption & Number of generated samples & Time consumption\
----
10
----
: Time consumed for generating samples, which is dominated by generating labels[]{data-label="time-dataAug"}
& 10 + 9,990 & 0.68 s &10,000 & 100 s\
----
20
----
: Time consumed for generating samples, which is dominated by generating labels[]{data-label="time-dataAug"}
& 10 + 149,990 & 10.27 s & 150,000 & 900 s\
----
30
----
: Time consumed for generating samples, which is dominated by generating labels[]{data-label="time-dataAug"}
& 10 + 399,990 & 30.6 s & 400,000 & 4000 s\
We can see from Table \[time-dataAug\] that the number of training samples required by FC-DNN for achieving an expected performance is identical for the training set with and without dataset augmentation. However, the time complexity of generating samples with dataset augmentation can be reduced by about 99% from that without dataset augmentation.
Conclusions {#sec: conclusion}
===========
In this paper, we constructed DNNs by sharing the weights among permutation invariant blocks and demonstrated how the proposed PINNs can adapt to the scales of wireless systems. We employed two case studies to illustrate how the PINNs can be applied, where the DNNs trained with and without supervision are used to learn the optimal solutions of predictive resource allocation and interference coordination, respectively. Simulation results showed that the numbers of model parameters of the PINNs are 1$/1000$ $\sim$ $1/10000$ of the fully-connected DNN when achieving the same performance, which leads to remarkably reduced sample and computational complexity for training. We also found that the property of permutation invariance can be utilized for dataset augmentation such that the time consumed to generate labels for supervised learning can be reduced drastically. The proposed DNNs are applicable to a broad range of wireless tasks, thanks to the general knowledge incorporated.
Proof of proposition \[pp: 1\] {#appendix: A}
==============================
We first prove the necessity. Assume that the function $f({\bf x})$ is permutation invariant to ${\bf x}$. If the $k$th block ${\bf x}_k$ in ${\bf x}=[{\bf x}_1, \cdots, {\bf x}_K]$ is changed to another position in ${\bf x}$ while the permutation of other blocks in ${\bf x}$ remains unchanged, i.e., $$\textstyle\tilde{\bf x}=[\underbrace{{\bf x}_1,\cdots,{\bf x}_{k-1}}_{(a)},\underbrace{{\bf x}_{k+1},\cdots,{\bf x}_K}_{(b)}],$$ where ${\bf x}_k$ may be in the blocks in $(a)$ or $(b)$, then $\tilde{\bf y}_k={\bf y}_{k-1}$ if ${\bf x}_k$ is in $(a)$ and $\tilde{\bf y}_k={\bf y}_{k+1}$ if ${\bf x}_k$ is in $(b)$, hence $\tilde{\bf y}_k\neq {\bf y}_k$. This indicates that the $k$th output block should change with the $k$th input block ${\bf x}_k$. On the other hand, if the position of ${\bf x}_k$ remains unchanged while the positions of other blocks ${\bf x}$ arbitrarily change, i.e., $\tilde{\bf x}=[{\bf x}_{N_1},\cdots,{\bf x}_{N_k-1},{\bf x}_k,{\bf x}_{N_{k}+1},\cdots,{\bf x}_{N_K}]$, then $\tilde{\bf y}=[{\bf y}_{N_1},\cdots,{\bf y}_{N_k-1},{\bf y}_k,{\bf y}_{N_k+1},\cdots,{\bf y}_{N_K}]$, and $\tilde{\bf y}_k={\bf y}_k$. This means that ${\bf y}_k$ is not affected by the permutation of the input blocks other than ${\bf x}_k$. Therefore, the function should have the form in .
We then prove the sufficiency. Assume that the function $f({\bf x})$ has the form in . If ${\bf x}$ is changed to $\tilde{\bf x}=[{\bf x}_{N_1},\cdots, {\bf x}_{N_K}]$, then the $k$th block of $\tilde{\bf y}$ is $\tilde{\bf y}_k = \eta(\psi({\bf x}_{N_k}),{\cal F}_{n=1,n\neq N_k}^K \phi({\bf x}_n))={\bf y}_{N_k}$. Hence, the output corresponding to $\tilde{\bf x}$ is $\tilde{\bf y}=[\tilde{\bf y}_1,\cdots,\tilde{\bf y}_K]=[{\bf y}_{N_1},\cdots,{\bf y}_{N_K}]$. According to Definition \[def: 1\], the function in is permutation invariant to ${\bf x}$.
[^1]: A part of this work is presented in conference version, which has been accepted by IEEE ICC 2020 [@ICC2020].
[^2]: When $K_{\max}=10$ or $20$, the majority of samples are generated in the scenarios with $K\leq 5$, and when $K_{\max}=30$, the majority of samples are generated with $K\leq 10$.
[^3]: The proposed PINNs cannot use the augmented samples for training, since the permutation invariance property has been used for constructing the architecture.
|
---
abstract: 'Motivated by the concept of Möbius aromatics in organic chemistry, we extend the recently introduced concept of fragile Mott insulators (FMI) to ring-shaped molecules with repulsive Hubbard interactions threaded by a half-quantum of magnetic flux ($hc/2e$). In this context, a FMI is the insulating ground state of a finite-size molecule that cannot be adiabatically connected to a single Slater determinant, i.e., to a band insulator, provided that time-reversal and lattice translation symmetries are preserved. Based on exact numerical diagonalization for finite Hubbard interaction strength $U$ and existing Bethe-ansatz studies of the one-dimensional Hubbard model in the large-$U$ limit, we establish a duality between Hubbard molecules with $4n$ and $4n+2$ sites, with $n$ integer. A molecule with $4n$ sites is an FMI in the absence of flux but becomes a band insulator in the presence of a half-quantum of flux, while a molecule with $4n+2$ sites is a band insulator in the absence of flux but becomes an FMI in the presence of a half-quantum of flux. Including next-nearest-neighbor-hoppings gives rise to new FMI states that belong to multidimensional irreducible representations of the molecular point group, giving rise to a rich phase diagram.'
author:
- Lukas Muechler$^1$
- 'Joseph Maciejko$^{2,3}$'
- Titus Neupert$^3$
- Roberto Car$^1$
bibliography:
- 'lit.bib'
title: Möbius molecules and fragile Mott insulators
---
Introduction
============
While the term *strong correlations* is commonly used to describe a broad range of interacting systems, one typically considers a fermionic system to be strongly correlated if the quasiparticle picture breaks down, i.e., there is no continuous evolution between the noninteracting system and its interacting counterpart. However, strong correlations are not in one-to-one correspondence with strong interactions. On the one hand, a system can be strongly interacting (large Hubbard $U$ or Hund’s coupling $J$) yet not be strongly correlated in the above sense, as commonly encountered in transition metal oxides.[@kuebler; @exchangeeffect; @dlmoxide; @Moxquasipart; @MnOlocalmoment; @georges2013strong] On the other hand, while in two and higher dimensions a Fermi liquid is stable against weak repulsive interactions, one-dimensional (1D) systems can exhibit a strongly correlated phase already at small values of $U$ and $J$, leading to a Luttinger liquid.[@luttliq] In quantum chemistry this distinction corresponds to the difference between dynamical and static correlation. Dynamically correlated systems interact strongly, yet there are well defined quasiparticles and perturbative methods can be applied. This description breaks down for statically correlated systems, a classic example of which is the dissociation of the $H_2$ molecule beyond the Coulson-Fischer point.[@coulsonfischer] At this point, a single Slater determinant is not sufficient to describe the correct physics even qualitatively.
The theoretical model most widely used to study correlation effects in fermionic systems is the Hubbard model. In particular, it is well known that the 1D Hubbard model exhibits distinct behavior for systems with $4n$ and $4n+2$ sites, where $n$ is an integer, when periodic boundary conditions (PBC) are assumed. This difference in behavior is also well studied in organic chemistry. Molecules of the form C$_N$H$_N$ are called aromatic if $N=4n+2$ and anti-aromatic if $N=4n$. Aromatic molecules, such as benzene (C$_6$H$_6$), have a unique chemistry due to their chemical stability as well as a complex response to magnetic fields due to the aromatic ring current.[@ringcurr_3; @*fowler2007aromaticity; @*steiner2001four] Anti-aromatic compounds can be as stable as aromatic compounds if the topology of the orbital arrangement is that of a Möbius band \[Fig. \[fig\_1\](b)\].[@Heilbronner] Remarkably, such Möbius aromatics have been successfully synthesized.[@moebius1; @*moebius2; @*moebius3; @*moebius4; @*moebius5] The Möbius topology of the orbital arrangement is equivalent to a ring with PBC but threaded by a half-quantum of magnetic flux $hc/2e$ (also known as a $\pi$ flux), or equivalently to a system with antiperiodic boundary conditions (aPBC) and zero flux.[@zoltan]\
![Orbital topology of (a) an aromatic molecule and (b) a Möbius aromatic.[]{data-label="fig_1"}](fig_1.png){width="\columnwidth"}
Perhaps surprisingly, the simplest member of the $4n$ family—the Hubbard square with $n=1$—forms an interesting strongly correlated state at half-filling, the fragile Mott insulator (FMI).[@steve_fragile] In general, a FMI is an insulator that cannot be adiabatically connected to a band insulator (BI) under the condition that time-reversal symmetry and certain point-group symmetries are preserved. The ground-state wave function of a BI is a single Slater determinant that must transform as the identity (trivial) representation of the point group, whereas the FMI is a correlated state whose ground-state wave function transforms as a nontrivial representation of the point group. For any $U > 0$, the ground state of the Hubbard square is unique and transforms as the $d_{x^2-y^2}$ representation of the $C_{4v}$ point group (i.e., the spatial symmetry group of the molecule as a whole) with a $C_4$ eigenvalue of $-1$.
In this paper, we use numerical and analytical methods to explore the interplay between interaction and correlation in more generic Hubbard molecules with time-reversal and point group symmetries. After a brief review of the concept of FMI (Sec. II), we extend this concept to Möbius molecules (Sec. III) and find both weakly correlated BI phases and strongly correlated FMI phases in two representative examples—molecules with $N=4$ and $N=6$ sites (Sec. IV). Results in the general cases of $N=4n$ and $N=4n+2$ are then inferred from existing Bethe ansatz studies (Sec. V). Adding a next-nearest-neighbor (NNN) hopping to the $N=4$ and $N=6$ molecules, we find an even richer set of FMI phases, some corresponding to higher-dimensional irreducible representations of the molecular point group (Sec. VI).
Fragile Mott insulators
=======================
In this section we give a brief review of the concept of FMIs.[@steve_fragile] We consider spinful fermions governed by a noninteracting Hamiltonian $\mathcal{H}_0$ that commutes with the antiunitary time-reversal symmetry operator $\mathcal{T}$ with $\mathcal{T}^2=-1$. The single-particle eigenstates of $\mathcal{H}_0$ are Kramers doublets $\ket{n}$ and $\mathcal{T} \ket{n} \equiv \ket{\tilde{n}}$ with $\braket{n|\tilde{n}} = 0$. In a second-quantized formulation where $c^\dagger_n$ creates a fermion in the single-particle state $|n\rangle$, a BI is a state in which the members of a Kramers doublet are either both occupied or both unoccupied, $$\ket{\mathrm{BI}} = \prod_{n,\tilde{n} \in \mathrm{occ}} c^{\dag}_n c^{\dag}_{\tilde{n}} \ket{0},$$ where $\ket{0}$ is the vacuum state with no fermions. We assume that $\mathcal{H}_0$ also possesses a unitary symmetry represented by the operator $\mathcal{R}$, i.e., $[\mathcal{H}_0,\mathcal{R}] = 0$. The single-particle states $\ket{n}$ can then be chosen to be eigenstates of the symmetry operator $\mathcal{R}$ with eigenvalues $\lambda_n$, $$\mathcal{R}\ket{n} = \lambda_n \ket{n}.$$ If the unitary symmetry $\mathcal{R}$ is such that $\mathcal{R}^N = 1$ for some integer $N\geq 1$, the eigenvalues of $\mathcal{R}$ lie on the unit circle in the complex plane, $$\lambda_n = e^{i2\pi\ell_n/N},\hspace{5mm}\ell_n =1,\ldots,N.$$ Furthermore, we assume that the symmetry operation $\mathcal{R}$ commutes with time-reversal symmetry $[\mathcal{R},\mathcal{T}]=0$. In this case the $\mathcal{R}$ eigenvalues of the Kramers partners are related by complex conjugation $\lambda^{\ast}_n = \lambda_{\tilde{n}}$, since $$\mathcal{R}\ket{\tilde{n}} = \mathcal{R}\mathcal{T}\ket{n} = \mathcal{T}\mathcal{R}\ket{n} = \mathcal{T}\lambda_n\ket{n} \\= \lambda^{\ast}_n \mathcal{T}\ket{n} = \lambda^{\ast}_n \ket{\tilde{n}},$$ which implies that the band-insulator ground state $\ket{\mathrm{BI}}$ transforms trivially under the symmetry $\mathcal{R}$, $$\label{BItrans}
\mathcal{R}\ket{\mathrm{BI}} = \prod_{n,\tilde{n} \in \mathrm{occ}} \lambda_n \lambda_{\tilde{n}} \ket{\mathrm{BI}} = \ket{\mathrm{BI}}.$$ By contrast, a FMI is an insulator such that its ground state $\ket{\mathrm{FMI}}$ transforms nontrivially under $\mathcal{R}$, $$\begin{aligned}
\label{FMItrans}
\mathcal{R}\ket{\mathrm{FMI}}=\lambda\ket{\mathrm{FMI}},\end{aligned}$$ with $\lambda\neq 1$. By virtue of Eq. (\[BItrans\]), a FMI must be a correlated state that cannot be described by a single Slater determinant. In the present context of spinful fermions hopping on a translationally invariant ring-shaped molecule with $N$ sites, $\mathcal{R}$ is the operator for a translation by one lattice site (which can also be considered as a $C_N$ rotation in the point group of the molecule as a whole).
Möbius molecules {#sec:mobius}
================
We consider the following second-quantized Hubbard Hamiltonian to model a ring molecule threaded by a magnetic flux $\Phi$, $$\begin{aligned}
\label{Hamiltonian}
\mathcal{H}(\Phi) &= -t \sum_{\sigma}\left( \sum^N_{j=1} e^{i\phi_j}c^{\dag}_{j\sigma} c_{j+1,\sigma} + \mathrm{H.c.} \right)\nonumber\\
&\hspace{5mm}+ U \sum_{j=1}^N c^{\dag}_{j\uparrow} c_{j\uparrow} c^{\dag}_{j\downarrow} c_{j\downarrow}, \end{aligned}$$ where $c_{j\sigma}^\dag$ ($c_{j\sigma}$) creates (annihilates) an electron of spin $\sigma$ on site $j$ and we define $c_{N+1,\sigma}\equiv c_{1\sigma}$, which corresponds to PBC, $t>0$ is the hopping amplitude, and $U>0$ is the strength of the repulsive interaction. The total flux threading the ring is $\Phi = \sum_j \phi_j$, and we fix the total electron number to be $N$, which is half filling.
All physical observables such as the total flux $\Phi$ are invariant under a local $U(1)$ gauge transformation $$c^{\dag}_{j\sigma}\rightarrow e^{i\alpha_j}c^{\dag}_{j\sigma},
\qquad \sigma=\uparrow\downarrow.$$ In contrast, the individual phases $\phi_j$ that appear in the hopping matrix elements of $\mathcal{H}(\Phi)$ are not invariant under this gauge transformation. While our results will not depend on the choice of gauge, we will work in the uniform gauge $\phi_j=\Phi/N\equiv\phi$ from here on to make the derivation more transparent. We denote by $\mathcal{H}_{\mathrm{uni}}(\Phi)$ the Hamiltonian in this uniform gauge.
The Hamiltonian $\mathcal{H}(\Phi)$ possesses a family of translational symmetries labeled by $\varphi\in[0,2\pi)$, that are represented by $\mathcal{R}_{\varphi}$ in the uniform gauge Hamiltonian $\mathcal{H}_{\mathrm{uni}}(\Phi)$ with $$\label{Rvarphi}
\mathcal{R}_{\varphi} c^{\dag}_{j\sigma}\mathcal{R}_{\varphi}^{-1}= e^{-i\varphi} c^{\dag}_{j+1,\sigma}.$$ For general $\Phi$, the Hamiltonian $\mathcal{H}(\Phi)$ is not time-reversal symmetric. Only for the special values $\Phi=0,\pi$, corresponding to the ring and the Möbius molecule, is it possible to define a time-reversal symmetry. For $\Phi=0$, time-reversal symmetry is represented by $\mathcal{T}=i\sigma_2\mathcal{K}$, where $\mathcal{K}$ stands for complex conjugation and $\sigma_2$ is the second Pauli matrix acting on the spin index $\sigma$. By contrast, for $\Phi=\pi$ time-reversal symmetry is represented by $\widetilde{\mathcal{T}}=U_{\pi} \mathcal{T}$ where $U_\pi$ is a unitary operator defined by $$U_{\pi}c^{\dag}_{j\sigma}U_{\pi}^{-1} = e^{-i2\pi j/N} c^{\dag}_{j\sigma} .$$ Of the family of translational symmetries $ \mathcal{R}_{\varphi}$, only $\mathcal{R}_{\phi}$ commutes with $U_\pi$, while for instance $[\mathcal{R}_0,\tilde{\mathcal{T}}] = e^{i 2 \pi /N}$. For that reason, we will focus on the translational symmetry $\mathcal{R}_{\phi}$ from here on, because the proof of Eq. (\[BItrans\]) relied on the assumption that time-reversal and lattice symmetries commute.
Consider now the Hamiltonian $\mathcal{H}_{\mathrm{uni}}(\pi)$ in the noninteracting limit $U=0$. The $\mathcal{R}_{\phi}$ eigenvalues $\tilde{\lambda}_{n_{\pi}}$ of the single-particle eigenstates $\ket{n_{\pi}}$ lie on the unit circle with $$\tilde{\lambda}_{n_{\pi}} = e^{i2\pi \left(\ell_{n_{\pi}} + 1/2 \right)/N},\hspace{5mm} \ell_{n_{\pi}} =1,\ldots,N,$$ because $\mathcal{R}_{\phi}^N =e^{-i\Phi}= -1$. Furthermore, because $[\mathcal{R}_{\phi},\tilde{\mathcal{T}}] = 0$ the eigenvalues satisfy $\tilde{\lambda}^{\ast}_{n_{\pi}} = \tilde{\lambda}_{\tilde{n}_{\pi}}$, $$\begin{aligned}
\mathcal{R}_{\phi}\ket{\tilde{n}_{\pi}} & = \mathcal{R}_{\phi} \tilde{\mathcal{T}}\ket{n_{\pi}}
= \tilde{\mathcal{T}}\mathcal{R}_{\phi}\ket{n_{\pi}}
= \tilde{\mathcal{T}}\tilde{\lambda}_{n_{\pi}}\ket{n_{\pi}} \nonumber\\
& = \tilde{\lambda}^{\ast}_{n_{\pi}} \tilde{\mathcal{T}}\ket{n_{\pi}}
= \tilde{\lambda}^{\ast}_{n_{\pi}} \ket{\tilde{n}_{\pi}}.\end{aligned}$$ Thus a band-insulator ground state at $\Phi=\pi$, $$\begin{aligned}
\ket{\mathrm{BI}_{\pi}} = \prod_{n_{\pi},\tilde{n}_{\pi} \in \mathrm{occ}} c^{\dag}_{n_{\pi}} c^{\dag}_{\tilde{n}_{\pi}} \ket{0},\end{aligned}$$ transforms trivially under the translation operator $\mathcal{R}_\phi$, $$\mathcal{R}_{\phi} \ket{\mathrm{BI}_{\pi}} = \prod_{n_{\pi},\tilde{n}_{\pi} \in \mathrm{occ}} \tilde{\lambda}_{n_{\pi}} \tilde{\lambda}_{\tilde{n}_{\pi}} \ket{\mathrm{BI}_\pi} = \ket{\mathrm{BI}_\pi},$$ which is the $\pi$-flux analog of Eq. (\[BItrans\]). By analogy with Eq. (\[FMItrans\]), this allows us to extend the concept of FMI to Möbius molecules. We will say that a Möbius molecule has an FMI ground state $\ket{\mathrm{FMI}_\pi}$ if it transforms nontrivially under $\mathcal{R}_{\phi}$, $$\begin{aligned}
\mathcal{R}_{\phi}\ket{\mathrm{FMI}_\pi}=\tilde{\lambda}
\ket{\mathrm{FMI}_\pi},\end{aligned}$$ with $\tilde{\lambda}\neq 1$.
Molecules with 4 and 6 sites
============================
![Ground state energy of the 4-site (red diamonds) and 6-site (blue circles) half-filled Hubbard model with zero flux (open markers) and $\pi$ flux (filled markers), obtained by ED. The BIs are lower in energy than the FMIs.[]{data-label="fig_2"}](fig_2.pdf){width="0.95\columnwidth"}
As an illustration of the concepts presented above, and before discussing the general case with $N$ sites, we perform exact numerical diagonalization (ED) studies of the Hamiltonian (\[Hamiltonian\]) for $N=4$ and $N=6$, and with $\Phi=0$ and $\Phi=\pi$. The low-energy spectrum (Fig. \[fig\_3\]) allows us to determine whether the system is a metal or an insulator, and an explicit computation of the ground-state eigenvalue of the appropriate translation operator allows us to determine whether the system is a BI or a FMI. As mentioned previously, the ground state of the 4-site Hubbard model with zero flux and $U>0$ is a FMI with an $\mathcal{R}_{\phi}\,(=\mathcal{R}_{0})$ eigenvalue of $-1$.[@steve_fragile] At $\Phi=\pi$ however, the $\mathcal{R}_{\phi}\,(=\mathcal{R}_{\pi/N})$ eigenvalue is $+1$ for all $U\geq 0$ and thus the ground state is a BI. Although strictly speaking the theorem (\[BItrans\]) assumed noninteracting electrons, we find no level crossing as a function of $U$ between the ground and first excited state, and the $\mathcal{R}_{\phi}$ eigenvalue of the ground state does not change. Therefore, to be more precise, one should say that the ground state at $U>0$ is adiabatically connected to the BI at $U=0$. On the other hand, we find that a ring with 6 sites at zero flux is a BI with a $\mathcal{R}_{\phi}$ eigenvalue of $+1$ for all $U\geq 0$. With a $\pi$ flux the situation reverses and the $\mathcal{R}_{\phi}$ eigenvalue is $-1$ for $U>0$, so the ground state is a FMI.
This behavior can can be understood qualitatively from the $U=0$ electronic structure. For 4 sites and 0 flux there are two degenerate single-particle states at the Fermi level, so the ground state cannot be a BI at half filling \[Fig. \[fig\_2\](a)\]. This is the molecular analog of a metallic state that turns into a Mott insulating state for $U >0$. For 6 sites, all single-particle states can be completely filled \[Fig. \[fig\_2\](b)\]. This is the molecular analog of a BI. In this case, interactions effects at small $U>0$ can be treated perturbatively and do not destabilize the state because of the single-particle gap. For a $\pi$ flux, the single-particle states for 4 sites can all be filled to give a band insulating state \[Fig. \[fig\_2\](c)\]. However, the 6-site system has degenerate single-particle states at the Fermi level and the situation is reversed with respect to the case of zero flux \[Fig. \[fig\_2\](d)\]. At $U=0$ the ground states of the BIs are unique, while the metallic states are six-fold degenerate. For both 4 and 6 sites, the ground states for $U>0$ are unique for both fluxes (Fig. \[fig\_3\]).[@Lieb_peierls; @nakano2000flux] At large $U$, the gap between ground and first excited state becomes very small. This can be understood from the fact that the large-$U$ limit of the Hubbard model is a Heisenberg model with exchange constant $J=4t^2/U$ that defines the energy scale.
![Single-particle energy levels for the 4- and 6-site rings at $U=0$. The insets give the $U>0$ ground-state eigenvalues $\lambda$ and $\tilde{\lambda}$ of the translation operators $\mathcal{R}_{0}$ and $\mathcal{R}_{\phi}$, respectively.[]{data-label="fig_3"}](fig_3.pdf){width="\columnwidth"}
In the next section, we generalize these results to molecules with any even number of sites $N\geq 4$. In analogy to Möbius aromatics, we now prove that rings with $4n$ sites at zero flux and rings with $4n+2$ sites at $\pi$ flux are FMIs, since there will always be degenerate states at the Fermi level for $U=0$ in analogy to the 4- and 6-site rings discussed above.
Bethe ansatz and $4n$ vs $4n+2$
===============================
The translations $\mathcal{R}_{\phi}$ are generated by the total gauge-invariant momentum, $$\begin{aligned}
\label{GImomentum}
P = \sum_{k\sigma} \left(k - \frac{\Phi}{N}\right) c^{\dag}_{k\sigma} c_{k\sigma},\end{aligned}$$ through $\mathcal{R}_{\phi}=e^{iP}$, where the sum over $k$ is over all momenta in the first Brillouin zone $-\pi<k\leq\pi$. At half filling, $\sum_{k\sigma}c^\dag_{k\sigma}c_{k\sigma}=N$ and we have $$\begin{aligned}
P = \sum_{k\sigma}k c^{\dag}_{k\sigma} c_{k\sigma}-\Phi,\end{aligned}$$ i.e., the total gauge-invariant momentum is obtained by simply shifting the total canonical (non-gauge-invariant) momentum by the total flux $\Phi$. In the $\pi$-flux case, $\mathcal{R}_{\phi}$ corresponds to a translation generated by $P = \sum_{k\sigma}k c^{\dag}_{k\sigma} c_{k\sigma}-\pi$ (see Appendix \[sec:app\] for details). Based on the $U \rightarrow \infty$ Bethe ansatz solution of the 1D Hubbard model[@Lieb_uniqueness; @*thesis_uinf; @*uinf_article] and due to the fact that the ground states are unique for all $U$ without any level crossing, we can prove that rings with $4n+2$ sites and a $\pi$ flux are FMIs characterized by a nontrivial eigenvalue of $e^{iP}$, which corresponds to a finite total momentum. Rings with $4n$ sites are FMIs that become BIs upon inserting a $\pi$ flux.
For $U \rightarrow \infty$ at half filling and PBC, the momenta obtained from the Bethe ansatz are given by $k_j = 2\pi I_j/N$, where the $I_j$ are half-odd integers ($j - \frac{N+1}{2}$) for $N = 4n+2$ and integers ($ j - \frac{N}{2}$) for $N=4n$.[@Betheapbc] The total momentum $\sum_j 2\pi I_j/N$ is thus zero for $N = 4n+2$ and $\pi$ for $N = 4n$. For aPBC in the Bethe ansatz, which corresponds to a $\pi$ flux, the $I_j$ are integers for $N = 4n+2$ and half-odd integers for $N = 4n$, thus the total momentum is shifted by $\pi$.
Next-nearest-neighbor hopping
=============================
![Phase diagram for $N = 6$ at 0-flux with nearest-neighbor hopping $t_1$ and NNN hopping $t_2$, obtained by ED. Ground states are labeled by a symbol $^{2S+1}\Gamma$ where $\Gamma$ denotes the irreducible representation of the molecular point group $C_{6v}$ according to which the ground state transforms and $S$ is the total spin in the ground state.[]{data-label="fig_4"}](fig_4.pdf){width="0.90\columnwidth"}
Since we have considered a Hamiltonian with only nearest-neighbor hoppings so far, the stability of the FMI phases at $\pi$ flux with respect to NNN hoppings $t_2$ is an important question, as such terms are present in all realistic materials. We consider a nnn hopping term of the form $$\begin{aligned}
\mathcal{H}_{\mathrm{nnn}} = -t_2 \sum_{\sigma}\left( \sum^N_{j=1} e^{i2\phi_j}c^{\dag}_{j\sigma} c_{j+2,\sigma} + \mathrm{H.c.} \right),\end{aligned}$$ that preserves all the symmetries of the systems considered before if $\phi_j$ equals $0$ or $\pi/N$. Varying $t_2$ allows us to explore the relationship between the geometry of the molecule and its electronic structure. For example, in the 4-site model at zero flux, there is a phase transition at a critical value of the NNN hopping $t_2 = t_1$ where the system acquires an enhanced tetrahedral symmetry.[@steve_fragile] Choosing the $N =6$ case as an example, we focus on the evolution of the single-particle levels at $U=0$, from which the behavior at $ U > 0$ can be understood.
The single-particle energies are given by $$\begin{aligned}
\varepsilon(k) =- 2t_1 \cos(k_j) - 2 t_2 \cos(2k_j),\end{aligned}$$ where $k_j= 2\pi j/N$ for 0 flux and $k_j= 2\pi(j+1/2)/N $ for $\pi$ flux where $j= -3,-2,\ldots,2$. For 0 flux and $t_2 = 0$, there are 6 single-particle levels (including spin) that can be completely filled and the system is a BI \[Fig. \[fig\_2\](b)\]. To introduce electronic frustration at 0 flux, the levels at $k = \pi$ and $ k = \pm\pi/3$ must cross, which happens at $t_2 = t_1$ where the system acquires an enhanced octahedral symmetry.
Figure \[fig\_4\] shows the phase diagram of the 6-site molecule at zero flux and $U > 0$ calculated by ED, with nnn hopping. Ground states are labeled by a symbol $^{2S+1}\Gamma$ where $\Gamma$ denotes the irreducible representation of the molecular point group $C_{6v}$ according to which the ground state transforms and $S$ is the total spin in the ground state. The BI phase $^1A_1$ is the ground state until $t_2 = t_1$ and for a large range of values of $U$. When the symmetry of the problem is close to octahedral, three nontrivial correlated phases emerge. The $^1B_2$ phase is a unique FMI ground state that occurs at large values of $U$, whereas the $^3A_2$ phase is a spin-triplet FMI state that occurs for $t_2 > t_1$. In the limit $t_2/t_1\rightarrow\infty$ the problem reduces to two decoupled staggered triangles. The intermediate $^1E_2$ phase is a doubly degenerate FMI with total momentum $P = \pm 2\pi/3$. For $U=0$, the ground state is unique for $t_1 > t_2$. At $t_1=t_2$ the ground state is 15-fold degenerate and 6-fold degenerate for $t_2 > t_1$.\
Figure \[fig\_5\] shows the phase diagram for the same system but with a $\pi$ flux. At $t_2 = 0$, the many-body ground state is 6-fold degenerate at $U = 0$. There is a crossing of the doubly degenerate single-particle levels at $k_j = \pm 5\pi/6 $ and $k_j = \pm \pi/2 $ for $t_2/t_1 = 1/\sqrt{3}$. As in the absence of nnn hopping, the $\pi$ flux interchanges the FMI ($^1\widetilde{B}_2$) and BI ($^1\widetilde{A}_1$) phases with respect to the zero flux case. Around the level crossing the two other phases $^1\widetilde{E}_2$ and $^3\widetilde{A}_2$ emerge, similarly to the zero flux case. We use $\widetilde{\Gamma}$ instead of $\Gamma$ to denote the irreducible point group representations in the $\pi$ flux case simply to indicate that the translation operator or, alternatively, $C_N$ rotation operator should be taken as $\mathcal{R}_{\pi/N}$ for a $\pi$ flux, while it is $\mathcal{R}_0$ for zero flux (see Sec. \[sec:mobius\]). The ground state at $U = 0$ is 6-fold degenerate for $ t_2/t_1 < 1/\sqrt{3}$ and $ t_2/t_1 > 1/\sqrt{3}$ . At the level crossing the ground-state degeneracy is 28.
![Phase diagram for $N = 6$ at $\pi$-flux with nearest-neighbor hopping $t_1$ and NNN hopping $t_2$, obtained by ED. Ground states are labeled by a symbol $^{2S+1}\widetilde{\Gamma}$ where $\widetilde{\Gamma}$ denotes the irreducible representation of the molecular point group $C_{6v}$ according to which the ground state transforms and $S$ is the total spin in the ground state. $\widetilde{\Gamma}$ differs from $\Gamma$ in the definition of the translation operator ($\mathcal{R}_{\pi/N}$ for $\pi$ flux and $\mathcal{R}_0$ for zero flux, see Sec. \[sec:mobius\]).[]{data-label="fig_5"}](fig_5.pdf){width="0.90\columnwidth"}
Conclusion
==========
We have shown that there is a duality between Hubbard rings with $N = 4n$ and $N = 4n+2 $ sites, which can be tuned by threading a magnetic flux through the ring. For zero flux, $4n$-membered rings are FMIs that cannot be adiabatically transformed into BIs if time-reversal symmetry and the molecular point group symmetry are preserved. Rings with $4n+2$ sites and a $\pi$ flux are also FMIs. All these FMI states do not break any symmetries and are characterized by a nonlocal order parameter, the total gauge-invariant momentum (\[GImomentum\]). This order parameter is nonlocal because the lattice Fourier transform of $k$, as opposed to that of the periodic function $\sin k$, has a slow power-law decay $\sim 1/x$ in real space. The FMI is an example of non-fractionalized featureless Mott insulator protected by lattice symmetries.[@parameswaran2013; @kimchi2013] In the $N \rightarrow \infty$ limit, the BI and the FMI become degenerate, since their distinction is due to boundary conditions (and the half-filled 1D Hubbard model has a gapless spin sector in the thermodynamic limit[@Giamarchi]).
From an organic chemistry point of view, the FMIs considered here are anti-aromatic molecules, whereas BIs are aromatic molecules. Anti-aromatic molecules typically appear as transition states, since BIs are lower in energy. A BI state is usually obtained by breaking the symmetry of the molecule through a spontaneous structural distortion. For example, the anti-aromatic molecule cyclooctatetraene (C$_8$H$_8$) cannot be isolated in a $D_{8h}$ symmetric structure since it adopts a tub configuration with lower $D_{2d}$ symmetry.[@cot1; @cot2] Therefore, our findings appear to be related to the Woodward-Hoffmann rules.[@hoffmann1965selection; @*woodward1969conservation] These rules only allow pericyclic reactions where the transition states are BI states, whereas reactions with an FMI as transition state are forbidden.
More broadly, our work illustrates the complexity of the relationship between interaction and correlation in fermionic systems. This is made most apparent by considering nnn hopping as a tuning parameter in our models, which mimics the effects of the geometric structure of the molecule on electronic properties. Weakly correlated BI phases do occur at small $U$ (e.g., the $^1A_1$ phase in Fig. \[fig\_4\]) as one would expect, but they also occur at strong $U$ (e.g., the $^1\widetilde{A}_1$ phase in Fig. \[fig\_5\]). Conversely, strongly correlated FMI phases do occur at large $U$ (e.g., the $^1B_2$ phase in Fig. \[fig\_4\]) as one would expect, but they can also occur at small $U$ (e.g., the $^3A_2$ phase in Fig. \[fig\_4\] and the $^1\widetilde{B}_2$ and $^3\widetilde{A}_2$ phases in Fig. \[fig\_5\]). For small $t_2$, $U$ is not the deciding parameter in the phase diagram because it is the geometry that gives rise to electronic frustration. For large enough $t_2$ however, the precise value of $U$ plays an important role in determining the phase diagram.
Experimentally, it is very challenging to discriminate the FMI and BI phase directly, for their order parameters are hard to access and the measurement itself may not break the rotational symmetry explicitly. However, indirect evidence for the quantum phase transitions between FMI and BI could be obtained from spectroscopic measurements. For example, one could envision scanning tunneling microscopy experiments on molecules that are deposited on a solid substrate. [@gomes2012designer; @pitters2011tunnel] If that substrate is a type-II superconductor, it is even conceivable that a $\pi$ flux can be trapped at the center of an arrangement of molecules in order to explore the phase diagrams presented in this work.
L.M. would like to thank Claudia Felser and Shoucheng Zhang for important comments and discussions in the early stages of this work. We thank Steven Kivelson for providing useful references as well as Elliott H. Lieb and Zoltán G. Soos for stimulating discussions. This work was supported in part by the Simons Foundation (J.M.), the Natural Sciences Engineering Research Council (NSERC) of Canada (J.M.), the DARPA grant SPAWARSYSCEN Pacific N66001-11-1-4110 (T.N.) as well as the Department of Energy grant DE-FG02-05ER46201 (L.M. and R.C.).
Total momentum as generator of translations {#sec:app}
===========================================
To find the explicit form of the momentum operator, i.e., the generator of translations, we first consider the ansatz $$\begin{aligned}
\mathcal{R} = e^{iP}=\exp\left(\sum_{nm} A_{nm} c^{\dag}_n c_m\right),\end{aligned}$$ for the translation operator at zero flux $\mathcal{R}\equiv\mathcal{R}_0$ \[see Eq. (\[Rvarphi\])\], where $n,m=1,\ldots,N$ are lattice site indices. Using the Baker-Campbell-Hausdorff formula $$\begin{aligned}
e^{X}Y e^{-X} &=Y+\left[X,Y\right]+\frac{1}{2!}[X,[X,Y]]\nonumber\\
&\hspace{5mm}+\frac{1}{3!}[X,[X,[X,Y]]]+\ldots, \end{aligned}$$ where $X = \sum_{nm} A_{nm} c^{\dag}_n c_m$ and $Y= c^{\dag}_l$, as well as the commutator $$\begin{aligned}
[c^{\dag}_n c_m, c^{\dag}_l] = \delta_{ml} c^{\dag}_n,\end{aligned}$$ we find $$\begin{aligned}
e^{X} c^{\dag}_l e^{-X} &= c^{\dag}_l + \sum_{n} A_{nl} c^{\dag}_n + \frac{1}{2!} \sum_{n} A^2_{nl} c^{\dag}_n
+ \frac{1}{3!} \sum_{n} A^3_{nl} c^{\dag}_n \nonumber\\
&\hspace{5mm}+ \ldots\nonumber \\
&= \left[\exp({{\boldsymbol{A}}}^T){{\boldsymbol{c}}}^{\dag}\right]_l,\end{aligned}$$ where ${{\boldsymbol{A}}}$ is the $N\times N$ matrix with elements $A_{nm}$ and ${{\boldsymbol{c}}}^\dag$ is a $N\times 1$ column vector with elements $c^\dag_l$. The action of the translation should be $\left[\exp({{\boldsymbol{A}}}^T){{\boldsymbol{c}}}^{\dag}\right]_l= c^{\dag}_{l+1}$, which can be represented as $\left[{{\boldsymbol{T}}}{{\boldsymbol{c}}}^{\dag}\right]_l = c^{\dag}_{l+1}$ where ${{\boldsymbol{T}}}$ is a translation matrix. This implies that ${{\boldsymbol{A}}}^T = \ln{{\boldsymbol{T}}}$. PBC imply that ${{\boldsymbol{T}}}^N=1$, hence the eigenvalues of ${{\boldsymbol{T}}}$ are of the form $\lambda_n = e^{i2\pi \ell_n/N}$ with $\ell_n=1,\dots,N$. Furthermore, ${{\boldsymbol{T}}}$ is a diagonalizable matrix and its matrix logarithm is given by $\ln {{\boldsymbol{T}}} = {{\boldsymbol{V}}} (\ln{{\boldsymbol{T}}}_{\mathrm{diag}}) {{\boldsymbol{V}}}^{-1}$. Because ${{\boldsymbol{T}}}$ is a circulant matrix, the matrix of eigenvectors ${{\boldsymbol{V}}}$ is given by the kernel of the discrete Fourier transformation, $$\begin{aligned}
(\ln{{\boldsymbol{T}}}_{\mathrm{diag}})_{kl} &= 2\pi i\left(\frac{\ell_k}{N}+\left[\frac{1}{2}-\frac{\ell_k}{N} \right] \right)\delta_{kl},\\
V_{kl} &= \frac{1}{\sqrt{N}} e^{2 \pi i k l/ N},\end{aligned}$$ where $[\cdots]$ denotes the floor function which ensures that the eigenvalues, which can be identified with the single-particle momenta $k_j$, are contained inside the first Brillouin zone $(-\pi,\pi]$. Thus it follows that $$\begin{aligned}
\sum_{nm} A^{T}_{mn} c^{\dag}_n c_m = i \sum_{k\sigma}k c^{\dag}_{k\sigma} c_{k\sigma} =iP.\end{aligned}$$ To compare the Bethe ansatz results of Ref. with our model we need to choose a gauge in which all hoppings are real. A $\pi$ flux in this gauge corresponds to a Hamiltonian $\mathcal{H}_{-}$ where all the hoppings are equal to $t$ except for the hopping between site $N$ and site $1$, which is equal to $-t$. Under the gauge transformation $G: \mathcal{H}(\phi)\rightarrow\mathcal{H}_-$, the electron creation operator transforms as $G^{-1} c^{\dag}_n G = e^{i\phi n } c^{\dag}_n$. The translation operator $\mathcal{R}_\phi$ is also transformed $\mathcal{R}_{-} = G^{-1} \mathcal{R}_{\phi}G$, and acts on the electron creation operator as $$\mathcal{R}_{-} c^{\dag}_n \mathcal{R}_{-}^{-1} = \begin{cases}
c^{\dag}_n &\text{if } n =1,\ldots,N-1, \\
-c^{\dag}_n &\text{if } n=N. \end{cases}$$ The order parameter remains the same in first quantization for PBC and aPBC, where a system with PBC and a $\pi$ flux is equivalent to a system with zero flux and aPBC by a large gauge transformation of the basis functions. The sum of the momenta $k$ obtained by the Bethe ansatz for periodic boundary conditions is thus related to the total momentum caculated by $\mathcal{R}$ whereas that same sum for aPBC can be identified with the total momentum calculated by $\mathcal{R}_{\phi}$.
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---
title: |
A Meta-Programming Approach to Realizing\
Dependently Typed Logic Programming
---
\[ Constraint and logic languages\] \[ Lambda calculus and related systems, Logic and constraint programming, Proof theory\]
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abstract: 'Previous methods on estimating detailed human depth often require supervised training with ‘ground truth’ depth data. This paper presents a self-supervised method that can be trained on YouTube videos without known depth, which makes training data collection simple and improves the generalization of the learned network. The self-supervised learning is achieved by minimizing a photo-consistency loss, which is evaluated between a video frame and its neighboring frames warped according to the estimated depth and the 3D non-rigid motion of the human body. To solve this non-rigid motion, we first estimate a rough SMPL model at each video frame and compute the non-rigid body motion accordingly, which enables self-supervised learning on estimating the shape details. Experiments demonstrate that our method enjoys better generalization and performs much better on data in the wild.'
author:
- |
Feitong Tan$^{1,*}$ Hao Zhu$^{2,}$[^1] Zhaopeng Cui$^{3}$ Siyu Zhu$^{4}$ Marc Pollefeys$^{3}$ Ping Tan$^{1}$\
$^{1}$ Simon Fraser University $^{2}$ Nanjing University\
$^{3}$ ETH Zürich $^{4}$ Alibaba AI Labs\
bibliography:
- 'egbib.bib'
title: 'Self-Supervised Human Depth Estimation from Monocular Videos'
---
[^1]: These authors contributed equally to this work.
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---
abstract: 'Jim Agler revolutionized the area of Pick interpolation with his realization theorem for what is now called the Agler-Schur class for the unit ball in $\mathbb C^d$. We discuss an extension of these results to algebras of functions arising from test functions and the dual notion of a family of reproducing kernels, as well as the related interpolation theorem. When working with test functions, one ideally wants to use as small a collection as possible. Nevertheless, in some situations infinite sets of test functions are unavoidable. When this is the case, certain topological considerations come to the fore. We illustrate this with examples, including the multiplier algebra of an annulus and the infinite polydisk.'
address:
- |
School of Mathematics and Statistics\
Merz Court,\
University of Newcastle upon Tyne\
Newcastle upon Tyne\
NE1 7RU\
UK
- |
Department of Mathematics\
University of Florida\
Box 118105\
Gainesville, FL 32611-8105\
USA
author:
- 'Michael A. Dritschel$^1$ and Scott McCullough$^2$'
bibliography:
- 'test\_functions.bib'
title: 'Test Functions, Kernels, Realizations and Interpolation'
---
Introduction {#sec:introduction}
============
Let $\mathbb D^d$ denote the $d$-polydisk in $\mathbb C^d$, $$\mathbb D^d=\left\{z=(z_1,\dots,z_d)\in\mathbb C^d: |z_j|<1, \
j=1,2,\dots,d \right\}.$$ The Agler-Schur class $\mathcal S_d$ consists of those functions $f:\mathbb D^d\to \mathbb C$ for which there exist positive (that is, positive semidefinite) kernels $\Gamma_j:\mathbb D^d\times \mathbb D^d
\to \mathbb C$ such that $$\label{eq:1}
1-f(z)f(w)^*=\sum_1^d \Gamma_j(z,w)(1-z_jw_j^* ).$$ Here $\zeta^*$ denotes the conjugate of the complex number $\zeta$.
A $\mathbb D^d$ unitary colligation is a pair $\Sigma=(U,\mathcal
E)$, where $\mathcal E=\bigoplus_1^d \mathcal E_j$ is an auxiliary Hilbert space and $$U=\begin{pmatrix} A & B\\ C & D\end{pmatrix} :
\begin{matrix} \mathcal E \\ \oplus \\ \mathbb C \end{matrix}
\rightarrow
\begin{matrix}\mathcal E \\ \oplus \\ \mathbb C \end{matrix}$$ is unitary. With respect to the decomposition of $\mathcal E$, let $Z=\bigoplus z_j I_{\mathcal E_j}$, the $d\times d$ block diagonal operator matrix with diagonal entries $z_jI_{\mathcal E_j}$. The transfer function $W_\Sigma:\mathbb D^d\to \mathbb C$ of the colligation $\Sigma$ is defined as $$W_\Sigma = D+ CZ(I-AZ)^{-1}B.$$
The *primum mobile* for contemporary work on multivariable realization with applications to interpolation theory is the following result of Agler, stated for what is now called the Agler-Schur class functions on the polydisk. See [@MR1207393], [@MR1665697], [@MR1882259], [@MR1846055]. This theorem has been generalized in many directions ([@MR1846055] [@MR1722812], [@MR1885440], [@MR1797710], [@MR2069781], to give a few instances). An operator $C$ on a Hilbert space $H$ is a strict contraction provided $\|C\|<1$.
\[thm:getstarted\] Suppose $f:\mathbb D^d\to \mathbb C$. The following are equivalent.
(i) If $k$ is a positive kernel on $\mathbb D^d$ and if $$(z,w)\mapsto (1- z_jw_j^*)k(z,w)$$ is a positive kernel on $\mathbb D^d$ for each $j=1,2,\dots,d$, then the kernel $$(z,w)\mapsto (1-f(z)f(w)^*)k(z,w)$$ is also positive;
(ii) $f\in\mathcal S_d$;
(iii) There is a unitary colligation $\Sigma$ so that $f=W_\Sigma$; and
(iv) For each tuple $T=(T_1,\dots,T_d)$ of commuting strict contractions on a Hilbert space $H$, $$\|f(T_1,\dots,T_d)\|\le 1.$$
In newer versions $Z$ is replaced by some matrix function on a domain in $\mathbb C^d$, the domain being determined by those values at which the norm of the function is less than $1$ ([@MR1885440], [@MR2069781]). As an example, choosing $$Z=\begin{pmatrix} z_1 & z_2 & \dots & z_d \end{pmatrix}$$ on the unit ball $\mathbb B^d=\{z\in\mathbb C^d: \|z\|<1\}$ in $\mathbb C^d$ gives rise to the Agler-Schur class of functions associated with the row contractions and the Nevanlinna-Pick kernel $k(z,w)=(1-\langle z,w\rangle)^{-1}$. Of course, choosing $Z$ to be the diagonal matrix with diagonal entries $z_j$ leads to the Agler-Schur class of Theorem \[thm:getstarted\].
In the paper [@DMM] Agler-Schur classes and Agler-Pick interpolation were considered in the very general setting of algebras of functions over semigroupoids. With hopes of mollifying those who might otherwise be put off by the algebraic formalism of parts of [@DMM], we drop the semigroupoid structure (or rather, work with a trivial case). The realization and interpolation theorems we present have proofs which in outline follow those found in [@DMM]. However some simplification is achieved in the present setting, and as a novelty we include a von Neumann type inequality along the lines of part (iv) of Theorem \[thm:getstarted\]. Furthermore, we show that strictly contractive functions can be replaced by a certain class $\mathbb F$ of representations of the algebra generated by the test functions. Representations in this family allow for nice approximations in terms of what we call “simple representations” — essentially representations involving only finitely many test functions. Regarding the interpolation theorem, the realization formula for the the interpolating function in the Agler-Pick interpolation theorem has certain additional structure which we highlight.
As Jim Agler first discovered ([@Ag-unpublished], see also [@MR1882259]), realization and interpolation problems in function algebras which might not be multiplier algebras for some reproducing kernel Hilbert space can be effectively handled by means of so-called test functions (in the realization theorem for the polydisk given above, these are the coordinate functions). The test functions are used to delineate the unit ball in the algebra by means of a duality with a family of reproducing kernels. The function algebra is then the intersection of the multiplier algebras of the reproducing kernel Hilbert spaces.
The work of Ambrozie [@MR2106336] highlighted a second duality at play in the Agler realization theorem, and in particular in the definition of the Agler-Schur class. A similar duality is also evident in the work of Agler and McCarthy [@MR1882259]. One can view the test functions as points in some abstract space. Then in , the sum is replaced by a single kernel $\Gamma$ multiplied by $1-E(z)E(w)^*$, where $E(z)$ is the evaluation functional at the point $z$. The principle difference between Ambrozie’s approach and that of Agler and McCarthy then comes down to whether one views the kernel $\Gamma$ as an element of a predual (Agler and McCarthy), or as an element of a dual space, along with the introduction of a more general notion of unitary colligation (Ambrozie). There is little to distinguish these when there are only finitely many test functions. But when there are infinitely many test functions (something looked at by both), the two approaches are quite distinct. We feel that the dual approach offers certain advantages, which hopefully will be made apparent in what follows.
Our initial motivation was an interest in variants of Theorem 13.8 from [@MR1882259] (see also [@Ag-unpublished]) which covers Agler-Pick interpolation on multiply connected domains in $\mathbb C$ (see also [@MR532320], [@MR2163865], [@MR1818066], [@MR1386331], [@MR1909298], and [@MR0188824]), and at the suggestion of John McCarthy [@MR1984460], versions of the Agler algebra for the infinite polydisk. While the examples we consider are replete with analytic structure, a noteworthy feature of the general theory is that no such structure need be imposed, as will be clear from the axioms for test functions given below.
Function algebras on multiply connected domains and the infinite polydisk are examples where infinite families of test functions are required. We look at these in some detail, concentrating on the annulus as the multiply connected domain since all of the salient features of more general domains are already evident in this example. In these two cases, the emphasis will be somewhat different. The infinite polydisk has a natural choice of test functions. However this collection is not compact, the consequence being that there are functions in the Agler-Schur class that are not simply represented by some infinite version of , despite the fact that our definition of the Agler-Schur class naturally reduces to the original version for the finite polydisk. In terms of transfer function representations for functions in the Agler-Schur class, this is manifested in the need for the inclusion of representations in the colligation — or rather a broader class of representations, since in fact the decomposition of $\mathcal E$ used in the construction of $Z$ in the transfer function is essentially a representation of a rather simple form.
The situation is reversed for the annulus in that we are handed a function algebra (the bounded analytic functions on the annulus), and the first step then is to find a good set of test functions. Roughly speaking, such a set should be as small as possible. The collection of test functions we construct in this example is compact, and we show that it is minimal in the sense that there is no closed subset of this collection which is also a collection of test functions for this algebra.
Preliminaries and Main Results {#sec:prel-main-results}
==============================
This section contains a discussion of the ingredients that go into the statement and proof of our generalization of Theorem \[thm:main\], and ends with a statement of the realization formula and Agler-Pick interpolation theorem.
Test Functions and Evaluations {#subsec:test-funct-eval}
------------------------------
For a finite subset $F$ of $X$, let $P(F)$ denote all complex-valued functions on $X$. By declaring the indicator functions of points in $X$ to be an orthonormal basis, $P(F)$ can be identified with the Hilbert space $\mathbb C^F$. For a collection $\Psi$ of functions on $X$, let ${\Psi|_F}=\{\psi|_F:\psi\in \Psi\} \subset P(F)$.
A collection $\Psi$ of functions on $X$ is a *collection of test functions* provided,
(i) For each $x\in X$, $$\sup\{|\psi(x)|:\psi \in \Psi\} <1; \text{ and}$$
(ii) for each finite set $F$, the unital algebra generated by ${\Psi|_F}$ is all of $P(F)$.
The second hypothesis, while not essential, simplifies the exposition. It implies, among other things, that the functions in $\Psi$ separate the points of $X$. As we see shortly, the first hypothesis allows us to use the test functions to define a Banach algebra norm on the algebra of functions over $X$ with addition and multiplication defined pointwise.
The function algebras we will be working with may be multiplier algebras for $H^2(k)$ for some reproducing kernel $k$ (this is what happens in the classical setting when one studies Nevanlinna-Pick interpolation), though this is but a special version of what we wish to consider here. Rather, test functions will allow us to manage in a broader context by means of a familiar dual construction. To this end, we introduce the algebra of bounded continuous functions over $\Psi$ with pointwise algebra operations, denoted by $\CT$.
Let $\Psi$ be a collection of test functions. This is a subspace of $B(X,\overline{\mathbb D})$, the collection of bounded functions from $X$ into the closed unit disk $\overline{\mathbb D}$ (equivalently, $\overline{\mathbb D}^X$), which we endow with the topology of pointwise convergence. By Tychonov’s theorem, this is a compact Hausdorff space. Now $\overline{\mathbb D}$ is a Tychonov space in the usual metric topology (that is, points are closed and for any closed set and point disjoint from it, there is a continuous function separating the two). Consequently, $\overline{\mathbb D}^X$ is Tychonov, as is any subspace. In particular, $\Psi$ is a Tychonov space.
*A priori* the set $X$ is not assumed to have a topology, though in fact it inherits one as a subspace of $\CT$. Since $\Psi$ separates the points of $X$, there is an injective mapping $E:X \to
C_b(\Psi)$, where $E(x) = e_x$ is the evaluation mapping $e_x(\psi) =
\psi(x)$. Hence we can view $X$ as a subset of the unit ball of $C_b(\Psi)$. The same arguments applied above to $\Psi$ now show that with this topology, $X$ is Tychonov. In particular, if a net $\{x_\alpha\}$ converges to $x$ in $X$, then $\psi(x_\alpha)$ converges to $\psi(x)$. Hence $\Psi$ is a subset of the unit ball of $C_b(X)$.
By the same token, for a collection of test functions $\Psi$, the evaluation mapping $F:\Psi\to C_b(\Psi)^*$ can be defined by $F(\psi)
= f_\psi$, where $f_\psi (g) = g(\psi)$ for all $g\in C_b(\Psi)$. The mapping $F$ is continuous in the weak-$*$ topology. Since $\Psi$ is Tychonov, the map $\Psi \mapsto F(\Psi)$ is a homeomorphism and $F(\Psi)$ is a subset of the closed unit ball of $C_b(\Psi)^*$, which is compact. The Stone-Čech compactification $\beta\Psi$ of $\Psi$ is then the weak-$*$ closure of $F(\Psi)$. Since $\beta\Psi$ is a compact function space, it is pointwise closed, and so contains the image of the pointwise closure $\overline{\Psi}$ of $\Psi$. But then $F$ extends to a homeomorphism of $\overline{\Psi}$ to $\overline{F(\Psi)}$. Therefore we can identify $\beta\Psi$ with $\overline{\Psi}$.
By the way, none of the above depends on the assumption that the test functions are complex valued, with the exception of the conclusion that $\CT$ is an algebra generated by $\{E(x): x\in X\}$. We could, for example, take our test functions to have values in a Hilbert $C^*$-module $\mathcal M$ which we may concretely view as a norm closed subalgebra of $B(\mathcal H, \mathcal K)$, the bounded operators between Hilbert spaces $\mathcal H$ and $\mathcal K$. This is a corner of a $C^*$-subalgebra $\mathcal C$ of $B(\mathcal H \oplus
\mathcal K)$ with the property that each element of $\mathcal M$ is the corner of some element of $\mathcal C$ with the same norm. Since this added generality might obscure the main thrust of the paper, we restrict our attention to the simpler setting of scalar valued test functions.
Kernel/test function duality and the Agler-Schur class {#subsec:kern-funct-dual}
------------------------------------------------------
Let $\mathcal B$ denote a $C^*$-algebra with Banach space dual $\mathcal B^*$. A positive (that is, positive semidefinite) kernel on a subset $Y$ of $X$ with values in $\mathcal B^*$ is a function $\Gamma:Y\times Y\to \mathcal B^*$ such that for any finite subset $F$ of $Y$ and function $f:F\to \mathcal B$ $$\sum_{a,b\in F} \Gamma(a,b)(f(b)^*f(a)) \ge 0.$$ Naturally the restriction of a positive kernel on $X$ to a subset $F$ is still positive. We use $M(F,\mathcal B^*)^+$ to denote the collection of positive definite kernels on $F\subseteq X$ with values in $\mathcal B^*$. In the case that $\mathcal B = \mathbb C$ the modifier *“with values in”* is dropped.
Given a collection $\Psi$ of test functions, write $\mathcal K_\Psi$ for the collection of positive kernels $k$ on $X$ such that for each $\psi\in \Psi$ the kernel, $$X\times X \ni (x,y) \mapsto (1-\psi(x)\psi(y)^*)k(x,y)$$ is positive. This is nonempty, since it contains the kernel which is identically $0$. Condition (i) in the definition of a collection of test functions implies that the so-called *Toeplitz kernel* $s$ with $s(x,x) = 1$ for all $x$ and $s(x,y) = 0$ if $y\neq x$ is also in $\mathcal K_\Psi$.
Say that a function $\varphi:X\to \mathbb C$ is in ${{H^\infty({\mathcal K}_\Psi)}}$ if there is a real number $C$ so that, for each $k\in\mathcal K_\Psi$, the kernel $$\label{eq:inHP}
X\times X \ni (x,y) \mapsto (C^2-\varphi(x)\varphi(y)^*)k(x,y)$$ is positive, in which case write $C_\varphi$ for the infimum over all such $C$ (so $C_\varphi$ is independent of $k$). Then $${\|\varphi \|_{{{H^\infty({\mathcal K}_\Psi)}}}} = C_\varphi$$ defines a Banach algebra norm on ${{H^\infty({\mathcal K}_\Psi)}}$. Since the Toeplitz kernel $s$ is in $\mathcal K_\Psi$, norm convergent sequences in ${{H^\infty({\mathcal K}_\Psi)}}$ converge pointwise, and since positivity of the kernels in is verified on finite sets, completeness is easily checked. By definition, the test functions are in the unit ball of ${{H^\infty({\mathcal K}_\Psi)}}$. Indeed, if $\mathcal K_\Psi$ consists solely of those kernels which are conjugate equivalent to the Toeplitz kernel (that is, each $k\in\mathcal K_\Psi$ has the form $k(x,y) = c(x)s(x,y)c(y)^*$ for some function $c$) then we then have ${{H^\infty({\mathcal K}_\Psi)}}= C_b(X)$
Replacing $\Psi$ by its pointwise closure (equivalently, $\beta\Psi$) adds nothing new. We end up with the same collection of kernels (that is, $\mathcal K_{\beta\Psi}=\mathcal K_\Psi$), and hence the same space of functions ${{H^\infty({\mathcal K}_\Psi)}}$ with the same topology.
A function $f:X \to \mathbb C$ is said to be in the *Agler-Schur class* if there exists a positive kernel $\Gamma:
X\times X \to \CT^*$ so that for all $x,y \in X$, $$1-f(x)f(y)^*= \Gamma(x,y)(1-E(x)E(y)^*).$$ In the case that $X = \mathbb D^d$ and $\Psi$ consists of the coordinate functions, we recover the original definition of the Agler-Schur class from the Introduction.
$\CT$-unitary colligations, transfer functions, and the class $\mathbb F$ {#subsec:ct-unit-coll}
-------------------------------------------------------------------------
For a collection of test functions $\Psi$, following [@MR2106336], define a *$\CT$-unitary colligation* $\Sigma$ to be a tuple $\Sigma= (U,\mathcal E,\rho)$, $\mathcal E$ a Hilbert space, $U$ unitary on $\mathcal E \oplus \mathbb C$, and $$\rho:\CT\to B(\mathcal E)$$ a unital $*$-representation. Writing $U = \begin{pmatrix} A & B \\ C
& D \end{pmatrix}$, the *transfer function* associated to $\Sigma$ is defined as $$\label{eq:transfer-bw2}
W_\Sigma(x)= D+CZ(x)(I-AZ(x))^{-1}B,$$ where $Z(x) = \rho(E(x))$. As observed earlier, by assumption (i) for test functions, $\|E(x)\| < 1$ for all $x\in X$, and since $\rho$ is unital, it is contractive. Hence $\|Z(x)\| < 1$ for all $x\in X$ and the definition of $W_\Sigma$ makes sense. Additionally, as the next lemma indicates, $W_\Sigma$ is contractive.
\[lem:tfs-are-contractive\] Let $\Sigma= (U,\mathcal E,\rho)$ be a $\CT$-unitary colligation with associated transfer function $W_\Sigma$. Then $\|W_\Sigma\|
\leq 1$.
This is a standard calculation. Using the relations between the elements of $U = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$ implied by the assumption that it is unitary, we find that for any $k\in \mathcal K_\Psi$, $$\begin{split}
&(1-W_\Sigma(x) W_\Sigma(y)^*)k(x,y) = k(x,y) -
W_\Sigma(x)k(x,y) W_\Sigma(y)^* \\
= &C(I-Z(x)A)^{-1}(k(x,y) - Z(x)k(x,y)Z(y)^*)
(I-Z(y)A)^{-1\,*}C^*.
\end{split}$$ Given a finite set $F$, the matrix $$(k(x,y) - E(x)k(x,y)E(y)^*)_{x,y\in F}$$ is positive, since its value at a test function $\psi$ is $$(k(x,y) - \psi(x)k(x,y)\psi(y)^*)_{x,y\in F}.$$ Since $\rho$ is contractive, the result follows.
A unital representation $\pi:{{H^\infty({\mathcal K}_\Psi)}}\to B(\mathcal H)$ is *weakly continuous* if whenever $\varphi_\alpha$ is a bounded net from ${{H^\infty({\mathcal K}_\Psi)}}$ which converges pointwise to a $\varphi\in{{H^\infty({\mathcal K}_\Psi)}}$, then $\pi(\varphi_\alpha)$ converges in the weak operator topology to $\pi(\varphi)$.
We say that $\pi$ is in the class $\mathbb F$ if $\pi$ is a representation of $H^\infty(\mathcal K_\Psi)$ on a Hilbert space $\mathcal H$ such that
(i) $\pi$ is weakly continuous; and
(ii) $\pi$ is contractive on test functions (that is, $\|\pi(\psi)\|\le 1$ for each $\psi\in\Psi$).
An example of such a representation is one for which $\|\pi(\psi)\| <
1$ for each $\psi\in\Psi$. However the class $\mathbb F$ includes somewhat more, since for example, it also contains the identity representation, $\pi(\psi) = \psi$ for all $\psi\in\Psi$. The main fact about this class is that these representations are automatically contractive and approximately respect transfer function representations.
Abstract realization and Agler-Pick interpolation {#subsec:mainresult}
-------------------------------------------------
The following theorem is the analogue of Theorem \[thm:getstarted\] and in fact contains that theorem as a special case when $X = \mathbb
D^d$ and $\Psi$ consists of the $d$ coordinate functions. More or less a corollary of this is the Agler-Pick interpolation theorem which follows it. With the exception of (iv) in Theorem \[thm:main\] and the concrete form of the space $\mathcal E$ for the unitary colligation in Theorem \[thm:APint\], they are in fact special cases of results to be found in [@DMM]. Note that a corollary of Theorem \[thm:main\] is that the representations in $\mathbb F$ are contractive (recall that *a priori* only their behavior on test functions is prescribed).
\[thm:main\] Suppose $\Psi$ is a collection of test functions. The following are equivalent:
- $\varphi\in{{H^\infty({\mathcal K}_\Psi)}}$ and ${\|\varphi \|_{{{H^\infty({\mathcal K}_\Psi)}}}}\le 1$;
- For each finite set $F\subset X$ there exists a positive kernel $\Gamma:F\times F\to \CT^*$ so that for all $x,y\in F$ $$1-\varphi(x)\varphi(y)^*= \Gamma(x,y)(1-E(x)E(y)^*);$$
- There exists a positive kernel $\Gamma:X\times X\to
\CT^*$ so that for all $x,y\in X$ $$1-\varphi(x)\varphi(y)^*= \Gamma(x,y)(1-E(x)E(y)^*);$$
- There is a colligation $\Sigma$ so that $\varphi=W_\Sigma$;
- For every representation $\pi$ of ${{H^\infty({\mathcal K}_\Psi)}}$ such that $\|\pi(\psi)\| < 1$ for all $\psi\in\Psi$, $\|\pi(\varphi)\| \leq
1$; and
- For every representation of ${{H^\infty({\mathcal K}_\Psi)}}$ in $\mathbb F$, $\|\pi(\varphi)\| \leq 1$.
The Agler-Pick interpolation theorem corresponding to Theorem \[thm:main\] is the following.
\[thm:APint\] Suppose $\TT$ is a collection of test functions. Let $F$, a finite subset of $X,$ and $\xi:F\to \mathbb D$ be given.
- There exists $\varphi\in{{H^\infty({\mathcal K}_\Psi)}}$ so that ${\|\varphi \|_{{{H^\infty({\mathcal K}_\Psi)}}}}\le
1$ and $\varphi|_F=\xi$;
- for each $k\in\mathcal K_\Psi$, the kernel $$F\times F \ni (x,y) \mapsto (1-\xi(x)\xi(y)^*)k(x,y)$$ is positive;
- there exists a positive kernel $\Gamma:F\times F\to
\CT^*$ so that for all $x,y\in F$ $$\label{eq:lurk}
1-\xi(x)\xi(y)^*= \Gamma(x,y)(1-E(x)E(y)^*).$$
Moreover, in this case we also have $\Psi$ is compact (say replacing $\Psi$ by its closure) there is a bounded positive measure $\mu$ on $\Psi$ so that the interpolant $\varphi$ has a transfer function realization of the form $$\varphi(x)=D+CE(x)(I-AE(x))^{-1}B$$ for a unitary $$U=\begin{pmatrix} A &B\\ C&D \end{pmatrix} :
\begin{matrix} \mathbb C^n\otimes L^2(\mu) \\ \oplus \\ \mathbb C
\end{matrix}\rightarrow
\begin{matrix} \mathbb C^n\otimes L^2(\mu) \\ \oplus \\ \mathbb C
\end{matrix},$$ where the $E(x)$ is interpreted as the multiplication operator on $\mathbb C^n\otimes L^2(\mu)$ given by $E(x) h\otimes f = h\otimes
E(x) f$.
(In the last theorem the representation $\rho$ in the definition of $\Psi$-unitary colligation is simply a multiple of the representation of $C(\Psi)$ as multiplication operators on $L^2(\mu)$.
Note that in Theorem \[thm:main\] it is certainly not the case that all positive $\Gamma$ give rise to a $\varphi$. Indeed, Theorem \[thm:APint\] says a $\Gamma$ corresponding to an Agler-Pick interpolation problem can be chosen with additional structure. This will be highlighted in our discussion of the annulus.
As a consequence of the realization theorem, we also get the following.
\[prop:a-s-class-closed\] The Agler-Schur class is closed in the topology of pointwise convergence. In particular, it contains the closure of the test functions.
Organization {#subsec:organization}
------------
The remainder of the paper is organized as follows. Section \[sec:preliminaryresults\] collects results needed to prove Theorem \[thm:main\], which is then given in Section \[subsec:proof\]. Section \[subsec:interpolate\] contains the proof of Theorem \[thm:APint\], the basic Agler-Pick interpolation result companion to Theorem \[thm:main\]. Examples are found in Section \[sec:examples\]. These include the annulus algebra and the Agler algebra of the infinite polydisk. The case of the annulus $\mathbb A$ is covered in the greatest detail, and it is shown that modulo natural equivalences, the set of scalar-valued inner functions defined on $\mathbb A$ with precisely two zeros in $\mathbb A$ and a particular normalization is a minimal collection of test functions for $H^\infty(\mathbb A)$. Further examples show that a certain amount of care is needed in working in the context presented here, especially when the collection of test functions is not finite.
Ingredients {#sec:preliminaryresults}
===========
This section collects results preliminary to the proofs of Theorems \[thm:main\] and \[thm:APint\].
Simple representations and the class $\mathbb F$ {#subsec:reps}
------------------------------------------------
When dealing with an infinite collection of test functions, it is often useful to approximate using a finite subset. This is the idea behind simple representations, which are central to our proof that (iii) implies (iv) in Theorem \[thm:main\], where we approximate the function $x\mapsto \rho(E(x))$ appearing in the transfer function realization.
To be more precise, given a collection of test functions $\Psi$ with $\psi_j\in \Psi$, $j=1,\ldots, N$, and orthogonal projections $P_j$ such that $\sum_{j=1}^N P_j = I$, define a *simple representation* $\rho:\CT \to B(\mathcal E)$ to have the form $$\rho(f) = \sum_{j=1}^N P_j f(\psi_j).$$ Clearly $\rho$ is unital.
Set $Z(x) = \sum_{j=1}^N P_j \psi_j(x) = \rho(E(x))$ and suppose $\pi$ is a representation of ${{H^\infty({\mathcal K}_\Psi)}}$ on $B(\mathcal H)$. Then it is natural to define $$\label{eq:defn-piZ}
\pi(Z) = \sum_{j=1}^N P_j \otimes \pi(\psi_j).$$
\[lem:simple\] If
(i) $\Sigma = (U,\mathcal E,\rho)$ is a $\CT$-unitary colligation;
(ii) the representation $\rho$ is simple;
(iii) $\pi \in \mathbb F$; and
(iv) $\varphi= D+CZ(I-A Z)^{-1}B$, where $Z = \rho(E)$,
then $\pi(\varphi)$ is a contraction.
Let $0<r<1$ and define $$\varphi_r = D+CrZ(I-rA Z)^{-1}B.$$
Fix $x\in X$. Since $E(x)$ is a strict contraction and $\rho$ is a contractive representation, $Z = \rho(E(x))$ is also a strict contraction. It follows that pointwise, $\sum_1^M (rAZ(x))^n$ converges in norm to $(I-rAZ(x))^{-1}$. Consequently, $$\varphi_r^M = D + C Z \sum_1^M (AZ)^n B,$$ converges pointwise with $M$ to $\varphi_r$ and so the sequence $\pi(\varphi_r^M)$ converges weakly to $\pi(\varphi_r)$.
Let $A_j=AP_j$ and for an $n$-tuple $\alpha=(\alpha_1,\dots,\alpha_n)$ with each $\alpha_j \in \{1,2,\dots,N\}$, let $A_\alpha = A_{\alpha_1} A_{\alpha_2} \cdots A_{\alpha_n}$. Define $\psi_\alpha$ similarly. Let $|\alpha|=n$. By expanding $\varphi_r^M$ we have $$\varphi_M = D + \sum_j\sum_{|\alpha|\le M} r^{|\alpha|+1} CP_j
A_\alpha B \psi_j \psi_\alpha.$$ Thus $$\begin{split}
\pi(\varphi_r^M) =& D\otimes I + \sum_j\sum_{|\alpha|\le M}
r^{|\alpha|+1} CP_j A_\alpha B \pi(\psi_j) \pi(\psi)_\alpha. \\
=& D\otimes I + (C\otimes I)r\pi(Z) \sum_{n=1}^M ((r A\otimes
I)\pi(Z))^n (B\otimes I),
\end{split}$$ according to the definition of $\pi(Z)$ in . The right side converges in norm with $M$ to $$D\otimes I +(C\otimes I)r\pi(Z) \left ( (I\otimes I)- r (A\otimes I)
\pi(Z) \right )^{-1} (B\otimes I),$$ giving a transfer function representation for $\pi(\varphi_r)$. The proof that $\pi(\varphi)$ is a contraction proceeds along the lines of that given for Lemma \[lem:tfs-are-contractive\] and makes use of the assumption (built into the definition of $\pi$) that each $\pi(\psi_j)$ is a contraction.
To complete the proof, note that $\varphi_r$ converges (with $r$) pointwise boundedly to $\varphi$ and thus $\pi(\varphi_r)$ converges WOT to $\pi(\varphi)$. Since each $\pi(\varphi_r)$ is contractive, so is $\pi(\varphi)$.
Suppose $\pi\in \mathbb F$. \[prop:pw-simple\] If
(i) $\Sigma = (U,\mathcal E,\rho)$ is a $\CT$-unitary colligation; and
(ii) $\varphi= D+CZ(I-A Z)^{-1}B$, where $Z = \rho(E)$,
then there exists a net of simple representations $\rho_\alpha:\CT
\to B(\mathcal E)$ such that the net $$\varphi_\alpha = D+CZ_\alpha(I-A Z_\alpha)^{-1}B,$$ converges pointwise to $\varphi$, where $Z_\alpha = \rho_\alpha(E)$. Consequently, $\pi(\varphi_\alpha)$ converges weakly to $\pi(\varphi)$, and so $\pi(\varphi)$ is a contraction.
Consider the collection $\mathcal F$ consisting of ordered pairs $(F,\epsilon)$ where $F$ is a finite subset of $X$ and $\epsilon >0$ ordered by $(F,\epsilon)\le (G,\delta)$ if $F\subset G$ and $\delta\le \epsilon$. With this order $\mathcal F$ is a directed set.
Given $\alpha=(F,\epsilon)\in\mathcal F$, by the compactness of $\beta\Psi$ there exists a finite collection $\mathcal
U=\{U_1,\dots, U_m\}$ of nonempty open sets which covers $\beta
\Psi$ with the property that for any two $\psi^\prime,\psi^{\prime\prime} \in U_j$ and any $x\in F$, $$\label{eq:simple-part}
|\psi^\prime(x)-\psi^{\prime\prime}(x)|<\epsilon.$$
We construct a partition $\Delta_\alpha=\{\Delta_1,\dots,\Delta_m\}$ of $\beta \Psi$ from $\mathcal U$ in the usual way. Let $$\Delta_m = U_m \setminus (\cup_{j=1}^{m-1} U_j).$$ Choose points $\psi_j^\alpha \in U_j \cap \Psi$. While $\psi_j^\alpha$ need not be in $\Delta_j$ it is the case that if $\psi\in\Delta_j$, then holds with $\psi^\prime =\psi_j$ and $\psi^{\prime\prime} = \psi$. Since, for $x\in F$, $$\label{eq:2}
\left\| \sum \psi_j^\alpha(x) Q(\Delta_j) -\int \psi(x)dQ(\psi)
\right\| \le \sum \left\| \int_{\Delta_j}
(\psi_j^\alpha(x)-\psi(x)) dQ(\psi) \right\| \le \epsilon$$
Let $Q$ denote spectral measure associated to $\rho$ so that $$\rho(f)=\int_{\beta\Psi} f\,dQ.$$ Define $$\rho_\alpha (f) =\sum_{j=1}^n f(\psi_j)Q(\Delta_j)$$ and let $Z_\alpha (x)=\rho`_\alpha (E(x))$. It follows by that for $\alpha = (F,\epsilon)$, $\|Z_\alpha(x)
-Z(x)\| \le \epsilon$ for $x\in F$ and this remains true if $\alpha$ is replaced by any $\beta \ge \alpha$. Since $\|E(x)\| < 1$, we have $0< \delta = \sup_{x\in F}(1-\|E(x)\|)/2$, and so for $r =
1-\delta/2$ and $\epsilon < \delta/2$, it follows that $\|Z_\alpha(x)\| < r = 1-\epsilon$.
By Lemma \[lem:tfs-are-contractive\], for any $\alpha$, $$\varphi_\alpha = D+C Z_\alpha (I- A Z_\alpha)^{-1} B.$$ is a contraction. Note that $$\begin{split}
&(AZ_\alpha)^n - (AZ)^n \\ = & A(Z_\alpha - Z) AZ_\alpha \cdots
AZ_\alpha + AZ A(Z_\alpha - Z) AZ_\alpha \cdots AZ_\alpha
+ \ldots +
AZ\cdots AZ A(Z_\alpha - Z),
\end{split}$$ and so $$\|(AZ_\alpha)^n - (AZ)^n\| \leq n \|A\| \epsilon r^{n-1}.$$ Hence for suitably chosen $\alpha$, $$\begin{split}
&\|Z_\alpha (I- A Z_\alpha)^{-1} - Z_ (I- A Z)^{-1} \| \\ = &
\|(Z_\alpha - Z)(I- A Z_\alpha)^{-1} + Z[ (I- A Z_\alpha)^{-1} -
Z_ (I- A Z)^{-1}] \| \\
\leq & \epsilon \left[\frac{1}{1-r} + \frac{r^2}{(1-r)^2}\right],
\end{split}$$ Thus the bounded net $\varphi_\alpha$ converges pointwise to $\varphi$.
As constructed each $\varphi_\alpha$ has a simple transfer function representation, and so by Lemma \[lem:simple\], $\pi(\varphi_\alpha)$ is a contraction. Since the net $\varphi_\alpha$ is bounded and converges pointwise to $\varphi$, then net $\pi(\varphi_\alpha)$ converges in the weak operator topology to $\pi(\varphi)$. Hence $\pi(\varphi)$ is a contraction.
Factorization {#subsec:factorization}
-------------
The engine powering the lurking isometry argument in the proof of (iiX) implies (iii) of Theorem \[thm:main\] is the factorization in the following proposition. A similar result may be found in [@MR2106336]. A detailed proof, which we hint at, is given in [@DMM]. See also the book [@MR1882259] Theorem 2.53 proof 1.
\[prop:factorization\] If $\Gamma:X\times X \to \CT^*$ is positive, then there exists a Hilbert space $\mathcal E$ and a function $L:X\to B(\CT, \mathcal
E)$ such that $$\Gamma(x,y)(fg^*)={{
\left<
L(x)f,L(y)g
\right>}}$$ for all $f,g\in \CT$.
Further, there exists a unital $*$-representation $\rho:\CT\to
B(\mathcal E)$ such that $L(x)ab=\rho(a)L(x)b$ for all $x\in X$, $a,b\in\CT$.
Let $V$ denote the vector space with basis $X$. On the vector space $V\otimes \CT$ introduce the positive semidefinite sesquilinear form induced by $${{
\left<
x \otimes f,y\otimes g
\right>}} =\Gamma(x,y)(g^* f),$$ where $x,y\in X$ and $f,g\in\CT$. This is positive semidefinite by the hypothesis that $\Gamma$ is positive semidefinite. Mod out by the kernel and complete to get the Hilbert space $\mathcal E$. Define $L$ by $L(x)a= x\otimes a$ and verify that this is a bounded operator.
The $*$-representation is induced by the left regular representation of $\CT$, $\rho:\CT \to B(\mathcal E)$ with $\rho(a)(x\otimes f)=
x\otimes af$. Then check that $\rho$ is a contractive unital representation of $\CT$ satisfying $L(x)ab=\rho(a)L(x)b$ for all $x\in X$, $a,b\in\CT$.
A closed cone {#subsec:closed-cone}
-------------
The proof of (i) implies (ii) in Theorem \[thm:main\] is based on a cone separation argument which, in order to work, requires that the cone be closed and have nonempty interior. We present some of the background material here.
Given a finite subset $F\subset X$, let $\CT_F^+$ denote the collection of positive kernels $\Gamma:F\times F\to \CT^*$. If $\Gamma\in\CT_F^+$ and $x\in F$, then $\Gamma(x,x)$ is a positive linear functional on the unital $C^*$-algebra $\CT$ and therefore, $\|\Gamma\| = \Gamma(1)$.
Details of the proof outlined below can be found in [@DMM].
\[lem:closedcone\] Let $\TT$ be a set of test functions. If for each $x\in X$, $\|E(x)\|_\infty <1$, then $$\label{eq:C_F}
\mathcal C_F=\{ \begin{pmatrix} \Gamma(x,y)(I-E(x)E(y)^*)
\end{pmatrix}_{x,y\in F}: \Gamma \in \CT_F^+\}$$ is a closed cone of $|F|\times |F|$ matrices $($where $|F|$ is the cardinality of $F)$.
Suppose $M = (\Gamma(x,y)(I-E(x)E(y)^*)) \in \mathcal C_F$. Since $\|E(x)\| < 1$, $1-E(x)E(x)^* > \epsilon 1$ for some $\epsilon > 0$. Hence $\frac{1}{\epsilon}M(x,x) \geq \Gamma(x,x) 1 =
\|\Gamma(x,x)\|$, and so $\|\Gamma(x,x)\| \leq
\frac{1}{\epsilon}\|M\|$. Finiteness of $F$ means that there is in fact a single $\epsilon$ which will do for all $x\in F$, while positivity of $\Gamma$ implies that for $g\in\CT$, $$2|\Gamma(x,y) g| \leq \Gamma(x,x) 1 + \Gamma gg^* \leq
\|\Gamma(x,x)\| + \|\Gamma(y,y)\|\|gg^*\|.$$ Consequently, $\|\Gamma(x,y)\| \leq \frac{1}{\epsilon}\|M\|$ for all $x,y\in F$.
Now suppose $M_j\in\mathcal C_F$ is a Cauchy sequence. For each $j$ there exists $\Gamma_j\in \CT_F^+$ so that $$M_j=\begin{pmatrix} \Gamma_j(x,y)(I-E(x)E(y)^*)
\end{pmatrix}_{x,y\in F}.$$ Since the $M_j$’s are uniformly bounded, $\Gamma_j(x,y)$ is uniformly bounded for all $x,y$ and $j$. Thus there is a subsequence $\Gamma_{j_\ell}$ such that $\Gamma_{j_\ell}(x,y)$ converges weak-$*$ to $\Gamma(x,y)$. Likewise, $\Gamma_{j_\ell}(x,y)E(x)E(y)^*$ converges weak-$*$ to $\Gamma(x,y)E(x)E(y)^*$. Hence $M=\lim_j M_j =
(\Gamma(x,y)(1-E(x)E(y)^*)_{x,y\in F}$. Positivity of $\Gamma$ is a consequence of the positivity of the $\Gamma_j$’s. We conclude that $\mathcal C_F$ is closed.
The next lemma gives an example of a positive kernel in $\CT_F^+$ which will be particularly useful in showing that the cone $\mathcal
C_F$ in has nonempty interior in the subsequent lemma.
\[lem:evalmu\] Let $\TT$ be a set of test functions for which $\|E(x)\|<1$ for all $x$. For each $\psi\in\TT$ the function $\Gamma_\psi:X\times X\to
\CT^*$ given by $$\Gamma_\psi(x,y)(f)=\frac{f(\psi)}{1-\psi(x)\psi(y)^*}$$ is a positive kernel. Here $f\in \CT$.
Note that $|\psi(x)|\le \|E(x)\|<1$ so that the formula makes sense and moreover, $$S(x,y)=\frac{1}{1-\psi(x)\psi(y)^*}$$ defines a positive kernel on $X$.
With $y=x$, $$|\Gamma_\psi(x,x)(f)|
=\frac{|f(\psi)|}{|1-|\psi(x)|^2} \le \|f\| \frac{1}{|1-|\psi(x)|^2}$$ so that $\Gamma_\psi(x,x)$ is indeed in $\CT^*$.
For a finite set $F\subset X$ and function $f:F\to \CT$ $$\sum_{x,y\in F} \Gamma_\psi(x,y)(f(x)f(y)^*)
=\sum_{x,y\in F} f(x)(\psi) f(y)(\psi)^* S(x,y) \ge 0,$$ since $S$ is a positive kernel on $X$. It follows that each $\Gamma_\psi(x,y)\in\CT^*$ and $\Gamma_\psi$ is a positive kernel on $X$.
\[lem:Kpos\] Let $\TT$ be a set of test functions, $F\subset X$ a finite set, and $\mathcal C_F$ the cone in . Then $\mathcal C_F$ contains all positive $|F|\times |F|$ matrices, and hence has nonempty interior.
Let $\Gamma_\psi$ denote the positive kernel from Lemma \[lem:evalmu\]. Then $$[1]=\Gamma_\psi(x,y)(I-E(x)E(y)^*) \in\mathcal C_F,$$ where $[1]$ is the matrix with all entries equal to $1$. For $P$ be a positive $|F|\times |F|$ matrix, $\tilde\Gamma$ defined by $\tilde\Gamma(x,y) = P(x,y) \Gamma_\psi(x,y)$ is a positive kernel, and so $P(x,y) = \tilde\Gamma(x,y)(I-E(x)E(y)^*)$.
\[lem:conjcone\] The cone $\mathcal C_F$ in is closed under conjugation by diagonal matrices; i.e., if $M=(M(x,y))\in\mathcal
C_F$ and $c:F\to \mathbb C$, then $cMc^*=(c(x)M(x,y)c(y)^*)
\in\mathcal C_F$.
Simply note that if $\Gamma:F\times F\to \CT^*$ is positive, then so is $c\Gamma c^*$ defined by $(c\Gamma c^*)(x,y)=c(x)c(y)^*
\Gamma(x,y)$.
The next proposition connects the closed unit ball of of ${{H^\infty({\mathcal K}_\Psi)}}$ with the cone $\mathcal C_F$. Further details of the proof sketched below can be found in Lemmas 5.5 and 3.4 of [@DMM].
\[prop:separation\] Let $F\subset X$ be finite and $\varphi\in{{H^\infty({\mathcal K}_\Psi)}}$. If $M_\varphi$ defined by $$M_\varphi(x,y) = 1 - \varphi(x) \varphi(y)^*,\qquad x,y\in F,$$ is not in $\mathcal C_F$, then there is a kernel $k\in \mathcal
K_\Psi$ such that the matrix $$(1 - \varphi(x) \varphi(y)^*) k(x,y))_{x,y\in F}$$ is not positive. That is, $\|\varphi\|_{{H^\infty({\mathcal K}_\Psi)}}> 1$.
Use a version of the Hahn-Banach theorem to find a linear functional $\lambda\neq 0$ on the selfadjoint $|F|\times |F|$ matrices such that $\lambda(M) \geq 0$ for all $M\in\mathcal C_F$ but $\lambda(M_\varphi) < 0$. We can find such a $\lambda$ since $\mathcal C_F$ is closed and has nonempty interior by Lemmas \[lem:closedcone\] and \[lem:Kpos\].
For $f,g\in P(F)$ (viewed as vectors in $\mathbb C^F$), define ${{
\left<
f,g
\right>}} = \lambda(fg^*)$. Since $\mathcal C_F$ contains all positive $|F|\times |F|$ matrices, this is positive. Mod out by the kernel and call the resulting space $\mathcal H$. Let $q$ be the quotient map. Show that $\lambda(M_\varphi) < 0$ implies $\lambda([1])> 0$, and hence $q(\delta_F) \neq 0$, where $\delta_F$ is the function in $P(F)$ which is identically $1$.
Let $\mu$ be a representation of $P(F)$ given by $\mu(g)q(f) =
q(fg)$, where the product $fg$ is defined pointwise. Verify that $\mu$ is contractive on test functions and that $\mu([1]-\varphi|F
\varphi^*|F) < 0$.
What is more, $\delta_F$ is a cyclic vector for $\mu$. Hence if $\xi_x\in P(F)$ is defined to be $1$ at $x$ and zero elsewhere, then $\{\ell_x = \mu(\xi_x)\delta_F\}_{x\in F}$ is a basis for $\mathcal
H$. Let $\{k_x\}$ be the dual basis, $k(x,y) = {{
\left<
k_x,k_y
\right>}}$. Then for $c\in \mathbb C$, $$\begin{split}
{{
\left<
\mu(c\xi_x)^* k_a,\ell_b
\right>}} & = c^*{{
\left<
k_a,\mu(\xi_x)\ell_b
\right>}}
\\
&= c^*{{
\left<
k_a,\mu(\xi_x)\mu(\xi_b)\delta_F
\right>}} \\
&=
\begin{cases}
c^* & \text{if }x=b=a\\
0 & \text{otherwise}
\end{cases}
\\
&= {{
\left<
c^* k_a,\ell_b
\right>}}.
\end{split}$$ So for $f\in P(F)$, $\mu(f)^* k_a = f(a)^* k_a$. If $f$ is the test function $\psi$, this yields that the matrix $$((1-\psi(x)\psi(y)^*) k(x,y))_{x,y\in F}$$ is positive, while with $f=\varphi$, it is strictly negative. Extend $k$ to all of $X\times X$ by setting $k(x,y) = 0$ if either $x$ or $y$ are not in $F$. We then have $((1-f(x)f(y)^*)
k(x,y))_{x,y\in X}$ is positive when $f$ is a test function (so that $k\in\mathcal K_\Psi$), but not positive for $f=\varphi$.
A compact set {#subsec:compactset}
-------------
The proof of (iiF) implies (iiX) in Theorem \[thm:main\] uses Kurosh’s theorem ([@MR1882259], Theorem 2.56), the application of which requires that certain sets be compact.
Fix $\varphi:X\to \mathbb C$ and a collection of test functions $\TT$. For $F\subset X$, let $$\Phi_F = \{\Gamma\in\CT_F^+ : 1-\varphi(x)\varphi(y)^*=
\Gamma(x,y)(1-E(x)E(y)^*) \text{ for } x,y\in F\}.$$ The set $\Phi_F$ is naturally identified with a subset of the product of $\CT^*$ with itself $|F|^2$ times.
\[lem:compact-set\] If for each $x\in X$, $\|E(x)\|<1$, then the set $\Phi_F$ is compact.
Let $\Gamma_\alpha$ be a net in $\Phi_F$. Arguing as in the proof of Lemma \[lem:closedcone\], we find each $\Gamma_\alpha(x,x)$ is a bounded net and thus each $\Gamma_{\alpha}(x,y)$ is also a bounded net. By weak-$*$ compactness of the unit ball in $\CT^*$ there exists a $\Gamma$ and subnet $\Gamma_\beta$ of $\Gamma_\alpha$ so that for each $x,y\in
F$, $\Gamma_{\beta}(x,y)$ converges to $\Gamma(x,y)$.
Proofs {#sec:proofs}
======
We are now set to prove the theorems stated in Subsection \[subsec:mainresult\].
Proof of Theorem \[thm:main\] {#subsec:proof}
-----------------------------
### Proof of (i) implies (iiF)
Let $\varphi\in{{H^\infty({\mathcal K}_\Psi)}}$. If we suppose (iiF) does not hold, then by Proposition \[prop:separation\], $\|\varphi\|_{{H^\infty({\mathcal K}_\Psi)}}> 1$.
### Proof of (iiF) implies (iiX)
The proof here uses Kurosh’s Theorem and in much the same way as in [@MR1882259].
The hypothesis is that for every finite subset $F\subset X$, $\Phi_F$, as defined in subsection \[subsec:compactset\] is not empty, and so by Lemma \[lem:compact-set\], $\Phi_F$ is compact. For finite set $F\subset G$, define $ \pi_F^G :\Phi_F \to
\Phi_G$ by $$\pi_F^G (\Gamma) =\Gamma|_{F\times F}.$$ Thus, with $\mathcal F$ equal to the collection of all finite subsets of $X$ partially ordered by inclusion, the triple $(\Phi_F,\pi_F^G,\mathcal F)$ is an inverse limit of nonempty compact spaces. Consequently, by Kurosh’s Theorem, for each $F\in\mathcal F$ there is a $\Gamma_F \in \Phi_F$ so that whenever $F,G\in\mathcal F$ and $F\subset G$, $$\label{eq:consistent}
\pi_F^G (\Gamma_G) =\Gamma_F.$$
Define $\Gamma:X\times X\to \CT^*$ by $\Gamma(x,y)=\Gamma_{F}(x,y)$ where $F\in\mathcal F$ and $x,y\in F$. This is well defined by the relation in equation (\[eq:consistent\]). If $F$ is any finite set and $f:F\to \CT$ is any function, then $$\sum_{x,y\in F} \Gamma(x,y)(f(x)f(y)^*) =\sum_{x,y\in
F}\Gamma_F(x,y)(f(x)f(y)^*) \ge 0$$ since $\Gamma_F\in \CT_F^+$. Hence $\Gamma$ is positive.
### Proof of (iiX) implies (iii)
Let $\Gamma$ denote the positive kernel of the hypothesis of (iiX). Apply Lemma \[prop:factorization\] to find $\mathcal E$, $L:X\to
B(\CT, \mathcal E)$, and $\rho:\CT\to B(\mathcal E)$ as in the conclusion of the lemma.
Rewrite condition (iiX) as $$1+{{
\left<
Z(x)L(x)1,Z(x)L(x)1
\right>}} =\varphi(x) \varphi(y)^* +
{{
\left<
L(x)1,L(x)1
\right>}},$$ where we use Proposition \[prop:factorization\] to express $L(x)E(x)
= Z(x)L(x)1$ with $Z(x) = \rho(E(x))$. From here the remainder of the proof is the standard lurking isometry argument.
Let $\mathcal E_d$ denote finite linear combinations of $$\begin{pmatrix} Z(x)L(x)1 \\ 1 \end{pmatrix} \in \begin{matrix}
\mathcal E \\ \oplus \\ \mathbb C \end{matrix}$$ and let $\mathcal E_r$ denote finite linear combinations of $$\begin{pmatrix} L(x)1 \\ \varphi(x) \end{pmatrix} \in \begin{matrix}
\mathcal E \\ \oplus \\ \mathbb C \end{matrix}.$$
Define $V:\mathcal E_d\to \mathcal E_r$ by $$V\begin{pmatrix} Z(x)L(x)1 \\ 1 \end{pmatrix} = \begin{pmatrix}
L(x)1 \\ \varphi(x) \end{pmatrix},$$ extend by linearity and show that $V$ is a well defined isometry on $\mathcal E_d$, and hence on $\overline{\mathcal E_d}$. This further extends to a unitary operator $$U=\begin{pmatrix} A & B\\ C & D \end{pmatrix}
: \begin{matrix} \mathcal H \\ \oplus \\ \mathbb C \end{matrix} \to
\begin{matrix} \mathcal H \\ \oplus \\ \mathbb C \end{matrix},$$ with $U$ restricted to $\mathcal E_d$ equal to $V$; that is $U\gamma=V\gamma$ for $\gamma\in\mathcal E_d$.
This gives the system of equations $$\begin{split}
A Z(x)L(x)1 + B &= L(x)1 \\
\nonumber C Z(x)L(x)1 + D &= \varphi(x),
\end{split}$$ which, when solved for $\varphi$, yields $$\varphi(x)=D+CZ(x)(I-AZ(x))^{-1}B,$$ as desired.
### Proof of (iii) implies (ivF)
This is a direct consequence of Proposition \[prop:pw-simple\].
### Proof that (ivF) is equivalent to (ivS)
This is trivial in one direction. In the other, it follows from the proof of Lemma \[lem:simple\].
### Proof of (ivF) implies (i)
Take $\pi$ to be the identity representation in Proposition \[prop:pw-simple\].
Agler-Pick Interpolation: Proof of Theorem \[thm:APint\] {#subsec:interpolate}
--------------------------------------------------------
It turns out that in the Agler-Pick interpolation setting more can be said about the transfer function realization of the interpolant. Suppose $\mu$ is a (positive) measure on $\Psi$. The functions $E(x)$ determine multiplication operators on $L^2(\mu)$ by the formula $(E(x)f)(\psi)=\psi(x)f(\psi)$. Abusing notation, for a positive integer $n$, let $E(x)$ also denote the operator $I_n\otimes E(x)$ on $\mathbb C^n \otimes L^2(\mu)$, or more precisely, the representation $\rho(E(x))=I_n\otimes E(x)$
If $\varphi$ exists, the implication (i) implies (iiF) of Theorem \[thm:main\] applied to $F$ establishes the existence of $\Gamma$.
Conversely, suppose a positive $\Gamma$ satisfying equation exists. View $\Gamma$ as an $n\times n$ matrix $$\Gamma =\begin{pmatrix} \Gamma(x_\ell,x_j)\end{pmatrix}_{j,\ell}$$ with entries from $\CT^*$. The converse can be proved using the factorization from Proposition \[prop:factorization\]. However the proof of the last part about the measure $\mu$ requires a somewhat more concrete factorization of $\Gamma$.
Choose a positive measure $\mu$ on $\TT$ so that each $\Gamma(x_\ell,x_j)$ is absolutely continuous with respect to $\mu$. We can without loss of generality assume that the measure is defined on the closure of $\TT$ (if this is not already closed), and hence we may assume that the measure $\mu$ is bounded. By Radon-Nikodym, there exist $L^\infty(\mu)$ functions $F_{j,\ell}$ so that $\Gamma(x_\ell,x_j)=F_{\ell,j}\, d\mu$. In particular, the matrix-valued function $F$ can be identified with an element of the $C^*$-algebra of $n\times n$ matrices with entries from $L^\infty(\mu)$. The fact that $\Gamma$ is positive implies that $F$ is (almost everywhere $\mu$) pointwise positive. Consequently, there exists vectors $H$ from $C^n\otimes L^\infty(\mu)$ so that $F_{\ell,j}=H(x_\ell)H(x_j)^*$. This gives the factorization, $$\Gamma =HH^* \, d\mu.$$
Observe $$\begin{split}
\Gamma(x_\ell,x_j)(1-&E(x_\ell)E(x_j)^*)\\
=& \int H(x_\ell) H(x_j)^*\, d\mu
-\int H(x_\ell) E(x_\ell)^*E(x_j) H(x_j)^* \, d\mu \\
=& \langle H(x_\ell),H(x_j) \rangle -\langle E(x_j)H(x_j),
E(x_\ell)H(x_\ell)\rangle.
\end{split}$$ Thus, equation (\[eq:lurk\]) becomes, $$\label{eq:lurkspecial}
1+ {{
\left<
E(x_j)H(x_j),E(x_\ell)H(x_\ell)
\right>}} =
\xi(x_j)\xi(x_\ell)^* + {{
\left<
H(x_\ell),H(x_j)
\right>}}.$$
A lurking isometry argument as in the proof of (iiX) implies (iii) for Theorem \[thm:main\] allows us to define a unitary operator $$U=\begin{pmatrix} A &B\\ C&D \end{pmatrix} :
\begin{matrix} \mathbb C^n\otimes L^2(\mu) \\ \oplus \\ \mathbb C
\end{matrix}\rightarrow
\begin{matrix} \mathbb C^n\otimes L^2(\mu) \\ \oplus \\ \mathbb C
\end{matrix}$$ with $$U
\begin{pmatrix}
E(x_j)H(x_j) \\ 1
\end{pmatrix}
=
\begin{pmatrix}
H(x_j) \\ \xi(x_j)
\end{pmatrix},$$ which can then be solved to give $$\xi(x_j)= D+CE(x_j)(I-AE(x_j))^{-1}B.$$ Define $$\varphi(x)=D+CE(x)(I-A E(x))^{-1}B$$ for $x\in X$. Then $\varphi$ extends $\xi$ and the implication (iii) implies (i) of Theorem \[thm:main\] completes the proof.
Proof of Proposition \[prop:a-s-class-closed\] {#subsec:closed-class}
----------------------------------------------
Let $\Psi$ be a collection of test functions, $\mathcal K_\Psi$ and ${{H^\infty({\mathcal K}_\Psi)}}$ as above, and suppose $\varphi_\alpha$ is a net in the Agler-Schur class of ${{H^\infty({\mathcal K}_\Psi)}}$. Then for all $\alpha$, $\|\varphi_\alpha\| \leq 1$, and so $\|\varphi\| \leq 1$. Fix $F\subset X$ finite. Then there is a $\Gamma_{F,\alpha} \geq 0$ such that $$1-\varphi_\alpha(x)\varphi_\alpha(y)^* =
\Gamma_{F,\alpha}(x,y)(1-E(x)E(y)^*), \qquad x,y\in F.$$ So the matrix $M_\alpha = (1-\varphi_\alpha(x)\varphi_\alpha(y)^*)
\in \mathcal C_F$. Since by Lemma \[lem:closedcone\] $\mathcal
C_F$ is closed, arguing as at the end of the proof of that lemma, we have a $\Gamma_F \geq 0$ such that $$1-\varphi(x)\varphi(y)^* = \Gamma_F (x,y)(1-E(x)E(y)^*), \qquad
x,y\in F.$$ Applying (iiF) implies (iiX) of Theorem \[thm:main\], it follows that $\varphi$ is in the Agler-Schur class of ${{H^\infty({\mathcal K}_\Psi)}}$. Finally, since the test functions are a subset of the Agler-Schur class, the last statement is obvious.
Examples {#sec:examples}
========
In this section we concentrate on two main examples where an infinite collection of test functions is required; the annulus and the infinite polydisk. We then close with a few further examples illustrating the necessity of various parts of our definitions of test functions.
The annulus {#subsec:annulus}
-----------
Fix $q\in (0,1)$ and write $\mathbb A=\mathbb A_q$ for the annulus $$\mathbb A=\{z\in\mathbb C: q<|z|<1\}.$$ Let $H^\infty(\mathbb A)$ denote the bounded analytic functions on $\mathbb A$. There is a collection of functions $\vartheta_t$ naturally parameterized by $t$ in the unit circle $\mathbb T$ with the property that each $\vartheta_t$ is unimodular on the boundary of $\mathbb A$ (and so extending analytically across the boundary) and has precisely two zeros in $\mathbb A$. Moreover, any function with these properties is, up to pre-composition with an automorphism of $\mathbb A$ and post-composition with an automorphism of $\mathbb D$, one of these $\vartheta_t$. We begin by constructing $\vartheta_t$ and showing that $\Theta = \{\vartheta_t: t\in\mathbb T\}$ is indeed a family of test functions for $H^\infty(\mathbb A)$.
Let $B_0=\{|z|=1\}$ and $B_1=\{|z|=q\}$ denote the boundary components of the boundary $B$ of $\mathbb A$. For normalization, fix a base point $b\in\mathbb A$ such that $|b|\ne \sqrt{q}$. Using Green’s functions (or otherwise), for each point $\alpha\in B$ there exists a unique positive harmonic function $h_\alpha$ whose boundary values come from the measure on $B$ with point mass at $\alpha$. If $h$ is any positive harmonic function on $\mathbb A$ there is a (positive) measure $\mu$ on $B$ so that $$\label{eq:harmonic}
h(z)=\int_B h_\gamma \, d\mu(\gamma)
=\int_{B_0} h_\alpha \, d\mu(\alpha) +
\int_{B_1} h_\beta \, d\mu(\beta).$$ The harmonic function $h$ is the real part of an analytic function $f$ if and only if $\mu(B_0)=\mu(B_1)$. In particular, given $\alpha\in
B_0$ and $\beta \in B_1,$ the function $h_\alpha+h_\beta$ is the real part of an analytic function $f_{\alpha,\beta}$ which we may normalize by requiring the imaginary part of $f_{\alpha,\beta}(b)=0$. Since both $h_\alpha$ and $h_\beta$ are nonnegative on the boundary they are both positive inside the annulus and so $f(b)>0$. Then because the boundary values for the $h_\alpha$’s are point masses, can be re-expressed as $$h(z)=\frac{2}{\mu(B)} \Re\int_{B_0} \int_{B_1} f_{\alpha,\beta}\,
d\mu(\beta)\, d\mu(\alpha),$$ and when $\mu(B_0)=\mu(B_1)$, this will be the real part of an analytic function $f$ with $f(b)> 0$.
Given $\alpha\in B_0$ and $\beta \in B_1$, let $$\label{eq:defpsi}
\psi_{\alpha,\beta}=\frac{f_{\alpha,\beta}
-f_{\alpha,\beta}(b)}{f_{\alpha,\beta}+f_{\alpha,\beta}(b)}.$$ Then $$f_{\alpha,\beta} = f_{\alpha,\beta}(b)
\frac{\psi_{\alpha,\beta}+1}{\psi_{\alpha,\beta}-1}.$$ Note that $\psi_{\alpha,\beta}$ is unimodular on $B$, takes the value $0$ at $b$, and in fact extends to an analytic function on a region containing $\overline{\mathbb A}$. Further, $\psi_{\alpha,\beta}$ takes the value $1$ on $B$ precisely at those points where $f_{\alpha,\beta}=\infty$; namely $\alpha$ and $\beta$. Thus, $\psi_{\alpha,\beta}$ is two to one, and by the Maximum Modulus Principle has two zeros in $\mathbb A$. Since, the product of the moduli of the zeros is $q$, the second zero is also on the circle $\{|z|=\frac{q}{|b|}\}$. The assumption that $b\neq
\sqrt{q}$ thus ensures that the zeros are distinct.
We claim that $\Theta^\prime=\{\psi_{\alpha,\beta}\}$ is a collection of test functions for $H^\infty(\mathbb A)$; that is, that the unit ball of $H^\infty(\mathbb A)$ is the same as the unit ball of $H^\infty(\Theta^\prime)$. One direction is nearly automatic. Since $\Theta^\prime$ is a subset of the unit ball of $H^\infty(\mathbb A)$ it follows that the Szegő kernel $s$ for $\mathbb A$ is in $\mathcal
K_{\mathcal K_{\Theta^\prime}}$. Thus, if $\varphi$ is in the unit ball of $H^\infty(\mathcal K_{\Theta^\prime})$, then $$\left((1-\varphi(x)\varphi(y)^*)s(x,y)\right) \geq 0.$$ Hence $\varphi$ is in the unit ball of $H^\infty(\mathbb A)$. (In general if $\Psi$ is contained in $\Psi^\prime$, then the unit ball of $H^\infty(\mathcal K_{\Psi^\prime})$ is contained in the unit ball of $H^\infty(\mathcal K_\Psi)$.)
To prove the converse inclusion, suppose $\xi:\mathbb A\to
\mathbb D$ is analytic and $\xi(b)$ is real. There exists a $\mu$ so that $$\begin{split}
\frac{1+\xi}{1-\xi}
=&\int_{B_0}\int_{B_1} f_{\alpha,\beta}\, d\mu(\beta)\, d\mu(\alpha)\\
=&\int_{B_0}\int_{B_1}
f_{\alpha,\beta}(b)\frac{\psi_{\alpha,\beta}+1}{\psi_{\alpha,\beta}-1}
\, d\mu(\beta)\, d\mu(\alpha).
\end{split}$$ For $z,w \in \mathbb A$, $$\begin{split}
\frac{1+\xi}{1-\xi}(z)+\frac{1+\xi}{1-\xi}(w)^*
&= 2 \frac{1}{1-\xi(z)} (1-\xi(z)\xi(w)^*)\frac{1}{1-\xi(w)^*}\\
&=\int_{B_0}\int_{B_1} f_{\alpha,\beta}(b)
\frac{1-\psi_{\alpha,\beta}(z)\psi_{\alpha,\beta}(w)^*}
{(1-\psi_{\alpha,\beta}(z))(1-\psi_{\alpha,\beta}(w)^*)}
\, d\mu(\beta)\, d\mu(\alpha).
\end{split}$$ Thus, there exist functions $H_{\alpha,\beta}(z,w)$, analytic in $z$, conjugate analytic in $w$ and continuous in $\alpha,\beta$ for fixed $z,w$ so that $$1-\xi(z)\xi(w)^* =\int_{B_0}\int_{B_1} H_{\alpha,\beta}(z,w)
(1-\psi_{\alpha,\beta}(z)\psi_{\alpha,\beta}(w)^*) d\mu(\beta)\,
d\mu(\alpha).$$ and the claim is proved.
There is some redundancy in our choice of test functions. Given $t\in\mathbb T$, let $\vartheta_t$ denote the function analytic in $\mathbb A$, unimodular on $B$ with zeros at $b$ and $\frac{qt}{b}$ and with $\vartheta_t(1)=1$. The collection $\Theta=\{\vartheta_t:t\in\mathbb T\}$ is uniformly continuous in $t$ and $z$. For each $\alpha,\beta$, there exist $t,\gamma$ in $\mathbb
T$ so that $\psi_{\alpha,\beta}=\gamma \vartheta_t$. Thus, $\Theta$ is a totally bounded collection of test functions for $H^\infty(\mathbb A)$. (As an alternate, use the parameterization of the unimodular functions with precisely two zeros in terms of theta functions [@MR0335789]).
The collection $\Theta$ is a collection of test functions for $H^\infty(\mathbb A)$ and is compact in the norm topology of $H^\infty(\mathbb A)$.
Similar results for triply connected domains may be found in [@MR2163865]. See also a comment in [@MR1909298]. The realization theorem, Theorem \[thm:main\], now reads as follows.
\[prop:annulusalgebra\] Suppose $\varphi:\mathbb A\to \mathbb C$. The following are equivalent.
(i) $\varphi\in H^\infty(\mathbb A)$ with norm less than or equal to one;
(ii) There is a positive kernel $\Gamma:\mathbb A\times \mathbb A
\rightarrow C(\mathbb T)^*$ so that $$1-\varphi(z)\varphi(w)^*=\Gamma(z,w)(1-E(z)E(w)^*)$$ where $E(z)(\vartheta_t)=\vartheta_t(z)$; and
(iii) there exists an auxiliary Hilbert space $\mathcal E$ and an analytic function $\Phi:\mathbb A\to B(\mathcal E)$ whose values $\{\Phi(z)\}$ are commuting normal contraction operators and a unitary $$U=\begin{pmatrix} A &B\\ C&D \end{pmatrix} :
\begin{matrix} \mathcal E \\ \oplus \\ \mathbb C
\end{matrix}\rightarrow
\begin{matrix} \mathcal E \\ \oplus \\ \mathbb C \end{matrix}$$ so that $\varphi$ has the unitary colligation transfer function realization $$\varphi(z)=D+C\Phi(z)(I-A\Phi(z))^{-1}B.$$
We were able to reduce our original collection of test functions for the annulus to $\Theta$. It is reasonable to wonder if it is possible to throw out even more. The next proposition shows that the answer is “no”. It resembles a result of [@MR1386327] which says that in a sense all of the Sarason/Abrahamse reproducing kernels for the annulus are needed for Nevanlinna-Pick interpolation on the annulus.
\[prop:needall\] No proper closed subset of $\Theta$ is a set of test functions for $H^\infty(\mathbb A)$.
Note that a dense subset of the test functions will also be a set of test functions, though in this case there is no real advantage to taking such a set. The situation will be quite different in the case of the infinite polydisk, as we shall see.
Suppose $C$ is a proper subset of $\Theta$, and that $\vartheta_0 =
\vartheta_{t_0}$ is not in $C$.
Let $k:\mathbb A\times \mathbb A\to \mathbb C$ denote the Szegő kernel for the annulus with respect to harmonic measure $\omega$ for the base point $b\in \mathbb A$ (recall that we assume $|b|\ne
\sqrt{q}$). Let $X$ denote the Schottky double of $\mathbb A$ and write $Jz$ for the twin of $z$ in the double. According to Fay [@MR0335789] (see also [@MR1386327]), for each $a\in\mathbb A$ the kernel $k_a = k(\cdot,a)$ is meromorphic on $X$ with exactly two poles with the exception $k_b = 1$ (so in particular, $k(b,z) = k(z,b) = 1$ for all $z$). Moreover, there exists a point $P$ in the complement of the closure of $\mathbb A$ so that $k_a$ has poles at $P$ (independent of $a$) and $Ja$.
The kernels $$\Delta_t(z,w)=(1-\vartheta_t(z)\vartheta_t(w)^*)k(z,w)$$ are positive and have rank two. To see this, observe that $M_t$, the operator of multiplication by $\vartheta_t$ on $H^2(k)$, is an isometry, so $1-M_t M_t^*$ is the projection onto $\ker M_j^*$. Furthermore, if $b$ and $a_t$ are the two zeros of $\vartheta_t$ (distinct since $|b|\ne \sqrt{q}$), then the identity $M_t^* k_w =
\vartheta_t(w)^* k_w$ implies that $\ker M_t^* = \mathcal K_t =
\mathrm{span}\,\{k_b, k_{a_t}\}$. If we choose $f_t = k_b = 1$ and $g_t = k_{a_t} - k_b = k_{a_t} - 1$, then $$\Delta_t = f_tf_t^* + g_tg_t^* = 1 + g_tg_t^*.$$ It is useful to remark for later use that $g_t$ has the same poles as $k_{a_t}$; namely $P$ and $Ja_t$.
Choose three distinct points $z_1,z_2,z_3\in\mathbb A$ and consider the Agler-Pick interpolation problem of finding a $\varphi\in
H^\infty(\mathbb A)$ so that $\varphi(z_t)=\vartheta_0(z_t)$. The fact that the $3\times 3$ matrix $$\begin{pmatrix} k(z_\ell,z_m)(1-\vartheta_0(z_\ell)\vartheta_0(z_m)^*)
\end{pmatrix}_{j,m=1}^3$$ has rank two implies that this interpolation problem has a unique solution, namely $\varphi=\vartheta_0$. On the other hand, there is a bounded positive measure $\mu$ on $C$ so that $\varphi$ has a realization of the form in Theorem \[thm:APint\] with $n=3$. The usual computations convert that realization to $$1-\vartheta_0(z)\vartheta_0(w)^*
= \int_C \sum_{\nu=1}^3 h_\nu(z,\vartheta)
h_\nu(w,\vartheta)^*(1-\vartheta(z)\vartheta(w)^*)\, d\mu(\vartheta)$$ for functions $h_t(z,\cdot)\in L^2(\mu)$. In particular, multiplying through by $k(z,w)$ gives $$\Delta_0(z,w) = \int_C \sum_\nu h_\nu(z,\vartheta) \Delta_\vartheta(z,w)
h_\nu(w,\vartheta)^* \, d\mu(\vartheta)$$ where $\Delta_\vartheta(z,w) = (1-\vartheta(z)\vartheta(w)^*)k(z,w)$.
Fix $z$. Since $\Delta_\vartheta(z,z) \geq 0$ $\mu$ a.s., given $\delta > 0$, there is a set $C'\subset C$ and a constant $c_\delta>0$ such that $\mu(C-C') < \delta$ and for all $z\in\mathbb
A$, $$\Delta_0(z,z) \geq c_\delta h_\nu(z,\vartheta) \Delta_\vartheta(z,w)
h_\nu(w,\vartheta)^*, \qquad \vartheta\in C'.$$ Then using the factorization of the $\Delta$’s given above, by Douglas’ lemma there are constants $c_k$, $k=1,2,3,4$, such that for fixed $\vartheta\in C'$, $$\begin{split}
h_\nu(\cdot,\vartheta) &= c_1 + c_2 g_{a_0} \\
h_\nu(\cdot,\vartheta)g_{a_t} &= c_3 + c_4 g_{a_0}.
\end{split}$$ Since the kernels extend meromorphically to $X$, the same is true for $h_\nu(\cdot,\vartheta)$ by the first equation. That equation also implies that either $h_\nu(\cdot,\vartheta)$ is constant or that it has the same poles as $g_{a_0}$; that is, simple poles at $P$ and $a_0$. If $h_\nu(\cdot,\vartheta)$ is not constant, then the left side of the second equation has a double pole at $P$, while the right only has a single pole. Hence $h_\nu(\cdot,\vartheta)$ must be constant. If it is a nonzero constant, the second equation would imply that the poles of $g_{a_t}$ and $g_{a_0}$ agree, and in particular, that $a_t = a_0$, contradicting the assumption that $\vartheta_0 \notin C$. Hence $h_\nu(\cdot,\vartheta) = 0$.
Taking $\delta$ going to $0$, we see that the subset of $C'$ on which $h_\nu(\cdot,\vartheta)$ is nonzero has $\mu$ measure zero, yielding a contradiction.
The infinite polydisk {#subsec:F}
---------------------
By the infinite polydisk ${\mathbb D}^\infty$, we mean the open unit ball of $C_b(\mathbb N)$. Thus, $${\mathbb D}^\infty =\{z:\mathbb N\to \mathbb D: \sup
\{|z(n)|:n\}<1\}.$$
Let $e_n$ denote the function $e_n :{\mathbb D}^\infty \to \mathbb C$ given by $e_n(z)=z(n)$. The set of test functions $\Psi=\{e_n:n\in\mathbb N\}$ is topologized by the inclusion $\Psi\subset B({\mathbb D}^\infty, \overline{\mathbb D})$. The spaces $\Psi$ and $\mathbb N$ are homeomorphic and hence $\beta\Psi$ is identified with $\beta \mathbb N$.
A $\chi\in \beta\mathbb N\backslash \mathbb N$ determines a function $\varphi_\chi :X \to \mathbb D$ given by $$\varphi_\chi (z) = z(\chi),$$ where we have identified $z\in X$ with its unique extension to a continuous function $z:\beta\mathbb N \to \mathbb C$. This identification follows from the general discussion of $\Psi$ and $\beta\Psi$ in subsection \[subsec:test-funct-eval\]. Further, $\varphi_\chi$ is in the unit ball of $H^\infty(\Psi)$.
Theorem \[thm:main\] now implies that there is a positive kernel with entries in $C(\overline{{\mathbb D}^\infty})^*$ such that $$1-\varphi(z)\varphi(w)^* = \Gamma(z,w) (1-E(z)E(w)^*)\geq 0.$$ In this case there is a clear choice for $\Gamma$; namely, $\Gamma(z,w) = \gamma$, where $\gamma(e) = e(\chi)$ for $e\in
C(\overline{{\mathbb D}}^{\mathbb N})$.
\[prop:infinite-polydisc-no-no\] There **do not** exist positive kernels $\Gamma_n :X\times X\to
\mathbb C$ such that $$1-\varphi_\chi(z)\varphi_\chi(w)^* =\sum_n \Gamma_n(z,w)
(1-z(n)w(n)^*).$$ Similarly, if $C$ is any closed subset of $\beta\mathbb N$ with $\chi\notin C$, then there **does not** exist a positive $\Gamma:X\times X\to C(C)^*$ such that $$1-\varphi_\chi(z)\varphi_\chi(w)^* =\Gamma(z,w)(I-E(z)E(w)^*).$$
For the first part observe that if $z(n)$ converges to $L$ as $n\to\infty$, then $\chi(z)=L$ for any $\chi\in \beta\mathbb
N\backslash\mathbb N$.
Choose $z(n)=\sqrt{\frac{1}{2}\left(1-\frac{1}{n+1}\right)}$ we have $\frac{1}{2} = z(\chi) > z(n)\ge 0$ for all $n$. Let $0$ denote the zero sequence. Suppose that the first representation in the proposition holds. Then $$\frac{1}{2} = \sum_n \Gamma_n(z,z)\tfrac{1}{2}\left(1 +
\tfrac{1}{n+1}\right) > \tfrac{1}{2}\sum_n \Gamma_n(z,z),$$ and so $\sum_n \Gamma_n(z,z) < 1$. Obviously $\sum_n \Gamma_n(z,0)
= \sum_n \Gamma_n(0,z) = 1$. Also, for each $n$, $$\begin{pmatrix} \Gamma_n(z,z) & \Gamma_n(z,0) \\
\Gamma_n(0,z) & \Gamma_n(0,0) \end{pmatrix} \geq 0,$$ so $$\sum_n \begin{pmatrix} \Gamma_n(z,z) & \Gamma_n(z,0) \\
\Gamma_n(0,z) & \Gamma_n(0,0) \end{pmatrix} =
\begin{pmatrix} \sum_n \Gamma_n(z,z) & 1 \\
1 & 1 \end{pmatrix}\geq 0,$$ and thus $\sum_n \Gamma_n(z,z) \geq 1$, a contradiction.
The second part of the Proposition is proved similarly, in this situation choosing a function $z$ with $z(C)=0$, $z(\chi)=\sqrt{\tfrac{1}{2}}$, and $0\le z \le
\sqrt{\tfrac{1}{2}}$ (such a function exists since $\beta\mathbb N$ is Tychonov).
By the way, if we define $P_n$ as the projection of $f\in {\mathbb
D}^{\mathbb N}$ onto its first $n$ components, then despite the fact that $P_n f$ converges pointwise to $f$, $\varphi(P_n f) = 0$, and so obviously does not converge to $\varphi(f)$ in general. However this does not contradict Proposition \[prop:pw-simple\], since this only says that there is some net of simple representations converging to $\varphi$, and obviously this is not one!
Further Examples {#subsec:further-examples}
----------------
We end with a few examples illustrating some of the pitfalls into which an unwary applicant of the results presented can fall.
The following example shows that it is sometimes important and natural to use the compactification of $\Psi$, and illustrates more simply the phenomena observed with the infinite polydisk.
### Example 1. {#subsubsec:example1}
Choose $X$ equal the unit disk $\mathbb D$ and let, for $n=1,2,\dots,$ $\psi_n(z)=\left(\sqrt{1-\frac{1}{n}}\right)z$. The collection $\Psi
= \{\psi_n:n\}$ is a set of test functions for $H^\infty(\mathbb D)$ and the function $\xi(z)=z$ is in $H^\infty(\mathbb D)$ with $\|\xi\|=1$.
There do not exist positive kernels $\Gamma_n$ so that $$\label{eq:noPsD}
1-zw^*=\sum_{n\in\mathbb N} \Gamma_n(z,w)(1-\psi_n(z)\psi_n(w)^*).$$
Suppose (\[eq:noPsD\]) holds for some positive kernels $\Gamma_j$. Divide through by $1-zw^*$ to obtain, $$1=\sum \Gamma_n(z,w) +\frac{1}{n}\frac{\Gamma_n(z,w)}{1-zw^*}.$$ Note that the left side is the rank one positive matrix $[1]$ consisting of all $1$’s, and also each term on the right side is positive. Hence each term on the right side is a nonnegative constant multiple of $[1]$, which is clearly a contradiction.
Interestingly, if we had used the function $\left(\sqrt{1-\frac{1}{n}}\right)z$ instead of $z$, then there is an obvious choice for the $\Gamma_k$’s; namely, $\Gamma_n = [1]$ and all others equal to $0$. Furthermore, for a finite set $F \subset X$, the matrices $$\left(1-\left(1-\tfrac{1}{n}\right)zw^*\right)_{z,w\in F}$$ converge to $$\left(1-zw^*\right)_{z,w\in F},$$ and so by the proof of Lemma \[lem:closedcone\], there must be a $\Gamma$ such that $1-zw = \Gamma(z,w)(1-E(z)E(w)^*)$ (the proof of the lemma makes no use, either explicit or implicit, of the compactness of the set of test functions).
The kernel $\Gamma$ has entries which are continuous functions over $C_b(\Psi)$, and this includes the point evaluations which we tried to use above. However there are point evaluations we have not considered — the ones coming from points in the Stone-Čech compactification $\beta\Psi$ of $\Psi$. In this case, this agrees with the one point compactification where we add the function $\psi_\infty(z) = z$. The functions $E(z)$ extend uniquely to $\Psi$, and if the positive linear functional $\gamma\in C_b(\Psi_0)^*$ is defined by $\gamma(e) =
e(\psi_\infty)$, then the choice $\Gamma(x,y) = \gamma$ for all $x,y$ gets us out of our quandary.
In this example, it is clear that there would be no harm (in fact, it would be to our advantage) to include the functions in the Stone-Čech compactification of the set of test functions, particularly since it does not much effect the size of the set of test functions. This is in stark contrast to the case of the infinite polydisk, where the compactification increases the set size from being countable to at least having cardinality of $2^{\mathfrak c}$.
### Example 2.
Let $X=\{x_1,x_2\}$ denote a two point set and define $\psi(x_1)=0$ and $\psi(x_2)=1$. The set $\Psi=\{\psi\}$ is a set of test functions for $C(X)$. However, the function $\tilde{\psi}=1-\psi$ is in the unit ball of $C(X)$, but there does not exist a positive $\Gamma$ such that $$\label{eq:noGamma3}
1-\tilde{\psi}(x)\tilde{\psi}(y)^* = \Gamma(x,y)(1-\psi(x)\psi(y)^*).$$ The remainder of this subsection is devoted to these assertions.
Suppose $k\in\mathcal K_\Psi$; that is, $k$ is a positive kernel and the $2\times 2$ matrix $$\begin{pmatrix} (1-\psi(x_1)\psi(x_1)^*)k(x_1,x_1) &
(1-\psi(x_1)\psi(x_2)^*)k(x_1,x_2)\\
(1-\psi(x_2)\psi(x_1)^*)k(x_2,x_1) &
(1-\psi(x_2)\psi(x_2)^*)k(x_2,x_2)
\end{pmatrix} =
\begin{pmatrix} k(x_1,x_1) & k(x_1,x_2)\\ k(x_2,x_1) &
0\end{pmatrix}$$ is positive. It follows that $k(x_1,x_1)$ and $k(x_2,x_2)$ are nonnegative and $k(x_1,x_2)=0=k(x_2,x_1)$. Now for $\varphi:X\to\mathbb C$ it is readily verified that $(1-\varphi(x)\varphi(y)^*)k(x,y)$ is positive for all such $k$ if and only if $|\varphi(x_j)|\le 1$ for $j=1,2$. Hence ${{H^\infty({\mathcal K}_\Psi)}}= C(X)$ and $\tilde{\psi}$ is contractive.
Since the $(x_2,x_2)$ entry of the left hand side of equation (\[eq:noGamma3\]) is $1$, but the same entry on the right hand side of this equation is $0$, no such $\Gamma$ exists. So what went wrong? To begin with, $\psi$ violates condition (i) for a test function. However this is not so serious in this case. More to the point, it also violates (ii), since if we choose the set $F = \{x_2\}$, $\Psi|F$ does not generate $P(F)$. We could fix this either by taking a quotient or by adding another test function. A natural choice is $\tilde\psi$; sadly this still violates condition (i).
|
---
abstract: |
In [@FockThomas], Vladimir Fock and the author introduced a new geometric structure on surfaces, called higher complex structure, whose moduli space is conjecturally diffeomorphic to Hitchin’s component. This would give a new geometric approach to higher Teichmüller theory. In this paper, we prove several steps towards this conjecture and give a precise picture what has to be done.
We show that higher complex structures can be deformed to flat connections. More precisely we show that the cotangent bundle of the moduli space of higher complex structures can be included into a 1-parameter family of spaces of flat connections.
address: 'Université de Strasbourg, IRMA UMR 7501, 67084 Strasbourg, France'
author:
- Alexander Thomas
title: Higher Complex Structures and Flat Connections
---
Introduction
============
In [@Hit.1], Nigel Hitchin describes, for a Riemann surface $S$, a connected component of the character variety $$\operatorname{Rep}(\pi_1(S), \operatorname{PSL}_n({\mathbb{R}})) = \operatorname{Hom}(\pi_1(S), \operatorname{PSL}_n({\mathbb{R}}))/\operatorname{PSL}_n({\mathbb{R}})$$ which he parametrizes by holomorphic differentials. These components are called **Hitchin components** and their study *higher Teichmüller theory*. His approach uses Higgs bundle theory, more precisely the hyperkähler structure of the moduli space of polystable Higgs bundles. These components can also be described by representation-theoretic methods.
For $\operatorname{PSL}_2({\mathbb{R}})$, Hitchin’s component is **Teichmüller space**, which is the moduli space of various geometric structures on the underlying smooth surface ${\Sigma}$, for example complex structures or hyperbolic structures. Thus, the question naturally arises *whether there is a geometric structure on ${\Sigma}$ whose moduli space gives Hitchin’s component for higher rank*.
In [@FockThomas] a candidate for such a geometric structure is constructed, called the **higher complex structure** or $n$-complex structure, since it generalizes the complex structure. The higher complex structure can be seen as a special ${\mathfrak}{sl}_n$-valued 1-form. In local coordinates it is given by $\Phi=\Phi_1dz+\Phi_2d\bar{z}$ where $(\Phi_1, \Phi_2)$ is a pair of commuting nilpotent matrices. The construction of the $n$-complex structure uses the punctual Hilbert scheme of the plane and is reviewed in section \[highercomplex\].
The group of symplectomorphisms of $T^*{\Sigma}$ acts on 1-forms, so on the higher complex structure. We denote by ${\bm\hat{{\mathcal}{T}}}^n$ the moduli space of higher complex structures. A prominent role is played by the *cotangent bundle ${T^*\bm\hat{{\mathcal}{T}}^n}$*. Its elements consist of a higher complex structure and a cotangent vector, described by a set of holomorphic differentials.
In this paper we prove several steps towards a canonical diffeomorphism between the moduli space of higher complex structures ${\bm\hat{{\mathcal}{T}}}^n$ and Hitchin’s component. Before giving the structure of the paper, we give a comparison to Hitchin’s approach, which motivates and clarifies our ideas.
Comparison to Hitchin’s approach
--------------------------------
Hitchin’s approach to construct components in the character variety is to use the hyperkähler structure of the moduli space of Higgs bundles ${\mathcal}{M}_H$. One starts from a Riemann surface $S$, i.e. a smooth surface ${\Sigma}$ equipped with a *fixed* complex structure. Then one considers **Higgs bundles** on $S$, i.e. pairs of a holomorphic bundle $V$ and a holomorphic $\operatorname{End}(V)$-valued 1-form $\Phi$, the **Higgs field**. We summarize this approach in one picture: the twistor space of ${\mathcal}{M}_H$, which encodes all Kähler structures at once.
To a hyperkähler manifold $M$ one associates the **twistor space** $X_M ={\mathbb{C}}P^1\times M$ endowed with the complex structure at the point $({\lambda}, m)$ given by $I_{{\lambda}, m}=(I_0, I_{\lambda})$ where $I_0$ is the standard structure of ${\mathbb{C}}P^1$ and $I_{\lambda}$ is the complex structure of $M$ associated to ${\lambda}\in {\mathbb{C}}P^1$. The projection $X_M \rightarrow {\mathbb{C}}P^1$ is holomorphic and a holomorphic section is called a **twistor line**. With some extra data, it is possible to *reconstruct the hyperkähler manifold $M$ as the space of all real twistor lines*. This is a result of [@HKLR] (theorem 3.3).
On the left hand side of figure \[HK\] we draw the twistor space of the moduli space of Higgs bundles ${\mathcal}{M}_H$. In one complex structure, say at ${\lambda}=\infty$, we have the moduli space of Higgs bundles ${\mathcal}{M}_{H}$ (with its complex structure coming from the one of $S$). For ${\lambda}=0$, we see the conjugated complex structure. In all other ${\lambda}$, we see the complex structure of the character variety $\operatorname{Rep}(\pi_1({\Sigma}), G^{{\mathbb{C}}})$, which can be seen as hamiltonian reduction of the space of all connections ${\mathcal}{A}$ by all the gauge transformations ${\mathcal}{G}$ (Atiyah-Bott reduction for unitary gauge). Going from ${\lambda}=0$ to ${\lambda}=1$ is the **non-abelian Hodge correspondence**. Finally, there is the Hitchin fibration going from ${\mathcal}{M}_{H}$ to a space of holomorphic differentials. This fibration admits a section whose monodromy, via the non-abelian Hodge correspondence, is in the split real form. For $G=\operatorname{SL}_n({\mathbb{C}})$, we get flat $\operatorname{PSL}_n({\mathbb{R}})$-connections.
(0,0) circle (1cm); plot ([cos()]{},[sin()/3]{}); plot ([cos()]{},[sin()/3]{}); (0,1) circle (0.04); (0,-1) circle (0.04); (0,1.2) node [$\overline{{\mathcal}{M}}_H$]{}; (0,1) node [$0$]{}; (0,-1.2) node [${\mathcal}{M}_H$]{}; (0,-1) node [$\infty$]{}; (0,-1.5) node [$\downarrow$]{}; (0,-1.9) node [$\bigoplus_{i=2}^n H^0(K^i)$]{}; plot ([0.33\*cos()]{}, [0.33\*sin()-1.5]{});
(1,-1.4) node [Hitchin]{}; (1,-1.62) node [section]{}; (-1.25,-1.4) node [Hitchin]{}; (-1.25,-1.65) node [fibration]{}; (1.85,0.15) node [$\operatorname{Rep}(\pi_1{\Sigma}, G^{{\mathbb{C}}})$]{}; (1.65,-0.18) node [$\cong {\mathcal}{A}//{\mathcal}{G}$]{};
(0,0) circle (1cm); plot ([cos()]{},[sin()/3]{}); plot ([cos()]{},[sin()/3]{}); (0,1) circle (0.04); (0,-1) circle (0.04); (0,1.2) node [$\overline{T^*{\mathcal}{T}^n}$]{}; (0,1) node [$0$]{}; (0,-1.2) node [${T^*\bm\hat{{\mathcal}{T}}^n}$]{}; (0,-1) node [$\infty$]{}; (0,-1.5) node [$\downarrow$]{}; (0,-1.8) node [${\bm\hat{{\mathcal}{T}}}^n$]{}; plot ([0.33\*cos()]{}, [0.33\*sin()-1.5]{});
(1,-1.4) node [zero-]{}; (1,-1.6) node [section]{}; (-1.25,-1.4) node [canonical]{}; (-1.25,-1.65) node [projection]{}; (2,0.15) node [$\operatorname{Rep}(\pi_1{\Sigma}, G^{{\mathbb{C}}}) \cong$]{}; (2,-0.18) node [$({\mathcal}{A}//{\mathcal}{P})//\operatorname{Symp}_0$]{};
In our approach, we start from a smooth surface ${\Sigma}$ which we equip with a higher complex structure (which can vary). This structure is locally given by a 1-form $\Phi=\Phi_1dz+\Phi_2d\bar{z}$ where $(\Phi_1, \Phi_2)$ is a pair of commuting nilpotent matrices. The role of ${\mathcal}{M}_H$ is played by the cotangent bundle ${T^*\bm\hat{{\mathcal}{T}}^n}$ to the moduli space of higher complex structures, which is conjecturally hyperkähler near the zero-section (see discussion around conjecture \[hkcotang\]).
On the right-hand side of figure \[HK\], we draw the conjectural twistor space of ${T^*\bm\hat{{\mathcal}{T}}^n}$. In complex structure at ${\lambda}=\infty$, we see the cotangent bundle ${T^*\bm\hat{{\mathcal}{T}}^n}$. At the opposite point ${\lambda}=0$ we see some conjugated structure $\overline{T^*{\mathcal}{T}^n}$. In all other complex structures, we see the character variety $\operatorname{Rep}(\pi_1({\Sigma}), G^{{\mathbb{C}}})$, this time obtained as a double reduction of the space of connections ${\mathcal}{A}$, by some parabolic subgroup ${\mathcal}{P}$ of all gauges, and then by higher diffeomorphisms $\operatorname{Symp}_0$ (details in section \[parabolicreduction\]). The analog of the Hitchin fibration is simply the projection map ${T^*\bm\hat{{\mathcal}{T}}^n}\rightarrow {\bm\hat{{\mathcal}{T}}}^n$ and the analog of the Hitchin section is the zero-section ${\bm\hat{{\mathcal}{T}}}^n\subset {T^*\bm\hat{{\mathcal}{T}}^n}$. A flat connection associated to a point of the zero-section ${\bm\hat{{\mathcal}{T}}}^n\subset {T^*\bm\hat{{\mathcal}{T}}^n}$ should have real monodromy.
We stress again that most of the right-hand side is conjectural. In this paper we
- describe the double reduction space $({\mathcal}{A}//{\mathcal}{P})//\operatorname{Symp}_0$ and show that it is a space of flat connections,
- include this space into a family of flat $h$-connections ($h={\lambda}^{-1}$)
- show that at the limit ${\lambda}\rightarrow \infty$ we get ${T^*\bm\hat{{\mathcal}{T}}^n}$,
- give partial results for the existence of twistor lines, i.e. a canonical deformation of ${T^*\bm\hat{{\mathcal}{T}}^n}$ to flat connections,
- prove the diffeomorphism between ${\bm\hat{{\mathcal}{T}}}^n$ and Hitchin’s component assuming the existence of twistor lines.
Summary and structure
---------------------
In section \[highercomplex\], we review the construction of the higher complex structure. In particular, we describe the cotangent bundle ${T^*\bm\hat{{\mathcal}{T}}^n}$ in subsection \[cotangs\]. Then we give some new aspects: we describe a bundle induced by the $n$-complex structure in \[indbundle\] and the conjugated structure in \[dualcomplexstructure\].
The space of flat connections is constructed in section \[parabolicreduction\] by a double hamiltonian reduction: starting from the space of all connections ${\mathcal}{A}$, we reduce with respect to a parabolic subgroup ${\mathcal}{P}$ of the gauge group, those which fix a given direction. We show that symplectomorphisms of $T^*{\Sigma}$ act by gauge on ${\mathcal}{A}//{\mathcal}{P}$ and that the double reduction is a space of flat connections (see corollary \[flatparaconnections\]). It can also be described as a space of pairs of commuting differential operators. We perform this double reduction for $h$-connections in section \[parabolicwithlambda\], such that in the limit when $h$ goes to zero we get ${T^*\bm\hat{{\mathcal}{T}}^n}$ (see theorem \[conditioncinconnection\]).
We then investigate how to get a flat connection from a higher complex structure. We first put the connections we look at in some standard form in \[standard-form\]. We then give partial results and ideas of the existence of a canonical deformation of ${T^*\bm\hat{{\mathcal}{T}}^n}$ to flat connections in \[flatconnectionlambda\]. Finally under the assumption that this canonical deformation exists, we prove that our moduli space ${\bm\hat{{\mathcal}{T}}}^n$ is diffeomorphic to Hitchin’s component in theorem \[mainthmm\].
We include three appendices: in the first appendix \[appendix:A\] we give some facts about the punctual Hilbert scheme of the plane. In appendix \[appendix:B\] and \[appendix:C\] we prove two technical points.
***Notations.*** Throughout the paper, $\Sigma$ denotes a smooth closed surface of genus $g \geq 2$. A complex local coordinate system on $\Sigma$ is denoted by $(z, \bar{z})$ and its conjugate coordinates on $T^{*\mathbb{C}}\Sigma$ by $p$ and $\bar{p}$. The canonical bundle is $K=T^{*(1,0)}{\Sigma}$. The space of sections of a bundle $B$ is denoted by $\Gamma (B)$. The hamiltonian reduction (or symplectic reduction, or Marsden-Weinstein quotient) of a symplectic manifold $X$ by a group $G$ is denoted by $X//G$ where the reduction is over the zero-coadjoint orbit. The equivalence class of some element $a$ is denoted by $[a]$.
***Acknowledgments.*** I warmly thank Vladimir Fock for all the ideas and discussions he shared with me.
Higher complex structures {#highercomplex}
=========================
We recall here the construction of higher complex structures, their moduli space and the cotangent bundle to its moduli space. All details can be found in [@FockThomas]. The main ingredient for the higher complex structure is the punctual Hilbert scheme of the plane (see also appendix \[appendix:A\]). We also briefly discuss a bundle induced by the higher complex structure and a conjugated structure.
Higher complex structures {#higher-complex-structures}
-------------------------
A complex structure on a surface is characterized by the **Beltrami differential** $\mu\in\Gamma(K^{-1}\otimes \bar{K})$ where $K$ is the canonical bundle. It determines the notion of a local holomorphic function $f$ by the condition $({\bar\partial}-\mu{\partial})f=0$. The Beltrami differential determines a linear direction in $T^{{\mathbb{C}}}{\Sigma}$, the direction generated by the vector ${\bar\partial}-\mu{\partial}$. Since ${\bar\partial}-\mu{\partial}$ and ${\partial}-\bar{\mu}{\bar\partial}$ have to be linearly independent, we get the condition $\mu\bar{\mu}\neq 1$. Replacing the tangent bundle $T{\Sigma}$ by the cotangent bundle $T^*{\Sigma}$, we can say that the complex structure is *entirely encoded in a section of ${\mathbb}{P}(T^{*{\mathbb{C}}}{\Sigma})$*. The idea of higher complex structures is to replace the linear direction by a polynomial direction, or more precisely a $n$-jet of a curve inside $T^{*{\mathbb{C}}}{\Sigma}$. To get a precise definition, we use the **punctual Hilbert scheme** of the plane, denoted by $\operatorname{Hilb}^n({\mathbb{C}}^2)$ which is defined by $$\operatorname{Hilb}^n({\mathbb{C}}^2)=\{I \text{ ideal of } {\mathbb{C}}[x,y] \mid \dim {\mathbb{C}}[x,y]/I = n\}.$$ A generic point in $\operatorname{Hilb}^n({\mathbb{C}}^2)$ is an ideal whose algebraic variety is a collection of $n$ distinct points in ${\mathbb{C}}^2$. A generic ideal can be written as $$\langle-x^n+t_1x^{n-1}+...+t_n, -y+\mu_1+\mu_2x+...+\mu_nx^{n-1} \rangle.$$ Moving around in $\operatorname{Hilb}^n({\mathbb{C}}^2)$ corresponds to a movement of $n$ particles in ${\mathbb{C}}^2$. But whenever $k$ particles collide the Hilbert scheme retains an extra information: the $(k-1)$-jet of the curve along which the points entered into collision. The **zero-fiber**, denoted by $\operatorname{Hilb}^n_0({\mathbb{C}}^2)$, consists of those ideals whose support is the origin. A generic point in $\operatorname{Hilb}^n_0({\mathbb{C}}^2)$ is of the form $$\langle x^n, -y+\mu_2x+\mu_3x^2+...+\mu_nx^{n-1}\rangle$$ which can be interpreted as a $(n-1)$-jet of a curve at the origin (see appendix \[appendix:A\] for details).
We can now give the definition of the higher complex structure:
A **higher complex structure** of order $n$ on a surface $\Sigma$, in short **$n$-complex structure**, is a section $I$ of $\operatorname{Hilb}^n_0(T^{*\mathbb{C}}\Sigma)$ such that at each point $z\in {\Sigma}$ we have $I(z)+\bar{I}(z)=\langle p, \bar{p} \rangle$, the maximal ideal supported at the origin of $T_z^{*\mathbb{C}}\Sigma$.
Notice that we apply the punctual Hilbert scheme pointwise, giving a Hilbert scheme bundle over ${\Sigma}$. The condition on $I+\bar{I}$ ensures that $I$ is a generic ideal, so locally it can be written as $$I(z,\bar{z})=\langle p^n, -\bar{p}+\mu_2(z, \bar{z})p+\mu_3(z, \bar{z}) p^2...+\mu_n(z, \bar{z})p^{n-1}\rangle.$$ The coefficients $\mu_k$ are called **higher Beltrami differentials**. A direct computation gives $\mu_k \in \Gamma(K^{1-k}\otimes \bar{K})$. The coefficient $\mu_2$ is the usual Beltrami differential. In particular for $n=2$ we get the usual complex structure.
The punctual Hilbert scheme admits an equivalent description as a space of pairs of commuting operators. To an ideal $I$ of ${\mathbb{C}}[x,y]$ of codimension $n$, one can associate the multiplication operators by $x$ and by $y$ in the quotient ${\mathbb{C}}[x,y]/I$, denoted by $M_x$ and $M_y$. This gives a pair of commuting operators. Conversely, to two commuting operators $(A,B)$ we can associate the ideal $I(A,B)=\{P\in{\mathbb{C}}[x,y] \mid P(A,B)=0\}$. For details see \[matrixviewhilb\] in the appendix. The zero-fiber $\operatorname{Hilb}^n_0({\mathbb{C}}^2)$ corresponds to nilpotent commuting operators.
From this point of view, a higher complex structure is a *gauge class of special matrix-valued 1-forms locally of the form $\Phi_1dz+\Phi_2d\bar{z}$ where $(\Phi_1, \Phi_2)$ is a pair of commuting nilpotent matrices with $\Phi_1$ principal nilpotent* (which means of maximal rank $n-1$).
To define a finite-dimensional moduli space of higher complex structures, we have to define some equivalence relation. It turns out that the good notion is the following:
A **higher diffeomorphism** of a surface $\Sigma$ is a hamiltonian diffeomorphism of $T^*\Sigma$ preserving the zero-section $\Sigma \subset T^*\Sigma$ setwise. The group of higher diffeomorphisms is denoted by $\operatorname{Symp}_0(T^*\Sigma)$.
Symplectomorphisms act on $T^{*{\mathbb{C}}}{\Sigma}$, so also on 1-forms. This is roughly how higher diffeomorphisms act on the $n$-complex structure, considered as the limit of an $n$-tuple of 1-forms.
We then consider higher complex structures modulo higher diffeomorphisms, i.e. two structures are equivalent if one can be obtained by the other by applying a higher diffeomorphism. Locally, all $n$-complex structures are equivalent:
\[loctriii\] The $n$-complex structure can be locally trivialized, i.e. there is a higher diffeomorphism which sends the structure to $(\mu_2(z,\bar{z}),...,\mu_n(z,\bar{z}))=(0,...,0)$ for all small $z\in {\mathbb{C}}$.
We define the **moduli space of higher complex structures**, denoted by ${\bm\hat{{\mathcal}{T}}}^n$, as the space of $n$-complex structures modulo higher diffeomorphisms. The main properties are given in the following theorem:
\[mainresultncomplex\] For a surface $\Sigma$ of genus $g\geq 2$ the moduli space ${\bm\hat{{\mathcal}{T}}}^n$ is a contractible manifold of complex dimension $(n^2-1)(g-1)$. Its cotangent space at any point $\mu=(\mu_2,...,\mu_n)$ is given by $$T^*_{\mu}\bm\hat{\mathcal{T}}^n = \bigoplus_{m=2}^{n} H^0(\Sigma,K^m).$$ In addition, there is a forgetful map $\bm\hat{\mathcal{T}}^n \rightarrow \bm\hat{\mathcal{T}}^{n-1}$ and a copy of Teichmüller space ${\mathcal}{T}^2\rightarrow {\bm\hat{{\mathcal}{T}}}^n$.
The forgetful map in coordinates is just given by forgetting the last Beltrami differential $\mu_n$. The copy of Teichmüller space is given by $\mu_3=...=\mu_n=0$ (this relation is unchanged under higher diffeomorphisms).
We notice the similarity to Hitchin’s component, especially the contractibility, the dimension and the copy of Teichmüller space inside. At the end of the paper in section \[finalstep\] we indicate how to link ${\bm\hat{{\mathcal}{T}}}^n$ to Hitchin’s component. Assuming a strong conjecture (an analog of the non-abelian Hodge correspondence in our setting), we prove that ${\bm\hat{{\mathcal}{T}}}^n$ is canonically diffeomorphic to Hitchin’s component in theorem \[mainthmm\].
Cotangent bundle of higher complex structures {#cotangs}
---------------------------------------------
The main object to link higher complex structures to character varieties is the total cotangent bundle ${T^*\bm\hat{{\mathcal}{T}}^n}$ which we describe here in detail. The punctual Hilbert scheme inherits a complex symplectic structure from ${\mathbb{C}}^2$. It can be described as follows: to an ideal $I \in \operatorname{Hilb}^n({\mathbb{C}}^2)$, associate the two multiplication operators $M_x$ and $M_y$. The symplectic structure $\omega$ is given by $$\omega = \operatorname{tr}dM_x \wedge dM_y.$$ The zero-fiber is an isotropic subspace of dimension $n-1$. The dimension of $\operatorname{Hilb}^n({\mathbb{C}}^2)$ being $2n$, the zero-fiber cannot be Lagrangian. The subspace $\operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$, called **reduced Hilbert scheme**, consisting of those ideals $I$ whose support has barycenter the origin (generically $n$ points with barycenter equal to the origin), is a symplectic submanifold of $\operatorname{Hilb}^n({\mathbb{C}}^2)$ and the zero-fiber is Lagrangian inside the reduced Hilbert scheme. Hence its cotangent bundle is isomorphic to its normal bundle (using the symplectic form): $$T^*\operatorname{Hilb}^n_0({\mathbb{C}}^2)\cong T^{normal}\operatorname{Hilb}^n_{0}({\mathbb{C}}^2) \approx \operatorname{Hilb}^n_{red}({\mathbb{C}}^2).$$ Near the zero-section, the normal bundle is isomorphic to the whole space, here the reduced Hilbert scheme.
There is a general fact stating that the cotangent bundle to a quotient space $X/G$ (where $X$ is a manifold and $G$ a Lie group) is a hamiltonian reduction: $T^*(X/G) \cong T^*X//G$. Using this we can compute $$\begin{aligned}
{T^*\bm\hat{{\mathcal}{T}}^n}&= T^*\left(\Gamma(\operatorname{Hilb}^n_0(T^{*\mathbb{C}}\Sigma))/\operatorname{Symp}_0(T^*\Sigma)\right) & \nonumber \\
&= \Gamma(T^*\operatorname{Hilb}^n_0(T^{*\mathbb{C}}\Sigma)) // \operatorname{Symp}_0(T^*\Sigma) \nonumber \\
&= \Gamma(T^{normal}\operatorname{Hilb}^n_0(T^{*\mathbb{C}}\Sigma))// \operatorname{Symp}_0(T^*\Sigma) \nonumber \\
&= \Gamma(\operatorname{Hilb}^n_{red}(T^{*\mathbb{C}}\Sigma)) // \operatorname{Symp}_0(T^*\Sigma) & \mod t^2. \label{hkquotientofmodulispace}\end{aligned}$$
We see that ${T^*\bm\hat{{\mathcal}{T}}^n}$ is obtained by a hamiltonian reduction of $\operatorname{Hilb}^n_{red}(T^{*{\mathbb{C}}}{\Sigma})$. An element of the latter Hilbert scheme bundle is an ideal $$I=\langle p^n-t_2p^{n-2}-...-t_n, -\bar{p}+\mu_1+\mu_2p+...+\mu_np^{n-1}\rangle.$$ The coefficient $\mu_1$ is an explicit function of the other variables. So the $2n-2$ variables $(t_k, \mu_k)_{2\leq k \leq n}$ form a coordinate system. The fact that the normal bundle is only the total space near the zero-section is expressed by “modulo $t^2$”, meaning that all quadratic or higher terms in the $t_k$ have to be dropped.
To compute the moment map, we have to understand with more detail the action of higher diffeomorphisms on the Hilbert scheme bundle. The ideal $I$ has two generators which we put into the form $p^n-P(p)$ and $-\bar{p}+Q(p)$ where $P(p)=t_2p^{n-2}+...+t_n$ and $Q(p)=\mu_2p+...+\mu_np^{n-1}$. A higher diffeomorphism generated by some Hamiltonian $H$ acts on $I$ by changing the two polynomials. Their infinitesimal variations $\delta P$ and $\delta Q$ are given by $$\begin{aligned}
\delta P &= \{H, p^n-P(p)\} \mod I \nonumber \\
\delta Q &= \{H, -\bar{p}+Q(p)\} \mod I. \label{idealvariation}\end{aligned}$$
One can easily show that only the class $H \mod I$ acts, i.e. $H_1$ and $H_2$ with $H_1 = H_2 \mod I$ have the same action. $\triangle$
Using these variation formulas, one can compute the moment map which gives:
\[conditionC\] The cotangent bundle to the moduli space of $n$-complex structures is given by $$\begin{aligned}
T^*\bm\hat{\mathcal{T}}^n= \Big\{& (\mu_2, ..., \mu_n, t_2,...,t_n) \mid \mu_k \in \Gamma(K^{1-k}\otimes \bar{K}), t_k \in \Gamma(K^k) \text{ and } \; \forall k\\
& (-\bar{\partial}\!+\!\mu_2\partial\!+\!k\partial\mu_2)t_{k}+\sum_{l=1}^{n-k}((l\!+\!k)\partial\mu_{l+2}+(l\!+\!1)\mu_{l+2}\partial)t_{k+l}=0 \Big\} \Big/\operatorname{Symp}_0(T^*{\Sigma})\end{aligned}$$
We call the condition coming from the moment map **condition $({\mathcal}{C})$**. It is a *generalized holomorphicity condition*: for $\mu_k=0$ for all $k$, we simply get ${\bar\partial}t_k =0$.
The punctual Hilbert scheme $\operatorname{Hilb}^n({\mathbb{C}}^2)$ inherits a hyperkähler structure from ${\mathbb{C}}^2$ (see [@Nakajima]). This should induce a hyperkähler structure on ${T^*\bm\hat{{\mathcal}{T}}^n}$:
\[hkcotang\] The cotangent bundle ${T^*\bm\hat{{\mathcal}{T}}^n}$ admits a hyperkähler structure near the zero-section.
There are three good reasons to believe in the conjecture:
- *Construction by hyperkähler quotient*: Equation \[hkquotientofmodulispace\] points towards a possible hyperkähler reduction. Indeed under some mild conditions a complex symplectic reduction $X//G^{\mathbb{C}}$ is isomorphic to a hyperkähler quotient $X///G^{\mathbb{R}}$. In our case $X=\Gamma(\operatorname{Hilb}^n_{red}(T^{*\mathbb{C}}\Sigma))$ is hyperkähler, since $\operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$ is. So it is plausible that ${T^*\bm\hat{{\mathcal}{T}}^n}$ can be obtained as HK quotient of $\Gamma(\operatorname{Hilb}^n_{red}(T^{*\mathbb{C}}\Sigma))$ by the real group $\operatorname{Symp}_0(T^*{\Sigma})$, so it gets a hyperkähler structure itself. Notice that the complexified Lie algebra of $\operatorname{Symp}_0(T^*{\Sigma})$, i.e. the space of smooth complex-valued functions on $T^*{\Sigma}$, has the same action on $X$ as the real Lie algebra since one can prove that a Hamiltonian $H$ acts the same as $H\mod I$.
- *Feix-Kaledin structure*: If Hitchin’s component and our moduli space ${\bm\hat{{\mathcal}{T}}}^n$ are diffeomorphic, Hitchin’s component gets a complex structure. With its Goldman symplectic structure, there is good hope to get a Kähler structure. A general result of Feix and Kaledin (see [@Feix] and [@Kaledin]) asserts that for a Kähler manifold $X$, there is a neighborhood of the zero-section in $T^*X$ which admits a hyperkähler structure.
- *Construction by twistor approach*: The 1-parameter deformation of ${T^*\bm\hat{{\mathcal}{T}}^n}$ described in this paper is a good candidate to be the twistor space of ${T^*\bm\hat{{\mathcal}{T}}^n}$ (see figure \[HK\]).
Induced bundle {#indbundle}
--------------
To any point in the Hilbert scheme bundle $\operatorname{Hilb}^n(T^{*{\mathbb{C}}}{\Sigma})$, we can canonically associate a vector bundle $V$ of rank $n$ whose fiber over a point $z$ is $\mathbb{C}[p,\bar{p}]/I(z)$. We can glue these fibers together which gives a $n$-dimensional vector bundle over $\Sigma$. Locally, there is basis of the form $(s,ps,...,p^{n-1}s)$ for some generic section $s$. Under a coordinate change $z\mapsto w(z)$, this basis transforms in a diagonal way since $p^k \mapsto (\frac{dw}{dz})^kp^k$, so we get a bundle which is a direct sum of line bundles.
The matrix viewpoint of the punctual Hilbert scheme gives a ${\mathfrak}{sl}_n$-valued 1-form $M_pdz+M_{\bar{p}}d\bar{z}$ which acts on this bundle (locally $M_p$ by multiplication by $p$, $M_{\bar{p}}$ by multiplication by $\bar{p}$). If $I$ is a higher complex structure, then we get the 1-form $\Phi_1dz+\Phi_2d\bar{z}$ where $(\Phi_1, \Phi_2)$ is a pair of commuting nilpotent matrices. In that case there is a *preferred direction in each fiber*: the common kernel of both $\Phi_1$ and $\Phi_2$. In our local basis it is generated by $p^{n-1}s$.
The idea of this paper is to deform the 1-form $\Phi_1dz+\Phi_2d\bar{z}$ to a flat connection. The bundle $V$ itself gets deformed under this procedure. This is how we will associate a flat connection to a higher complex structure.
Conjugated higher complex structures {#dualcomplexstructure}
------------------------------------
There is a natural notion of conjugated space to ${T^*\bm\hat{{\mathcal}{T}}^n}$ using the natural complex conjugation on the complexified cotangent bundle $T^{*{\mathbb{C}}}\Sigma$. We associate to an ideal $I\in \operatorname{Hilb}^n_{red}(T^{*{\mathbb{C}}}{\Sigma})$ the ideal $\bar{I}$ and then take $\operatorname{Symp}_0$-equivalence classes.
In coordinates, we start from $$I=\langle p^n-t_2p^{n-2}-...-t_n, -\bar{p}+\mu_1+\mu_2p+...+\mu_np^{n-1} \rangle.$$ To get the conjugated structure, we have to express $\bar{I}$ in the same form as $I$, i.e. as $$\begin{aligned}
\bar{I} &=\langle \bar{p}^n-\bar{t}_2\bar{p}^{n-2}-...-\bar{t}_n, -p+\bar{\mu}_1+\bar{\mu}_2\bar{p}+...+\bar{\mu}_n\bar{p}^{n-1} \rangle \\
&= \langle p^n-{}_2tp^{n-2}-...-{}_nt, -\bar{p}+{}_1\mu+{}_2\mu p+...+{}_n\mu p^{n-1} \rangle.\end{aligned}$$ where $({}_kt, {}_k\mu)$ are the parameters of the conjugate to ${T^*\bm\hat{{\mathcal}{T}}^n}$. It is possible to explicitly express the conjugated coordinates $({}_kt, {}_k\mu)_k$ in terms of $(t_k, \mu_k)_k$. For example one gets ${}_2\mu=\frac{1}{\bar{\mu}_2}$ and ${}_nt = \bar{\mu}_2^n\bar{t}_n$.
Parabolic connections and reduction {#parabolicreduction}
===================================
In this section, we describe the generic fiber of the twistor space of ${T^*\bm\hat{{\mathcal}{T}}^n}$ from figure \[HK\] which is a space of flat connections. The idea about the deformation of ${T^*\bm\hat{{\mathcal}{T}}^n}$ is to replace the polynomial functions on $T^*{\Sigma}$ by differential operators. The higher complex structure is given by two polynomials (the generators of $I$), so in the deformation one gets a pair of differential operators.
The space of pairs of differential operators can be obtained by a reduction of all connections by some specific parabolic gauge. This procedure was first introduced by Bilal, Fock and Kogan in [@BFK]. In that paper, the authors also describe some ideas for generalized complex and projective structures. Our higher complex structures are the mathematically rigorous version of their ideas. Our treatment of the parabolic reduction is independent of their paper and follows some other notation. The question about how to impose the commutativity condition on the differential operators remained open in their paper. We show that the answer is given by a second reduction with respect to the group of higher diffeomorphisms.
Atiyah-Bott reduction
---------------------
Before going to the parabolic reduction, we recall the classical reduction of connections by gauge transforms, developed by Atiyah and Bott in their famous paper [@AtBott].
Let $\Sigma$ be a surface and $G$ be a semisimple Lie group with Lie algebra ${\mathfrak{g}}$. Let $E$ be a trivial $G$-bundle over $\Sigma$. Denote by $\mathcal{A}$ the space of all ${\mathfrak{g}}$-connections on $E$. It is an affine space modeled over the vector space of ${\mathfrak{g}}$-valued 1-forms $\Omega^1(\Sigma, {\mathfrak{g}})$. Further, denote by $\mathcal{G}$ the space of all gauge transforms, i.e. bundle automorphisms. We can identify the gauge group with $G$-valued functions: $\mathcal{G}=\Omega^0(\Sigma,G)$.
On the space of all connections $\mathcal{A}$, there is a natural symplectic structure given by $$\bm\hat{\omega} = \int_{\Sigma} \operatorname{tr}\;\delta A \wedge \delta A$$ where tr denotes the Killing form on ${\mathfrak{g}}$ (the trace for matrix Lie algebras). Since $\mathcal{A}$ is an affine space, its tangent space at every point is canonically isomorphic to $\Omega^1(\Sigma, {\mathfrak{g}})$. So given $A \in \mathcal{A}$ and $A_1, A_2 \in T_A\mathcal{A} \cong \Omega^1(\Sigma, {\mathfrak{g}})$, we have $\bm\hat{\omega}_A(A_1, A_2) = \int_{\Sigma} \operatorname{tr}\; A_1\wedge A_2$. Note that $\bm\hat{\omega}$ is constant (independent of $A$) so $d\bm\hat{\omega} = 0$. Further, the 2-form $\bm\hat{\omega}$ is clearly antisymmetric and non-degenerate (since the Killing form is). Remark finally that this construction only works on a surface.
We can now state the famous theorem of Atiyah-Bott (see end of chapter 9 in [@AtBott] for unitary case, see section 1.8 in Goldman’s paper [@Goldman.2] for the general case): *the action of gauge transforms on the space of connections is hamiltonian and the moment map is the curvature*. Thus, the hamiltonian reduction $\mathcal{A}//\mathcal{G}$ is the moduli space of flat connections.
Let us explain the moment map with more detail: the moment map $m$ is a map from $\mathcal{A}$ to $\operatorname{Lie}(\mathcal{G})^*$. The Lie algebra $\operatorname{Lie}(\mathcal{G})$ is equal to $\Omega^0(\Sigma, {\mathfrak{g}})$, so its dual is isomorphic to $\Omega^2(\Sigma, {\mathfrak{g}})$ via the pairing $\int_{\Sigma} \operatorname{tr}$. On the other hand, given a connection $A$, its curvature $F(A)$ is a ${\mathfrak{g}}$-valued 2-form, i.e. an element of $\Omega^2(\Sigma,{\mathfrak{g}})$. Hence, the map $m$ is well-defined.
Parabolic reduction
-------------------
### Setting and coordinates {#settingparab}
In subsection \[indbundle\] we have seen how to associate a rank $n$-bundle $V$ over ${\Sigma}$ to a higher complex structure. Moreover we have seen that there is a privileged direction in each fiber, the common kernel of $\Phi_1$ and $\Phi_2$. This gives a line-subbundle $L$ in $V$. We want to mimic the Atiyah-Bott reduction with the extra constraint of fixing $L$. That is why we consider the subspace of gauge transformations fixing the subbundle $L$ (more precisely its dual).
Let us take the same setting as for the Atiyah-Bott reduction with $G=\operatorname{SL}_n(\mathbb{C})$. But instead of all gauge transforms $\mathcal{G}$, we consider the subgroup $\mathcal{P}\subset \mathcal{G}$ consisting of matrices of the form $$\begin{pmatrix}
* & \cdots & * & * \\
\vdots & & \vdots & \vdots \\
* & \cdots & *& * \\
0 & \cdots& 0 & * \\
\end{pmatrix}$$ i.e. preserving the last direction in the dual space. We want to compute and analyze the hamiltonian reduction $\mathcal{A}//\mathcal{P}$, which we call **space of parabolic connections**.
The reason to consider a fixed direction in the dual bundle and not in the bundle itself is purely of technical advantage. $\triangle$
Since $\mathcal{P}\subset \mathcal{G}$, we know by the Atiyah-Bott theorem that the action of $\mathcal{P}$ on the space of connections $\mathcal{A}$ is hamiltonian with moment map $m: A\mapsto i^*F(A)$ where $i: \mathcal{P} \hookrightarrow \mathcal{G}$ is the inclusion and $i^*: \operatorname{Lie}(\mathcal{G})^* \twoheadrightarrow \operatorname{Lie}(\mathcal{P})^*$ the induced surjection on the dual Lie algebras. Since $G=\operatorname{SL}_n(\mathbb{C})$, the map $i^*$ is explicitly given by forgetting the first $n-1$ entries in the last column. This means that $m^{-1}(\{0\})$ is the space of all $A \in \mathcal{A}$ such that the curvature $F(A)$ is of the form $$\begin{pmatrix}
0& \cdots & 0 & \xi_n\\
\vdots & & \vdots &\vdots\\
\vdots & & \vdots & \xi_2\\
0& \cdots &0 & 0 \\
\end{pmatrix}.$$
In order to give a description in coordinates of the hamiltonian reduction $\mathcal{A}//\mathcal{P}$, we fix a reference complex structure on the surface $\Sigma$. We take a connection $A\in \mathcal{A}$ and decompose it into its holomorphic and anti-holomorphic parts: $A=A_1 + A_2$. As a covariant derivative, we set $\nabla = \partial + A_1$ and $\bar{\nabla} = \bar{\partial}+ A_2$. Using the parabolic gauge, it is possible to reduce $A_1$ locally to a companion matrix: $$\label{firstmatrix}
A_1 \sim
\begin{pmatrix}
& & & \bm\hat{t}_n\\
1 & & &\vdots \\
& \ddots & &\bm\hat{t}_2 \\
& & 1 & 0 \\
\end{pmatrix}dz.$$ The existence of such a gauge is proven in the appendix \[appendix:B\]. Reducing $A_1$ to the above form means that we choose a basis of the form $B= (s, \nabla s, \nabla^2 s,..., \nabla^{n-1}s)$. This takes all the gauge freedom. A connection in $\mathcal{A}//\mathcal{P}$ verifies $[\nabla, \bar{\nabla}]\nabla^i s = 0$ for $i=0,1,...,n-2$ since $[\nabla, \bar{\nabla}] = F(A)$ is the curvature which is concentrated on the last column. It follows that $\bar{\nabla}\nabla^i s = \nabla^i \bar{\nabla} s$ for all $i=1,...,n-1$. Thus, the connection is fully described by $\nabla^n s$ and $\bar{\nabla}s$. We can write these expressions in the basis $B$: $$\label{eqqq1}
\nabla^n s = \bm\hat{t}_{n}s + \bm\hat{t}_{n-1} \nabla s+...+\bm\hat{t}_{2}\nabla^{n-2}s = \bm\hat{P}(\nabla)s$$ $$\label{eqqq2}
\bar{\nabla}s = \bm\hat{\mu}_1 s + \bm\hat{\mu}_2 \nabla s + ... + \bm\hat{\mu}_n \nabla^{n-1}s = \bm\hat{Q}(\nabla)s.$$ Notice that $\bm\hat{t}_1=0$ since $\operatorname{tr}A_1=0$.
The second part $A_2$ is uniquely determined by its first column given by equation . Since $\bar{\nabla}\nabla^i s = \nabla^i \bar{\nabla} s$ for $i=1,...,n-1$, the $i$-th column of $A_2$ is given by applying $(i-1)$ times $\nabla$ to the first column. We get a 1-form of the following type: $$\label{secondmatrix}
A_2 \sim
\begin{pmatrix}
\bm\hat{\mu}_1& \partial \bm\hat{\mu}_1+\bm\hat{\mu}_n \bm\hat{t}_n & \cdots\\
\bm\hat{\mu}_2 & \bm\hat{\mu}_1+\partial \bm\hat{\mu}_2+\bm\hat{\mu}_n \bm\hat{t}_{n-1} & \cdots \\
\vdots & \vdots & \vdots \\
\bm\hat{\mu}_{n-1} & \bm\hat{\mu}_{n-2}+\partial \bm\hat{\mu}_{n-1}+\bm\hat{\mu}_n \bm\hat{t}_2 & \cdots \\
\bm\hat{\mu}_n & \bm\hat{\mu}_{n-1} +\partial \bm\hat{\mu}_n & \cdots
\end{pmatrix}d\bar{z}.$$
Notice that modulo $\partial$ (meaning that you drop all terms with a partial derivative), equations and become the relations of $p^n$ and $\bar{p}$ in a generic ideal of $\operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$. So $A_1$ and $A_2$ become the multiplication operators by $p$ and $\bar{p}$ respectively. $\triangle$
The functions $(\bm\hat{\mu}_2, ..., \bm\hat{\mu}_n, \bm\hat{t}_{2}, ..., \bm\hat{t}_{n})$ completely parameterize $\mathcal{A}//\mathcal{P}$ since it is possible to express $\bm\hat{\mu}_{1}$ in terms of these using that the second matrix is traceless. We call an element of $\mathcal{A}//\mathcal{P}$ a **parabolic connection**. We consider ${\mathcal}{A}//{\mathcal}{P}$ as a subspace of ${\mathcal}{A}$ by using the representative $A_1 + A_2$ with $A_1$ of the local form \[firstmatrix\] and $A_2$ like in \[secondmatrix\]. Its parabolic curvature is concentrated on the last column: $[\nabla, \bar{\nabla}]\nabla^{n-1}s = \xi_n s + \xi_{n-1} \nabla s + ...+ \xi_2 \nabla^{n-2}s$. The following proposition allows to compute the parabolic curvature easily.
\[thmcourbure\] $[\nabla^n,\bar{\nabla}]s = \sum_{k=2}^n \xi_k\nabla^{n-k}s.$
Since the first $n-1$ columns of the curvature $F(A)$ are 0, we have $[\nabla, \bar{\nabla}]\nabla^is = 0$ for $i=0,1,...,n-2$. Using Leibniz’s rule and induction on $k$, we can prove that $[\nabla^k, \bar{\nabla}]s = 0$ for $k=1,...,n-1$. Indeed, it is true for $k=1$ and we have $[\nabla^{k+1}, \bar{\nabla}]s = \nabla [\nabla^k,\bar{\nabla}]s + [\nabla,\bar{\nabla}]\nabla^k s=0$ whenever $k\leq n-2$.
Therefore, we get $$[\nabla^n, \bar{\nabla}]s = \nabla[\nabla^{n-1}, \bar{\nabla}]s + [\nabla, \bar{\nabla}]\nabla^{n-1}s = [\nabla, \bar{\nabla}]\nabla^{n-1}s = \sum_{k=2}^n \xi_k\nabla^{n-k}s$$ by the last column of the curvature.
Inside the non-commutative ring of differential operators, we define the left-ideal $\bm\hat{I}=\langle \nabla^n-\bm\hat{P}, -\bar{\nabla}+\bm\hat{Q} \rangle$ where $\bm\hat{P}$ and $\bm\hat{Q}$ are defined in equations and respectively. We can express the previous proposition as $$[\nabla^n,\bar{\nabla}]=\sum_{k=2}^n \xi_k\nabla^{n-k} \mod \bm\hat{I}.$$
Notice finally that our coordinates $(\bm\hat{\mu}_2, ..., \bm\hat{\mu}_n, \bm\hat{t}_{2}, ..., \bm\hat{t}_{n})$ do not behave like tensors under coordinate change $z\mapsto w(z)$. We will see in the following section \[parabolicwithlambda\] that if we introduce a parameter ${\lambda}$ we get at the semiclassical limit tensors out of our coordinates.
### Example n=2 {#casen2}
Consider a parabolic $\operatorname{SL}(2, {\mathbb{C}})$-connection locally written as $A=A_1dz+A_2d\bar{z}$. The first matrix $A_1$ is a companion matrix of the form $\left( \begin{smallmatrix} 0 & \bm\hat{t}_2 \\ 1 & 0 \end{smallmatrix} \right)$.
Let us compute the transformed matrix $A_2$. It is the image of the operator $\bar{\nabla}$ in a basis $(s,\nabla s)$. Put $\bar{\nabla}s = \bm\hat{\mu}_1 s+ \bm\hat{\mu}_2 \nabla s.$ The second column can be computed using $\bar{\nabla}\nabla s = \nabla \bar{\nabla}s - [\nabla,\bar{\nabla}]s = \nabla \bar{\nabla}s$ and $\nabla^2 s = \bm\hat{t}_2s$. Since the trace of the matrix is zero, we get $\bm\hat{\mu}_1 = -\frac{1}{2}\partial \bm\hat{\mu}_2$. Hence $$A_2 = \left( \begin{array}{cc}
-\frac{1}{2}\partial \bm\hat{\mu}_2 & -\frac{1}{2}\partial^2 \bm\hat{\mu}_2+\bm\hat{t}_2\bm\hat{\mu}_2 \\
\bm\hat{\mu}_2 & \frac{1}{2}\partial \bm\hat{\mu}_2
\end{array} \right).$$ The curvature is of the form $\left( \begin{smallmatrix} 0 & \xi_2 \\ 0 & 0 \end{smallmatrix} \right)$ where $$\xi_2 = (\bar{\partial}-\bm\hat{\mu}_2\partial-2\partial\bm\hat{\mu}_2)\bm\hat{t}_2+\frac{1}{2}\partial^3\bm\hat{\mu}_2.$$ Suppose that the curvature $\xi_2$ is 0. We can then look for flat sections $\Psi = (\psi_1,\psi_2)$. The first condition $(\partial+A_1)\Psi=0$ gives $\psi_1=-\partial \psi_2$ and $$(\partial^2-\bm\hat{t}_2)\psi_2 = 0.$$ The second condition $(\bar{\partial}+A_2)\Psi= 0$ only gives one extra condition: $$(\bar{\partial}-\bm\hat{\mu}_2\partial+\frac{1}{2}\partial\bm\hat{\mu}_2)\psi_2 = 0.$$ For $\bm\hat{\mu}_2=0$ this just means that $\psi_2$ is holomorphic and we get an ordinary differential equation $(\partial^2-\bm\hat{t}_2)\psi_2 = 0$. For $\bm\hat{\mu}_2\neq 0$, the second condition is still a holomorphicity condition, but with respect to another complex structure.
For general $n$, a flat section $\Psi=(\psi_k)_{1\leq k \leq n}$ is of the form $\psi_{n-k}={\partial}^k\psi_n$ and there are two equations on $\psi_n$. The first equation comes from the last column in $A_1$, so directly generalizes to $(\partial^n-\bm\hat{t}_1\partial^{n-1}-...-\bm\hat{t}_n)\psi_n = 0$. The generalized holomorphicity condition comes from the last row in $A_2$: $$\label{diffops}
(-\bar{\partial}+\bm\hat{\alpha}_{nn}+\bm\hat{\alpha}_{n,n-1}\partial+...+\bm\hat{\alpha}_{n,1}\partial^{n-1})\psi_n=0$$ where $\bm\hat{\alpha}_{ij}$ denote the entries of $A_2$ which have an explicit but complicated expression in terms of the $\bm\hat{\mu}_k$ and $\bm\hat{t}_k$.
Higher diffeomorphisms and flat connections {#symponconnections}
-------------------------------------------
To get from parabolic connections to flat connections, we define an action of higher diffeomorphisms on the space of parabolic connections $\mathcal{A}//\mathcal{P}$. We prove that this action is hamiltonian and show that the double reduction $\mathcal{A}//\mathcal{P}//\operatorname{Symp}_0(T^*\Sigma)$ is a space of flat connections.
### Action of higher diffeomorphisms
Recall that the description in coordinates of the space of parabolic connections relies on a basis $B$ of the form $(s, \nabla s, ..., \nabla^{n-1}s)$. A variation $\delta s$ of the section $s$ can be expressed in this basis: $$\delta s = v_1 s+v_2\nabla s +...+v_{n}\nabla^{n-1}s = \bm\hat{H}s$$ where $\bm\hat{H}=v_1+v_2\nabla+...+v_{n}\nabla^{n-1}$ is a differential operator of degree $n-1$.
The Lie algebra of higher diffeomorphisms $\operatorname{Lie}(\operatorname{Symp}_0(T^*\Sigma))$ is the space of functions on $T^*{\Sigma}$ which can be deformed to differential operators on ${\Sigma}$. The infinitesimal action of higher diffeomorphisms on parabolic connections is given by a base change induced by $s \mapsto s+\varepsilon \delta s$ such that the basis $B$ preserves its form. More specifically, to a higher diffeomorphism generated by $H=v_2p+v_3p^2+...+v_{n}p^{n-1}$ we associate the variation $\bm\hat{H}=v_1+v_2\nabla+v_3\nabla^2+...+v_{n}\nabla^{n-1}$ where $v_1$ is uniquely determined by the other $v_i$ by the condition that the infinitesimal gauge transform is of trace zero.
This only defines the infinitesimal action. The question about how to integrate the action to the whole group $\operatorname{Symp}_0$, or maybe to a deformation of it, has to be worked out. $\triangle$
Let us describe how to compute the matrix $X$ describing the infinitesimal base change induced by a higher diffeomorphism. Write the base change as $$(s,\nabla s,...,\nabla^{n-1}s) \mapsto (s,\nabla s,...,\nabla^{n-1}s)+\varepsilon (\delta s,\nabla \delta s,...,\nabla^{n-1}\delta s).$$ So the first column of $X$ is just given by $Xs = \delta s = v_1s+v_2\nabla s+...+v_n\nabla^{n-1}s$. The second is given by $X\nabla s = \nabla \delta s = \nabla(v_1s+v_2\nabla s+...+v_n\nabla^{n-1}s)$. We notice that the construction of this matrix $X$ is exactly the same as for the matrix $A_2$ (see equation ) with the only difference that the variables in $A_2$ are called $\bm\hat{\mu}_k$ instead of $v_k$. Since both matrices are traceless, even the terms $v_1$ and $\bm\hat\mu_1$ coincide. Notice that if $X$ is a parabolic gauge, i.e. the $(n-1)$ first entries of the last column are zero, then $v_k=0 \;\forall k$, so $X=0$.
\[computeX\] The matrix $X$ of the gauge coming from a higher diffeomorphism is given by $$X=A_2 \mid_{\bm\hat{\mu}_k\mapsto v_k}.$$
Let us indicate how to compute the action of a higher diffeomorphism on our coordinates $(\bm\hat{t}_k, \bm\hat{\mu}_k)$. The coordinates $\bm\hat{t}_k$ are given by the relation $\nabla^n s = \bm\hat{P}s$ where $\bm\hat{P}=\bm\hat{t}_2\nabla^{n-2}+...+\bm\hat{t}_n$. The variation $\delta \bm\hat{P}$ satisfies $$\nabla^n (s+\varepsilon \bm\hat{H}s) = (\bm\hat{P}+\varepsilon \delta \bm\hat{P})(s+\varepsilon \bm\hat{H}s)$$ which gives $$\label{variation-hat-t}
\delta \bm\hat{P} = [\bm\hat{H}, -\nabla^n+\bm\hat{P}] \mod \bm\hat{I}$$ where $\bm\hat{I} = \langle \nabla^n-\bm\hat{P}, -\bar{\nabla}+\bm\hat{Q} \rangle$ is a left ideal of differential operators.
Similarly, the coordinates $\bm\hat{\mu}_k$ are given by $\bar{\nabla}s = \bm\hat{Q}s$ where $\bm\hat{Q} = \bm\hat{\mu}_1+\bm\hat{\mu}_2\nabla...+\bm\hat{\mu}_n\nabla^{n-1}$. We can easily compute the variation of $\bm\hat{Q}$ to be $$\label{variation-hat-mu}
\delta \bm\hat{Q} = [\bm\hat{H}, -\bar{\nabla}+\bm\hat{Q}] \mod \bm\hat{I}.$$
In the case of the Hilbert scheme, we used the variation formula from symplectic geometry $\delta f = \frac{df}{dt} = \{H,f\}$ (see equation ). Here we find the deformed version of this: $\delta \bm\hat{P} = [\bm\hat{H}, \bm\hat{P}]$ where $\bm\hat{H}$ is the quantum Hamiltonian and $\bm\hat{P}$ some operator. In the next section, we introduce a deformation parameter $h$ and we get $\delta \bm\hat{P}(h) = \frac{1}{h}[\bm\hat{H}(h), \bm\hat{P}(h)]$. $\triangle$
### Double reduction to flat connections
We have just seen that higher diffeomorphisms act on the space of parabolic connections by gauge transforms. Since we see ${\mathcal}{A}//{\mathcal}{P}$ as a subset of ${\mathcal}{A}$ and since the gauge action on ${\mathcal}{A}$ is hamiltonian, we see that the action of higher diffeomorphisms on $\mathcal{A}//\mathcal{P}$ is also hamiltonian. It is not surprising that the moment map is nothing else than the parabolic curvature:
The infinitesimal action of higher diffeomorphisms $\operatorname{Symp}_0(T^*\Sigma)$ on the space of parabolic connections $\mathcal{A}//\mathcal{P}$ is hamiltonian with moment map $$m(\bm\hat{t}_i,\bm\hat{\mu}_j).(v_2,...,v_n) = \int_{\Sigma} \sum_{i=1}^n x_{n,n+1-i}\xi_i$$ where $x_{i,j}$ are the matrix elements of the gauge $X$ and $\xi_i$ is the parabolic curvature of the parabolic connection described by $(\bm\hat{t}_i,\bm\hat{\mu}_i)_{2\leq i \leq n}$.
Some explanation for the moment map is necessary: $m$ goes from the space $\mathcal{A}//\mathcal{P}$, which is described by coordinates $(\bm\hat{t}_i,\bm\hat{\mu}_j)$, into $\operatorname{Lie}(\operatorname{Symp}_0)^*$, the dual to the Lie algebra of higher diffeomorphisms. The Lie algebra of higher diffeomorphisms is described by Hamiltonians of the form $v_2p+...+v_np^{n-1}$. To such a function, we compute the associated matrix $X$ (see proposition \[computeX\]) from which we take the last row for computing $m$. All elements of $X$ are functions depending on the $v_k$ and the $\bm\hat{t}_k$. The parabolic curvature described by the $\xi_i$ is a function of $(\bm\hat{t}_i,\bm\hat{\mu}_j)$.
Our computation is analogous to the Atiyah-Bott reduction. An infinitesimal gauge transform given by $X$ affects $A_1$ and $A_2$ by $$\begin{aligned}
\chi(A_1) = [X,A_1]-\partial X \\
\chi(A_2) = [X,A_2] -\bar{\partial}X\end{aligned}$$
The symplectic form on $\mathcal{A}//\mathcal{P}$ is the restriction of the one on $\mathcal{A}$, so we can compute $$\begin{aligned}
\iota_{\chi}\omega_{\mathcal{A}//\mathcal{P}} &= \int \operatorname{tr}\left(\chi(A_1)\delta A_2-\chi(A_2)\delta A_1 \right)\\
&= \int \operatorname{tr}([X,A_1]-\partial X )\delta A_2-([X,A_2] -\bar{\partial}X)\delta A_1 \\
&= \int \operatorname{tr}([A_1,\delta A_2]+\delta\partial A_2-[A_2,\delta A_1]-\delta\bar{\partial}A_1)X \\
&= \int \operatorname{tr}\delta (\partial A_2-\bar{\partial}A_1+[A_1,A_2])X \\
&= \delta \int \operatorname{tr}F(A)X \\
&= \delta \int \sum_{i=1}^n x_{n,n+1-i}\xi_i.\end{aligned}$$
Therefore $$m= \int_{\Sigma} \sum_{i=1}^n x_{n,n+1-i}\xi_i.$$
\[flatparaconnections\] The double reduction $\mathcal{A}//\mathcal{P}//\operatorname{Symp}_0$ gives a space of flat connections: $$\mathcal{A}//\mathcal{P}//\operatorname{Symp}_0 \cong \{A_1+A_2\in \mathcal{A}//\mathcal{P} \mid \xi_i=0 \; \forall i\} / \operatorname{Symp}_0$$ with $A_1$ locally of the form \[firstmatrix\] and $A_2$ like in \[secondmatrix\].
The corollary directly follows from the previous theorem since $m(\bm\hat{t}_i, \bm\hat{\mu}_j)(v_2,...,v_n)=0$ for all $v_2,...,v_n$ implies $\xi_i=0$ for all $i$.
We call the double reduction space $\mathcal{A}//\mathcal{P}//\operatorname{Symp}_0$ the space of **flat parabolic connections**. For $n=2$ we get those flat $\operatorname{SL}_2({\mathbb{C}})$-connections whose monodromy is the developing map of a complex projective structure on ${\Sigma}$. For general $n$, we probably get a complicated subset of the space of all flat connections.
Parabolic reduction of h-connections {#parabolicwithlambda}
====================================
In this section, we study the parabolic reduction on $h$-connections to get the twistor space description from figure \[HK\]. The main idea is the following: a point in ${T^*\bm\hat{{\mathcal}{T}}^n}$ is a $\operatorname{Symp}_0$-equivalence class of ideals of the form $$I=\langle -p^n+t_2p^{n-2}+...+t_n, -\bar{p}+\mu_1+\mu_2 p+...+\mu_n p^{n-1} \rangle.$$ Replace the polynomials by $h$-connections using the rule $p \mapsto \nabla=h{\partial}+ A_1(h)$ and $\bar{p} \mapsto \bar{\nabla}=h{\bar\partial}+A_2(h)$ where $h$ is a formal parameter. This corresponds to the deformation of a higher complex structure $\Phi$ to $\Phi+hd+hA+h^2\Phi^*=h(d+{\lambda}\Phi+A+{\lambda}^{-1}\Phi^*)$ where ${\lambda}=h^{-1}$. For $h\neq 0$ we divide the connection by $h$ to get a usual connection with parameter ${\lambda}$. For all ${\lambda}\in {\mathbb{C}}^*$ fixed, we get the same space as described in the previous section \[parabolicreduction\], i.e. the space of flat parabolic connections. For ${\lambda}\rightarrow \infty$ we get the cotangent bundle ${T^*\bm\hat{{\mathcal}{T}}^n}$. For ${\lambda}\rightarrow 0$ we get the space of conjugated structures $[({}_kt, {}_k\mu)]$ (see subsection \[dualcomplexstructure\]).
Parametrization
---------------
Take ${\mathcal}{A}({\lambda})={\lambda}\Phi + A + {\lambda}^{-1}\Phi^*$ where the $*$-operator is the hermitian conjugate and $\Phi$ is in the Hilbert scheme bundle $\operatorname{Hilb}^n_0(T^{*{\mathbb{C}}}\Sigma)$. Recall that this means that locally $\Phi(z,\bar{z})=\Phi_1(z,\bar{z})dz+\Phi_2(z,\bar{z})d\bar{z}$ with $\Phi_1 \in {\mathfrak}{sl}_n$ is a principal nilpotent element and $\Phi_2$ is in the centralizer of $\Phi_1$, i.e. $[\Phi_1,\Phi_2]=0$. We define ${\mathcal}{A}_1({\lambda})={\lambda}\Phi_1 + A_1 + {\lambda}^{-1}\Phi_2^*$ and ${\mathcal}{A}_2({\lambda})={\lambda}\Phi_2 + A_2 + {\lambda}^{-1}\Phi_1^*$, i.e. the $(1,0)$-part and $(0,1)$-part of ${\mathcal}{A}({\lambda})$. We also define $\nabla=\partial + {\mathcal}{A}_1({\lambda})$ and $\bar{\nabla}=\bar{\partial}+{\mathcal}{A}_2({\lambda})$.
As for the case without parameter, there is a parabolic gauge which transforms ${\mathcal}{A}({\lambda})$ locally to $$\label{paragauge}
\left(\begin{array}{cccc}
& & & \bm\hat{t}_n({\lambda}) \\
1& & & \vdots \\
& \ddots & & \bm\hat{t}_2({\lambda}) \\
& & 1& 0
\end{array}\right)
dz + \left(\begin{array}{cc}
\bm\hat{\mu}_1({\lambda}) & \\
\bm\hat{\mu}_2({\lambda})& \bm\hat{\alpha}_{ij}({\lambda})\\
\vdots & \\
\bm\hat{\mu}_n({\lambda}) &
\end{array}\right) d\bar{z}$$ where $\bm\hat{\alpha}_{ij}({\lambda})$ and $\bm\hat{\mu}_1({\lambda})$ are explicit functions of the other variables. Thus, the space is parametrized by $(\bm\hat{t}_i({\lambda}), \bm\hat{\mu}_i({\lambda}))_{i=2,...,n}$.
This local representative comes from a basis of the form $(s, \nabla s, ..., \nabla^{n-1}s)$ for some section $s$. We then get our coordinates by $$\label{paraboliccoord1}
\nabla^n s = \bm\hat{t}_{n}({\lambda})s + \bm\hat{t}_{n-1}({\lambda}) \nabla s+...+\bm\hat{t}_{2}({\lambda})\nabla^{n-2}s$$ $$\label{paraboliccoord2}
\bar{\nabla}s = \bm\hat{\mu}_1({\lambda}) s + \bm\hat{\mu}_2({\lambda}) \nabla s + ... + \bm\hat{\mu}_n({\lambda}) \nabla^{n-1}s.$$
You can compute the $\bm\hat{\alpha}_{ij}({\lambda})$ using $\bar{\nabla}\nabla^ks = \nabla^k\bar{\nabla}s$ for $k\leq n-1$ which holds since the curvature $[\nabla, \bar{\nabla}]$ is concentrated in the last column.
\[examplen2\] Take $n=2$ and consider $\Phi_1 = \left(\begin{smallmatrix} 0 & 0 \\ b_1 & 0\end{smallmatrix}\right)$, $A_1 = \left(\begin{smallmatrix} a_0 & a_1 \\ a_2 & -a_0\end{smallmatrix}\right)$, $\Phi_2=\mu_2\Phi_1$ and $A_2 = -A_1^{\dagger}$. So we have $${\mathcal}{A}_1({\lambda})=\begin{pmatrix} a_0 & a_1+{\lambda}^{-1}\bar{\mu}_2\bar{b}_1 \\ a_2+{\lambda}b_1 & -a_0\end{pmatrix}\; \text{ and } \;{\mathcal}{A}_2({\lambda})=\begin{pmatrix} -\bar{a}_0 & -\bar{a}_2+{\lambda}^{-1}\bar{b}_1 \\ -\bar{a}_1+{\lambda}\mu_2b_1 & \bar{a}_0\end{pmatrix}.$$ We look for $P=\left(\begin{smallmatrix} p_1 & p_2 \\ 0 & 1/p_1\end{smallmatrix}\right)$ such that $$P{\mathcal}{A}_1({\lambda})P^{-1}+ P\partial P^{-1} = \begin{pmatrix} 0 & \bm\hat{t}_2({\lambda}) \\ 1 & 0\end{pmatrix}.$$ Multiplying by $P$ from the right, one can solve the system. One finds $p_1=({\lambda}b_1+a_2)^{1/2}$ and $p_2=-\frac{a_0}{p_1}+\frac{\partial p_1}{p_1^2}$. Hence $$\bm\hat{t}_2({\lambda})={\lambda}a_1b_1+ \text{ constant term }+{\lambda}^{-1}\bar{\mu}_2 a_2\bar{b}_1.$$ Transforming ${\mathcal}{A}_2({\lambda})$ with $P$ we get $$\bm\hat{\mu}_2({\lambda}) = \frac{-\bar{a}_1+{\lambda}\mu_2 b_1}{{\lambda}b_1+a_2} = \frac{-\bar{a}_1+{\lambda}\mu_2 b_1}{{\lambda}b_1-\bar{\mu}_2\bar{a}_1}$$ where we used $a_2=-\bar{\mu}_2\bar{a}_1$ coming from the flatness of ${\mathcal}{A}({\lambda})$.
For ${\lambda}\rightarrow \infty$, we can develop the rational expression of $\bm\hat{\mu}_2({\lambda})$ to get $$\bm\hat{\mu}_2({\lambda})=\mu_2+(\mu_2\bar{\mu}_2-1)\sum_{k=1}^\infty \frac{\bar{\mu}_2^{k-1}\bar{a}_1^k}{b_1^k}{\lambda}^{-k}.$$ For ${\lambda}\rightarrow 0$, we get $$\bm\hat{\mu}_2({\lambda})=\frac{1}{\bar{\mu}_2}+(1-\mu_2\bar{\mu}_2)\sum_{k=1}^\infty \frac{b_1^k}{\bar{\mu}_2^{k+1}\bar{a}_1^k}{\lambda}^{k}.$$ Notice that we get ${}_2\mu=1/\bar{\mu}_2$ as leading term (see section \[dualcomplexstructure\]). $\triangle$
The example shows several phenomena which are true in general:
\[parametrisationlambda\] The $\bm\hat{\mu}_k({\lambda})$ are rational functions in ${\lambda}$. The highest term in ${\lambda}$ when ${\lambda}\rightarrow \infty$ is ${\lambda}^{2-k}\mu_k$ where $\mu_k$ is the higher Beltrami differential from the $n$-complex structure. For ${\lambda}\rightarrow 0$ we get as lowest term ${\lambda}^{k-2}{}_k\mu$ where ${}_k\mu$ is the conjugated $n$-complex structure.
The $\bm\hat{t}_k({\lambda})$ are also rational functions in ${\lambda}$. For ${\lambda}\rightarrow \infty$, the highest term is given by ${\lambda}^{k-1}t_k$, and the lowest term for ${\lambda}\rightarrow 0$ is given by ${\lambda}^{1-k}{}_kt$ where $$t_k=\operatorname{tr}A_1\Phi_1^{k-1} \text{ and } {}_kt = \operatorname{tr}A_1 (\Phi_2^*)^{k-1}.$$
We will see later that $(\mu_k, t_k)$ is a point of the cotangent bundle ${T^*\bm\hat{{\mathcal}{T}}^n}$, and that $({}_k\mu, {}_kt)$ is the conjugated structure, which justifies the notation.
The whole point is to analyze equations and in detail. Let us start with $$\bar{\nabla}s = \bm\hat{\mu}_1({\lambda}) s + \bm\hat{\mu}_2({\lambda}) \nabla s + ... + \bm\hat{\mu}_n({\lambda}) \nabla^{n-1}s.$$ Since $\bar{\nabla}s=(\bar{\partial}+{\lambda}\Phi_2+A_2+{\lambda}^{-1}\Phi_1^*)s$ the highest ${\lambda}$-term is ${\lambda}\Phi_2s={\lambda}\mu_2\Phi_1s+...+{\lambda}\mu_n\Phi_1^{n-1}s$. On the other side, the highest term of $\nabla^ks$ is ${\lambda}^k\Phi_1^ks$ for $0\leq k \leq n-1$. For generic $s$ the set $(s, \Phi_1s, ..., \Phi_1^{n-1}s)$ is a basis. Hence, we can compare the highest terms and deduce that for ${\lambda}\rightarrow \infty$: $$\bm\hat{\mu}_k({\lambda}) = {\lambda}^{2-k}\mu_k+\text{ lower terms}.$$
Similarly, the set $(s, \Phi_2^*s, ..., \Phi_2^{*(n-1)}s)$ is generically a basis. Comparing highest terms and using $\Phi_1^*={}_2\mu \Phi_2^*+...+{}_n\mu\Phi_2^{*(n-1)}$, we get for ${\lambda}\rightarrow 0$: $$\bm\hat{\mu}_k({\lambda}) = {\lambda}^{k-2}{}_k\mu+\text{ higher terms}.$$
In any case, we can decompose $\nabla^ks$ and $\bar{\nabla}s$ in the basis $(s,\Phi_1s,...,\Phi_1^{n-1}s)$ and notice that the defining equations for $\bm\hat{\mu}_k$ is a quotient of two polynomials in ${\lambda}$, i.e. $\bm\hat{\mu}_k$ is a rational function in ${\lambda}$. The same decomposition gives that $\bm\hat{t}_k$ is a rational function in ${\lambda}$.
The last thing is to study the asymptotic behavior of $\bm\hat{t}_k$. For that, we have to study $$\nabla^n s = \bm\hat{t}_{n}({\lambda})s + \bm\hat{t}_{n-1}({\lambda}) \nabla s+...+\bm\hat{t}_{2}({\lambda})\nabla^{n-2}s.$$ The highest term of $\nabla^ns$ is not ${\lambda}^n \Phi_1^n$ since $\Phi_1^n=0$. The next term is given by $${\lambda}^{n-1}\sum_{l=0}^{n-1}\Phi_1^l \circ(\partial+A_1)\circ\Phi_1^{n-1-l}s$$ where $\circ$ denotes the composition of differential operators. On the other side, the highest terms are given by $\bm\hat{t}_k{\lambda}^{n-k}\Phi_1^{n-k}s$. When ${\lambda}$ goes to infinity, we compare coefficients in the basis $(s, \Phi_1s, ..., \Phi_1^{n-1}s)$ as before. Using Dirac’s “bra-ket” notation, we get $$\begin{aligned}
{\lambda}^{n-k}\bm\hat{t}_k =& {\lambda}^{n-1} \langle \Phi_1^{n-k}s \mid \sum_{l=0}^{n-1}\Phi_1^l \circ(\partial+A_1)\circ\Phi_1^{n-1-l} \mid s \rangle \\
=& {\lambda}^{n-1}\sum_{l=0}^{n-k} \langle \Phi_1^{n-k-l}s\mid (\partial +A_1)\circ\Phi_1^{n-1-l}\mid s\rangle \\
=& {\lambda}^{n-1}\sum_{l=0}^{n-k} \langle \Phi_1^{n-k-l}s\mid (\partial +A_1)\circ\Phi_1^{k-1}\mid \Phi_1^{n-k-l}s\rangle \\
=& {\lambda}^{n-1}\operatorname{tr}((\partial+A_1)\circ\Phi_1^{k-1}) \\
=& {\lambda}^{n-1} \operatorname{tr}(A_1\Phi_1^{k-1}).\end{aligned}$$ In the last line, we used that $\operatorname{tr}{\partial}\circ \Phi_1^{k-1} = 0$ since $\Phi_1$ is strictly lower triangular which is preserved under derivation. This precisely gives the expression for $t_k$ as stated in the proposition. The same analysis goes through for ${\lambda}\rightarrow 0$.
At the end of subsection \[settingparab\] we have noticed that $\bm\hat{t}_k$ and $\bm\hat{\mu}_k$ do not transform as tensors. We now show that the highest terms, $t_k$ and $\mu_k$, are tensors. Recall that $K=T^{*(1,0)}\Sigma$ is the canonical bundle and that $\Gamma(.)$ denotes the space of sections.
\[highesttermtensor\] We have $t_i \in \Gamma(K^i)$ and $\mu_i \in \Gamma(K^{1-i}\otimes \bar{K})$.
Consider a holomorphic coordinate change $z\mapsto w(z)$. We compute how $\mu_i(z)$ and $t_i(z)$ change.
For $\mu_i$, notice that $\Phi_1dz \mapsto \Phi_1 \frac{dz}{dw}dw$, so using $$\Phi_2d\bar{z} = \mu_2(z)\Phi_1dz+...+\mu_n\Phi_1^{n-1}dz^{n-1}$$ we easily get $\mu_i(z)=\frac{d\bar{z}/d\bar{w}}{(dz/dw)^{i-1}}\mu_i(w)$.
For $t_i$, we use $t_i=\operatorname{tr}(\Phi_1^{i-1}A_1)$ where $\Phi_1$ and $A_1$ are both $(1,0)$-forms, thus $t_i$ is a $(i,0)$-form, i.e. a section of $K^i$.
Action of higher diffeomorphisms
--------------------------------
In \[symponconnections\] we have described an infinitesimal action of $\operatorname{Symp}_0(T^*\Sigma)$ on the space of parabolic connections ${\mathcal}{A}//{\mathcal}{P}$. Recall that to write a representative of an element of ${\mathcal}{A}//{\mathcal}{P}$, we use a basis of the form $(s,\nabla s,..., \nabla^{n-1}s)$. A higher diffeomorphism changes the section $s$ and thus the whole basis. The same action holds for the parabolic $h$-connections. In particular, corollary \[flatparaconnections\] about flat parabolic connections stays true. Here we analyze the infinitesimal action of $\operatorname{Symp}_0$ on ${\mathcal}{A}(h)//{\mathcal}{P}$, in particular what it does on the highest terms $\mu_k$ and $t_k$. There are two steps: a local analysis and a global analysis.
### Local analysis
We prove that the action of higher diffeomorphisms on the highest terms $\mu_k$ of the parabolic reduction is precisely the action on the $n$-complex structure. So we can trivialize it locally.
Take a change of section $\delta s = \bm\hat{v}_1s+\bm\hat{v}_2\nabla s+...+\bm\hat{v}_{n}\nabla^{n-1}s=\bm\hat{H}s$. We have previously seen in equation that the change of coordinates $\delta \bm\hat{\mu}_k$ can be computed by $$\delta \bm\hat{Q}=[\bm\hat{H}, \bm\hat{Q}] \mod \bm\hat{I}$$ where $\bm\hat{I}=\langle -\nabla^n+\bm\hat{t}_2\nabla^{n-2}+...+\bm\hat{t}_n, -\bar{\nabla}+\bm\hat{\mu}_1+\bm\hat{\mu}_2\nabla+...+\bm\hat{\mu}_n\nabla^{n-1} \rangle$ is a left-ideal in the space of differential operators.
Since we have a parameter ${\lambda}$ in our setting, the variations $\bm\hat{v}_k$ also depend on ${\lambda}$. More precisely, for $k\geq 2$ we have that $\bm\hat{v}_k({\lambda})$ is a rational function in ${\lambda}$ with highest term ${\lambda}^{2-k}v_k$ when ${\lambda}\rightarrow \infty$. Notice that $\bm\hat{v}_1$ is not a free parameter, but depends on the others. It assures that the trace of the gauge transform is zero. One can compute that $\bm\hat{v}_1$ has highest term of degree 0.
It is not clear for the moment how to determine the precise expression for $\bm\hat{v}_k({\lambda})$ from a higher diffeomorphism generated by some Hamiltonian $H$. The highest ${\lambda}$-terms in $\bm\hat{v}_k({\lambda})$ are given by the coefficients of $H=v_2p+...+v_np^{n-1}$. $\triangle$
We can now state:
\[actionsymponlambdaconn\] The infinitesimal action of $\operatorname{Symp}_0(T^*\Sigma)$ on the highest terms $\mu_k$ of the coordinates $\bm\hat{\mu}_k({\lambda})$ of the space of parabolic connections with parameter is the same as the infinitesimal action of higher diffeomorphisms on the $n$-complex structure.
The reason for the theorem to be true is roughly speaking that the Poisson bracket is the semi-classical limit of commutators of differential operators. The strategy of the proof is the following: we prove the theorem first for $\mu_2$, and then for $\mu_k$ ($k>2$) supposing $\mu_2=...=\mu_{k-1}=0$ which simplifies the computations. From [@FockThomas] proposition 3, we know that the infinitesimal action of a Hamiltonian $H=v_2p+...+v_np^{n-1}$ on the higher Beltrami differentials is given by $$\delta \mu_2 = (\bar{\partial}-\mu_2\partial+\partial\mu_2)v_2$$ for $\mu_2$ and for $\mu_k$, supposing $\mu_2=...=\mu_{k-1}=0$, we simply have $$\delta \mu_k = \bar{\partial}v_{k}.$$
First, we compute the variation of $\mu_2$ using equation : $$\delta \bm\hat{\mu}_1+\delta \bm\hat{\mu}_2\nabla+...+\delta \bm\hat{\mu}_n\nabla^{n-1}=[\bm\hat{v}_1+\bm\hat{v}_2\nabla +...+\bm\hat{v}_{n}\nabla^{n-1}, -\bar{\nabla}\!+\!\bm\hat{\mu}_1\!+\!\bm\hat{\mu}_2\nabla+...+\bm\hat{\mu}_n\nabla^{n-1}] \mod \bm\hat{I}.$$ Since the highest ${\lambda}$-term of $\bm\hat{\mu}_2$ is of degree 0, we are interested in the part of degree $0$ of the coefficient of $\nabla$ in $[\bm\hat{v}_1+\bm\hat{v}_2\nabla +...+\bm\hat{v}_{n}\nabla^{n-1}, -\bar{\nabla}+\bm\hat{\mu}_1+\bm\hat{\mu}_2\nabla+...+\bm\hat{\mu}_n\nabla^{n-1}] \mod \bm\hat{I}$.
We first look on contributions coming from $[\bm\hat{v}_k\nabla^{k-1}, \bm\hat{\mu}_l\nabla^{l-1}]$ for $k, l\geq 2$: If $k+l-3<n$ then we do not reduce modulo $\bm\hat{I}$, so the highest term in ${\lambda}$ is of degree $4-(k+l)$. Since we have $k, l\geq 2$, the highest term comes from $k=l=2$, which gives $v_2{\partial}\mu_2-\mu_2{\partial}v_2$. If $k+l-3\geq n$, we can have terms with $\nabla^m$ with $n\leq m \leq k+l-3$. So we have to use $\bm\hat{I}$ to reduce it. This reduction gives $\nabla^m = c({\lambda})\nabla+\text{ other terms}$, and the highest term of $c({\lambda})$ is of degree $m-2 \leq k+l-5$. Hence, the highest term for $[\bm\hat{v}_k\nabla^{k-1}, \bm\hat{\mu}_l\nabla^{l-1}]$ is $4-(k+l)+k+l-5=-1$.
The contributions from $\bm\hat{\mu}_1$ and $\bm\hat{v}_1$ also have degree at most -1. There is one more contribution in degree 0 coming from $[\bm\hat{v}_2\nabla, -\bar{\nabla}]$, which gives ${\bar\partial}v_2$. Therefore, we have $$\delta \mu_2 = (\bar{\partial}-\mu_2\partial+\partial\mu_2)v_2.$$
Now, suppose $\mu_2=...=\mu_{k-1}=0$ and compute the variation $\delta \mu_k$ under an action generated by $\bm\hat{v}_k\nabla^{k-1}+...+\bm\hat{v}_n\nabla^{n-1}$. From $$\delta \bm\hat{\mu}_k\nabla^{k-1}+...+\delta \bm\hat{\mu}_n\nabla^{n-1}=[\bm\hat{v}_k\nabla^{k-1} +...+\bm\hat{v}_{n}\nabla^{n-1}, -\bar{\nabla}+\bm\hat{\mu}_1+\bm\hat{\mu}_2\nabla+...+\bm\hat{\mu}_n\nabla^{n-1}] \mod \bm\hat{I}$$ we can analyze as above the contribution to the term of degree $2-k$ of the coefficient of $\nabla^{k-1}$. Since $\bm\hat{v}_l$ is of degree at most $2-l$ and $\bm\hat{\mu}_l$ of degree at most $1-l$ for $l<k$ (since we suppose that $\mu_l=0$), we can see that $[\bm\hat{v}_l\nabla^{l-1}, \bm\hat{\mu}_m\nabla^{m-1}]$ cannot contribute to the highest degree. The only contribution comes from the term with $-\bar{\nabla}$. Thus, $$\delta \mu_k = \bar{\partial}v_k.$$ This concludes the proof since the action of higher diffeomorphisms on the $n$-complex structure has the same expression.
Under the action of higher diffeomorphisms, we can locally render $\Phi_2=0$.
The corollary directly follows from the previous theorem and the fact that the higher complex structure can be locally trivialized (theorem \[loctriii\]), i.e. we can render $\mu_2=...=\mu_n=0$ locally and since $\Phi_2=\mu_2\Phi_1+...+\mu_n\Phi_1^{n-1}$ this implies $\Phi_2=0$.
We see that a term $\bm\hat{v}_k\nabla^{k-1}$ can influence $\bm\hat{\mu}_i$ with $i<k$ (unlike the case higher complex structures where $H$ acts like $H \mod I$), but it does not influence the highest term $\mu_i$. In the same vein, a term $\bm\hat{v}_k\nabla^{k-1}$ with $k>n$ acts on parabolic connections, but not on the highest terms. $\triangle$
### Global analysis
We show that the highest term in ${\lambda}$ in the zero-curvature condition relates $(\mu_k, t_k)$ to the cotangent bundle $T^*\bm\hat{{\mathcal}{T}}^n$.
We know that the moment map of the hamiltonian action of $\operatorname{Symp}_0(T^*\Sigma)$ on ${\mathcal}{A}//{\mathcal}{P}$ is given by $\xi_k=0$, i.e. the remaining curvature of a parabolic connection has to vanish. For connections with parameter ${\lambda}$, this gives $\xi_k({\lambda})=0$.
\[conditioncinconnection\] The highest term in ${\lambda}$ of $\xi_k({\lambda})=0$ gives the condition $({\mathcal}{C})$ of the cotangent bundle $T^*\bm\hat{{\mathcal}{T}}^n$ (see theorem \[conditionC\]).
The proof strategy is to reduce the analysis of the highest term in the parabolic curvature to the expression $\xi_k \mod \bm\hat{t}^2 \mod \partial^2$. The following lemma shows that we then get condition $({\mathcal}{C})$.
\[curvaturemodmod\] The parabolic curvature modulo $\bm\hat{t}^2$ and ${\partial}^2$ gives condition $(\mathcal{C})$ on $T^*\bm\hat{\mathcal{T}}^n$: $$\xi_k = (\bar{\partial}\!-\!\bm\hat{\mu}_2\partial\!-\!k\partial\bm\hat{\mu}_k)\bm\hat{t}_k-\sum_{l=1}^{n-k}\left((l\!+\!k)\partial\bm\hat{\mu}_{l+2}+(l\!+\!1)\bm\hat{\mu}_{l+2}{\partial}\right)\bm\hat{t}_{k+l} \mod \bm\hat{t}^2 \mod \partial^2.$$
You find the proof of this technical lemma in appendix \[appendix:C\]. Using the lemma, we can prove theorem \[conditioncinconnection\]:
From the explicit expression of $\xi_k({\lambda})$, we know that only derivatives, $\bm\hat{t}_k$’s and $\bm\hat{\mu}_k$’s appear. Since we are only interested in the highest term, we can replace $\bm\hat{t}_k$ by ${\lambda}^{k-1}t_k$ and $\bm\hat{\mu}_k$ by ${\lambda}^{2-k}\mu_k$. Hence, we get an expression which is a tensor, since both $t_k$ and $\mu_k$ are tensors (by proposition \[highesttermtensor\]). Since one term is $\bar{\partial}t_k$, we know that the highest term of $\xi_k({\lambda})$ is a section of $K^k\otimes \bar{K}$ and is of degree $k-1$ in ${\lambda}$.
In addition, we know that every term in $\xi_k$, apart from $\bar{\partial}t_k$, has at least one partial derivative $\partial$, which adds a $K$-factor to the tensor. The rest is thus at most of type $K^{k-1}\otimes \bar{K}$. The $\bar{K}$-factor comes from a unique $\mu_m$ in each term. Once this $\mu_m$ fixed, only partial derivatives $\partial$ and $t_k$’s contribute to the $K$-factor.
Since $t_k$ comes with a factor ${\lambda}^{k-1}$, we see that whenever there is a term with a factor $t_it_j$, the contribution in ${\lambda}$ is ${\lambda}^{i+k-2}$ which is not optimal, since $t_{i+j}$ would contribute with ${\lambda}^{i+j-1}$. In the same vein, whenever there is a term with at least two $\partial$, so that the rest is a tensor of type at most $K^{k-2}\otimes \bar{K}$, this term does not have an optimal contribution in ${\lambda}$.
Therefore, the highest term in $\xi_k({\lambda})$ is the same as in $\xi_k({\lambda}) \mod \bm\hat{t}^2 \mod \partial^2$. Finally, the statement of the previous lemma \[curvaturemodmod\] concludes the proof of theorem \[conditioncinconnection\].
With the previous theorem, we now understand the global meaning of the highest terms $(\mu_k, t_k)$: the $\mu_k$ are the higher Beltrami differentials coming from the higher complex structure, whereas the $t_k$ are a cotangent vector to that higher complex structure. We can say that the *semi-classical limit of ${\mathcal}{A}//{\mathcal}{P}//\operatorname{Symp}_0$ is ${T^*\bm\hat{{\mathcal}{T}}^n}$*, which confirms the twistor space picture \[HK\].
We have seen in proposition \[parametrisationlambda\] that $t_k = \operatorname{tr}\Phi_1^{k-1}A_1$. The previous theorem applied for trivial $n$-complex structure $\mu_k=0 \;\forall k$ gives ${\bar\partial}t_k = 0$. It can be checked directly that ${\bar\partial}\operatorname{tr}\Phi_1^{k-1}A_1 = 0$ using the flatness of ${\mathcal}{A}({\lambda})$. $\triangle$
The question remains how to determine the coefficients of lower degree in $\bm\hat{\mu}_k$ and $\bm\hat{t}_k$. This will be discussed in \[finalstep\] below. Before, we push the similarity to Higgs bundles further by choosing a special gauge.
Higgs gauge {#Higgsgauge}
-----------
Up to now, we have seen the flat connection ${\mathcal}{A}({\lambda})$ in two gauges. The first, which we call *symmetric gauge*, is the form ${\mathcal}{A}({\lambda})={\lambda}\Phi+A+{\lambda}^{-1}\Phi^*$ where $A_2=-A_1^*$ and the $*$-operator is the hermitian conjugate. The second, which we call *parabolic gauge* and which in the literature is sometimes called *$W$-gauge* or *Drinfeld-Sokolov gauge*, is the form described in equation where our parameters $\tilde{t}_k({\lambda})$ and $\tilde{\mu}_k({\lambda})$ appear. The existence of parabolic gauge (see subsection \[existence-para-gauge\]) assures that one can go from the symmetric to the parabolic gauge. In Higgs theory, there is a third gauge used, which we call *Higgs gauge*, characterized by $A_2=0$ and by the fact that $\Phi_1$ is a companion matrix. Here we show that for trivial higher complex structure, there exists the Higgs gauge in our setting. We start with the existence of the Higgs gauge for trivial higher complex structure. We denote by ${\mathcal}{E}_-$ the sum of the negative simple roots, i.e. ${\mathcal}{E}_-=\left(\begin{smallmatrix} 0&&& \\ 1 &0&&\\ &\ddots &\ddots& \\ &&1& 0\end{smallmatrix}\right)$.
For $\mu=0$ and a flat connection ${\lambda}\Phi+A+{\lambda}^{-1}\Phi^*$ in symmetric gauge, there is a gauge $P$ which is lower triangular transforming $\Phi_1$ to ${\mathcal}{E}_-$ and $A_2$ to 0.
The statement is equivalent to the following two equations: $$P\Phi_1={\mathcal}{E}_-P \;\text{ and }\; PA_2-{\bar\partial}P = 0.$$ The first matrix equation allows to express all entries $p_{i,j}$ of $P$ in terms of the last row $(p_{n,k})_{1\leq k \leq n}$.
We then put $\Phi_1 = P^{-1}{\mathcal}{E}_-P$ into the flatness equation $0={\bar\partial}\Phi_1+[A_2, \Phi_1]$. After some manipulation, we get $$0=[{\mathcal}{E}_-, ({\bar\partial}P)P^{-1}-PA_2P^{-1}].$$ We know that the centralizer of ${\mathcal}{E}_-$ are polynomials in ${\mathcal}{E}_-$. Hence we get $${\bar\partial}P-PA_2=\begin{pmatrix} 0&&&\\ w_2&0&&\\ \vdots&\ddots&\ddots&\\ w_n &\cdots &w_2 &0 \end{pmatrix}P.$$ Looking at the $n$ equations given by the last row, we can choose $(p_{n,k})_{1\leq k \leq n}$ such that $w_2=...=w_n=0$. Therefore ${\bar\partial}P =PA_2$, i.e. $A_2$ is transformed to 0.
In the Higgs gauge, our flat connection takes the following form:
We suppose $\mu=0$. The flat connection ${\mathcal}{A}({\lambda})$ in Higgs gauge is locally given by $$({\lambda}{\mathcal}{E}_-+A)dz+{\lambda}^{-1}{\mathcal}{E}_-^*d\bar{z}$$ where the $*$-operation is given by $M^*=HM^{\dagger}H^{-1}$ for some hermitian matrix $H$. Further, we have $\operatorname{tr}{\mathcal}{E}_-^kA = t_{k+1}$ and $A=-({\partial}H) H^{-1}$.
From the existence of Higgs gauge, we know that $\Phi_1={\mathcal}{E}_-$ and $A_2=0$. Since $\mu=0$, we also have $\Phi_2=0$. A direct computation shows that if $P$ denotes the matrix from the Higgs gauge, the matrix $\Phi_1^*$ transforms to $PP^{\dagger}{\mathcal}{E}_-^{\dagger}(PP^{\dagger})^{-1}$. So $H=PP^{\dagger}$ which is indeed a hermitian matrix.
Since $P$ is lower triangular, $t_{k+1}=\operatorname{tr}\Phi_1^kA_1$ transforms to $t_{k+1}=\operatorname{tr}{\mathcal}{E}_-^kA$.
Finally, since $A_2=P^{-1}{\bar\partial}P$ and $A_2=-A_1^{\dagger}$, we get $A_1=-({\partial}P^{\dagger})P^{\dagger \;-1}$ which transforms under $P$ to $A=-{\partial}(PP^{\dagger})(PP^{\dagger})^{-1}=-({\partial}H)H^{-1}$.
We see that ${\mathcal}{A}({\lambda})$ in the Higgs gauge becomes close to a Higgs bundle. But in our setting the *holomorphic differentials are in $A$, and not in the Higgs field*. We illustrate the similarity for $n=2$.
For $n=2$ and $\mu=0$, we will see in subsection \[n2n3\] that in symmetric gauge, our connection reads $${\mathcal}{A}({\lambda})=\begin{pmatrix} -\frac{{\partial}\varphi}{2} & t_2e^{-\varphi} \\ {\lambda}e^\varphi & \frac{{\partial}\varphi}{2} \end{pmatrix}dz+\begin{pmatrix} \frac{{\bar\partial}\varphi}{2} & {\lambda}^{-1} e^{\varphi} \\ -\bar{t}_2 e^{-\varphi} & -\frac{{\bar\partial}\varphi}{2} \end{pmatrix}d\bar{z}.$$ The flatness condition is equivalent to the $\cosh$-Gordon equation ${\partial}{\bar\partial}\varphi = e^{2\varphi}+t_2\bar{t}_2e^{-2\varphi}$.
A direct computation gives the form in parabolic gauge: $${\mathcal}{A}({\lambda})=\begin{pmatrix} 0 & \bm\hat{t}_2({\lambda}) \\ 1 & 0 \end{pmatrix}dz+\begin{pmatrix} -\frac{1}{2}{\partial}\bm\hat{\mu}_2 & -\frac{1}{2}{\partial}^2 \bm\hat{\mu}_2+\bm\hat{t}_2\bm\hat{\mu}_2 \\ \bm\hat{\mu}_2({\lambda}) & \frac{1}{2}{\partial}\bm\hat{\mu}_2 \end{pmatrix}d\bar{z}$$ where $\bm\hat{t}_2({\lambda})={\lambda}t_2 + ({\partial}\varphi)^2-{\partial}^2\varphi$ and $\bm\hat{\mu}_2({\lambda})=-{\lambda}^{-1}\bar{t}_2e^{-2\varphi}$.
In Higgs gauge, we get $${\mathcal}{A}({\lambda})=\begin{pmatrix} -{\partial}\varphi-t_2p_2e^{-\varphi/2} & t_2 \\ {\lambda}-a_1 & {\partial}\varphi+t_2p_2e^{-\varphi/2}\end{pmatrix}dz+\begin{pmatrix} -{\lambda}^{-1}p_2e^{3\varphi/2} & {\lambda}^{-1} e^{2\varphi} \\ -{\lambda}^{-1} p_2^2 e^{\varphi} & {\lambda}^{-1}p_2e^{3\varphi/2} \end{pmatrix}d\bar{z}$$ where $a_1=({\partial}p_2+\frac{3}{2}p_2{\partial}\varphi+t_2p_2^2e^{-\varphi/2}) e^{-\varphi/2}$ and $p_2$ comes from the matrix of the Higgs gauge and satisfies ${\bar\partial}p_2=-\bar{t}_2e^{-3\varphi/2}+p_2\frac{{\bar\partial}\varphi}{2}$.
Finally, we can compare to the non-abelian Hodge correspondence which gives $${\mathcal}{A}({\lambda})=\begin{pmatrix} -{\partial}\varphi & 0 \\ {\lambda}& {\partial}\varphi\end{pmatrix}dz+\begin{pmatrix} 0 & {\lambda}^{-1} e^{2\varphi} \\ 0 & 0 \end{pmatrix}d\bar{z}.$$ The flatness condition is equivalent to Liouville’s equation ${\partial}{\bar\partial}\varphi=e^{2\varphi}$. Notice that we get this connection in our setting in the Higgs gauge for $t_2=0$ (then $p_2=0$). $\triangle$
For non-trivial $n$-complex structure $\mu\neq 0$, there is no Higgs gauge. Even for $n=2$, one can check that there is no $P$ satisfying $P\Phi_1={\mathcal}{E}_-P$ and $PA_2-{\bar\partial}P = 0$.
Conjectural geometric approach to Hitchin components {#finalstep}
====================================================
In this section, we try to construct an analog to the non-abelian Hodge correspondence in our setting: the existence and uniqueness of real twistor lines. We give partial results and conjectures. Assuming the existence of real twistor lines, we prove a canonical diffeomorphism between higher complex structures and Hitchin components.
Consider ${\mathcal}{A}({\lambda})={\lambda}\Phi + A + {\lambda}^{-1}\Phi^*$ where $\Phi=\Phi_1+\Phi_2$ is given by an $n$-complex structure. Now, we look at ${\mathcal}{A}({\lambda})$ as a twistor line, i.e. a section of the twistor space. We impose the reality condition $$-{\mathcal}{A}(-1/\bar{{\lambda}})^*={\mathcal}{A}({\lambda}).$$ Notice that $-1/\bar{{\lambda}}$ is the diametrically opposed point of ${\lambda}$ in ${\mathbb{C}}P^1$. For trivial $n$-complex structure the $*$-operator is the hermitian conjugate $M^*=M^\dagger=\bar{M}^\top$. Intrinsically, the operation $A \mapsto -A^*$ is an antiholomorphic involution which corresponds to the compact real form of ${\mathfrak}{sl}_n$.
For general higher complex structure, the real structure $*$ has to be defined in such a way that $\operatorname{Symp}_0$ preserves it. Notice also that we need a hermitian structure on the bundle. $\triangle$
Standard form
-------------
We start with ${\mathcal}{A}({\lambda})={\lambda}\Phi + A + {\lambda}^{-1}\Phi^*$ as above and reduce it to a standard form.
\[phi1lower\] There is a unitary gauge such that $\Phi_1$ becomes lower triangular with entries of coordinates $(i+1,i)$ given by positive real numbers of the form $e^{\varphi_i}$ for all $i=1,...,n-1$.
The gauge acts by conjugation on $\Phi_1(z)$. Since $\Phi_1(z)$ is nilpotent, for every $z\in {\Sigma}$, there is an invertible matrix $G(z)\in \operatorname{GL}_n({\mathbb{C}})$ such that $G\Phi_1G^{-1}$ is strictly lower triangular. Since $\Phi(z)$ varies smoothly with $z$, so does $G(z)$. We omit the dependence in $z$ in the sequel of the proof.
We decompose $G$ as $G=TU$ where $T$ is lower triangular (not strict) and $U$ is unitary (Gram-Schmidt). Then the matrix $U\Phi_1U^{-1} = T^{-1}(G\Phi_1G^{-1})T$ is already lower triangular. So we have conjugated $\Phi_1$ to a lower triangular matrix via a unitary gauge.
Finally, we use a diagonal unitary gauge to change the arguments of the matrix elements with coordinates $(i+1,i)$ to zero. Since $\Phi_1$ is principal nilpotent, all these elements are non-zero, so strictly positive real numbers which can be written as $e^{\varphi_i}$ with $\varphi_i \in {\mathbb{R}}$.
Notice that the unitary gauge preserves the operation $*$, so the form ${\lambda}\Phi+A+{\lambda}^{-1}\Phi^*$ is preserved. Now, we show that for $\mu=0$, the matrix $A_1$ is upper triangular. Notice the importance of $\Phi_1$ being principal nilpotent.
\[a1upper\] For $\Phi_2=0$ (trivial higher complex structure) and $\Phi_1$ lower triangular, the flatness of ${\mathcal}{A}({\lambda})$ implies that $A_1$ is upper triangular.
We write $A_1=A_l + A_u$ where $A_l$ and $A_u$ are respectively the strictly lower and the (not strictly) upper part of $A_1$. Thus we have $A_2 = -A_l^{*}-A_u^{*}$.
The flatness condition at the term ${\lambda}$ gives $$0=\bar{\partial}\Phi_1 + [\Phi_1,A_u^{*}]+[\Phi_1,A_l^{*}].$$ Since the first two terms are lower triangular (the operation $*$ exchanges upper and lower triangular matrices), so is the third term $[\Phi_1,A_l^{*}]$.
A simple computation shows that a commutator between a principal nilpotent lower triangular matrix and a non-zero strictly upper triangular matrix can never be strictly lower triangular. Thus, $A_l=0$.
Case n=2 and n=3 {#n2n3}
----------------
Let us study the examples of smallest rank, those with $n=2$ and $n=3$. We work locally, so we can suppose that the $n$-complex structure is trivial, i.e. $\mu_k=0$ for $k=2, 3$. We use the standard form from subsection \[standard-form\].
For $n=2$, write $\Phi_1 = \left(\begin{smallmatrix} 0 & 0 \\ e^{\varphi} & 0\end{smallmatrix}\right)$, $A_1 = \left(\begin{smallmatrix} a_0 & a_1 \\ a_2 & -a_0\end{smallmatrix}\right)$ and $A_2 = -A_1^{\dagger}$. So we have $${\mathcal}{A}({\lambda})=\begin{pmatrix} a_0 & a_1 \\ a_2+{\lambda}e^{\varphi} & -a_0\end{pmatrix}dz+ \begin{pmatrix} -\bar{a}_0 & -\bar{a}_2+{\lambda}^{-1}e^{\varphi} \\ -\bar{a}_1 & \bar{a}_0\end{pmatrix}d\bar{z}.$$ Notice that this is example \[examplen2\] with $\mu_2=0$ and $b_1=e^{\varphi}$. The flatness equation gives $$\left \{\begin{array}{cl}
a_2 e^{\varphi} &= \; 0 \\
\bar{\partial}\varphi &= \; -2\bar{a}_0 \\
\bar{\partial}a_1 &=\; 2\bar{a}_0a_1 \\
\partial \bar{a}_0+\bar{\partial}a_0 &= \; -a_1\bar{a}_1-e^{2\varphi}.
\end{array}\right.$$ The first equation gives $a_2=0$, the second $a_0=-\frac{{\partial}\varphi}{2}$, the third is automatic once we write $a_1=t_2e^{-\varphi}$, where $t_2=\operatorname{tr}\Phi_1A_1$ is the holomorphic quadratic differential. Finally, the last equation gives $$\partial\bar{\partial} \varphi = e^{2\varphi} + t_2\bar{t}_2e^{-2\varphi}$$ which is the so-called $\mathbf{\cosh}$**-Gordon equation**, which is elliptic for small $t_2$. So we see that the flat connection is uniquely determined by $\mu_2=0, t_2$ and a solution to the $\cosh$-Gordon equation. More details for this case can be found in [@Fock], in particular a link to minimal surface sections in ${\Sigma}\times {\mathbb{R}}$.
For $n=3$, take $\Phi_1 = \left(\begin{smallmatrix} & & \\ c_1 & & \\ b_2 & c_2 &\end{smallmatrix}\right)$. As for $n=2$ the matrix $A_1$ is upper triangular. Thus, we get $${\mathcal}{A}({\lambda})=\begin{pmatrix} a_0 & b_0 & c_0 \\ {\lambda}c_1 & a_1 & b_1 \\ {\lambda}b_2 & {\lambda}c_2 & a_2 \end{pmatrix}dz+ \begin{pmatrix} -\bar{a}_0 & {\lambda}^{-1}\bar{c}_1 & {\lambda}^{-1}\bar{b}_2 \\ -\bar{b}_0 & -\bar{a}_1 & {\lambda}^{-1}\bar{c}_2 \\ -\bar{c}_0 & -\bar{b}_1 & -\bar{a}_2\end{pmatrix}d\bar{z}.$$ With a diagonal gauge, we can suppose $c_1=e^{\varphi_1}, c_2=e^{\varphi_2} \in {\mathbb{R}}_+$. Further, the expressions for the holomorphic differentials are $t_3=\operatorname{tr}\Phi_1^2A_1 = c_0c_1c_2$ and $t_2= \operatorname{tr}\Phi_1A_1 = b_0c_1+b_1c_2+b_2c_0$, hence $c_0=t_3e^{-\varphi_1-\varphi_2}$ and $b_1=-e^{\varphi_1-\varphi_2}b_0-b_2t_3e^{-2\varphi_2-\varphi_1}$.
The flatness condition and the zero trace condition then give $a_0=-\frac{2}{3}\partial\varphi_1-\frac{1}{3}\partial\varphi_2$, $a_1=\frac{1}{3}\partial\varphi_1-\frac{1}{3}\partial\varphi_2$ and $a_2=-a_0-a_1$.
Let us consider the case where $t_2=t_3=0$. Then $c_0=0$ and $b_1=-e^{\varphi_1-\varphi_2}b_0$. The remaining equations of the flatness are $$\left \{\begin{array}{cl}
\bar{\partial}b_2 &= \; b_2(\bar{\partial}\varphi_1+\bar{\partial}\varphi_2)-\bar{b}_0(e^{\varphi_2}+e^{2\varphi_1-\varphi_2}) \\
-\bar{\partial}b_0 &= \; b_0\bar{\partial}\varphi_1+\bar{b}_2e^{\varphi_2} \\
2\partial\bar{\partial}\varphi_1 &= \; 2e^{2\varphi_1}-e^{2\varphi_2}+b_2\bar{b}_2+b_0\bar{b}_0(2-e^{2\varphi_1-2\varphi_2}) \\
2\partial\bar{\partial}\varphi_2 &= \; 2e^{2\varphi_2}-e^{2\varphi_1}+b_2\bar{b}_2+b_0\bar{b}_0(-1+2e^{2\varphi_1-2\varphi_2}).
\end{array}\right.$$
For $b_0=b_2=0$ we get the **Toda integrable system** for ${\mathfrak}{sl}_3$. This is the same solution as the one obtained from the non-abelian Hodge correspondence applied to the principal nilpotent Higgs field. We see that we need some extra data in order to impose $b_0=b_2=0$. The two variables $b_0$ and $b_2$ are solutions to a system of differential equations. Thus, we only need some initial conditions.
For $t_2=0$ and $t_3\neq 0$, if we impose $b_0=b_1=b_2=0$ and $\varphi_1=\varphi_2=\varphi$, the flatness becomes **Ţiţeica’s equation** $$\label{Titeica}
2{\partial}{\bar\partial}\varphi = e^{2\varphi}+t_3\bar{t}_3e^{-4\varphi}.$$ From [@Loftin], we know that Ţiţeica’s equation is linked to affine spheres, minimal embeddings and Hitchin representations.
General case {#flatconnectionlambda}
------------
Set $t=(t_2,...,t_n)$ and $\mu=(\mu_2,...,\mu_n)$. We discuss the cases when $t=0$ or $\mu=0$.
***Case $t=0$ and $\mu=0$.*** For the trivial structure we find the following result, generalizing the observations for $n=2$ and $n=3$ from the previous subsection \[n2n3\].
\[linktohiggs\] For $\Phi_2=0$ and $t=0$, the flat connection ${\mathcal}{A}({\lambda})$ is uniquely determined up to some finite initial data. There is a choice of initial data such that the flatness equations are equivalent to the Toda integrable system. In particular ${\mathcal}{A}({\lambda})$ is the same as the connection given by the non-abelian Hodge correspondence applied to a principal nilpotent Higgs field.
Using lemmas \[phi1lower\] and \[a1upper\], we can write ${\mathcal}{A}_1({\lambda})$ in the following form: $${\mathcal}{A}_1({\lambda})=a_0+a_1T+...+a_nT^n$$ where $a_i$ are diagonal matrices and $T$ is given by $$\label{matrixT}
T=\begin{pmatrix} & 1 & & \\ &&\ddots & \\ &&& 1 \\ {\lambda}&&& \end{pmatrix}.$$ We denote by $a_{i,j}$ the $j$-th entry of the diagonal matrix $a_i$ and $a_i'$ the shifted matrix with $a'_{i,j} = a_{i,j+1}$. We write $a^{(k)}$ for the shift applied $k$ times. Notice that $aT=Ta^{(n-1)}$. We can then write $${\mathcal}{A}_2({\lambda})= a_0^*+T^{-1}a_1^*+...+T^{-n}a_n^*$$ where $a^*_{i,j}=\pm \bar{a}_{i,j}$, the sign depends on whether the coefficient comes with a ${\lambda}$ or not in ${\mathcal}{A}_2({\lambda})$.
By the standard form (lemma \[phi1lower\]) we can further impose $a_{n,i}=e^{\varphi_i}$ for $i=1,...,n-1$ and $a_{n,0}=0$ since $0=t_n=\prod_i a_{n,i}$. One of the flatness equations gives ${\bar\partial}a_n = a_n (a_0^{(n-1)}-a_0)$. Together with the condition that the trace is 0, we can compute $a_0$. We get $$\label{a0i}
a_{0,i}= \sum_{k=1}^{i-1}\frac{k}{n}{\partial}\varphi_k-\sum_{k=i}^{n-1}\frac{n-k}{k}{\partial}\varphi_k.$$ The other equations give a system of differential equations in $a_1, ..., a_{n-1}$ which is quadratic. It allows the solution $a_i=0$ for all $i=1,...,n-1$. In that case, using a diagonal gauge $\operatorname{diag}(1,\lambda, ..., \lambda^{n-1})$ the connection ${\mathcal}{A}({\lambda})$ becomes $$\label{mu0t0}
{\mathcal}{A}({\lambda})=\begin{pmatrix} *&&& \\ e^{\varphi_1} &* && \\ & \ddots &* & \\ && e^{\varphi_{n-1}} & * \end{pmatrix}dz+\begin{pmatrix}* & e^{\varphi_1} && \\ &* & \ddots & \\ &&* & e^{\varphi_{n-1}} \\ &&&* \end{pmatrix}d\bar{z}$$ where on the diagonals are the $a_{0,i}$ and $-\bar{a}_{0,i}$ given by equation . This is precisely the form of the Toda system. It is known that the Hitchin equations for a principal nilpotent Higgs field are the Toda equations for ${\mathfrak}{sl}_n$ (see [@AF], proposition 3.1).
Notice that in particular the gauge class of the connection ${\mathcal}{A}({\lambda})$ is independent of ${\lambda}\in {\mathbb{C}}^*$ (i.e. we have a variation of Hodge structure). This is an intrinsic property which might be used to fix the initial data.
Putting in parabolic gauge, we get the following explicit formula for our coordinates $\tilde{t}({\lambda})$ and $\tilde{\mu}({\lambda})$ (see also proposition 3.1 and 4.4 in [@AF]):
For $\mu=0$ and $t=0$, one can choose initial conditions such that $\tilde{\mu}_k({\lambda})=0$ and $\tilde{t}_k({\lambda})=w_k$ for all $k$, where the $w_k$ are given by $\det ({\partial}-A_1)=\prod_i({\partial}-a_{0,i}) = {\partial}^n+w_2{\partial}^{n-2}+...+w_n$ (a “Miura transform”). Furthermore, $A_1$ is diagonal given by equation and the parabolic gauge is upper triangular.
***Case $t=0$.*** We get the following result:
\[monodromyreal\] For $t=0$, the connection ${\mathcal}{A}({\lambda})$ is determined by the flatness condition and by some initial conditions. Its monodromy is in $\operatorname{PSL}_n({\mathbb{R}})$.
The idea of the proof is the following: locally, one can trivialize the higher complex structure, so we are led to $\mu=0$ and $t=0$. Thus ${\mathcal}{A}({\lambda})$ is given by the non-abelian Hodge correspondence and we can apply Hitchin’s strategy to prove real monodromy, which is a local argument.
By theorem \[actionsymponlambdaconn\], we know that we can locally render $\Phi_2=0$ by trivializing the $n$-complex structure. Thus we can choose the initial conditions such that ${\mathcal}{A}({\lambda})$ is given by the non-abelian Hodge correspondence applied to the nilpotent Higgs field $\Phi_1$ (see proposition \[linktohiggs\]).
In [@Hit.1], Hitchin constructs a real form $\tau$, associated with the split real form, which for ${\mathfrak}{sl}_n$ is given by a rotation of the matrix by $180$ degrees composed with complex conjugation. He shows that $\tau^*\Phi_1=\Phi_1^*$ and that $\tau^*{\mathcal}{A}({\lambda})={\mathcal}{A}({\lambda})$. This is a local statement, therefore the monodromy of ${\mathcal}{A}({\lambda})$ has to be in the fixed point set of $\tau$, so in $\operatorname{PSL}_n({\mathbb{R}})$.
***Case $\mu=0$.*** For trivial $n$-complex structure, the standard form from lemmas \[phi1lower\] and \[a1upper\] allow to consider ${\mathcal}{A}({\lambda})$ as an affine connection with special properties. We denote by ${\mathcal}{L}({\mathfrak}{sl}_n)$ the loop algebra of ${\mathfrak}{sl}_n$. It is defined by ${\mathcal}{L}({\mathfrak}{sl}_n) = {\mathfrak}{sl}_n \otimes {\mathbb{C}}[\lambda,\lambda^{-1}]$, the space of Laurent polynomials with matrix coefficients. There is another way to think of elements of ${\mathcal}{L}({\mathfrak}{sl}_n)$: as an infinite periodic matrix $(M_{i,j})_{i,j \in \mathbb{Z}}$ with $M_{i,j}=M_{i+n,j+n}$ and finite width (i.e. $M_{i,j}=0$ for all $\left| i+j \right|$ big enough). The isomorphism is given as follows: to $\sum_{i=-N}^N N_i{\lambda}^i$ we associate $M_{i,j}=(N_{k_j-k_i})_{r_i,r_j}$ where $i=k_in+r_i$ and $j=k_jn+r_j$ are the Euclidean divisions of $i$ and $j$ by $n$ (so $0\leq r_i,r_j <n$), see also figure \[affine-matrix\].
(0,0)–(3,0); (0,1)–(3,1); (0,2)–(3,2); (0,3)–(3,3); (0,0)–(0,3); (1,0)–(1,3); (2,0)–(2,3); (3,0)–(3,3);
(-0.3,1.5) node [$\hdots$]{}; (3.3,1.5) node [$\hdots$]{}; (1.5,-0.3) node [$\vdots$]{}; (1.5,3.3) node [$\vdots$]{}; (-0.3,3.3) node [$\ddots$]{}; (3.3,-0.3) node [$\ddots$]{};
(-0.3,2.3)–(2.3,-0.3); (0.7,3.3)–(3.3,0.7);
(0.5,1.5) circle (0.3); (1.5,0.5) circle (0.3); (1.5,2.5) circle (0.3); (2.5,1.5) circle (0.3);
(1.5,1.5) node [$N_0$]{}; (2.5,0.5) node [$N_0$]{}; (0.5,2.5) node [$N_0$]{}; (1.5,2.5) node [$N_1$]{}; (2.5,1.5) node [$N_1$]{}; (2.5,2.5) node [$N_2$]{}; (0.5,1.5) node [$N_{-1}$]{}; (1.5,0.5) node [$N_{-1}$]{}; (0.5,0.5) node [$N_{-2}$]{};
plot ([0.2+3.5\*cos()]{},[1.4+5.5\*sin()]{}); plot ([2.7+3.5\*cos()]{},[1.4+5.5\*sin()]{});
In the second viewpoint, a connection ${\lambda}\Phi+A+{\lambda}^{-1}\Phi^*$ with $\Phi_1$ lower triangular, $\Phi_2=0$ and thus $A_1$ upper triangular, is precisely an *infinite matrix with period $n$ and width $n$* (shown in figure \[affine-matrix\] by dashed lines). The $(1,0)$-part ${\mathcal}{A}_1({\lambda})$ is upper triangular ($\Phi_1$ is lower triangular but ${\lambda}\Phi_1$ is upper triangular in the infinite matrix) and the $(0,1)$-part ${\mathcal}{A}_2({\lambda})$ is lower triangular.
Thus, the flatness of ${\mathcal}{A}({\lambda})$ is a *generalized Toda system*, replacing the tridiagonal property by “width equal to periodicity”. For $t_i=0$ for $i=2,...,n-1$ but $t_n\neq 0$, we should get the usual affine Toda system for ${\mathcal}{L}({\mathfrak}{sl}_n)$.
In order to describe $h$-connections, we can include parameters into ${\mathcal}{L}({\mathfrak}{sl}_n)$ by considering its central extension $\widehat{{\mathfrak}{sl}}_n$ or central coextension. $\triangle$
Since for $t=0$ we get an elliptic system, the system stays elliptic for at least small $t\neq 0$, since ellipticity is an open condition (Cauchy-Kowalewskaya theorem). So the generalized Toda system can be solved for small $t$.
The study of this generalized Toda system is subject of future research.
***General case.*** For $\mu\neq 0$ and $t\neq 0$, the system is still elliptic at least for small $t$, since it is for $t=0$. We should get a generalized Toda system with differentials $t_k$ satisfying the higher holomorphicity condition $({\mathcal}{C})$.
We conjecture that the connection ${\mathcal}{A}({\lambda})={\lambda}\Phi+A+{\lambda}^{-1}\Phi^*$ is uniquely determined by $\mu$ and $t$. To be more precise:
\[nahc\] Given an element $[(\mu_k, t_k)]\in T^*\bm\hat{{\mathcal}{T}}^n$ and some finite extra data (initial conditions to differential equations), there is a unique (up to unitary gauge) flat connection ${\mathcal}{A}({\lambda})={\lambda}\Phi+A+{\lambda}^{-1}\Phi^*$ satisfying
1. Locally, $\Phi=\Phi_1 dz+\Phi_2 d\bar{z}$ with $\Phi_1$ principal nilpotent and $\Phi_2=\mu_2\Phi_1+...+\mu_n\Phi_1^{n-1}$
2. Reality condition: $-{\mathcal}{A}(-1/\bar{{\lambda}})^*={\mathcal}{A}({\lambda})$
3. $t_k = \operatorname{tr}\Phi_1^{k-1}A_1$.
In addition, if $t_k=0$ for all $k$, then the monodromy of ${\mathcal}{A}({\lambda})$ is in $\operatorname{PSL}_n({\mathbb{R}})$.
Assuming this conjecture, we get the desired link to Hitchin’s component:
\[mainthmm\] If conjecture \[nahc\] holds true, there is a canonical diffeomorphism between our moduli space ${\bm\hat{{\mathcal}{T}}}^n$ and Hitchin’s component ${\mathcal}{T}^n$.
With conjecture \[nahc\] we get a canonical way to associate a flat connection ${\mathcal}{A}({\lambda}=1)$ to a point in ${T^*\bm\hat{{\mathcal}{T}}^n}$. By proposition \[monodromyreal\] the monodromy of ${\mathcal}{A}({\lambda})$ for $t=0$ is in $\operatorname{PSL}_n({\mathbb{R}})$. Following Hitchin’s argument from theorem 7.5 in [@Hit.1], we prove that the zero-section in ${T^*\bm\hat{{\mathcal}{T}}^n}$ where $t=0$ describes a connected component of $\operatorname{Rep}(\pi_1({\Sigma}), \operatorname{PSL}_n({\mathbb{R}}))$.
Since ${\bm\hat{{\mathcal}{T}}}^n$ is closed in ${T^*\bm\hat{{\mathcal}{T}}^n}$, the image of the map $s:{\bm\hat{{\mathcal}{T}}}^n \rightarrow \operatorname{Rep}(\pi_1({\Sigma}), \operatorname{PSL}_n({\mathbb{R}}))$ is a closed submanifold. Furthermore both spaces have the same dimension by theorem \[mainresultncomplex\]. Therefore the image of $s$ is an open and closed submanifold, i.e. a connected component.
Finally, for $\mu=0$ we get the same connection ${\mathcal}{A}({\lambda})$ as by the non-abelian Hodge correspondence of the principal nilpotent Higgs field. So the component described by ${\bm\hat{{\mathcal}{T}}}^n$ and Hitchin’s component ${\mathcal}{T}^n$ coincide.
Notice that the map between ${\mathcal}{T}^n$ and ${\bm\hat{{\mathcal}{T}}}^n$ is something like an exponential map. For $n=2$ Hitchin’s description of Teichmüller space is exactly via the exponential map identifying a fiber of the cotangent bundle $T^*_\mu{\mathcal}{T}^2$ to ${\mathcal}{T}^2$.
Hitchin’s component has a natural complex structure. Further, there is a natural action of the mapping class group on it.
The first statement follows from theorem \[mainresultncomplex\] since we explicitly know the cotangent space at a point. The second simply follows by the description of Hitchin’s component as moduli space of some geometric structure on the surface. Labourie describes this action in [@Lab.3] and shows that it is properly discontinuous using cross ratios.
Punctual Hilbert schemes revisited {#appendix:A}
==================================
In this appendix, we review some aspects of the punctual Hilbert scheme of the plane. Main references are Nakajima’s book [@Nakajima] and Haiman’s paper [@Haiman].
Definition
----------
Consider $n$ points in the plane $\mathbb{C}^2$ as an algebraic variety, i.e. defined by some ideal $I$ in $\mathbb{C}[x,y]$. Its function space $\mathbb{C}[x,y]/I$ is of dimension $n$, since a function on $n$ points is defined by its $n$ values. So the ideal $I$ is of codimension $n$. The space of all such ideals, or in more algebraic language, the space of all zero-subschemes of the plane of given length, is the punctual Hilbert scheme:
The **punctual Hilbert scheme** $\operatorname{Hilb}^n(\mathbb{C}^2)$ of length $n$ of the plane is the set of ideals of $\mathbb{C}\left[x,y\right]$ of codimension $n$: $$\operatorname{Hilb}^n(\mathbb{C}^2)=\{I \text{ ideal of } \mathbb{C}\left[x,y\right] \mid \dim(\mathbb{C}\left[x,y\right]/I)=n \}.$$ The subspace of $\operatorname{Hilb}^n(\mathbb{C}^2)$ consisting of all ideals supported at the origin, i.e. whose associated algebraic variety is $(0,0)$, is called the **zero-fiber** of the punctual Hilbert scheme and is denoted by $\operatorname{Hilb}^n_0(\mathbb{C}^2)$.
A theorem of Grothendieck and Fogarty asserts that $\operatorname{Hilb}^n(\mathbb{C}^2)$ is a smooth and irreducible variety of dimension $2n$ (see [@Fogarty]). The zero-fiber $\operatorname{Hilb}^n_0(\mathbb{C}^2)$ is an irreducible variety of dimension $n-1$, but it is in general not smooth.
A generic element of $\operatorname{Hilb}^n({\mathbb{C}}^2)$, geometrically given by $n$ distinct points, is given by $$I=\left\langle -x^n+t_1x^{n-1}+\cdots+t_n,-y+\mu_1+\mu_2x+...+\mu_nx^{n-1}\right\rangle.$$ The second term can be seen as the Lagrange interpolation polynomial of the $n$ points.
A generic element of the zero-fiber is given by $$I=\left\langle x^n,-y+\mu_2x+...+\mu_nx^{n-1}\right\rangle.$$ In particular, we see that for $n=2$, we get projective space:$$\operatorname{Hilb}^2_0({\mathbb{C}}^2)\cong {\mathbb}{P}({\mathbb{C}}^2) = {\mathbb{C}}P^1.$$
Given an ideal $I$ of codimension $n$, we can associate its support, the algebraic variety defined by $I$, which is a collection of $n$ points (counted with multiplicity). The order of the points does not matter, so there is a map, called the **Chow map**, from $\operatorname{Hilb}^n(\mathbb{C}^2)$ to $\operatorname{Sym}^n(\mathbb{C}^2) := (\mathbb{C}^2)^n/\mathcal{S}_n$, the configuration space of $n$ points ($\mathcal{S}_n$ denotes the symmetric group). A theorem of Fogarty asserts that the punctual Hilbert scheme is a *minimal resolution of the configuration space*.
Matrix viewpoint {#matrixviewhilb}
----------------
To an ideal $I$ of codimension $n$, we can associate two matrices: the multiplication operators $M_x$ and $M_y$, acting on the quotient ${\mathbb{C}}[x,y]/I$ by multiplication by $x$ and $y$ respectively. To be more precise, we can associate a conjugacy class of the pair: $[(M_x,M_y)]$.
The two matrices $M_x$ and $M_y$ commute and they admit a cyclic vector, the image of $1 \in {\mathbb{C}}[x,y]$ in the quotient (i.e. 1 under the action of both $M_x$ and $M_y$ generate the whole quotient).
\[bijhilbert\] There is a bijection between the Hilbert scheme and conjugacy classes of certain commuting matrices: $$\operatorname{Hilb}^n(\mathbb{C}^2) \cong \{(A,B) \in {\mathfrak}{gl}_n^2 \mid [A,B]=0, (A,B) \text{ admits a cyclic vector}\} / GL_n$$
The inverse construction goes as follows: to a conjugacy class $[(A,B)]$, associate the ideal $I=\{P \in {\mathbb{C}}[x,y] \mid P(A,B)=0\}$, which is well-defined and of codimension $n$ (using the fact that $(A,B)$ admits a cyclic vector). For more details see [@Nakajima].
The zero-fiber of the Hilbert scheme corresponds to *nilpotent* commuting matrices.
Reduced Hilbert scheme
----------------------
We wish to define a subspace of $\operatorname{Hilb}^n({\mathbb{C}}^2)$ corresponding to matrices in ${\mathfrak}{sl}_n$ in the matrix viewpoint. A generic point should be a pair of points in the Cartan subalgebra ${\mathfrak{h}}$ of ${\mathfrak}{sl}_n$ modulo order. This corresponds to $n$ points in the plane with barycenter 0.
The **reduced Hilbert scheme** $\operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$ is the space of all elements of $\operatorname{Hilb}^n({\mathbb{C}}^2)$ whose image under the Chow map ($n$ points with multiplicity modulo order) has barycenter 0.
With this definition, we get
$$\operatorname{Hilb}^n_{red}(\mathbb{C}^2) \cong \{(A,B) \in {\mathfrak}{sl}_n^2 \mid [A,B]=0, (A,B) \text{ admits a cyclic vector}\} / SL_n.$$
Finally, it can be proven that the reduced Hilbert scheme is symplectic and that the zero-fiber $\operatorname{Hilb}^n_0({\mathbb{C}}^2)$ is a Lagrangian subspace of $\operatorname{Hilb}^n_{red}({\mathbb{C}}^2)$.
Existence of parabolic gauge {#appendix:B}
============================
In this rather technical appendix, we prove the existence of a parabolic gauge (see subsection \[settingparab\]).
\[existence-para-gauge\] For a generic connection $A=A_1+A_2$, there is a gauge $P\in \mathcal{C}^{\infty}(\Sigma,\operatorname{SL}(n,\mathbb{C}))$ with last row zero except for the last entry (parabolic gauge) such that $A_1$ is locally transformed into a companion matrix.
We begin by setting up notations. The matrix $P$ we look for, is of the following form:
$$P= \left( \begin{array}{cccc}
p_{11} & ... & p_{1,n-1} &p_{1n} \\
\vdots & & \vdots & \vdots \\
p_{n-1,1} & ...& p_{n-1,n-1}& p_{n-1,n} \\
0 & 0 & 0 & p_{nn}
\end{array} \right) = \left( \begin{array}{cc}
L_1 & p_{1n} \\
\vdots & \vdots \\
L_{n-1} & p_{n-1,n} \\
0 & p_{nn}
\end{array} \right)$$ where $L_k$ denotes the $k$^th^ line in the matrix without the last entry (i.e. a row vector of length $n-1$). Denote the entries of $A_1$ by $a_{ij}$. We adopt Einstein’s summation convention in this section (automatic summation over repeated indices).
We want that $P$ transforms $A_1$ into a companion matrix under gauge transform $PA_1P^{-1}+P\partial(P^{-1})$. Since $P$ is of determinant 1, $A_1$ stays traceless. Using $P\partial{P^{-1}} = -\partial P P^{-1}$, we get $$\label{ppara}
PA_1-\partial P = \left( \begin{array}{cccc}
& & & *\\
1 & & & \vdots \\
& \ddots & & * \\
& & 1& 0
\end{array} \right) P = \left( \begin{array}{cc}
0 & *\\
L_1 & * \\
\vdots & \vdots \\
L_{n-1}& *
\end{array} \right).$$ This gives $n^2-n$ equations by the first $n-1$ columns.
Our strategy is the following: we express $p_{ij}$ for $1\leq i \leq n-1$ and $1\leq j \leq n-1$ in terms of $a_{ij}$ (the “constants”) and the $(p_{kn})_{k=1,...,n}$ (and their derivatives). Then we get an expression of $p_{nn}$ in terms of $a_{ij}$. Finally we compute $p_{kn}$ for $k=1,...,n-1$.
The matrix equation above gives $$\sum_{k=1}^n p_{ik}a_{kj}-\partial p_{ij} = p_{i-1,j} \label{auxmat}$$ $\forall 1\leq i \leq n$ and $\forall 1\leq j\leq n-1$ where we have put $p_{0j}= 0$.
Setting $i=n$ (and $j<n$), we get $p_{n-1,j} = a_{nj}p_{nn}$, i.e. $L_{n-1}$ is $p_{nn}$ times the last row of $A_1$. Setting $i=n-1$ (and $j<n$), we get $p_{n-2,j}=p_{n-1,k}a_{kj}-\partial p_{n-1,j} = p_{nn}(a_{nk}a_{kj}-\partial a_{nj})-a_{nj}\partial p_{nn} +p_{n-1,n}a_{nj}$.
By continuing, we see that for $2\leq i \leq n$, we get our first goal: the equations express the $p_{ij}$ for $1\leq i \leq n-1$ and $1\leq j \leq n-1$ in terms of $a_{ij}$ and the $(p_{kn})_{k=1,...,n}$.
To achieve our second goal, we prove the following:
Denote by $P_0$ the square-submatrix of the $p_{ij}$ for $1\leq i \leq n-1$ and $1\leq j \leq n-1$. We then have $$\det(P_0) = A p_{nn}^{n-1}$$ where $A$ is some constant only depending on the $a_{ij}$.
We interpret the equation for $PA_1-\partial P$ as a condition on the covariant derivative $\nabla = (-\partial+A_1)$ on $P$, acting from the right. The factor $p_{nn}$ is interpreted as a scalar denoted by $f$. Put $a=(a_{n1},a_{n2},...,a_{n,n-1})$ the last row of $A_1$ which we consider as a row vector. We already noticed that $L_{n-1}=fa$. In what follows we write $\nabla a$ for $-{\partial}a + aA_1$. The other equations of now successively give $$\begin{aligned}
L_{n-2} &= \nabla(fa)+p_{n-1,n}a \nonumber \\
L_{n-3} &= \nabla L_{n-2} + p_{n-2,n}a = \nabla(\nabla(fa)+p_{n-1,n}a ) + p_{n-2,n}a \nonumber\\
L_{n-k} &= \nabla L_{n-k+1}+p_{n-k+1,n}a \label{parabdetail}\end{aligned}$$ Thus, we can write $L_{n-k} = \sum_{l=0}^{k-1} \alpha_{l,k}\nabla^l a$ with $\alpha_{k-1,k} = f$. The other $\alpha_{l,k}$ are functions of $a_{ij}$ and the $(p_{kn})_{k=1,...,n-1}$.
With this expression for $L_{n-k}$, we see that $P_0$ in the basis $(\nabla^{n-2}a,..., \nabla a, a)$ is upper-triangular with $f$ on the diagonal. Hence, its determinant is $$\det(P_0) = f^{n-1}\det(\nabla^{n-2} a, ..., \nabla a,a) = p_{nn}^{n-1} A.$$
Since $1=\det P = p_{nn} \det P_0 $, we get $$p_{nn} = A^{-\frac{1}{n}}.$$
We see in particular the condition under which the parabolic gauge exists: we need that $A=\det(\nabla^{n-2} a,...,\nabla a, a) \neq 0$.
To finish, take equations for $i=1$ which give $$0=\nabla L_1 + p_{1,n}a = f\nabla^{n-1}a + \sum_{l=0}^{n-2}\alpha_{l,n}\nabla^l a \label{auxmat2}$$ with $\alpha_{l,n} = p_{l+1,n} +$ terms with $p_{k,n}$ with $k>l+1$. Then, we express $\nabla^{n-1}a$ in the basis $(a,\nabla a, ..., \nabla^{n-2}a)$: $\nabla^{n-1}a = \beta_0 a+\beta_1\nabla a +...+\beta_{n-2}\nabla^{n-2}a$ (thus the $\beta_k$ depend only on the $a_{ij}$). By the freedom of $(a,\nabla a, ..., \nabla^{n-2}a)$, we get out of $$\alpha_{l,n}+f\beta_l = 0.$$ Therefore, we can successively express $p_{1n}, p_{2n}$, ... up to $p_{n-1,n}$ in terms of $a_{ij}$ and $p_{nn}$ (which we already expressed in terms of the $a_{ij}$).
This proves the existence of the parabolic gauge.
Proof of lemma \[curvaturemodmod\] {#appendix:C}
==================================
The parabolic curvature modulo $\bm\hat{t}^2$ and ${\partial}^2$ gives condition $(\mathcal{C})$ on $T^*\bm\hat{\mathcal{T}}^n$: $$\xi_k = (\bar{\partial}\!-\!\bm\hat{\mu}_2\partial\!-\!k\partial\bm\hat{\mu}_k)\bm\hat{t}_k-\sum_{l=1}^{n-k}\left((l\!+\!k)\partial\bm\hat{\mu}_{l+2}+(l\!+\!1)\bm\hat{\mu}_{l+2}{\partial}\right)\bm\hat{t}_{k+l} \mod \bm\hat{t}^2 \mod \partial^2.$$
The proof is a combination of several formulas:
1. Proposition \[thmcourbure\] together with the expression of the differential operators (see ) give $$[\partial^n-\bm\hat{t}_2\partial^{n-2}-...-\bm\hat{t}_n,-\bar{\partial}+\bm\hat{\alpha}_{nn}+\bm\hat{\alpha}_{n,n-1}\partial+...+\bm\hat{\alpha}_{n,1}\partial^{n-1}] = \sum_{k=2}^n \xi_k\partial^{n-k} \mod \bm\hat{I}$$ where $\bm\hat{\alpha}_{ij}$ are the entries of the matrix $A_2$.
2. Link between Poisson bracket and commutator: $$\begin{aligned}
& \{p^n-\bm\hat{t}_2p^{n-2}-...-\bm\hat{t}_n,-\bar{p}+\bm\hat{\mu}_1+\bm\hat{\mu}_2p+...+\bm\hat{\mu}_np^{n-1}\} \\
&= \lim_{h \rightarrow 0}\frac{1}{h}[h^n\partial^n-\bm\hat{t}_2h^{n-2}\partial^{n-2}-...-\bm\hat{t}_n,-h\bar{\partial}+\bm\hat{\mu}_1+\bm\hat{\mu}_2h\partial+...+\bm\hat{\mu}_nh^{n-1}\partial^{n-1}]\Big|_{\substack{h\partial\mapsto p \\ h\bar{\partial}\mapsto \bar{p}}}.\end{aligned}$$
3. Link between $\mod \partial^2$ and brackets: for $h$-connections $D_1(h)$ and $D_2(h)$, we have $$\lim_{h \rightarrow 0} \frac{1}{h}[D_1(h),D_2(h)] = \frac{1}{h}[D_1(h),D_2(h)] \mod \partial^2.$$
4. The following formula from proposition 5 in [@FockThomas] linking the Poisson bracket to condition $({\mathcal}{C})$: $$\begin{aligned}
& \{p^n-\bm\hat{t}_2p^{n-2}-...-\bm\hat{t}_n,-\bar{p}+\bm\hat{\mu}_1+\bm\hat{\mu}_2p+...+\bm\hat{\mu}_np^{n-1}\} \\
= &\sum_{k=2}^n\left( (\bar{\partial}\!-\!\bm\hat{\mu}_2\partial\!-\!k\partial\bm\hat{\mu}_k)\bm\hat{t}_k-\sum_{l=1}^{n-k}\left((l\!+\!k)\partial\bm\hat{\mu}_{l+2}+(l\!+\!1)\bm\hat{\mu}_{l+2}{\partial}\right)\bm\hat{t}_{k+l} \right)p^{n-k} \mod \bm\hat{t}^2, I.\end{aligned}$$
Now, we are ready to conclude. By a direct computation, we can see that modulo $\bm\hat{t}^2, \partial^2$ we can replace $\bm\hat{\alpha}_{n,n+1-l}$ by $\bm\hat{\mu}_l$ in point 2. Define $D_1(h) = h^n\partial^n-\bm\hat{t}_2h^{n-2}\partial^{n-2}-...-\bm\hat{t}_n$ and $D_2(h)=-h\bar{\partial}+\bm\hat{\mu}_1+\bm\hat{\mu}_2h\partial+...+\bm\hat{\mu}_nh^{n-1}\partial^{n-1}$. Then using 1. to 4. and computing modulo $\bm\hat{t}^2$ and ${\partial}^2$, we get: $$\begin{aligned}
\sum_{k=2}^n \xi_k p^{n-k} &= \sum_{k=2}^n \xi_k (h\partial)^{n-k} \Big|_{h\partial\mapsto p} \\
&= \left(\frac{1}{h}[D_1(h),D_2(h)] \right)\Big|_{\substack{h\partial\mapsto p \\h\bar{\partial}\mapsto\bar{\partial}}} \mod \bm\hat{I}\\
&= \lim_{h \rightarrow 0}\frac{1}{h}[D_1(h),D_2(h)]\Big|_{\substack{h\partial\mapsto p \\h\bar{\partial}\mapsto\bar{\partial}}} \mod \bm\hat{I}\\
&= \{p^n-\bm\hat{t}_2p^{n-2}-...-\bm\hat{t}_n,-\bar{p}+\bm\hat{\mu}_1+\bm\hat{\mu}_2p+...+\bm\hat{\mu}_np^{n-1}\} \mod I\\
&= \sum_{k=2}^n \left((\bar{\partial}\!-\!\bm\hat{\mu}_2\partial\!-\!k\partial\bm\hat{\mu}_k)\bm\hat{t}_k-\sum_{l=1}^{n-k}\left((l\!+\!k)\partial\bm\hat{\mu}_{l+2}+(l\!+\!1)\bm\hat{\mu}_{l+2}{\partial}\right)\bm\hat{t}_{k+l} \right)p^{n-k}\end{aligned}$$ Comparing coefficients, we get the lemma.
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|
---
abstract: 'In this paper we address practical aspects of the implementation of the $0$-$1$ test for chaos in deterministic systems. In addition, we present a new formulation of the test which significantly increases its sensitivity. The test can be viewed as a method to distill a binary quantity from the power spectrum. The implementation is guided by recent results from the theoretical justification of the test as well as by exploring better statistical methods to determine the binary quantities. We give several examples to illustrate the improvement.'
author:
- 'Georg A. Gottwald [^1]'
- 'Ian Melbourne [^2]'
title: 'On the Implementation of the $0$–$1$ Test for Chaos'
---
Introduction {#sec-intro}
============
Being able to distinguish between regular and chaotic dynamics in a deterministic system is an important question with applications ranging from cardiac arrhythmias to the stability of our solar system. Much progress has been made in developing tests for chaos [@Kantz; @Laskar93; @Skokos01; @Fouchard02; @Cincotta03; @Barrio05]. Recently we have introduced a binary test for chaos, the $0$–$1$ test, designed for the analysis of deterministic dynamical systems [@GM04; @GM05]. The test distinguishes between regular and chaotic dynamics for a deterministic system. The nature of the dynamical system is irrelevant for the implementation of the test; it is applicable to data generated from maps, ordinary differential equations and partial differential equations. The test has been applied to noisy numerical data [@GM05], experimental data [@FGMW07], quasiperiodically forced systems and strange nonchaotic attractors [@Dawes07], Hamiltonian systems [@Skokos04], nonsmooth systems [@Litak07] and fluid dynamics [@JulienWeiss06].
The usual test of whether a deterministic dynamical system is chaotic or nonchaotic involves the calculation of the maximal Lyapunov exponent $\lambda$ [@Kantz]. A positive maximal Lyapunov exponent indicates chaos: if $\lambda>0$, then nearby trajectories separate exponentially and if $\lambda\le0$, then nearby trajectories remain in a close neighbourhood of each other. This approach has been widely used for dynamical systems whose equations are known. If the equations are not known or one wishes to examine experimental data, then $\lambda$ may be estimated using the phase space reconstruction method of Takens [@Takens81], by approximating the linearisation of the evolution operator [@SanoSawada], or by the “direct method” [@Rosenstein93].
In contrast our test does not depend on phase space reconstruction but rather works directly with the time series given. The main advantages of our test are (i) it is binary (minimizing issues of distinguishing small positive numbers from zero), (ii) the nature of the vector field as well as its dimensionality does not pose practical limitations, and (iii) it does not suffer from the difficulties associated with phase space reconstruction [@Kantz].
In this paper, we describe in detail how to implement the $0$–$1$ test for chaos. In addition, we carry out modifications to the test that greatly improve the previous versions in [@GM04; @GM05].
Throughout the paper, we use the logistic map to illustrate our claims, with the exception of Section \[sec-cont\] where we use the Lorenz attractor as an example of a continuous time system. The reader can verify that our results apply equally well to other systems, including those considered in our previous papers [@GM04; @GM05].
Recipe for the $0$–$1$ test {#sec-recipe}
---------------------------
We briefly review how the test is implemented. Given an observation $\phi(j)$ for $j=1,\ldots ,N$ we perform the following sequence of steps:
1. For $c\in(0,\pi)$, we compute the translation variables $$\begin{aligned}
\label{eq-p}
p_c(n) =\sum_{j=1}^n \phi(j) \cos jc, \quad
q_c(n) =\sum_{j=1}^n \phi(j) \sin jc\end{aligned}$$ for $n=1,2,\ldots,N$. Typical plots of $p$ and $q$ for regular and chaotic dynamics are given in Fig. \[fig-pq\].
![ Plot of $p$ versus $q$ for the logistic map $x_{n+1}=\mu
x_n(1-x_n)$. Left: Regular dynamics at $\mu=3.55$; Right: Chaotic dynamics at $\mu=3.9$. We used $5000$ data points. []{data-label="fig-pq"}](Fig3a.eps "fig:"){width=".4\textwidth" height=".4\textwidth"} ![ Plot of $p$ versus $q$ for the logistic map $x_{n+1}=\mu
x_n(1-x_n)$. Left: Regular dynamics at $\mu=3.55$; Right: Chaotic dynamics at $\mu=3.9$. We used $5000$ data points. []{data-label="fig-pq"}](Fig3b.eps "fig:"){width=".4\textwidth" height=".4\textwidth"}
2. The diffusive (or non-diffusive) behaviour of $p_c$ and $q_c$ can be investigated by analyzing the mean square displacement $M_c(n)$. The theory behind our test assures that if the dynamics is regular then the mean square displacement is a bounded function in time, whereas if the dynamics is chaotic then the mean square displacement scales linearly with time. In Section \[sec-MSQ\] we look at expressions for the mean square displacement and describe how one may use analytical expressions derived in [@GMprep] to conveniently modify the expression for the mean square displacement.
3. We then compute the asymptotic growth rate $K_c$ of the mean square displacement. Methods for the most effective estimation of this are discussed in Section \[sec-Kc\].
4. Steps 1–3 are performed for $N_c$ values of $c$ chosen randomly in the interval $(0,\pi)$. In practice, $N_c=100$ is sufficient. The choice of $c$ is discussed further in Section \[sec-c\]. We then compute the median of these $N_c$ values of $K_c$ to compute the final result $K={\rm{median}}(K_c)$. Our test states that a value of $K\approx 0$ indicates regular dynamics, and $K\approx 1$ indicates chaotic dynamics.
In this paper, we explore practical issues arising in the implementation of the above algorithm. Various issues associated with steps 2–4 are discussed in respectively. In Section \[sec-confidence\] we examine finite data size effects. In particular we look at weak chaos. In Section \[sec-cont\] we consider continuous time systems where oversampled data can lead to small values of $K$ despite an underlying chaotic dynamics. In Section \[sec-noise\] we investigate the issue of measurement noise.
\[rmk-first\] In the first version of our test, introduced in [@GM04], we defined $p_c(n)$ and $q_c(n)$ by iterating the extended system $$\begin{aligned}
p_c(n+1) &= p_c(n) + \phi(n) \cos(\vartheta_c(n)) \nonumber\\
q_c(n+1) &= q_c(n) + \phi(n) \sin(\vartheta_c(n)) \nonumber\\
\vartheta_c(n+1) &= \vartheta_c(n)+c+\alpha \, \phi(n)\; .\end{aligned}$$ The current version of the test corresponds to the case $\alpha=0$. As shown in [@GM05], the test with $\alpha=0$ is less sensitive to measurement noise.
\[rmk-valid\] It can be rigorously shown that (i) $p_c(n)$ and $q_c(n)$ are bounded if the underlying dynamics is regular, i.e. periodic or quasiperiodic and (ii) $p_c(n)$ and $q_c(n)$ behave asymptotically like Brownian motion for large classes of chaotic dynamical systems.
In [@GM04] we used results of [@FMT03; @MN04; @NMA01] to prove this for the case $\alpha \neq 0$ in Remark \[rmk-first\]. In the case $\alpha=0$ these results are not applicable; nevertheless in [@GMprep] we cover the case $\alpha=0$ under even weaker assumptions on the underlying dynamics.
\[rmk-third\] In [@GM04; @GMprep] it was shown that the results are valid for almost all observables $\phi$. Of course, the choice of the observable $\phi$ influences the rate of convergence (but not the limiting value $K=0$ or $K=1$). From a practical point of view we found that changing the observable does not greatly alter the computed value of $K$.
Computation of the mean square displacement {#sec-MSQ}
===========================================
For a given time series $\phi(j)$ with $j=1,\ldots ,N$, we compute the mean square displacement of the translation variables $p_c(n)$ and $q_c(n)$ defined in for several values of $c\in(0,\pi)$. The mean square displacement is defined as $$\begin{aligned}
M_c(n) = \lim_{N\to\infty}\frac{1}{N}
\sum_{j=1}^{N}[p_c(j+n)-p_c(j)]^2\, + \,
[q_c(j+n)-q_c(j)]^2 \; .
\label{e.MSQ}\end{aligned}$$ Note that this definition requires $n\ll N$. In [@GM05] we calculated the mean square displacement using directly the definition (\[e.MSQ\]). The limit is assured by calculating $M_c(n)$ only for $n\le n_{\rm{cut}}$ where $n_{\rm{cut}}\ll N$. In practice we find that $n_{\rm{cut}}=N/10$ yields good results.
The test for chaos is based on the growth rate of $M_c(n)$ as a function of $n$. In the following, we use analytical expressions derived in [@GMprep] to formulate a modified mean square displacement $D_c(n)$ which exhibits the same asymptotic growth as $M_c(n)$ but with better convergence properties.
Under mild assumptions on the underlying dynamical system, described in Remark \[rmk-rho\] below, for each $c\in(0,\pi)$, $$\begin{aligned}
\label{e-Mcn}
M_c(n) = V\!(c)\, n + V_{\rm{osc}}(c,n) + e(c,n)\;,\end{aligned}$$ where $e(c,n)/n \to0$ as $n\to\infty$ uniformly in $c\in(0,\pi)$ and $$\begin{aligned}
V_{\rm{osc}}(c,n) = (E\phi)^2 \frac{1-\cos nc}{1-\cos c}\;.\end{aligned}$$ The expectation $E\phi$ is given by $$\begin{aligned}
E\phi=\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N \phi(j)\; .\end{aligned}$$
The form (\[e-Mcn\]) suggests an improvement for the test: We can subtract the explicit term $V_{\rm{osc}}(c,n)$ from the mean square displacement and introduce $$\begin{aligned}
D_c(n) = M_c(n)-V_{\rm{osc}}(c,n)\; .
\label{e.MSQNEW}\end{aligned}$$ Note that the asymptotic growth rates of $M_c(n)$ and $D_c(n)$ are the same.\
![ Plot of mean square displacement versus $n$ for the logistic map with $\mu=3.91$ corresponding to chaotic dynamics. The oscillating (green) curve is the original mean square displacement $M_c(n)$ as defined in (\[e.MSQ\]); the straighter (red) curve is the modified mean square displacement $D_c(n)$ as defined in (\[e.MSQNEW\]). We used $2000$ data points and computed $M_c(n)$ and $D_c(n)$ for $n=1,\ldots,200$ and $c=1.0$.[]{data-label="fig-MSQ"}](Fig1N.eps){width=".495\textwidth"}
In Fig. \[fig-MSQ\] we show the two mean square displacements $M_c(n)$ and $D_c(n)$ for the logistic map $x_{n+1}=\mu x_n(1-x_n)$ with $\mu=3.91$ (which corresponds to chaotic dynamics) and an arbitrary value of $c=1.0$. Evidently, the subtraction of the oscillatory term $V_{\rm{osc}}(c,n)$ regularizes the linear behaviour of $M_c(n)$. This allows a much better determination of the asymptotic growth rate $K_c$.
\[rmk-rho\] The autocorrelation function for the observation $\phi(j)$ is given by $$\rho(k)=E(\phi(1)\phi(k+1))-(E\phi)^2, \enspace k=0,1,2\ldots$$ Provided the autocorrelations are absolutely summable (that is, $\sum_{k=0}^\infty |\rho(k)|<\infty$) then equation (\[e-Mcn\]) is valid, and moreover, the error term $e(c,n)$ decays uniformly in $c\in(0,\pi)$ (see for example [@GMprep]). It is for this reason that the test based on $D_c(n)$ greatly outperforms the test based on $M_c(n)$.
Furthermore, the absolute summability condition guarantees [@GMprep] that $$\begin{aligned}
\label{eq-S}
V(c)=\sum_{k=-\infty}^\infty e^{ikc}\rho(|k|)
=\lim_{n\to\infty}\frac1n E \Bigl|\sum_{j=0}^{n-1}e^{ijc} \phi(j) \Bigr|^2\end{aligned}$$ for all $c\in(0,2\pi)$. This result follows from the Birkhoff ergodic theorem, the Wiener-Khintchine theorem, and standard calculations. In particular, the slope $V(c)$ of the mean square displacement is identified with the power spectrum.
For nonmixing systems, the error term $e(c,n)$ no longer decays to zero and there are further oscillatory terms in addition to $V_{\rm{osc}}(c,n)$. Nevertheless, the identification remains valid for nonmixing systems under very weak conditions [@MGapp].
More importantly from the point of view of the test for chaos, working with $D_c(n)$ remains highly advantageous even for nonmixing systems. This is illustrated for the logistic map in Fig. \[fig-TEST\] later in this paper,
Computation of $K_c$ {#sec-Kc}
====================
Having calculated the modified mean square displacement $D_c(n)$ for $n=1,2,\ldots,n_{\rm{cut}}$, the next step is to estimate the asymptotic growth rate $K_c$. We have tried out two different methods: a [*regression*]{} method and a [*correlation*]{} method, described in subsections \[sec-regress\] and \[sec-corr\] below.
Regression method {#sec-regress}
-----------------
The regression method consists of linear regression for the log-log plot of the mean square displacement. In [@GM05] we used the original mean square displacement $M_c(n)$, so the asymptotic growth rate $K_c$ is given by the definition $$\begin{aligned}
K_c=\lim_{n\to\infty}\frac{\log {M}_c(n)}{\log n} \; .\end{aligned}$$ Numerically, $K_c$ is determined by fitting a straight line to the graph of $\log M_c(n)$ versus $\log n$ through minimizing the absolute deviation [@NRinC].
In Section \[sec-MSQ\], we demonstrated the superiority of the modified mean square displacement $D_c(n)$ when compared to $M_c(n)$, so it is natural to apply the regression method to $D_c(n)$. Whereas $M_c(n)$ is strictly positive, $D_c(n)$ may be negative due to the subtraction of the oscillatory term $V_{\rm{osc}}(c,n)$. Hence, we set $$\begin{aligned}
\tilde D_c(n)=D_c(n)-\min_{n=1,\dots,n_{\rm cut}}D_c(n)\;,
$$ and obtain the asymptotic growth rate $$\begin{aligned}
K_c=\lim_{n\to\infty}\frac{\log \tilde D_c(n)}{\log n}\; .
$$ Again, $K_c$ can be determined numerically by regression (minimizing the absolute deviation) for the graph of $\log \tilde D_c(n)$ versus $\log n$.
![ Plot of $\log \tilde D_c(n)$ as a function of $\log n$ for the logistic map at $\mu=3.62$. We used $N=2000$ and calculated the mean square displacement up to $n_{\rm{cut}}=N/10$.[]{data-label="fig-logM"}](Fig_logMN2.eps){width=".495\textwidth"}
\[rmk-regress\] Minimizing the absolute deviation is preferable when compared to the usual least square method as the latter assigns a higher weight to outliers. Since the linear behaviour of the mean square displacement is only given asymptotically, one typically encounters outliers for small values of $n$. We find that it is usually sufficient to use the absolute deviation for estimating $K_c$, and that it is not necessary to employ more complicated higher-order regression methods such as the method by Yohai [@Yohai].
The finite value of the $o(n)$-term $e(c,n)$ in the definition of $M_c(n)$ in (\[e-Mcn\]) leads to a distortion for small values of $n$. In such situations, one typically observes a flattening of the slope of $\log {M}_c(n)$ or $\log \tilde D_c(n)$ as illustrated in Fig. \[fig-logM\]. It is those values for small $n$ of $\log
{M}_c(n)$ (or $\log \tilde D_c(n)$) which would be overestimated in a least square fit.
Correlation method {#sec-corr}
------------------
We now present an alternative method for determining $K_c$ from the mean square displacement. (The method is described in terms of $D_c(n)$, but we could use $M_c(n)$ instead.)
Form the vectors $\xi=(1,2,\dots,n_{\rm cut})$ and $\Delta=(D_c(1),D_c(2),\dots,D_c(n_{\rm cut}))$. Given vectors $x$, $y$ of length $q$, we define covariance and variance in the usual way: $$\begin{aligned}
& {\rm cov}(x,y) = \frac1q\sum_{j=1}^q (x(j)-\bar x)(y(j)-\bar y),
\quad \text{where} \enspace
\bar x = \frac1q\sum_{j=1}^q x(j)\; , \\
& {\rm var}(x) ={\rm cov}(x,x)\; .\end{aligned}$$ Now define the correlation coefficient $$\begin{aligned}
K_c = {\rm corr}(\xi,\Delta)=
\frac{{\rm{cov}}(\xi,\Delta)}{\sqrt{{{\operatorname{var}}(\xi)}{{\operatorname{var}}(\Delta)}}}\in[-1,1]\;.\end{aligned}$$ This quantity measures the strength of the correlation of $D_c(n)$ with linear growth. Again, it can be shown rigorously [@GMprep] that under weak conditions on the underlying dynamics (as described in Remark \[rmk-valid\]) we obtain $K_c=0$ for regular dynamics and $K_c=1$ for chaotic dynamics.
![ Plot of $K$ versus $\mu$ for the logistic map with $3.5\le \mu\le 4$ increased in increments of $0.001$. We used $2000$ data points. The darker (red) lines are obtained by using the original definition of the mean square displacement $M_c(n)$ in (\[e.MSQ\]). The lighter (green) lines are obtained by using the modified mean square displacement $D_c(n)$ in (\[e.MSQNEW\]). The resulting values of $K$ are shown for the regression method (left) and the correlation method (right). The horizontal lines (blue and magenta) indicate the cases $K=0$ and $K=1$. We used $N_c=100$ values of $c$. []{data-label="fig-TEST"}](Fig4a_N.eps "fig:"){width=".495\textwidth"} ![ Plot of $K$ versus $\mu$ for the logistic map with $3.5\le \mu\le 4$ increased in increments of $0.001$. We used $2000$ data points. The darker (red) lines are obtained by using the original definition of the mean square displacement $M_c(n)$ in (\[e.MSQ\]). The lighter (green) lines are obtained by using the modified mean square displacement $D_c(n)$ in (\[e.MSQNEW\]). The resulting values of $K$ are shown for the regression method (left) and the correlation method (right). The horizontal lines (blue and magenta) indicate the cases $K=0$ and $K=1$. We used $N_c=100$ values of $c$. []{data-label="fig-TEST"}](Fig4b_N.eps "fig:"){width=".495\textwidth"}
In practical terms, the correlation method greatly outperforms the regression method. This is evident from Fig. \[fig-TEST\] which compares the regression method and the correlation method (using both $M_c(n)$ and $D_c(n)$) for the logistic map.
Choice of $c$ and determination of $K$ {#sec-c}
======================================
In Fig. \[fig-cscan\] we show the asymptotic growth rate $K_c$ as a function of $c$ for regular and chaotic dynamics. In the case of periodic dynamics, most values of $c$ yield $K_c=0$ as expected, but there are isolated values of $c$ for which $K_c$ is large. (For the regression method, $K_c\approx2$ at these resonant points.) These resonances are easily explained as follows: Equation (\[eq-p\]) shows that if the Fourier decomposition of the observation $\phi$ contains a term proportional to $\exp (-i \omega k)$, then there is a resonance at $c=\omega$ where $p_c(n)\sim n$, and hence $M_c(n) \sim
n^2$, irrespective of whether the dynamics is regular or chaotic. For the plots in Fig. \[fig-cscan\], we have calculated the asymptotic growth rate using both the regression method described in Section \[sec-regress\] and the correlation method described in Section \[sec-corr\].
![ Plot of $K_c$ versus $c$ for the logistic map calculated using the regression method (top) and correlation method (bottom). We used here $N=5000$ data points, and $1000$ equally spaced values for $c$. Left: $\mu=3.55$ corresponding to regular dynamics; Middle: $\mu=3.9$ corresponding to chaotic dynamics; Right: $\mu=3.6$ corresponding to chaotic but non-mixing dynamics.[]{data-label="fig-cscan"}](Fig5a.eps "fig:"){width=".33\textwidth"} ![ Plot of $K_c$ versus $c$ for the logistic map calculated using the regression method (top) and correlation method (bottom). We used here $N=5000$ data points, and $1000$ equally spaced values for $c$. Left: $\mu=3.55$ corresponding to regular dynamics; Middle: $\mu=3.9$ corresponding to chaotic dynamics; Right: $\mu=3.6$ corresponding to chaotic but non-mixing dynamics.[]{data-label="fig-cscan"}](Fig5b.eps "fig:"){width=".33\textwidth"} ![ Plot of $K_c$ versus $c$ for the logistic map calculated using the regression method (top) and correlation method (bottom). We used here $N=5000$ data points, and $1000$ equally spaced values for $c$. Left: $\mu=3.55$ corresponding to regular dynamics; Middle: $\mu=3.9$ corresponding to chaotic dynamics; Right: $\mu=3.6$ corresponding to chaotic but non-mixing dynamics.[]{data-label="fig-cscan"}](Fig5c.eps "fig:"){width=".33\textwidth"}
![ Plot of $K_c$ versus $c$ for the logistic map calculated using the regression method (top) and correlation method (bottom). We used here $N=5000$ data points, and $1000$ equally spaced values for $c$. Left: $\mu=3.55$ corresponding to regular dynamics; Middle: $\mu=3.9$ corresponding to chaotic dynamics; Right: $\mu=3.6$ corresponding to chaotic but non-mixing dynamics.[]{data-label="fig-cscan"}](Fig5a_d.eps "fig:"){width=".33\textwidth"} ![ Plot of $K_c$ versus $c$ for the logistic map calculated using the regression method (top) and correlation method (bottom). We used here $N=5000$ data points, and $1000$ equally spaced values for $c$. Left: $\mu=3.55$ corresponding to regular dynamics; Middle: $\mu=3.9$ corresponding to chaotic dynamics; Right: $\mu=3.6$ corresponding to chaotic but non-mixing dynamics.[]{data-label="fig-cscan"}](Fig5b_d.eps "fig:"){width=".33\textwidth"} ![ Plot of $K_c$ versus $c$ for the logistic map calculated using the regression method (top) and correlation method (bottom). We used here $N=5000$ data points, and $1000$ equally spaced values for $c$. Left: $\mu=3.55$ corresponding to regular dynamics; Middle: $\mu=3.9$ corresponding to chaotic dynamics; Right: $\mu=3.6$ corresponding to chaotic but non-mixing dynamics.[]{data-label="fig-cscan"}](Fig5c_d.eps "fig:"){width=".33\textwidth"}
The occurrence of resonances for isolated values of $c$ suggests using the median of the computed values of $K_c$. (We use the median rather than the mean, since the median is robust against outliers associated with resonances.)
In Fig. \[fig-cscan\]c, $K_c$ is shown as a function of $c$ for $\mu=3.6$ where the dynamics is chaotic but not mixing on the whole interval $[0,1]$. The actual dynamics in the logistic map oscillates between two disjoint sets, each of which is mixing, and there is a resonance at $c=\pi$. At resonance, $p_c(n)\sim n$ and $M_c(n)\sim
n^2$ as before. Close to resonance, the $p$-$q$ plot eventually behaves like Brownian motion, but in practice one sees only a small part of this motion and so $K_c\approx 0$.
\[rmk-c\] Naturally, the choices of $c$ are equally spaced in Fig. \[fig-cscan\], whereas in applying the test (and throughout the paper with the exception of Fig. \[fig-cscan\]) we choose randomly sampled values of $c$.
To avoid that resonances distort the statistics, we further restrict the range of randomly sampled values for $c$ to $c\in(\pi/5 ,4 \pi/5)$ for all our computations. The resonance at $c=0$ is inherent to our test, but it may leak through adjacent values of $c$ as seen in Fig. \[fig-cscan\]c. The further restriction to exclude $\pi$ is not necessary, but we found it helpful. (A typical route to chaos is the Feigenbaum route via period doubling. Here, the parameter ranges for fixed points and period two points are largest.)
In Fig. \[fig-c-Nc\], we show how the result for $K$ depends on the number $N_c$ of different values of $c$. Here we use the correlation version of the test to calculate $K_c$ as described in Section \[sec-corr\]. There is no measurable gain in increasing $N_c$ from $100$ to $1000$ and we find that generally $N_c = 100$ different values of $c$ is sufficient.
Finite size problems {#sec-confidence}
====================
There are three types of finite size effects. First, the time series needs to be long enough to explore and sample the relevant phase space area (i.e. the attractor). This is an inherent problem affecting all tests for chaos. Second, the definition of the mean square displacement involves a limit which requires $n \ll N$. Accordingly, we have chosen $n\le n_{\rm{cut}}=N/10$.
Third, the theory developed in [@GM04; @GMprep] makes statements about the asymptotic behaviour of $D_c(n)$ (or $M_c(n)$) and as such requires $n_{\rm cut}$, and hence $N$, to be sufficiently large. Finite size effect in this context means that for small $n$ the asymptotic linear growth is not yet dominating, see Fig. \[fig-logM\]. This finite size effect is explored in the remainder of this section. From now on, we work exclusively with the modified mean square displacement $D_c(n)$ and the correlation method.
In Fig. \[fig-K\_of\_N\] we show how the value of $K$ depends on the amount of data used. We can see clearly the convergence towards the asymptotic values $K=0$ and $K=1$ for regular and chaotic underlying dynamics, respectively. (For values of $\mu$ corresponding to stronger chaotic dynamics well within the chaotic range, the convergence towards $K=1$ is even more rapid.)
![ Plot of $K$ versus the available amount of data $N$ for the logistic map. Left: $\mu=3.55$ corresponding to regular dynamics; Right: $\mu=3.6$ corresponding to chaotic dynamics.[]{data-label="fig-K_of_N"}](Fig7a_N.eps "fig:"){width=".495\textwidth"} ![ Plot of $K$ versus the available amount of data $N$ for the logistic map. Left: $\mu=3.55$ corresponding to regular dynamics; Right: $\mu=3.6$ corresponding to chaotic dynamics.[]{data-label="fig-K_of_N"}](Fig7b_N.eps "fig:"){width=".495\textwidth"}
In the case of “weak chaos”, close to the so called “edge of chaos”, longer data sets are required to obtain $K=1$. Weak chaos is characterized by a slow decay of correlations. This has consequences for the modified mean square displacement $D_c(n)=V(c) n + o(n)$. For systems whose auto-correlation function is slowly decaying, it may be the case that the $o(n)$ term dominates for the available data. We illustrate this problem in the context of the logistic map. The bifurcation parameter $\mu$ takes the value $\mu=\mu_{\infty}=3.569945672\dots$ at the edge of chaos and for $\mu=\mu_{\infty}+0.001$ one observes weak chaos.
It has been erroneously claimed that our test cannot detect weak chaos, see [@HuTung05; @GMsub]. In fact, there are two methods whereby we can distinguish between regular dynamics and weak chaos:
- By visual inspection of the plot in the $p$-$q$ plane generated as in Fig. \[fig-pq2\]. (Note that for longer data sets the dynamics in the $p$-$q$ plane in Fig. \[fig-pq2\]b would look just like Fig. \[fig-pq\]b.)
- By looking at the dependence of $K$ as a function of $N$. As illustrated in Fig. \[fig-Kn\], we can distinguish weakly chaotic from regular dynamics even when the value of $K$ is very small – note that $K=0.027$ for $N=2000$ in the weakly chaotic case.
![ Plot of $p$ versus $q$ for the logistic map. Left: $\mu=\mu_{\infty}$; Right: $\mu=\mu_{\infty}+0.001$. We used $5000$ data points.[]{data-label="fig-pq2"}](Fig_pq_8a.eps "fig:"){width=".4\textwidth" height=".4\textwidth"} ![ Plot of $p$ versus $q$ for the logistic map. Left: $\mu=\mu_{\infty}$; Right: $\mu=\mu_{\infty}+0.001$. We used $5000$ data points.[]{data-label="fig-pq2"}](Fig_pq_8b.eps "fig:"){width=".4\textwidth" height=".4\textwidth"}
![ Plot of $K$ as a function of $N$ for the logistic map at $\mu=\mu_\infty$ (left) and $\mu=\mu_\infty+0.001$ (right). Although the value of $K$ is small in both cases, the behaviour of $K$ as a function of $N$ distinguishes the two cases.[]{data-label="fig-Kn"}](Fig8a_N.eps "fig:"){width=".475\textwidth"} ![ Plot of $K$ as a function of $N$ for the logistic map at $\mu=\mu_\infty$ (left) and $\mu=\mu_\infty+0.001$ (right). Although the value of $K$ is small in both cases, the behaviour of $K$ as a function of $N$ distinguishes the two cases.[]{data-label="fig-Kn"}](Fig8b_N.eps "fig:"){width=".495\textwidth"}
Continuous time systems {#sec-cont}
=======================
In the previous sections, the $0$–$1$ test was formulated for discrete time systems. For continuous time series $\phi(t)$, there is a well-known oversampling issue that must be addressed. In this section, we discuss this difficulty and how to overcome it.
Given $0<t_1<t_2<t_3<\cdots$ we obtain a discrete time series $\phi(t_1)$, $\phi(t_2)$, $\phi(t_3),\ldots$ to which the test for chaos may be applied as in previous sections. (The sequence $t_j$, $j\ge1$, should be chosen in a deterministic manner so that the time series $\phi(t_j)$ is deterministic.) One method of choosing the $t_j$ is as the intersection times with a cross-section, so the time series $\phi(t_j)$ corresponds to observing a Poincaré map. In this situation, there are no issues with oversampling.
A second, perhaps more usual, approach is to take $t_j=j\tau_s$ where $\tau_s>0$ is the sampling time. The time series $\phi(t_j)=\phi(j\tau_s)$ corresponds to observing the “time-$\tau_s$” map associated with the underlying continuous time system. If $\tau_s$ is too small, then the system is [*oversampled*]{} and this often leads to incorrect results. To illustrate the issue of oversampling we study the $3$-dimensional Lorenz system $$\begin{aligned}
\nonumber
\dot x &= {\textstyle10}(y-x)\\
\dot y &= {\textstyle30}\,x-y-xz
\label{lorenz}
\\
\dot z &= xy - {\textstyle\frac83} z \; ,
\nonumber\end{aligned}$$ which exhibits robust chaos. We have integrated this system with a time step of $\Delta t=0.001$ and recorded $100,000$ data points (ie.$100$ time units).
Fig. \[fig-sampleT\] shows an oversampled and a sufficiently coarsely sampled observable for the Lorenz system (\[lorenz\]). The finely sampled time series ($\tau_s=0.005$) yields $K\approx0$ even for $N=100,000$ whereas the coarsely sampled data ($\tau_s=0.05$) yields $K\approx 1$ already for $N=5,000$.
![Plot of the observable $\phi(t)=x(t)$ for the Lorenz system (\[lorenz\]). The finely sampled data (red) are sampled at $\tau_s=0.005$ time units. The coarsely sampled data (green filled circles) are sampled at $\tau_s=0.05$ time units.[]{data-label="fig-sampleT"}](Fig19.eps){width=".695\textwidth"}
A good choice of the sampling time $\tau_s$ can often be obtained by visual inspection as in Fig. \[fig-sampleT\]. A more refined method is to use the first minimum of the [*mutual information*]{} [@FraserSwinney86; @Kantz]. For the data depicted in Fig. \[fig-sampleT\] this method yields $\tau_s=0.17$ (roughly a quarter of the oscillation period). Note however that in this particular instance the smaller sampling time $\tau_s=0.05$ already gives $K\approx 1$ and extracts a longer time series from the data in Fig. \[fig-sampleT\]. In general, the optimal sampling time will depend on the dynamical system and the time series under consideration. We refer the reader to [@Kantz] for a discussion on optimal time delays in the context of phase space reconstruction.
Although oversampling is a practical problem for data series of finite size, it should be emphasized that theoretically the test works for all sampling times $\tau_s$ in the limit $N\to \infty$.
Oversampling and power spectra
------------------------------
For continuous time systems, the mean square displacement is defined as $$\begin{aligned}
M_c(t) = \lim_{T\to\infty}\frac{1}{T} \int_0^T(p_c(t+\tau)-p_c(\tau))^2\, + \,
(q_c(t+\tau)-q_c(\tau))^2\; d\tau \; .\end{aligned}$$ For a time series sampled with sample time $\tau_s$ this can be approximated by $$\begin{aligned}
M_c(n) = \lim_{N\to\infty}\frac{1}{N} \sum_{j=1}^{N}
\left( [p_{c\tau_s}(j+n)-p_{c\tau_s}(j)]^2\, + \, [q_{c\tau_s}(j+n)-q_{c\tau_s}(j)]^2 \right)
\tau_s^2\; .\end{aligned}$$ Similarly the power spectrum for the time-continuous case discretizes to $$\begin{aligned}
\label{eq-Sc}
S(\nu)=\lim_{n\to\infty}\frac1n
E\Bigl|\sum_{j=0}^{n-1}e^{2 \pi i\frac{\nu}{\nu_s} j}\phi(j)\Bigr|^2
\tau_s^2, $$ where $\nu_s=1/\tau_s$ is the sample frequency. The power spectrum consists of discrete peaks if the underlying system is regular, and is nowhere zero for a large class of chaotic systems [@MGapp]. However, for chaotic systems the power spectrum decays for large frequencies $\nu$, and so for frequencies larger than some $\nu_{\rm{max}}$ the power spectrum is zero for all practical purposes.
Comparing (\[eq-Sc\]) with the power spectrum (\[eq-S\]) for discrete-time data, we identify $$c = 2 \pi \frac{\nu}{\nu_s}, \quad \nu \in[0,\nu_{\rm{max}}]\; .$$ Sampling at the Nyquist rate with $\nu_s^\star=2
\nu_{\rm{max}}$ yields $c\in(0,\pi)$ as before. However, oversampling at a higher frequency $\nu_s>\nu_s^\star$, restricts the effective choices of $c$ to $c\in(0,c^\star)$ where $c^\star=\frac{\nu_s^\star}{\nu_s} \pi <\pi$. There is now a positive probability that the test for chaos will incorrectly yield $K=0$ since it is possible that more than half of the randomly chosen values of $c\in(0,\pi)$ will lie in $(c^\star,\pi)$.
We illustrate the previous argument using the Lorenz system (\[lorenz\]) sampled with $\tau_s$ ranging from $\tau_s=\Delta t$ up to $\tau_s=300\Delta t$. In Fig. \[fig-sampleK\] the median of the asymptotic growth rate $K$ is shown as a function of the sample time. For data that is too finely sampled, we obtain $K=0$ although the dynamics is actually chaotic.
![Plot of $K$ as a function of the sample time $\tau_s$ for the Lorenz system (\[lorenz\]). The sample time is measured in units of $\Delta t=0.001$.[]{data-label="fig-sampleK"}](Fig15.eps){width=".495\textwidth"}
Fig. \[fig-sampleKc\] illustrates how the range of effective values of $c$ depends on the sampling time $\tau_s$.
![Plot of $K_c$ as a function of the frequency $c$ for the Lorenz system (\[lorenz\]). From left to right we used $\tau_s=5
\Delta t$, $\tau_s=10 \Delta t$, $\tau_s=20 \Delta t$, $\tau_s=30
\Delta t$, $\tau_s=50 \Delta t$, $\tau_s=70 \Delta t$. The linear scaling of the range of $c$ for which $K_c\approx 1$ is evident in the relative spacing of the respective lines. []{data-label="fig-sampleKc"}](Fig14.eps){width=".495\textwidth"}
Noise contaminated data {#sec-noise}
=======================
Real-world data is invariably contaminated with noise. Any method for distinguishing regular from chaotic dynamics can only succeed if the noise-level is sufficiently small. There are various standard noise reduction techniques [@Kantz] that may be applied in advance of applying any given test for chaos. In addition, the test itself may be modified. Below we indicate a modification of the $0$–$1$ test for chaos that makes it more robust to the presence of noise.
In [@GM05] we introduced a version of the $0$–$1$ test that works well for data contaminated with measurement noise. (This is the test as presented in Sections \[sec-MSQ\] and \[sec-Kc\], but using $M_c(n)$ instead of $D_c(n)$ and using the regression method instead of the correlation method.) We showed that the $0$–$1$ test clearly outperforms tangent space methods and compares favourably to “direct methods” based on phase space reconstruction. The improvements in this paper have made our test extremely sensitive to weak chaos. However, an unavoidable consequence is an increased sensitivity also to noise (see Fig. \[fig-noise\] below).
It turns out that the success of the version of the test in [@GM05] is due to the oscillatory term $V_{\rm{osc}}(c,n) =
(E\phi)^2 \frac{1-\cos nc}{1-\cos c}$ that we subtracted in Section \[sec-MSQ\] to define the modified mean-square-displacement $D_c(n)=M_c(n)-V_{\rm{osc}}(c,n)$. This term desensitizes the test and damps the ability to detect slow growth of the mean-square-displacement for time-series data of moderate length. Instead of reintroducing this term we adopt a more flexible approach, defining $$\begin{aligned}
D^\star_c(n)=D_c(n) + \alpha V_{\rm damp}(n), \quad
V_{\rm damp}(n)=(E\phi)^2 \sin(\sqrt{2}n)\; .\end{aligned}$$ (The frequency $\sqrt{2}$ was chosen arbitrarily.) For $\alpha$ large, we expect $K=0$. The amplitude $\alpha$ of the term $V_{\rm
damp}(n)$ controls the sensitivity of the test to weak noise and simultaneously to weak chaos. This trade-off is unavoidable in any test for chaos.
As an illustration, we consider the logistic map with measurement noise. Take as observable $\phi(n)=x_n$ and write $$\tilde\phi(n)=\phi(n)(1 + \frac{\epsilon}{100}\eta_n)$$ where $\eta_n$ are i.i.d. random variables drawn from a uniform distribution on $[-1,1]$ and $\epsilon$ is the noise-level in percent. Fig. \[fig-noise\] shows how the undamped version of the test in this paper copes with a noise level of $10\%$ and the improvement that is obtained by using the damped mean-square-displacement $D_c^*(n)$. We obtain similar results for normally distributed noise.
![ Plot of $K$ versus $\mu$ for the logistic map increased in increments of $0.001$. The darker (red) lines were computed using clean data. The lighter (green) lines were computed after addition of $10\%$ uniformly distributed measurement noise. Both lines were computed using the undamped mean-square-displacement $D_c(n)$. Left: $N=1000$ using the undamped mean-square-displacement $D_c(n)$, Right: $N=5000$ using $D_c(n)$. Bottom: $N=5000$ using the damped mean-square-displacement $D_c^*(n)$ with $\alpha=2.5$. []{data-label="fig-noise"}](Fig9a.eps "fig:"){width=".495\textwidth"} ![ Plot of $K$ versus $\mu$ for the logistic map increased in increments of $0.001$. The darker (red) lines were computed using clean data. The lighter (green) lines were computed after addition of $10\%$ uniformly distributed measurement noise. Both lines were computed using the undamped mean-square-displacement $D_c(n)$. Left: $N=1000$ using the undamped mean-square-displacement $D_c(n)$, Right: $N=5000$ using $D_c(n)$. Bottom: $N=5000$ using the damped mean-square-displacement $D_c^*(n)$ with $\alpha=2.5$. []{data-label="fig-noise"}](Fig9b.eps "fig:"){width=".495\textwidth"}
![ Plot of $K$ versus $\mu$ for the logistic map increased in increments of $0.001$. The darker (red) lines were computed using clean data. The lighter (green) lines were computed after addition of $10\%$ uniformly distributed measurement noise. Both lines were computed using the undamped mean-square-displacement $D_c(n)$. Left: $N=1000$ using the undamped mean-square-displacement $D_c(n)$, Right: $N=5000$ using $D_c(n)$. Bottom: $N=5000$ using the damped mean-square-displacement $D_c^*(n)$ with $\alpha=2.5$. []{data-label="fig-noise"}](Fig20.eps){width=".495\textwidth"}
Under the assumption that the noise is diffusive and not correlated with the dynamics, the mean square displacement for data contaminated with measurement noise may be written as $$D_c(n)=(V_{\rm{dyn}}(c)+V_{\rm{noise}}(c))n + o(n)\;,$$ where for a given value of $c$, $V_{\rm{dyn}}(c)$ is the variance associated with the deterministic dynamics and $V_{\rm{noise}}(c)$ the variance associated with the measurement noise. Consider an idealized situation where the value of $V_{\rm{noise}}(c)$ is roughly constant as a parameter $\lambda$ is varied. Suppose further that the dynamics is known to be regular at $\lambda=\lambda_0$. Then we may estimate $V_{\rm{noise}}(c)$ by making a gauge-measurement at $\lambda=\lambda_0$, applying the correlation method to $D_c(n)-Vn$. The unique value $V=V_c(\lambda_0)$ which yields $K_c=0$ is our estimate for $V_{\rm{noise}}(c)$. For other values of $\lambda$ we may now apply the correlation method to $D_c(n)-V_c(\lambda_0)n$.
Discussion {#sec-Discussion}
==========
We have presented a guide for the implementation of the $0$–$1$ test for chaos. At the same time, we have introduced an improved version of the test which uses analytical expressions derived in [@GMprep]. Issues such as oversampling for continuous-time data and the presence of noise have been discussed. We hope that this guide will be helpful for scientists who would like to use the test.
There are numerous methods in the literature for distinguishing between deterministic and chaotic dynamics. In our previous papers [@GM04; @GM05], we made a careful comparison of the $0$–$1$ test with methods for computing the maximal Lyapunov exponent. Another method is to use the power spectrum for which there are efficient computational techniques. It should be pointed out however that these techniques generally rely on the Wiener-Khintchine theorem which assumes summable decay of correlations and hence excludes periodic and quasiperiodic dynamics. Hence to use power spectra as a test for chaos, it seems necessary to avoid the Wiener-Khintchine theorem and to work directly with the expression $\lim_{n\to\infty}\frac1n E \Bigl|\sum_{j=0}^{n-1}e^{ijc} \phi(j)
\Bigr|^2$. Equation shows the relationship between the $0$–$1$ test and power spectra, and our test can be viewed as a way of condensing the information relevant for chaoticity or regularity contained in the power spectrum into a single binary number.
#### Acknowledgments
We would like to thank Ramon Xulvi-Brunet for pointing us towards the correlation method, and for explaining us the problem of oversampling in power spectra. We would like to thank Michael Breakspear and John Dawes for providing encouraging feedback on an earlier version of the manuscript. GAG is partly supported by ARC grant DP0452147 and DP0667065. The research of IM was partly supported by EPSRC grant EP/D055520/1 and by a Leverhulme Research Fellowship. IM is grateful to the University of Sydney for its hospitality.
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[^1]: Mathematics and Statistics, University of Sydney, NSW 2006, Australia. gottwald@maths.usyd.edu.au
[^2]: Mathematics and Statistics, University of Surrey, Guildford, Surrey GU2 7XH, UK. ism@math.uh.edu
|
---
abstract: 'We present an analytic formalism that describes the evolution of the stellar, gas, and metal content of galaxies. It is based on the idea, inspired by hydrodynamic simulations, that galaxies live in a slowly-evolving equilibrium between inflow, outflow, and star formation. We argue that this formalism broadly captures the behavior of galaxy properties evolving in simulations. The resulting equilibrium equations for the star formation rate, gas fraction, and metallicity depend on three key free parameters that represent ejective feedback, preventive feedback, and re-accretion of ejected material. We schematically describe how these parameters are constrained by models and observations. Galaxies perturbed off the equilibrium relations owing to inflow stochasticity tend to be driven back towards equilibrium, such that deviations in star formation rate at a given mass are correlated with gas fraction and anti-correlated with metallicity. After an early gas accumulation epoch, quiescently star-forming galaxies are expected to be in equilibrium over most of cosmic time. The equilibrium model provides a simple intuitive framework for understanding the cosmic evolution of galaxy properties, and centrally features the cycle of baryons between galaxies and surrounding gas as the driver of galaxy growth.'
author:
- |
\
\
$^1$ Astronomy Department, University of Arizona, Tucson, AZ 85721, USA\
$^2$ Hubble Fellow; Physics Department, University of California, Santa Barbara, CA 93106, USA\
$^3$ Veni Fellow; Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, Netherlands
title: 'An Analytic Model for the Evolution of the Stellar, Gas, and Metal Content of Galaxies'
---
Introduction
============
Galaxy formation involves a wide range of diverse physical processes operating on stellar to cosmological scales, including the hierarchical growth of structure, star formation, black hole accretion, and a plethora of poorly-understood feedback processes that strongly modulate galaxy growth. Given this complexity, it is surprising that galaxies display simple and tight scaling relations between many of their key constituents. These include well-established relations between the bulge velocity dispersion and central black hole mass [e.g. @gul09; @gra11], circular velocity and luminosity [@tul77], star formation rate and stellar mass [e.g. @dav08; @gon10 and references therein], and metallicity and stellar mass [e.g. @tre04; @erb06]. Each has low scatter and evolves roughly independently of mass. The simplicity of these relations hints at an underlying uniformity in galaxy evolution that is not immediately evident from the complexity of current hierarchical galaxy formation models.
In the longstanding canonical scenario for galaxy formation, galaxies form as angular momentum-conserving disks cooling from hot gas bound within dark matter halos, and these disks subsequently merge to form larger and earlier-type galaxies [@ree77; @whi78; @whi91; @mo98]. This scenario is well-situated within currently favored hierarchical cosmologies, and analytic models based on it have been quite successful at reproducing many observed galaxy properties [see review by @ben10]. However, the present generation of such models (often called “semi-analytic" models) are typically enormously complex, with a host of free parameters describing various interrelated physical phenomena. Numerical simulations that explicitly track gas dynamical processes enable a more ab initio calculation, but still require many “sub-grid" parameters for key physical processes and are in practice limited by dynamic range and numerical uncertainties. In either case, the complexity of such models makes it difficult to extract simple physical intuition for what drives the evolution of basic galaxy properties.
In this paper, we present an analytic framework for understanding the evolution of the stellar, gas, and metal content of galaxies. This framework is based on intuition gained from hydrodynamic simulations of galaxy formation. In such models, galaxies are fed primarily by cold ($\sim 10^4$ K) streams connecting to filamentary large-scale structure [@ker05; @dek09], outflows are strong and ubiquitous [@spr03b; @opp08], and outflowing material commonly returns to galaxies . Hence in this framework, galaxy evolution is governed by the cycle of baryons exchanging matter and energy between galaxies and surrounding intergalactic gas.
Our framework is an attempt to distill the insights gained from such hydrodynamic simulations into an analytic formalism that both describes the results of simulations and provides intuition into the key physical drivers. It is based primarily on the formalism presented in @fin08, with key extensions, and shares features with various recent works [e.g. @ras06; @bou10; @dut10; @kru11], indicating a groundswell towards this “baryon cycling"[^1] view of galaxy formation. We demonstrate that the star formation rate, gas content, and metallicity of galaxies can be described by simple equations that depend on three parameters that are directly related to inflows, outflows, and wind recycling. These parameters are poorly constrained in both value and functional form, and hence the number of free parameters in this model may be much greater than three, pending observations that can better constrain them. Together, these parameters quantify the impact of baryon cycling on galaxy growth. We give examples of how these equations lead to straightforward intuitive explanations, often differing from traditional ones, for the results seen in recent observations and models of galaxy evolution.
This paper begins in §\[sec:basics\] by describing the basis for our analytic framework, namely the equilibrium condition, and discusses the physical constraints on its various terms that ultimately govern stellar growth. In §\[sec:mgr\] we present an expression for gas fractions and explore some implications. §\[sec:mzr\] discusses what governs galaxy metallicities, and relates this to wind recycling. §\[sec:equil\] gives some brief examples of how these equilibrium relations yield straightforward intuition into what governs basic galaxy properties. §\[sec:sample\] discusses a preliminary implementation of an equilibrium model, and explores some parameter variations. §\[sec:scatter\] considers what happens when galaxies depart from equilibrium owing both to stochastic fluctuations and more permanent departures. §\[sec:zeq\] discusses when galaxies first attain equilibrium in the early universe. Finally, we summarize and discuss broader implications of our framework in §\[sec:summary\].
The Equilibrium Condition {#sec:basics}
=========================
Star-forming galaxies in hydrodynamic simulations are usually seen to lie near the [*equilibrium condition*]{} [see e.g. Figure 13 of @fin08 and @dut10 [@bou10]]: $$\label{eqn:equil}
\dot{M}_{\rm in} = \dot{M}_{\rm out} + \dot{M}_{\rm *},$$ where the terms are the mass inflow rate, mass outflow rate, and star formation rate (SFR), respectively. Inflow and outflow refer to gas motion in and out of the galaxy’s star-forming region, i.e. the interstellar medium (ISM). Star-forming galaxies fluctuate around this relation but are generally driven back to it on short timescales, as we discuss in §\[sec:scatter\]; this “self-regulating" behavior is why we dub this model the [*equilibrium model.*]{}
The equilibrium condition is close to an expression for mass conservation, except that it importantly does not contain a term describing a gas reservoir. A key ansatz of this formalism is that the rate of change in the gas reservoir is small compared to the other terms in Equation \[eqn:equil\]. The motivation for this ansatz is that in @fin08, we found this to be explicitly true in hydrodynamic simulations of galaxy formation. We note that this scenario has also been referred to as a “reservoir" or “bath tub" model [@bou10; @kru11].
Defining the mass loading factor $\eta\equiv \dot{M}_{\rm
out}/\dot{M}_{\rm *}$, we can rewrite the equilibrium condition as $$\label{eqn:sfr}
{\rm SFR} = \dot{M}_{\rm in}/(1+\eta).$$ Hence in this scenario, a galaxy’s star formation history over cosmic timescales is determined by the evolution of $\dot{M}_{\rm in}$ and $\eta$.
Let us consider inflow first. $\dot{M}_{\rm in}$ can be broadly separated into three terms:
- $\dot{M}_{\rm grav}=$ Baryonic inflow into galaxy’s halo, which is primarily set by the assumed cosmology. Here we employ the form forwarded by @dek09: $$\label{eqn:Min}
\frac{\dot{M}_{\rm grav}}{M_{\rm halo}}
= 0.47 f_b \Bigl(\frac{M_{\rm halo}}{10^{12} M_\odot}\Bigr)^{0.15} \Bigl(\frac{1+z}{3}\Bigr)^{2.25}\;{\rm Gyr}^{-1}.$$ @fak10 presented a different parameterization of $25.3 M_{\rm
halo}^{0.1}(1+1.65z)\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}\;
M_\odot/$yr, while @fau11 found $33.6 M_{\rm
halo}^{0.06}(1+0.91z)\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}\;
M_\odot/$yr; these yield similar results over most of cosmic time.
- $\dot{M}_{\rm prev}=$ The amount of the gas entering the halo that is [*prevented*]{} from reaching the ISM (to be subtracted from the halo infall). This is material that ends up in the gaseous halo of the galaxy, and can also be regarded as the rate of growth of halo gas. We characterize this by defining a [*preventive feedback parameter*]{} $$\label{eqn:zeta}
\zeta\equiv 1-\dot{M}_{\rm prev}/\dot{M}_{\rm grav}.$$
- $\dot{M}_{\rm recyc}=$ Gas infalling that has previously been ejected in outflows, along with gas returned to the ISM via stellar evolution. This provides an extra component in addition to the baryons associated with the gravitational infall of dark matter (i.e. $\dot{M}_{\rm grav}$).
Expressing $\dot{M}_{\rm in}$ in these terms, we obtain $$\label{eqn:Minsplit}
\dot{M}_{\rm in} = \dot{M}_{\rm grav}-\dot{M}_{\rm prev}+\dot{M}_{\rm recyc} = \zeta \dot{M}_{\rm grav} + \dot{M}_{\rm recyc}$$
Halo infall ($\dot{M}_{\rm grav}$) is driven by gravity, hence is mostly independent of feedback processes [@vdv11; @fau11] and is determined primarily by cosmology and the halo merger rate [e.g. @nei06; @nei08; @mcb09; @genel10]. In contrast, $\dot{M}_{\rm prev}$ and $\dot{M}_{\rm recyc}$ are direct consequences of feedback processes. We will discuss $\dot{M}_{\rm recyc}$ further in §\[sec:mzr\], when we will relate it to the metallicity of the infalling gas. This leaves the preventive feedback parameter $\zeta$, which we now consider.
There are a number of sources of preventive feedback, each with its own dependence on halo mass and redshift, the products of which comprise the total $\zeta$:
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- $\zeta_{\rm photo}$ represents suppression of inflow owing to photo-ionisation heating. This operates at low masses, and approaches zero below a [*photo-suppression mass*]{} that increases from a halo mass of $M_\gamma\sim 10^{8}M_\odot$ during reionisation to $M_\gamma\sim {\rm few}\times 10^{9}M_\odot$ at the present epoch [@gne00; @oka08].
- $\zeta_{\rm quench}$ is associated with whatever physical process(es) quench star formation in massive halos and prevents cooling flows, probably related to feedback from supermassive black holes [e.g. @som08]. It drops to zero above the [*quenching mass*]{} $M_{\rm q}\sim 10^{12}M_\odot$ [e.g. @cro06; @gab11], which may be higher at high-$z$ [@dek09]. This may further depend on the merger history of galaxies [@hop08].
- $\zeta_{\rm grav}$ reflects suppression of inflow by ambient gas heating owing to gravitational structure formation via the formation of virial shocks. Using hydrodynamic simulations with no outflows, @fau11 determined $$\label{eqn:zetagrav}
\zeta_{\rm grav} \approx 0.47\Bigl(\frac{1+z}{4}\Bigr)^{0.38}
\Bigl(\frac{M_{\rm halo}}{10^{12}M_\odot}\Bigr)^{-0.25}.$$ We note that those simulations did not include metal-line cooling, which may be an important effect for heating gas in virial shocks [@dek06; @ocv08]; nonetheless, we show in Figure \[fig:ssfrhalo\] that our simulations including metal-line cooling yield similar results.
- $\zeta_{\rm winds}$ is associated with additional heating of surrounding gas provided by energetic input from winds. This tends to affect lower-mass systems more, but is highly dependent on the physics of how outflows interact with surrounding gas, which is poorly understood. Recent results [@opp10; @vdv11; @fau11] have demonstrated that this can be a significant effect in plausible (though somewhat extreme) wind models. Note that although both $\zeta_{\rm winds}$ and $\eta$ arise from winds, the former is a preventive feedback parameter, whereas $\eta$ is an ejective feedback parameter; the two are not necessarily related in a simple way, so we keep them separate.
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Multiplying these terms together, we obtain $\zeta(M_{\rm halo})$ that is schematically illustrated in Figure \[fig:zeta\]. It is small at low and high halo masses owing to photo-suppression and quenching respectively, and approaches unity at intermediate halo masses which is where vigorous star formation can occur. This depiction is intended only to illustrate broad trends, as the actual values and functional forms of the various $\zeta$ terms are at best only qualitatively known. The general shape of this curve with a cutoff in accretion at low and high masses is long known [e.g. @tho96; @ker05; @cro06], although the quantitative masses for the cutoffs are debated to this day [for recent work on this see @bou10; @cat11].
There may be other sources of energetic preventive feedback that retard accretion such as cosmic rays, stellar winds, quasar outflows, local photo-ionisation, and magnetic fields, but their importance for global galaxy evolution has not yet been firmly established. There may also be subtle “amplification" effects by which two or more preventive mechanisms serve to strengthen each other beyond their individual impact [@paw09; @fin11b]. Hence the list of individual $\zeta$’s above is intended to illustrate the sort of physical processes contributing to preventive feedback, and how they might manifest in the overall shape of $\zeta$. The actual trend of $\zeta(M_{\rm halo})$ may involve more complicated and subtle effects than described here.
Figure \[fig:ssfrhalo\] illustrates the impact of these various inflow and feedback terms on the specific star formation rate (sSFR$\equiv$SFR$/M_*$) as a function of $M_{\rm halo}$, using large-scale cosmological hydrodynamic simulations with various outflow models [see @dav11 for details]. The upper left panel shows the case without outflows. If all gas entering into the halo ended up in the ISM, i.e. $\dot{M}_{\rm in}=\dot{M}_{\rm grav}$, then the relation would be as shown by the solid line, having a positive slope. Even without any feedback, gravitational heating results in a negative slope (dashed line). The results from our simulations are in good agreement with Equation \[eqn:zetagrav\] [from @fau11], since the simulations themselves are quite similar; the main difference is that ours include metal-line cooling, but this is not very important until hot gaseous halos form at $M_{\rm halo}\ga 10^{12}M_\odot$ [@ker05; @gab11] since cold accretion is generally limited by the infall time rather than the cooling time.
The simulations in the two right panels assume $\eta=2$. If $\dot{M}_{\rm in}$ is unchanged, then one expects both the SFR and $M_*$ to be lowered by a factor of three (eq. \[eqn:sfr\]), resulting in no change in sSFR from the no-wind case (the no-wind cyan curve is reproduced in all panels for comparison). But clearly there is a change, which reflects the impact of outflows on $\dot{M}_{\rm in}$. At large masses, wind recycling ($\dot{M}_{\rm
recyc}$) returns material to the galaxy rapidly once the wind speed (680 km/s in the upper right, 340 km/s in the lower right) drops below the escape velocity [@opp10]. Hence at large masses $\eta$ is effectively 0 [@fin08], and sSFR jumps owing to $\dot{M}_{\rm recyc}$. For the case of the slower 340 km/s winds (lower right), the jump occurs at a factor of eight lower in mass as expected from the factor of two difference in wind speeds. Winds also affect sSFR at low masses, where it is suppressed relative to no winds. This reflects $\zeta_{\rm winds}$, which is as expected stronger in the case of the higher wind speed. In the lower left we show simulations using momentum-driven wind scalings of (approximately) $\eta\propto M_{\rm halo}^{-1/3}$. This flattens sSFR$(M_{\rm halo})$, in addition to exhibiting different behaviors for wind recycling and suppression.
These examples illustrate how star formation rates at a given mass in simulations are impacted by the various ejective and preventive feedback processes described above. For instance, the fact that observations find no sudden increase in sSFR at any characteristic mass [e.g. @sal07; @dad07] suggests that galaxies do not eject material at a characteristic wind velocity.
@bou10, @dut10, and @kru11 also present empirical models based on accretion-driven star formation, focusing on the form and evolution of the sSFR. Interestingly, @bou10 demonstrates that a model in which $\zeta=1$ for $10^{11}<M_{\rm
halo}<10^{12.2} M_\odot$ and zero elsewhere nicely reproduces some key observed properties of high-redshift star-forming galaxies, including the observed lack of sSFR amplitude evolution from $z\sim
7\rightarrow 2$. Simulations, in contrast, predict that sSFR evolution tracks the accretion rate, and thus continues to rise to high redshifts [@dav08]; analytic models that do not include such a low-mass cutoff show similar behavior [@dut10]. If this low-mass cutoff for accretion were true, it would suggest that there are additional feedback processes affecting small systems beyond what is shown in Figure \[fig:zeta\], namely, metagalactic photo-ionisation. An alternative explanation that does not employ a sharp mass cutoff but still reproduces the observed sSFR behavior is to invoke an observationally-motivated metallicity-dependent star formation law, as explored by @kru11. Pushing such observations of sSFR out to higher redshifts and lower masses [e.g. with the Cosmic Assembly Near-Infrared Deep Legacy Survey; @gro11; @koe11] should provide interesting constraints on the physical mechanisms regulating early, low-mass galaxy growth.
Gas fractions {#sec:mgr}
=============
A galaxy’s gas fraction is defined here as $$\label{eqn:fgas}
{f_{\rm gas}}\equiv \frac{{M_{\rm gas}}}{{M_{\rm gas}}+M_*} = \frac{1}{1+({t_{\rm dep}}{\rm sSFR})^{-1}}$$ where in the second equality we have employed the [*depletion time*]{} ${t_{\rm dep}}\equiv {M_{\rm gas}}/$SFR. We argue below that the latter formulation offers the intuitive advantage that it splits ${f_{\rm gas}}$ into a term that is fairly insentive to feedback (${t_{\rm dep}}$) and a term that depends strongly on feedback (sSFR).
The depletion time measures the timescale over which gas, when present in the ISM, gets converted into stars. This is expected to be primarily determined by the star formation law, such as the observed @ken98 relation between gas surface density ($\Sigma_{\rm gas}$) to SFR surface density ($\Sigma_{\rm SFR}$). Indeed, simulations by @dav11b [hereafter DFO11; see their Figure 4] assuming a Kennicutt-Schmidt law show that ${t_{\rm dep}}$ is essentially independent of outflows, and scales as $${t_{\rm dep}}\propto t_H M_*^{-0.3},$$ where $t_H$ is the Hubble time.
We can derive this scaling of ${t_{\rm dep}}$ directly from the star formation law. The temporal scaling can be most easily understood using a formulation of the star formation law given by SFR$\approx
0.02 {M_{\rm gas}}/t_{\rm dyn}$, where $t_{\rm dyn}$ is the dynamical time of the star formation region [e.g. @sil97; @kru07; @gen10]. This then gives ${t_{\rm dep}}\propto t_{\rm dyn}$, which in a canonical disk model scales as the Hubble time $t_H$ [@mo98]. Meanwhile, the stellar mass dependence arises from the Kennicutt law plus the ISM gas profile. The Kennicutt law states that $\Sigma_{\rm SFR}\propto \Sigma_{\rm
gas}^N$ (where $N\approx 1.4$), from which it is straightforward to show that $t_{\rm dep}\propto \Sigma_{\rm gas}^{1-N}$. In simulations, $\Sigma_{\rm gas}\propto M_*^{3/4}(1+z)^{2}$ (DFO11), which gives rise to the weak anti-correlation with $M_*$ quoted above. We caution that these dependences may be somewhat different in the real Universe since these simulations lack the resolution to properly model the internal structure of galaxies.
The evolution of ${f_{\rm gas}}$ depends on the evolution of ${t_{\rm dep}}$ and sSFR. The former evolves with $t_H$ (e.g. as $(1+z)^{-1.5}$ in the matter-dominated regime), while the latter is generally driven by cosmic inflow (eq. \[eqn:Min\]) which scales as $(1+z)^{2.25}$ if driven by gravitional infall. Combining these, galaxy gas fractions are predicted to evolve with time roughly as $\sim (1+z)^{2.25}t_H$ (if ${f_{\rm gas}}$ is not near unity), which increases slowly with redshift, qualitatively consistent with observations [@tac10; @gea11]. Hence in the equilibrium scenario, galaxy gas fractions represent a competition between supply and consumption, such that [*galaxies become less gas-rich with time because the gas supply rate drops faster than the gas consumption rate.*]{}
Metallicities {#sec:mzr}
=============
The global metallicity within the ISM is given by the enrichment rate, which is the yield $y$ times SFR, divided by the mass inflow rate $\dot{M}_{\rm in}$ that must be enriched. As derived in @fin08, if the inflow is pre-enriched there is an additional term that depends on $\alpha_Z\equiv Z_{\rm in}/Z_{\rm ISM}$, where $Z_{\rm in}$ and $Z_{\rm ISM}$ are the metallicities of the inflowing and ambient ISM gas, respectively: $$\label{eqn:mzr}
Z_{\rm ISM} = y\frac{{\rm SFR}}{\dot{M}_{\rm in}} = \frac{y}{1+\eta} \frac{1}{1-\alpha_Z}.$$ Hence the mass-metallicity relation and its evolution are established by a the mass and redshift dependence of $\eta$ and $\alpha_Z$. In our currently favored outflow model [@dav11], $\eta$ has a significant mass dependence but little or no redshift dependence, while $\alpha_Z$ is generally small but has a significant redshift dependence. Equation \[eqn:mzr\] would then suggest that the shape of the mass-metallicity relation is primarily established by $\eta(M_*)$, while its evolution is driven by $\alpha_Z$; this was demonstrated for simulations in DFO11. Hence in this scenario, [*the shape of the mass-metallicity relation is modulated by the fraction of inflow that forms stars, while its evolution is governed by the enrichment level of infalling gas.*]{}
Conspicuously absent in this scenario is any explicit reference to potential wells of galaxies, or any consideration of outflow velocities versus escape velocities. These processes are canonically believed to govern the mass-metallicity relation [e.g. @dek86; @tre04]; the phrase “metals can more easily escape from the shallower potential wells of small galaxies" is oft-repeated. However, in our scenario, it is instead the net mass outflow rate that is the key determinant of the mass-metallicity relation, and the potential well depth is at most only indirectly implicated.
Since the vast majority of metals in the IGM are deposited there by outflows [e.g. @opp08; @opp11], the infalling gas metallicity is a direct measure of $\dot{M}_{\rm recyc}$. $Z_{\rm in}$ is given by the metal mass arriving in the form of recycled winds, divided by the total mass inflow rate, i.e. $$\label{eqn:Zin}
Z_{\rm in} = Z_{\rm recyc} \frac{\dot{M}_{\rm recyc}}{\dot{M}_{\rm recyc}+\zeta \dot{M}_{\rm grav}},$$ where the denominator is $\dot{M}_{\rm in}$ from Equation \[eqn:Minsplit\].
Under the typical case of highly mass-loaded outflows, the outflowing metallicity must be similar to the ambient ISM metallicity, i.e. $Z_{\rm out}\approx Z_{\rm ISM}$. Furthermore, since galaxies evolve slowly in metallicity [e.g. @bro07 DFO11] and wind recycling times are typically of order a Gyr [@opp10], the galaxy metallicity has probably not evolved strongly from when the gas was ejected to when it is being re-accreted, and hence $Z_{\rm
recyc}\approx Z_{\rm out}$. Substituting $Z_{\rm recyc}=Z_{\rm
ISM}$ into Equation \[eqn:Zin\] and solving for $\dot{M}_{\rm
recyc}$ yields $$\label{eqn:Mrecyc}
\dot{M}_{\rm recyc} = \frac{\alpha_Z}{1-\alpha_Z} \zeta \dot{M}_{\rm grav}.$$ This relates the mass recycling term in the inflow equation (eq. \[eqn:sfr\]) to the metallicity infalling into the ISM. The advantage of formulating recycling in terms of $\alpha_Z$ is that it is in principle an observable quantity via absorption or emission measures in the outskirts of galaxies. In contrast, $\dot{M}_{\rm recyc}$ is not directly measurable since it is not clear how to distinguish recycled wind inflow from other inflow, or even how to measure galaxy inflow rates at all. Note that since galaxy metallicities evolve slowly upwards with time, Equation \[eqn:Mrecyc\] will tend to slightly underestimate $\dot{M}_{\rm recyc}$ for a given $\alpha_Z$. The ejection of winds from one galaxy (typically a satellite) being accreted onto another (typically the associated central) could also affect $\alpha_Z$, which would also cause an underestimate in $\dot{M}_{\rm recyc}$ since the satellites are generally smaller and hence lower metallicity.
The Equilibrium Relations & Implications {#sec:equil}
========================================
We can substitute Equation \[eqn:Mrecyc\] into Equation \[eqn:Minsplit\] to obtain $$\label{eqn:sfrfull}
{\rm SFR} = \frac{\zeta \dot{M}_{\rm grav}}{(1+\eta)(1-\alpha_Z)}.$$ This is the key equation that delineates how galaxy star formation rates are governed by accretion and feedback processes, i.e. baryon cycling. This equation, together with Equations \[eqn:fgas\] and \[eqn:mzr\], represent the [*equilibrium relations*]{} that govern the stellar, gas, and metal content of galaxies across cosmic time. Galaxies will tend to lie around these relations owing to a balance of inflow, outflow, and star formation.
The equilibrium relations depend on three parameters: $\eta$, $\zeta$, and $\alpha_Z$, representing ejective feedback (i.e. outflows), preventive feedback, and wind recycling. Additionally, the star formation law governs ${t_{\rm dep}}$, $\dot{M}_{\rm grav}$ is set by cosmology, and $y$ is set by nucleosynthetic processes. Assuming those are well-established, the mass and redshift (and possibly environmental) dependence of $\eta$, $\zeta$, and $\alpha_Z$ govern the evolution of the global SFR, ${f_{\rm gas}}$, and $Z_{\rm ISM}$ of galaxies. Note that since the mass and redshift dependence of these parameters are not fully known, the actual number of free parameters can be significantly larger than three.
There are many possible ways to characterize simulation results into an analytic formalism [e.g. @nei11]. One virtue of our particular parameterization is that the parameters involved are, at least in principle, directly observable. This provides an optimally direct connection from observations to constraints on galaxy formation models. Unfortunately, measuring these parameters is challenging, but preliminary constraints have already been obtained.
For instance, $\eta$ has been constrained in high-$z$ galaxies to have a value of order unity or more [e.g. @ste10; @gen11]. $\alpha_Z$ can be constrained by examining metallicities in the outskirts of low-$z$ galaxies [e.g. @bre09; @mor11]. Constraining $\zeta$ by direct observations would require an accurate census of all halo gas which is highly challenging, but aside from $\zeta_{\rm winds}$, its main terms can be constrained using a combination of relatively straightforward numerical work ($\zeta_{\rm photo}$ and $\zeta_{\rm
grav}$) and empirical arguments ($\zeta_{\rm quench}$).
The equilibrium relations have some interesting implications for the behavior of SFR, ${f_{\rm gas}}$, and $Z$. For instance, hydrodynamic simulations indicate that the star formation history of galaxies is insensitive to the assumed star formation law [@kat96; @sch10]. This seems paradoxical at first, but is straightforwardly seen from Equation \[eqn:sfrfull\], since there is no dependence here (or in the metallicity equation) on the star formation law. The star formation law only affects the gas fractions, via ${t_{\rm dep}}$. This can be regarded as a self-regulation mechanism [@sch10], in which gas collects in galaxies as required in order to achieve the star formation rate set by the balance of inflows and outflows.
Another straightfoward prediction of the equilibrium relations is that if one desires the mass-metallicity relation to scale as $Z\propto M_*^{1/3}$ at small masses as observed [@tre04; @lee06], then Equation \[eqn:mzr\] directly implies $\eta\propto M_*^{-1/3}$ (assuming $\alpha_Z\ll 1$), roughly as expected for momentum-driven winds [@mur05]. Indeed, simulations assuming such a scaling appear to provide a good match to mass-metallicity relation observations [@fin08 DFO11].
It is instructive to combine Equations \[eqn:mzr\] and \[eqn:sfrfull\] to give $$\zeta = \frac{\rm SFR}{\dot{M}_{\rm grav}} \frac{y}{Z_{\rm ISM}}$$ The first ratio is the [*halo star formation efficiency*]{} (SFE), i.e., the fraction of gravitational infall into a halo that ends up forming into stars[^2], while the second ratio quantifies the metal retention fraction within galaxies. If the halo mass and metal yield can be determined, measuring SFR$/Z_{\rm ISM}$ provides a quantitative constraint on $\zeta$. This can be done at least at $z\sim 0$ with existing data from e.g. the Sloan Digital Sky Survey.
A Sample Equilibrium Model {#sec:sample}
==========================
The equilibrium model can be used to quickly explore parameter space and obtain intuition about the governing physics for galaxy properties of interest. We illustrate this here by presenting results for the evolution of galaxies in a full equilibrium model.
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Figure \[fig:equil\] shows the evolution of the SFR, halo SFE, $Z$, and ${f_{\rm gas}}$ for four galaxies spanning the indicated range of final ($z=0$) halo masses. These are computed using Equations \[eqn:fgas\], \[eqn:mzr\], and \[eqn:sfrfull\], tracking the stellar and halo mass growth starting at an early epoch when the halo is at the photo-suppression mass. We take $\dot{M}_{\rm
grav}$ from Equation \[eqn:Min\], parameterize ${t_{\rm dep}}=0.4 t_H
(M_*/10^{10} M_\odot)^{-0.3}$ as discussed in §\[sec:mgr\], and take $\alpha_Z=(0.5-0.1z)(M_*/10^{10}M_\odot)^{0.25}$ (with $\alpha_Z\geq0$) as a crude parameterization of simulation results from DFO11. We choose $\zeta$ as described in Figure \[fig:zeta\], and define $\eta$ as indicated in the upper right of the Figure: solid lines approximately represent momentum-driven wind scalings, while dotted lines represent energy-conserving wind scalings. We also include instantaneous recycling of 18% of star formation back into gas as expected for a @cha03 IMF, but do not include further stellar mass loss. We reiterate that these “base model" parameter choices are at some level arbitrary, and are intended only to illustrate how parameter variations influence observables.
The green line represents a Milky Way-sized halo of $10^{12} M_\odot$. At $z=0$, it has SFR$\approx 2.5 M_\odot$/yr, $M_*\approx 5\times
10^{10} M_\odot$, ${f_{\rm gas}}\approx 0.1$, and $Z\approx Z_\odot$, in fair agreement with measured values and showing that our parameter choices are reasonable. Larger galaxies form stars more vigorously at earlier epochs and for shorter intervals, which is qualitatively similar to the behavior in the empirical “staged" galaxy formation model of @noe07. The peak SFRs are 50–100 $M_\odot/$yr at $z\sim 2-3$, which is lower than the observed values for the largest main sequence galaxies at that epoch by a factor of a few, reiterating the issue noted in simulations by @dav08 that observed galaxy SFRs at that epoch approach or exceed their cosmic accretion rate; the resolution to this quandary remains unclear [see e.g. @bou10; @kru11 for ideas].
The halo SFE is plotted in the second panel. For star-forming galaxies, this efficiency is roughly one-third to one-half over most of cosmic time. Higher mass halos show a marked drop in efficiency once they grow above the quenching mass, here assumed to be $2\times 10^{12} M_\odot$ at all epochs. Going to a steeper scaling of $\eta(M_h)\propto M_h^{-2/3}$ (dotted lines) more strongly suppresses low-mass galaxy growth, which is favored in semi-analytic models to reproduce the observed faint-end slope of the stellar mass function [e.g. @som08].
Metallicity evolution is very rapid in all galaxies early on, and galaxies typically enrich to $\ga 0.1Z_\odot$ within the first Gyr, as seen in simulations [@dav06; @fin11]. Evolution is slow thereafter. Galaxies tend to follow parallel tracks, showing that the shape of the mass-metallicity relation does not evolve much (DFO11). A steeper scaling of $\eta(M_h)$ strongly steepens the mass-metallicity relation in smaller galaxies at all redshifts, as expected from Equation \[eqn:mzr\].
${f_{\rm gas}}$ drops slowly with time since the accretion rate drops faster than the consumption rate, as discussed in §\[sec:mgr\]. Here we include some observational comparisons (large symbols), from CO measurements by @tac10 and @gea11. As @tac10 noted, the relatively slow evolution requires continual replenishment, as our model naturally predicts. However, the $z\sim 1-2$ gas fractions are higher than the model predictions, because the observed galaxies are best compared to the green and magenta lines representing final halo masses of $\sim 10^{12}-10^{13} M_\odot$. The equilibrium model can quickly test whether parameter variations can reconcile this discrepancy.
According to Equation \[eqn:fgas\], higher ${f_{\rm gas}}$ values can be achieved by raising sSFR and/or ${t_{\rm dep}}$. Raising sSFR e.g. by lowering $\eta$ would concurrently alleviate the discrepancy with observed $z\sim
2$ SFRs, but would likely overproduce stars globally. Delaying star formation by accumulating gas could explain the discrepancy, though it seems somewhat contrived to have galaxies at all masses suddenly start vigorously consuming gas around $z\sim 2$. A steeper scaling of $\eta(M_h)$ works in this direction, but the dotted lines show that this only helps marginally; a redshift-dependent $\eta$ that is higher at high-$z$ may work better, but the physical motivation is not obvious. Another avenue is to raise ${t_{\rm dep}}$ by appealing to the observed lower gas consumption rate in low-metallicity systems [@bol11]. An illustration of this is shown as the dashed lines in Figure \[fig:equil\], where we add a dependence of $Z^{-2}$ to ${t_{\rm dep}}$ when below solar metallicity. Within this model, we find that this steep metallicity dependence is required in order to sufficiently raise gas fractions at high-$z$; however, it is not physically-motivated, and does not necessarily reflect recently proposed metallicity-dependent star formation laws [e.g. @kru11]. The delay in star formation results in higher gas fractions at $z\sim 2$ in better agreement with data, but at lower redshifts when metallicities approach solar this makes less difference. Another avenue explored by @bou10 is to postulate that star formation can only occur in halos above $10^{11}M_\odot$, which has the advantage of producing higher has fractions in smaller systems at earlier epochs; they demonstrate that this can broadly match the observed evolution of galaxy gas fractions.
This exercise illustrates how an equilibrium model can be utilized to gain intuition about various physical processes in galaxy evolution. The relatively small number of parameters and their direct connection to observable physical processes makes this type of model easier to interpret than modern simulations or semi-analytic models.
Departures From Equilibrium {#sec:scatter}
===========================
Stochastic variations in $\dot{M}_{\rm in}$, including mergers, can cause departures from equilibrium. This generates scatter about the equilibrium relations. Generically, departures from equilibrium tend to return galaxies towards equilibrium. This self-regulating behavior is why we dub this scenario the equilibrium model.
To illustrate this behavior, consider a galaxy experiencing an upward fluctation in $\dot{M}_{\rm in}$. Its ${f_{\rm gas}}$ increases, and perhaps $M_*$ as well if there is a small galaxy accompanying the infall, while $Z_{\rm ISM}$ decreases since either fresh infall or a lower-mass galaxy will have lower metallicity. This moves the galaxy off the equilibrium relations. The increased gas content immediately stimulates more vigorous star formation, which over time enriches the galaxy as it consumes the excess gas. This then lowers its gas content and increases its metallicity, moving the galaxy back towards equilibrium.
Conversely, a temporary lull in accretion will make a galaxy more gas-poor and metal-rich as it consumes its existing gas. Quantitatively, its metallicity will evolve following the relation $\Delta Z_{\rm
ISM}\approx y\Delta M_*/M_{\rm gas}$ [see eq. 19 of @fin08 in the case of zero accretion], which is steeper than the MZR and hence will move the galaxy above the MZR. In time, hierarchical growth will bring in fresh gas, lowering the metallicity and increasing the gas content to return the galaxy towards equilibrium. In this way, galaxies oscillate around the equilibrium relations, constantly being perturbed from them and driven back owing to fluctuations in infall.
An inevitable prediction of this scenario is that departures from the equilibrium relations will correlate with star formation rate, gas fraction, and metallicity. From the above scenarios, one can see that [*at a given mass, galaxies that are gas-rich (gas-poor) and metal-poor (metal-rich) will have higher (lower) star formation rates.*]{} These trends are qualitatively consistent with observations [@ell08; @lar10; @man10; @pee10].
This trend is illustrated for gas fractions and star formation rates by the coloured points in Figure \[fig:ssfrhalo\], showing that at a given mass, high SFR and high ${f_{\rm gas}}$ go hand in hand. Figure 1 of DFO11 analogously shows that high SFR accompanies low $Z_{\rm
gas}$ at a given $M_*$. Note that these second-parameter trends do not arise from outflows, being present even in simulations without winds. Instead, it is a direct and unavoidable consequence of equilibrium, and results from galaxies’ self-regulating response to fluctuations in inflow rather than any feedback process.
Quantitatively, the scatter around the equilibrium relations depends on how quickly galaxies can return to equilibrium after being perturbed. To return, there must be sufficient infall to re-equilibrate the galaxy. The timescale for this to happen can be quantified by the dilution time [@fin08]: $$\label{eqn:tdil}
t_{\rm dil}\equiv\frac{M_{\rm gas}}{\dot{M}_{\rm in}}=(1+\eta)^{-1} \frac{M_{\rm gas}}{\dot{M}_*}=(1+\eta)^{-1} t_{\rm dep}.$$ If the dilution time is small compared to the inflow fluctuation timescale, then the scatter will be small [@fin08]. The inflow fluctuation timescale likely depends on mass and environment, with small galaxies generally suffering (relatively) larger perturbations. Equation \[eqn:tdil\] shows that the dilution time depends on $\eta$ and ${t_{\rm dep}}$, and hence observations of the scatter versus $M_*$ provides an independent constraint on $\eta(M_*)$ and ${t_{\rm dep}}(M_*)$.
@dut10 argues, based on an analytic model of accretion-driven galaxy formation similar to this one, that the scatter in the observed $M_*-$SFR relation must be driven by fluctuations in inflow, since assuming zero scatter in the relation between $\dot{M}_{\rm
in}$ and $M_{\rm halo}$ results in a relation that is too tight compared to observations. However, hydrodynamic simulations that implictly include inflow fluctations also yield a small scatter [e.g. @dav08; @fin11]. Therefore it is not solely inflow fluctuations that govern the scatter, it is the more complex relationship between inflow fluctuations and the dilution time. Equation \[eqn:tdil\] suggests that this, in turn, depends on the outflow rate ($\eta$) and gas consumption rate (${t_{\rm dep}}$). The fact that feedback regularizes galaxy properties may have other effects on galaxy properties. For instance, it has long been suggested that the low scatter in the Tully-Fisher relation arises owing to feedback processes, since fluctuations in halo growth alone would naively predict a scatter that is large compared to data [@eis96].
Satellite galaxies lie permanently off the equilibrium relations, because the inflowing filaments bypass them and flow to the centers of halos. Hence they are expected to end up with lower gas content and higher metallicities than centrals of the same mass; this is as observed [@pee09]. How far they lie off the equilibrium relations depends on their gas reservoir at the time they are cut off from their supply, which simulations indicate is typically $\sim 1$ Gyr after falling into a hot gas-dominated halo [@sim09]. Note that some satellite galaxies fall in along the inflowing filaments, and in those cases they are not bypassed since they are actually part of the inflow. However, these satellites are expected to quickly merge into the central galaxy, and so will not typically end up as part of the long-lived satellite population.
Major mergers are another population lying far out of equilibrium. It is not merely that such systems represent a particularly large inflow perturbation, it is that the induced torques drive gas flows that fuel central star formation [@mih96] making cosmological inflow mostly irrelevant during the merger event. In a global context, major mergers are seen to be responsible for only a small fraction of overall cosmic star formation [e.g. @jog09], although they may be where much of the central black hole mass growth occurs [e.g. @dim05]. Overall, the equilibrium model is broadly valid for central galaxies that are quiescently forming stars, i.e. main-sequence galaxies [@noe07], that dominate cosmic star formation.
Before Equilibrium: The Gas Accumulation Phase {#sec:zeq}
==============================================
Since the infall rate has a steeper redshift dependence than the consumption rate (e.g. $\sim (1+z)^{2.25}$ vs. $t_H^{-1}$), at sufficiently early epochs galaxies will be in a [*gas accumulation phase*]{} during which galaxies cannot process gas into stars as fast as they receive it [@bou10; @kru11]. Only when consumption can keep up with supply will equilibrium be achieved. During the gas accumulation phase, gas fractions are expected to be higher and metallicities lower than predicted by equilibrium.
-0.4in
-3.2in
We can estimate the redshift $z_{\rm eq}$ where gas accumulation ends and equilibrium is attained. Star formation must be able to occur fast enough to satisfy SFR$= \dot{M}_{\rm in}/(1+\eta)$ (eq. \[eqn:sfr\]). We assume that at early times, $\zeta\approx
1$ and $\alpha_Z\approx 0$, so that $\dot{M}_{\rm in}\approx
\dot{M}_{\rm grav}$ (eq. \[eqn:Min\]). We further assume that SFR=$0.02 M_{\rm gas}/t_{\rm dyn}$, where $t_{\rm dyn}\approx 10^8
(1+z)^{-3/2}$ yr is the disk dynamical time. Taking $M_{\rm gas}\approx {f_{\rm gas}}f_b M_{\rm halo}$, we obtain an equation for $z_{\rm eq}$: $$\label{eqn:zeq}
1+z_{\rm eq}\approx
\Bigl[5 {f_{\rm gas}}(1+\eta) \Bigr]^{4/3} \Bigr(\frac{M_{\rm halo}}{10^{12} M_\odot}\Bigl)^{-0.2}.$$
We illustrate these trends in Figure \[fig:zeq\]. For sizeable high-$z$ galaxies with ${f_{\rm gas}}\approx 0.4-0.5$ [@tac10] and no outflows ($\eta=0$), we obtain $z_{\rm eq}\approx 1.5-2.5$, in agreement with @kru11. But the superlinear dependence on ${f_{\rm gas}}$ and $(1+\eta)$ means that the results are quite sensitive to these values. For instance, even a modest outflow rate of $\eta=1$ yields $z_{\rm eq}\approx 5-7$. Effectively, outflows lower the amount of inflow that needs to be processed into stars, allowing for an earlier equilibration epoch. There is a weak halo mass dependence as well such that smaller galaxies equilibrate earlier. Note that this derivation of $z_{\rm eq}$ depends on the star formation law: if the star formation law was different in early galaxies owing e.g. to metallicity effects, then this would also impact when galaxies achieve equilibrium. As such, a precise prediction of $z_{\rm eq}$ is sensitive to poorly known factors. Nevertheless, with realistic outflows, it is likely that $z_{\rm
eq}\gg 2$, and hence galaxies live in equilibrium over the vast majority of cosmic time.
Observationally, @pap10 used the star formation rates, masses, and (estimated) gas contents of high-$z$ Lyman break galaxies to infer that gas accumulation occurs down to $z_{\rm eq}\sim 4$, after which accretion and star formation track each other as expected in equilibrium. Hence there is some direct empirical support for an early gas accumulation epoch. Constraining $z_{\rm eq}$ more precisely will provide quantitative constraints on gas processing rates in early galaxies.
Summary and Discussion {#sec:summary}
======================
We have presented a simple formalism for understanding the evolution of the stellar, gaseous, and metal content of galaxies, inspired by intuition gained from cosmological hydrodynamic simulations. This formalism is encapsulated by the equilibrium relations: $$\begin{aligned}
\label{eqn:full}
{\rm SFR} &=& \frac{\zeta \dot{M}_{\rm grav}}{(1+\eta)(1-\alpha_Z)}, \\
{f_{\rm gas}}&=& \frac{1}{1+({t_{\rm dep}}{\rm sSFR})^{-1}},\\
Z_{\rm ISM} &=& \frac{y}{1+\eta} \frac{1}{1-\alpha_Z}.\end{aligned}$$ These relations are established by a balance between inflows and outflows, and evolve on cosmological timescales over which inflow and outflow rates slowly vary. They are primarily governed by three baryon cycling parameters that describe ejective feedback ($\eta$), preventive feedback ($\zeta$), and the re-accretion of ejected material ($\alpha_Z$); each of these parameters is in principle directly observable, but at present has poorly known dependences on mass and redshift (and perhaps other properties). Additionally, they depend on $\dot{M}_{\rm
grav}$, the gravitational infall rate of baryons into the halo set by $\Lambda$CDM, ${t_{\rm dep}}$, the ISM gas depletion time, and $y$, the metal yield. These relations capture, to first order, the behavior of galaxies in modern hydrodynamic simulations that incorporate these processes dynamically within a hierarchical structure formation scenario.
The equilibrium model is broadly valid for quiescently star-forming central galaxies, at epochs where star formation is able to keep up with inflow (i.e. past the gas accumulation epoch; Figure \[fig:zeq\]), and when averaged over timescales longer than stochastic fluctuations in the inflow rate. On shorter timescales, galaxies oscillate around the equilibrium relations such that more (less) rapidly star-forming galaxies at a given mass having higher (lower) gas fractions and lower (higher) metallicities. The scatter about the relations is governed by a competition between the dilution time $t_{\rm dil}=(1+\eta)^{-1}{t_{\rm dep}}$ and the inflow stochasticity timescale. This model is [*not*]{} valid for satellite galaxies disconnected from feeding filaments, or for galaxies undergoing a major merger where gas feeding is temporarily driven by internal dynamical processes. Nonetheless, observations indicate that quiescently star-forming galaxies along the so-called galaxy main sequence dominate cosmic star formation at all epochs where measured [e.g. @noe07; @rod11], and hence this model describes how the bulk (but not all) of the stars in the Universe formed.
Most current galaxy formation models, both hydrodynamic and semi-analytic, already include inflow and outflow processes within growing large-scale structure. Hence there is no new physics in the equilibrium model. What is notable is not what this scenario contains, but rather what it [*doesn’t*]{} contain. In particular, there is no explicit mention of mergers, disks, environment, cooling radii, or virial radii— all central elements in the canonical scenario for galaxy formation. Such elements are automatically accounted for in hydrodynamic simulations, which form disks, merge them, and implictly include environmental effects within growing large-scale structure. Yet the equilibrium relations well describe such simulations without reference to these elements, suggesting that they are not of primary importance for the evolution of galaxies’ SFR, ${f_{\rm gas}}$, and $Z$.
In a broader context, the usefulness of the equilibrium scenario is that it lays bare the overwhelming complexity of modern galaxy formation models, and isolates those aspects that are critical for governing global galaxy evolution, thereby providing a simpler intuitive view for how galaxies grow. Although the canonical “halo-merger" view of disks cooling within halos and merging to drive galaxy evolution is not incorrect (i.e. these processes do happen), such a view obfuscates the primary driver of global galaxy evolution, namely the balance between inflows, outflows, and star formation. Indeed, the very notion of a galaxy halo, which is central to the classical view of galaxy formation, is only a second-order effect in the equilibrium scenario: The equilibrium relations are driven by the total inflow rate, and the “lumpiness" of that inflow owing to individual halos merely manifests as scatter around these relations.
We have argued that the equilibrium model provides a reasonable description of sophisticated simulations, but this in no way guarantees that it accurately describes the real Universe. It is encouraging that certain unavoidable predictions such as continual gas replenishment and the second-parameter dependence of the mass-metallicity relation seem to be in broad agreement with observations. But much work remains to be done in order to fully test this scenario. In particular, the equilibrium model centrally invokes a continual cycle of baryons flowing in and out of galaxies as a key moderator of galaxy evolution. But direct observational evidence for such processes is currently scant [see e.g. @rub11]. Critically testing and constraining these baryon cycling processes, particularly within circum-galactic gas where such processes are likely to be most prominent, will be a key contribution from upcoming multi-wavelength observational facilities.
The equilibrium model provides a re-parameterized framework for understanding certain key governing aspects of galaxy evolution, but is far from a full solution to the problem. To fully solve galaxy evolution, we must at minimum understand the physics that governs $\eta$, $\zeta$, and $\alpha_Z$, which will require concerted efforts on both observational and theoretical fronts. Furthermore, halo accretion rates, metal yields, and the star formation law remain uncertain, particularly in regimes such as small low-metallicity galaxies. Perhaps most importantly, this scenario in its current form explicitly does not address many interesting aspects of galaxy evolution such as the establishment of the Hubble sequence and the growth of central black holes. It also does not include processes that may be central to the evolution of certain classes of galaxies; for instance, it does not account for stellar (“dry") mergers which are important for the late-time growth of large passive systems. Hence much work remains to be done in order to comprehensively understand how galaxies evolve from primordial fluctuations into their present state. It is hoped that the equilibrium model provides, in its simplicity, a useful guiding framework for understanding the increasingly complex problem of galaxy evolution.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank N. Bouché, A. Dekel, R. Genzel, N. Katz, D. Kereš, A. Loeb, N. Murray, C. Papovich, E. Quataert, J. Schaye, L. Tacconi, and D. Weinberg for helpful discussions. This work was supported by the National Science Foundation under grant numbers AST-0847667 and AST-0907998. Computing resources were obtained through grant number DMS-0619881 from the National Science Foundation.
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[^1]: A term coined in the Astro2010 Decadal Survey Report: New Worlds, New Horizons
[^2]: This is distinguished from the ISM SFE, which is how much gas is converted into stars over some characteristic galaxy timescale, or the “cosmological" SFE, which is the galaxy stellar mass divided by the cosmologically-expected halo baryon mass (i.e. $M_*/f_bM_{\rm halo}$).
|
---
author:
- 'Yu-Pin Hsu, Eytan Modiano, and Lingjie Duan [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'ref.bib'
title: 'Scheduling Algorithms for Minimizing Age of Information in Wireless Broadcast Networks with Random Arrivals: The No-Buffer Case'
---
In recent years there has been a growing research interest in an *age of information* [@age:kaul]. The age of information is motivated by a variety of network applications requiring *timely* information. Examples range from information updates for *network users*, e.g., live traffic, transportation, air quality, and weather, to status updates for *smart systems*, e.g., smart home systems, smart transportation systems, and smart grid systems.
Fig. \[fig:motivation\] shows an example network, where network users $u_1, \cdots, u_N$ are running applications that need some timely information (e.g., user $u_1$ needs both traffic and transportation information for planning the best route), while at some epochs, snapshots of the information are generated at the sources and sent to the users in the form of packets over wired or wireless networks. The users are being updated and keep the latest information only. Since the information at the end-users is expected to be as timely as possible, the age of information is therefore proposed to capture the *freshness of the information at the end-users*; more precisely, it measures the elapsed time since the generation of the information. In addition to the timely information for the network users, the smart systems also need timely status (e.g., locations and velocities in smart transportation systems) to accomplish some tasks (e.g., collision-free smart transportation systems). As such, the age of information is a good metric to evaluate these networks supporting age-sensitive applications.
![Timely information updates for network users.[]{data-label="fig:motivation"}](motivation.eps){width=".4\textwidth"}
Next, we characterize the age-sensitive networks in two aspects. First, while packet delay is usually referred to as the elapsed time from the generation to its delivery, the age of information includes not only the packet delay but also the inter-delivery time, because the age of information keeps increasing until the information at the end-users is updated. We hence need to jointly consider the two parameters so as to design an age-optimal network. Moreover, while traditional relays (i.e., intermediate nodes) need buffers to keep all packets that are not served yet, the relays in the network of Fig. \[fig:motivation\] for timely information at most store the latest information and discard out-of-date packets. The buffers for minimizing the age here are no longer as useful as those in traditional relay networks.
In this paper, we consider a wireless broadcast network, where a base-station (BS) is updating many network users on timely information. The new information is *randomly* generated at its source. We assume that the BS can serve at most one user for each transmission opportunity. Under the transmission constraint, a transmission scheduling algorithm manages how the channel resources are allocated for each time, depending on the packet arrivals at the BS and the ages of the information at the end-users. The scheduling design is a critical issue to optimize network performance. In this paper we hence develop scheduling algorithms for minimizing the long-run average age.
Contributions
-------------
We study the age-optimal scheduling problem in the wireless broadcast network without the buffers at the BS. Our main contributions lie at designing novel scheduling algorithms and analyzing their age-optimality. For the case when the arrival statistics are available at the BS as prior information, we develop two *offline* scheduling algorithms, leveraging a Markov decision process (MDP) and the Whittle index. However, the MDP and the Whittle index in our problem will be difficult to analyze as they involve *long-run average cost optimization problems* with *infinite state spaces* and *unbounded immediate costs* [@MDP:Bertsekas]. Moreover, it is in general very challenging to obtain the Whittle index in closed form. By investigating some *structural results*, we not only successfully resolve the issues but also simplify the calculation of the Whittle index. It turns out that our index scheduling algorithm is very simple. When the arrival statistics are unknown, we develop *online* versions of the two offline algorithms. We show that both offline and online MDP-based scheduling algorithms are asymptotically age-optimal, and the index scheduling algorithm is age-optimal when the information arrival rates for all users are the same. Finally, we compare these algorithms via extensive computer simulations, and further investigate the impact of the buffers storing the latest information.
Related works
-------------
The general idea of *age* was proposed in [@data-age:cho] to study how to refresh a local copy of an autonomous information source to maintain the local copy up-to-date. The age defined in [@data-age:cho] is associated with *discrete* events at the information source, where the age is zero until the source is updated. Differently, the age of information in [@age:kaul] measures the age of a sample of *continuous* events; therefore, the sample immediately becomes old after generated. Many previous works, e.g., [@age:kaul; @age:kam2; @age:sun; @age:yates2; @age:bacinoglu; @age:Costa], studied the age of information for a single link. The papers [@age:kaul; @age:kam2] considered buffers to store all unserved packets (i.e., out-of-date packets are also stored) and analyzed the long-run average age, based on various queueing models. They showed that neither the throughput-optimal sampling rate nor the delay-optimal sampling rate can minimize the average age. The paper [@age:sun] considered a *smart* update and showed that the *always update* scheme might not minimize the average age. Moreover, [@age:yates2; @age:bacinoglu] developed power-efficient updating algorithms for minimizing the average age. The model in [@age:Costa] considered no buffer or a buffer to store the latest information.
Of the most relevant works on scheduling multiple users are [@age:he; @joowireless; @sun2018age; @age:igor; @yatesage]. The works [@age:he; @joowireless; @sun2018age] considered buffers at a BS to store all out-of-date packets. The paper [@age:igor] considered a buffer to store the latest information with *periodic* arrivals, while information updates in [@yatesage] can be generated *at will*. In contrast, our work is the first to develop both offline and online scheduling algorithms for *random* information arrivals, with the purpose of minimizing the long-run average age.
System overview {#section:system}
===============
![A BS updates $N$ users $u_1, \cdots, u_N$ on information of sources $s_1, \cdots, s_N$, respectively.[]{data-label="fig:model"}](model.eps){width=".3\textwidth"}
Network model
-------------
We consider a wireless broadcast network in Fig. \[fig:model\] consisting of a base-station (BS) and $N$ wireless users $u_1, \cdots, u_N$. Each user $u_i$ is interested in timely information generated by a source $s_i$, while the information is transmitted through the BS in the form of packets. We consider a discrete-time system with slot . Packets from the sources (if any) arrive at the BS at the *beginning* of each slot. The packet arrivals at the BS for different users are independent of each other and also independent and identically distributed (i.i.d.) over slots, following a Bernoulli distribution. Precisely, by $\Lambda_i(t)$, we indicate if a packet from source $s_i$ arrives at the BS in slot $t$, where $\Lambda_i(t)=1$ if there is a packet; otherwise, $\Lambda_i(t)=0$. We denote the probability $P[\Lambda_i(t)=1]$ by $p_i$.
Suppose that the BS can successfully transmit at most one packet during each slot, i.e., the BS can update at most one user in each slot. By $D(t) \in \{0, 1, \cdots N\}$ we denote a decision of the BS in slot $t$, where $D(t)=0$ if the BS does not transmit any packet and $D(t)=i$ for $i=1, \cdots, N$ if user $u_i$ is scheduled to be updated in slot $t$.
In this paper we fosus on the scenario without depolying any buffer at the BS, where an arriving packet is discarded if it is not transmitted in the arriving slot. The *no-buffer network* is not only simple to implement for practical systems, but also was shown to achieve good performance in a single link (see [@age:Costa]). In Section \[section:simulation\], we will numerically study networks with buffers in general.
Age of information model
------------------------
We initialize the ages of all arriving packets *at the BS* to be zero. The age of information *at a user* becomes one on receiving a new packet, due to one slot of the transmission time. Let $X_i(t)$ be the age of information at user $u_i$ in slot $t$ *before* the BS makes a scheduling decision. Suppose that the age of information at a user increases linearly with slots if the user is not updated. Then, the dynamics of the age of information for user $u_i$ is $$\begin{aligned}
X_i(t+1)=\left\{
\begin{array}{ll}
1 & \text{if $\Lambda_i(t)=1$ and $D(t)=i$;} \\
X_i(t)+1 & \text{else,}
\end{array}
\right.
\label{eq:age-dynamic}\end{aligned}$$ where the age of information in the next slot is one if the user gets updated on the new information; otherwise, the age increases by one. Let $\mathbf{X}(t)=(X_1(t), \cdots, X_N(t))$ be the age vector in slot $t$.
Let $\mathbf{X}$ be the set consisting of all age vectors $(x_1, \cdots, x_N)$ where the ages satisfy $x_i \geq 1$ for all $i$ and $x_i\neq x_j$ for all $i \neq j$. Since the BS can update at most one user in each slot, if an initial age vector $\mathbf{X}(0)$ is outside $\mathbf{X}$, then eventually age vector $\mathbf{X}(t)$ will enter $\mathbf{X}$ and stay in $\mathbf{X}$ onwards; otherwise, someone is never updated and its age approaches infinity. In other words, the age vector outside $\mathbf{X}$ is *transient*. Without loss of generality, we assume that initial age vector $\mathbf{X}(0)$ is within $\mathbf{X}$. Later, in the proof of Lemma \[thm:stationary\], we will show that any transmission decision before the age vector enters $\mathbf{X}$ will not affect the minimum *average age* (defined in the next section).
Problem formulation
-------------------
A *scheduling algorithm* $\pi=\{D(0), D(1), \cdots\}$ specifies a transmission decision for each slot. We define the *average age* under scheduling algorithm $\pi$ by $$\begin{aligned}
\limsup_{T \rightarrow \infty} \frac{1}{T+1} E_{\pi} \left[ \sum_{t=0}^T \sum_{i=1}^N X_i(t)\right], \end{aligned}$$ where $E_{\pi}$ represents the conditional expectation, given that scheduling algorithm $\pi$ is employed. We remark that this paper focuses on the total age of users for delivering clean results; whereas our design and analysis can work perfectly for the *weighted sum* of the ages. Our goal is to develop *age-optimal* scheduling algorithms, defined below.
A scheduling algorithm $\pi$ is *age-optimal* if it minimizes the average age.
In this paper, we will develop two offline scheduling algorithms and two online scheduling algorithms. Leveraging Markov decision process (MDP) techniques and Whittle’s methodology, we develop two offline scheduling algorithms in Sections \[section:mdp\] and \[section:whittle\], respectively, when the arrival statistics are available to the BS; later, two online versions of the offline algorithms are proposed in Section \[section:online\] for the case when the arrival statistics are oblivious to the BS.
A structural MDP scheduling algorithm {#section:mdp}
=====================================
Our first scheduling algorithm is driven by the MDP techniques. To that end, we formulate our problem as an MDP $\Delta$ with the components [@MDP:Puterman] below.
- **States**: We define the state $\mathbf{S}(t)$ of the MDP in slot $t$ by $\mathbf{S}(t)=(X_1(t), \cdots, X_N(t),$ $\Lambda_1(t), \cdots, \Lambda_N(t))$. Let $\mathbf{S}$ be the state space consisting of all states $(x_1, \cdots, x_N, \lambda_1, \cdots, \lambda_N)$ where
- $(x_1, \cdots, x_N) \in \mathbf{X}$ or $x_i> N$ for all $i$;
- $\lambda_i \in \{0,1\}$ for all $i$.
The state space includes some transient age vectors. That is used to fit *truncated states* in Section \[subsection:finite-state\]. We will show later in Lemma \[thm:stationary\] that adding these transient states will not change the minimum average age. Note that $\mathbf{S}$ is a *countable infinite* set because the ages are possibly unbounded.
- **Actions**: We define the action of the MDP in slot $t$ to be $D(t)$. Note that the action space is finite.
- **Transition probabilities**: By $P_{\mathbf{s},\mathbf{s}'}(d)$ we denote the transition probability of the MDP from state $\mathbf{s}=(x_1, \cdots, x_N, \lambda_1, \cdots, \lambda_N)$ to state $\mathbf{s}'=(x_1', \cdots, x_N',$ $\lambda_1', \cdots, \lambda_N')$ under action $D(t)=d$. According to the age dynamics in Eq. (\[eq:age-dynamic\]) and the i.i.d. assumption of the arrivals, we can describe the non-zero $P_{\mathbf{s},\mathbf{s}'}(d)$ as $$\begin{aligned}
P_{\mathbf{s},\mathbf{s}'}(d)=
\prod_{i:\lambda'_i=1} p_i \prod_{i:\lambda'_i=0} (1-p_i), \end{aligned}$$ if $x_i'=(x_i+1)-x_i\mathbbm{1}_{i =d \text{\,\,and\,\,} \lambda_i= 1}$ for all $i=1, \cdots, N$, where $\mathbbm{1}$ is the indicator function.
- **Cost**: Let $C(\textbf{S}(t), D(t)=d)$ be the *immediate cost* of the MDP if action $D(t)=d$ is taken in slot $t$ under state $\mathbf{S}(t)$, representing the resulting total age in the next slot: $$\begin{aligned}
C(\textbf{S}(t), D(t)=d) \triangleq & \sum^N_{i=1} X_i(t+1)\\
=&\sum_{i=1}^N (X_i(t)+1)-X_d(t)\cdot \Lambda_d(t), \end{aligned}$$ where we define $X_0(t)=0$ and $\Lambda_0(t)=0$ for all $t$ (for the no update case $d=0$), while the last term indicates that user $u_d$ is updated in slot $t$.
The objective of the MDP $\Delta$ is to find a *policy* $\pi$ (with the same definition as the scheduling algorithm) that minimizes the *average cost* $V(\pi)$ defined by $$\begin{aligned}
V(\pi) = \limsup_{T \rightarrow \infty} \frac{1}{T+1} E_{\pi} \left[ \sum_{t=0}^T C(\mathbf{S}(t),D(t))\right].\end{aligned}$$
A policy $\pi$ of the MDP $\Delta$ is *$\Delta$-optimal* if it minimizes the average cost $V(\pi)$.
Then, a $\Delta$-optimal policy is an age-optimal scheduling algorithm. Moreover, policies of the MDP can be classified as follows. A policy of the MDP is *history dependent* if $D(t)$ depends on $D(0), \cdots, D(t-1)$ and $\mathbf{S}(0) \cdots, \mathbf{S}(t)$. A policy is *stationary* if $D(t_1)=D(t_2)$ when $\mathbf{S}(t_1)=\mathbf{S}(t_2)$ for any $t_1, t_2$. A *randomized* policy specifies a probability distribution on the set of decisions, while a *deterministic* policy makes a decision with certainty. A policy in general belongs to one of the following sets [@MDP:Puterman]:
- $\Pi^{\text{HR}}$: a set of randomized history dependent policies;
- $\Pi^{\text{SR}}$: a set of randomized stationary policies;
- $\Pi^{\text{SD}}$: a set of deterministic stationary policies.
Note that $\Pi^{\text{SD}} \subseteq \Pi^{\text{SR}} \subseteq \Pi^{\text{HR}}$ [@MDP:Puterman], while the complexity of searching a $\Delta$-optimal policy increases from left to right. According to [@MDP:Puterman], there may exist neither $\Pi^{\text{SR}}$ nor $\Pi^{\text{SD}}$ policy that is $\Delta$-optimal. Hence, we target at exploring a regime under which a $\Delta$-optimal policy lies in a smaller policy set $\Pi^{\text{SD}}$, and investigating its structures.
Characterization of the $\Delta$-optimality {#section:characterization}
-------------------------------------------
To characterize the $\Delta$-optimality, we start with introducing an infinite horizon *$\alpha$-discounted cost*, where $0 < \alpha < 1$ is a discount factor. We subsequently connect the discounted cost case to the average cost case, because structures of a $\Delta$-optimal policy usually depend on its discounted cost case.
Given initial state $\mathbf{S}(0)=\mathbf{s}$, the *expected total $\alpha$-discounted cost* under scheduling algorithm $\pi$ (that can be history dependent) is $$\begin{aligned}
V_{\alpha}(\mathbf{s}; \pi) =\lim_{T \rightarrow \infty} E_\pi \left[ \sum_{t=0}^T \alpha^t C(\mathbf{S}(t), D(t)) |\mathbf{S}(0)=\mathbf{s}\right]. \end{aligned}$$
A policy $\pi$ of the MDP $\Delta$ is *$\Delta_{\alpha}$-optimal* if it minimizes the expected total $\alpha$-discounted cost $V_{\alpha}(\mathbf{s}; \pi)$. In particular, we define $$\begin{aligned}
V_{\alpha}(\mathbf{s})=\min_{\pi}V_{\alpha}(\mathbf{s};
\pi).\end{aligned}$$
Moreover, by $h_{\alpha}(\mathbf{s})=V_{\alpha}(\mathbf{s})-V_{\alpha}(\mathbf{0})$ we define the *relative cost function*, which is the difference of the minimum discounted costs between state $\mathbf{s}$ and a reference state $\mathbf{0}$. We can arbitrarily choose the reference state, e.g., $\mathbf{0}=(1,2,\cdots,N, 1, \cdots, 1)$ in this paper. We then introduce the *discounted cost optimality equation* of $V_{\alpha}(\mathbf{s})$ below.
\[lemma:optimality-eq\] The minimum expected total $\alpha$-discounted cost $V_{\alpha}(\mathbf{s})$, for initial state $\mathbf{s}$, satisfies the following *discounted cost optimality equation*: $$\begin{aligned}
V_\alpha(\mathbf{s}) &= \min_{d \in \{0,1, \cdots, N\}} C(\mathbf{s},d) + \alpha E[V_\alpha(\mathbf{s}') ], \label{eq:discounte-optimality-equation}\end{aligned}$$ where the expectation is taken over all possible next state $\mathbf{s}'$ reachable from the state $\mathbf{s}$, i.e., $E[V_\alpha(\mathbf{s}')]=\sum_{\mathbf{s}' \in \mathbf{S}} P_{\mathbf{s},\mathbf{s}'}(d)V_{\alpha}(\mathbf{s}')$. A deterministic stationary policy that realizes the minimum of the right-hand-side (RHS) of the discounted cost optimality equation in Eq. (\[eq:discounte-optimality-equation\]) is a $\Delta_{\alpha}$-optimal policy. Moreover, we can define a value iteration $V_{\alpha, n}(\mathbf{s})$ by $V_{\alpha, 0}(\mathbf{s})= 0$ and for any $n \ge 0$, $$\begin{aligned}
V_{\alpha, n+1}(\mathbf{s}) &= \min_{d \in \{0,1, \cdots, N\}} C(\mathbf{s},d) + \alpha E[V_{\alpha, n}(\mathbf{s}') ]. \label{eq:discount-itr}\end{aligned}$$ Then, $V_{\alpha, n}(\mathbf{s}) \rightarrow V_{\alpha}(\mathbf{s})$ as $n \rightarrow \infty$, for every $\mathbf{s}$ and $\alpha$.
Please see Appendix \[appendix:lemma:optimality-eq\].
The value iteration in Eq. (\[eq:discount-itr\]) is helpful for characterizing $V_{\alpha}(\mathbf{s})$, e.g., showing that $V_{\alpha}(\mathbf{s})$ is a non-decreasing function in the following.
\[lemma:monotone\] $V_{\alpha}(x_i, \mathbf{x}_{-i}, \boldsymbol{\lambda})$ is a non-decreasing function in $x_i$, for given $\mathbf{x}_{-i}=(x_1, \cdots, x_N) - \{x_i\}$ and $\boldsymbol{\lambda}=(\lambda_1, \cdots, \lambda_N)$.
Please see Appendix \[appendix:lemma:monotone\].
Using Propositions \[lemma:optimality-eq\] and \[lemma:monotone\] for the discounted cost case, we show that the MDP $\Delta$ has a deterministic stationary $\Delta$-optimal policy, as follows.
\[thm:stationary\] There exists a deterministic stationary policy that is $\Delta$-optimal. Moreover, there exists a finite constant $V^*=\lim_{\alpha \rightarrow 1}(1-\alpha)V_{\alpha}(\mathbf{s})$ for every state $\mathbf{s}$ such that the minimum average cost is $V^*$, independent of initial state $\mathbf{s}$.
Please see Appendix \[appendix:thm:stationary\].
We want to further elaborate on Lemma \[thm:stationary\].
- First, note that there is no condition for the existence of a deterministic stationary policy that is $\Delta$-optimal. In general, we need some conditions to ensure that the reduced Markov chain by a deterministic stationary policy is positive recurrent. Intuitively, we can think of the age of our problem as an age-queuing system, consisting of an age-queue, input to the queue, and a server. The input rate is one per slot since the age increases by one for each slot, while the server can serve an *infinite* number of age-packets for each service opportunity. As such, we always can find a scheduling algorithm such that the average arrival rate is less than the service rate and thus the reduced Markov chain is positive recurrent. Please see the proof in Appendix \[appendix:thm:stationary\] for details.
- Second, since our MDP $\Delta$ involves a *long-run average cost optimization* with a *countably infinite* state space and *unbounded* immediate cost, a $\Delta$-optimal policy of such an MDP might not satisfy the average cost optimaility equation like Eq. (\[eq:discounte-optimality-equation\]) (see [@cavazos1991counterexample] for a counter-example), even though the optimality of a deterministic stationary policy is established in Lemma \[thm:stationary\].
In addition to the optimality of deterministic stationary policies, we show that a $\Delta$-optimal policy has a nice structure. To investigate such structural results not only facilitates the scheduling algorithm design in Section \[section:mdp\], but also simplifies the calculation of the Whittle index in Section \[section:whittle\].
A *switch-type* policy is a special deterministic stationary policy of the MDP $\Delta$: for given $\mathbf{x}_{-i}$ and $\boldsymbol{\lambda}$, if the action of the policy for state $\mathbf{s}=(x_i, \mathbf{x}_{-i}, \boldsymbol{\lambda})$ is $d_{\mathbf{s}}=i$, then the action for state $\mathbf{s}'=(x_i+1,\mathbf{x}_{-i},\boldsymbol{\lambda})$ is $d_{\mathbf{s}'}=i$ as well.
In general, showing that a $\Delta$-optimal policy satisfies a structure relies on an optimality equation; however, as discussed, the average cost optimality equation for the MDP $\Delta$ might not be available. To resolve this issue, we first investigate the discounted cost case by the well-established value iteration in Eq. (\[eq:discount-itr\]), and then extend to the average cost case.
\[theorem:optimal-switch\] There exists a switch-type policy of the MDP $\Delta$ that is $\Delta$-optimal.
First, we start with the discounted cost case, and show that a $\Delta_{\alpha}$-optimal scheduling algorithm is switch-type. Let $\nu_{\alpha}(\mathbf{s};d)=C(\mathbf{s},d) + \alpha E[V_\alpha(\mathbf{s}')]$. Then, $V_{\alpha}(\mathbf{s})=\min_{d \in \{0, 1, \cdots, N\}} \nu_{\alpha}(\mathbf{s};d)$. Without loss of generality, we suppose that a $\Delta_{\alpha}$-optimal action at state $\mathbf{s}=(x_1,\mathbf{x}_{-1}, \boldsymbol{\lambda})$ is to update the user $u_1$ with $\lambda_1=1$. Then, according to the optimality of $d^*_{(x_1, \mathbf{x}_{-1}, \boldsymbol{\lambda})}=1$, $$\begin{aligned}
\nu_{\alpha}(x_1,\mathbf{x}_{-1},\boldsymbol{\lambda};1) - \nu_{\alpha}(x_1,\mathbf{x}_{-1},\boldsymbol{\lambda};j) \leq 0, \end{aligned}$$ for all $j \neq 1$.
Let $\mathbf{1}=(1, \cdots, 1)$ be the vector with all entries being one. Let $\mathbf{x}_i=(0, \cdots, x_i, \cdots,0)$ be the zero vector except for the $i$-th entry being replaced by $x_i$. To demonstrate the switch-type structure, we consider the following two cases:
1. *For any other user $u_j$ with $\lambda_j=1$*: Since $V_{\alpha}(x_1, \mathbf{x}_{-1},\boldsymbol{\lambda})$ is a non-decreasing function in $x_1$ (see Proposition \[lemma:monotone\]), we have $$\begin{aligned}
&\nu_{\alpha}(x_1+1, \mathbf{x}_{-1},\boldsymbol{\lambda};1) - \nu_{\alpha}(x_1+1, \mathbf{x}_{-1},\boldsymbol{\lambda};j) \\
= & x_j-(x_1+1)+\alpha E[V_{\alpha}(1, \mathbf{x}_{-1}+\mathbf{1},\boldsymbol{\lambda}')\\
&- V_{\alpha}(x_1+2, \mathbf{x}_{-1}+\mathbf{1}-\mathbf{x}_j,\boldsymbol{\lambda}') ]\\
\leq & x_j-x_1+\alpha E[V_{\alpha}(1, \mathbf{x}_{-1}+\mathbf{1},\boldsymbol{\lambda}')\\
&- V_{\alpha}(x_1+1, \mathbf{x}_{-1}+\mathbf{1}-\mathbf{x}_j,\boldsymbol{\lambda}') ]\\
= & \nu_{\alpha}(x_1, \mathbf{x}_{-1},\boldsymbol{\lambda};1) - \nu_{\alpha}(x_1, \mathbf{x}_{-1},\boldsymbol{\lambda};j) \leq 0,\end{aligned}$$ where $\boldsymbol{\lambda}'$ is the next arrival vector.
2. *For any other user $u_j$ with $\lambda_j=0$*: Similarly, we have $$\begin{aligned}
&\nu_{\alpha}(x_1+1, \mathbf{x}_{-1},\boldsymbol{\lambda};1) - \nu_{\alpha}(x_1+1, \mathbf{x}_{-1},\boldsymbol{\lambda};j) \\
= &-(x_1+1)+\alpha E[V_{\alpha}(1, \mathbf{x}_{-1}+\mathbf{1},\boldsymbol{\lambda}')\\
&- V_{\alpha}(x_1+2, \mathbf{x}_{-1}+\mathbf{1},\boldsymbol{\lambda}') ]\leq 0.\end{aligned}$$
Considering the two cases, a $\Delta_{\alpha}$-optimal action for state $(x_1+1, \mathbf{x}_{-1},\boldsymbol{\lambda})$ is still to update $u_1$, yielding the switch-type structure.
Then, we preceed to establish the optimality for the average cost case. Let $\{\alpha_n\}_{n=1}^{\infty}$ be a sequence of the discount factors. According to [@stationary-policy:Sennott], if the both conditions in Appendix \[appendix:thm:stationary\] hold, then there exists a subsequence $\{\beta_n\}_{n=1}^{\infty}$ such that a $\Delta$-optimal algorithm is the limit point of the $\Delta_{\beta_n}$-optimal policies. By induction on $\beta_n$ again, we obtain that a $\Delta$-optimal is switch-type as well.
Finite-state MDP approximations {#subsection:finite-state}
-------------------------------
The classical method for solving an MDP is to apply a value iteration method [@MDP:Puterman]. However, as mentioned, the average cost optimality equation might not exist. Even though average cost value iteration holds like Eq. (\[eq:discount-itr\]), the value iteration cannot work in practice, as we need to update an infinite number of states for each iteration. To address the issue, we propose a sequence of finite-state approximate MDPs. In general, a sequence of approximate MDPs might not converge to the original MDP according to [@MDP:Sennott]. Thus, we will rigorously show the convergence of the proposed sequence.
Let $X^{(m)}_i(t)$ be a *virtual age* of information for user $u_i$ in slot $t$, with the dynamic being $$\begin{aligned}
X^{(m)}_{i}(t+1)=\left\{
\begin{array}{ll}
1 & \text{if $\Lambda_i(t)=1$, $D(t)=i$} ;\\
\left[X^{(m)}_i(t)+1\right]^+_m & \text{else},
\end{array}
\right.\end{aligned}$$ where we define the notation $[x]^+_m$ by $[x]^+_m=x$ if $x \leq m$ and $[x]^+_m=m$ if $x >m$, i.e., we truncate the real age by $m$. This is different from Eq. (\[eq:age-dynamic\]). While the real age $X_i(t)$ can go beyond $m$, the virtual age $X^{(m)}_i(t)$ is at most $m$. Here, we reasonably choose the truncation $m$ to be greater than the number $N$ of users, i.e., $m > N$. Later, in Appendix \[appendix:theorem:finite-approximation\] (see Remark \[remark to truncation\]), we will discuss some mathematical reasons for the choice.
By $\{\Delta^{(m)}\}_{m=N+1}^{\infty}$ we define a sequence of approximate MDPs for $\Delta$, where each MDP $\Delta^{(m)}$ is the same as the original MDP $\Delta$ except:
- **States**: The state in slot $t$ is $S^{(m)}(t)=(X^{(m)}_1(t),$ $\cdots, X^{(m)}_N(t), \Lambda_1(t), \cdots,\Lambda_N(t))$. Let $\mathbf{S}^{(m)}$ be the state space.
- **Transition probabilities**: Under action $D(t)=d$, the transition probability $P^{(m)}_{\mathbf{s},\mathbf{s}'}(d)$ of the MDP $\Delta^{(m)}$ from state $\mathbf{s}=(x_1, \cdots, x_N, \lambda_1, \cdots, \lambda_N)$ to state $\mathbf{s}'=(x_1', \cdots, x_N',$ $\lambda_1', \cdots, \lambda_N')$ is $$\begin{aligned}
P_{\mathbf{s},\mathbf{s}'}(d)=
\prod_{i:\lambda'_i=1} p_i \prod_{i:\lambda'_i=0} (1-p_i), \end{aligned}$$ if $x_i'=[(x_i+1)-x_i\mathbbm{1}_{i =d \text{\,\,and\,\,} \lambda_i= 1}]^+_m$ for all $i=1, \cdots, N$.
Remember that the state space $\mathbf{S}$ of the MDP $\Delta$ includes some transient age vectors, e.g., $(N, \cdots, N)$. That is because, if not, the truncated state space $\mathbf{S}^{(m)}$ would not be a subset of original state space $\mathbf{S}$.
Next, we show that the proposed sequence of approximate MDPs converges to the $\Delta$-optimum.
\[theorem:finite-approximation\] Let $V^{(m)*}$ be the minimum average cost for the MDP $\Delta^{(m)}$. Then, $V^{(m)*} \rightarrow V^*$ as $m \rightarrow \infty$.
Please see Appendix \[appendix:theorem:finite-approximation\].
Structural MDP scheduling algorithm
-----------------------------------
Now, for a given truncation $m$, we are ready to propose a practical algorithm to solve the MDP $\Delta^{(m)}$. The traditional *relative value iteration algorithm* (RVIA), as follows, can be applied to obtain an optimal deterministic stationary policy for $\Delta^{(m)}$: $$\begin{aligned}
V^{(m)}_{n+1}(\mathbf{s}) = \min_{d \in \{0,1, \cdots, N\}} C(\mathbf{s},d) + E[V^{(m)}_{n}(\mathbf{s}') ] - V^{(m)}_n(\mathbf{0}), \label{eq:rvia}\end{aligned}$$ for all $\mathbf{s} \in \mathbf{S}^{(m)}$ where the initial value function is $V^{(m)}_0(\mathbf{s})=0$. For each iteration, we need to update actions for *all* virtual states by minimizing the RHS of Eq. (\[eq:rvia\]) as well as update $V^{(m)}(\mathbf{s})$ for *all* $\mathbf{s} \in \mathbf{S}^{(m)}$. As the size of the state space is $O(m^N)$, the computational complexity of updating all virtual states in each iteration of Eq. (\[eq:rvia\]) is more than $O(m^N)$. The complexity primarily results from the truncation $m$ of the MDP $\Delta^{(m)}$ and the number $N$ of users. In this section, we focus on dealing with large values of $m$ for the case of fewer users. In next section we will solve the case of more users.
To develop a low-complexity scheduling algorithm for fewer users, we propose *structural RVIA* in Alg. \[alg:offline\], which is an improved RVIA by leveraging the switch-type structure. In Alg. \[alg:offline\], we seek an optimal action $d^*_{\mathbf{s}}$ for each virtual state $\mathbf{s} \in \mathbf{S}^{(m)}$ by iteration. For each iteration, we update both the optimal action $d^*_{\mathbf{s}}$ and $V^{(m)}(\mathbf{s})$ for all virtual states. If the switch property holds[^2], we can determine an optimal action *immediately* in Line \[alg:offline-switch\]; otherwise we find an optimal action according to Line \[alg:offline-regular-update\]. By $V_{\text{tmp}}(\mathbf{s})$ in Line \[alg:offline-value-update-tmp\] we temporarily keep the updated value, which will replace $V^{(m)}(\mathbf{s})$ in Line \[alg:offline-value-update\]. Using the switch structure to prevent from the minimum operations on all virtual states in the conventional RVIA, we can reduce the computational complexity resulting from the size $m$. Next, we establish the optimality of the structural RVIA for the approximate MDP $\Delta^{(m)}$.
\[theorem:truncation\] For MDP $\Delta^{(m)}$ with a given $m$, the limit point of $d^*_{\mathbf{s}}$ in Alg. \[alg:offline\] is a $\Delta^{(m)}$-optimal action for every virtual state $\mathbf{s} \in \mathbf{S}^{(m)}$. In particular, Alg. \[alg:offline\] converges to the $\Delta^{(m)}$-optimum in a finite number of iterations.
(Sketch) According to [@MDP:Puterman Theorem 8.6.6], we only need to verify that the truncated MDP is *unichain*. Please see Appendix \[appendix:theorem:truncation\] for details.
$V^{(m)}(\mathbf{s}) \leftarrow 0$ for all virtual states $\mathbf{s}\in \mathbf{S}^{(m)}$;\
Based on the structural RVIA in Alg. \[alg:offline\], we propose the *structural MDP scheduling algorithm*: Given the actions for all state $\mathbf{s} \in \mathbf{S}^{(m)}$ from Alg. \[alg:offline\], for each slot $t$ the scheduling algorithm makes a decision according to the *virtual age* $X^{(m)}_i(t)$ for all $i$, instead of the *real age* $X_i(t)$. Then, combining Theorems \[theorem:finite-approximation\] and \[theorem:truncation\] yields that the proposed algorithm is asymptotically $\Delta$-optimal as $m$ approaches infinity.
An index scheduling algorithm {#section:whittle}
=============================
By mean of the MDP techniques, we have developed the structural MDP scheduling algorithm. The scheduling algorithm not only reduces the complexity from the traditional RVIA, but also was shown to be asymptotically age-optimal. However, the scheduling algorithm might not be feasible for many users; thus, a low-complexity scheduling algorithm for many users is still needed. To fill this gap, we investigate the scheduling problem from the perspective of *restless bandits* [@gittins2011multi]. A restless bandit generalizes a classic bandit by allowing the bandit to keep evolving under a *passive* action, but in a distinct way from its continuation under an *active* action.
The restless bandits problem, in general, is PSPACE-hard [@gittins2011multi]. Whittle hence investigated a relaxed version, where a constraint on the number of active bandits for each slot is replaced by the expected number. With this relaxation, Whittle then applied a Lagrangian approach to decouple the multi-armed bandit problem into multiple sub-problems, while proposing an index policy and a concept of *indexability*. The index policy is optimal for the relaxed problem; moreover, in many practical systems, the low-complexity index policy performs remarkably well, e.g., see [@larranaga2015stochastic].
With the success of the Whittle index policy to solve the restless bandit problem, we apply the Whittle’s approach to develop a low-complexity scheduling algorithm. However, to obtain the Whittle index in closed form and to establish the indexability can be very challenging [@gittins2011multi]. To address the issues, we simplify the derivation of the Whittle index by investigating structural results like Section. \[section:mdp\].
Decoupled sub-problem
---------------------
We note that each user in our scheduling problem can be viewed as a restless bandit. Then, applying the Whittle’s approach, we can decouple our problem into $N$ sub-problems. Each sub-problem consists of a single user $u_i$ and adheres to the network model in Section \[section:system\] with $N=1$, except for an additional cost $c$ for updating the user. In fact, the cost $c$ is a scalar Lagrange multiplier in the Lagrangian approach. In each decoupled sub-problem, we aim at determining whether or not the BS updates the user in each slot, for striking a balance between the updating cost and the cost incurred by age. Since each sub-problem consists of a single user only, hereafter in this section we omit the index $i$ for simplicity.
Similarly, we cast the sub-problem into an MDP $\Omega$, which is the same as the MDP $\Delta$ in Section \[section:mdp\] with a single user except:
- **Actions**: Let $A(t) \in \{0, 1\}$ be an action of the MDP in slot $t$ indicating the BS’s decision, where $A(t)=1$ if the BS decides *to update* the user and $A(t)=0$ if the BS decides *to idle*. Note that the action $A(t)$ is different from the scheduling decision $D(t)$. The action $A(t)$ is used for the decoupled sub-problem. In Section. \[subsection:index\], we will use the MDP $\Omega$ to decide $D(t)$.
- **Cost**: Let $C(\mathbf{S}(t), A(t))$ be an immediate cost if action $A(t)$ is taken in slot $t$ under state $\mathbf{S}(t)$, with the definition as follows. $$\begin{aligned}
&C\Bigl(\textbf{S}(t)=(x, \lambda), A(t)=a\Bigr) \nonumber\\
\triangleq& (x+1- x \cdot a\cdot \lambda) +c\cdot a, \label{eq:cost}\end{aligned}$$ where the first part $x+1- x \cdot a\cdot \lambda$ is the resulting age in the next slot and the second part is the incurred cost for updating the user.
A *policy* $\mu=\{A(0), A(1), \cdots\}$ of the MDP $\Omega$ specifies an action $A(t)$ for each slot $t$. The *average cost* $J(\pi)$ under policy $\mu$ is defined by $$\begin{aligned}
J(\mu)=\limsup_{T \rightarrow \infty} \frac{1}{T+1} E_{\mu} \left[ \sum_{t=0}^T C(\mathbf{S}(t), A(t))\right]. \end{aligned}$$
Again, the objective of the MDP $\Omega$ is to find an $\Omega$-optimal policy defined as follows.
A policy $\mu$ of the MDP $\Omega$ is *$\Omega$-optimal* if it minimizes the average cost $J(\mu)$.
Traditionally, the Whittle index might be obtained by solving the optimality equation of $J(\mu)$, e.g. [@gittins2011multi; @index:igor]. However, as discussed, the average cost optimality equation for the MDP $\Omega$ might not exit, and even if it exists, solving an optimality equation might be tedious. To look for a simpler way for obtaining the Whittle index, we investigate structures of an $\Omega$-optimal policy instead, by looking at its discounted case again. It turns out that our structural results will further simplify the derivation of the Whittle index.
Characterization of the $\Omega$-optimality
-------------------------------------------
First, we show that an $\Omega$-optimal policy is stationary deterministic as follows.
\[theorem:stationary-omgea\] There exists a deterministic stationary policy that is $\Omega$-optimal, independent of the initial state.
Please see Appendix \[appendix:theorem:stationary-omgea\].
Next, we show that an $\Omega$-optimal policy is a special type of deterministic stationary policies.
A *threshold-type* policy is a special deterministic stationary policy of the MDP $\Omega$. The action for state $(x,0)$ is to idle, for all $x$. Moreover, if the action for state $(x,1)$ is to update, then the action for state $(x+1,1)$ is to update as well. In other words, there exists a threshold $\bar{X} \in \{1,2, \cdots\}$ such that the action is to update if there is an arrival and the age is greater than or equal to $\bar{X}$; otherwise, the action is to idle.
\[theorem:threshold\] If the update cost $c\geq 0$, then there exists a threshold-type policy that is $\Omega$-optimal.
It is obvious that an optimal action for state $(x,0)$ is to idle if $c \geq 0$. To establish the optimality of the threshold structure for state $(x,1)$, we need the *discounted cost optimality equation* for $J_{\alpha}(\mathbf{s})$, similar to Proposition \[lemma:optimality-eq\]: $$\begin{aligned}
J_{\alpha}(\mathbf{s})=\min_{a \in \{0,1\}} C(\mathbf{s}, a)+\alpha E[J_{\alpha}(\mathbf{s}')].\end{aligned}$$
Similar to the proof of Theorem \[theorem:optimal-switch\], we can focus on the discounted cost case and show that an $\Omega_{\alpha}$-optimal policy is the threshold type. Let $\mathscr{J}_{\alpha}(\mathbf{s};a)=C(\mathbf{s}; a)+\alpha E[J_{\alpha}(\mathbf{s}')]$. Then, $J_{\alpha}(\mathbf{s})=\min_{a \in \{0,1\}} \mathscr{J}_{\alpha}(\mathbf{s};a)$. Moreover, an $\Omega_{\alpha}$-optimal action for state $\mathbf{s}$ is ${\operatornamewithlimits{arg\,min}}_{a \in \{0,1\}} \mathscr{J}_{\alpha}(\mathbf{s};a)$. Suppose that an $\Omega_{\alpha}$-optimal action for state $(x,1)$ is to update, i.e., $$\begin{aligned}
\mathscr{J}_{\alpha}(x,1;1)-\mathscr{J}_{\alpha}(x,1;0) \leq 0.\end{aligned}$$ Then, an $\Omega_{\alpha}$-optimal action for state $(x+1,1)$ is still to update since $$\begin{aligned}
&\mathscr{J}_{\alpha}(x+1,1;1)-\mathscr{J}_{\alpha}(x+1,1;0)\\
=&\left(1+c+\alpha E[J_{\alpha}(1,\lambda')] \right) -\left(x+2+\alpha E[J_{\alpha}(x+2,\lambda')] \right)\\
\mathop{\leq}^{(a)} & \left(1+c+\alpha E[J_{\alpha}(1,\lambda')] \right) -\left(x+1+\alpha E[J_{\alpha}(x+1,\lambda')] \right)\\
=&J_{\alpha}(x,1;1)-J_{\alpha}(x,1;0) \leq 0,\end{aligned}$$ where (a) results from the non-decreasing function of $J_{\alpha}(x,\lambda)$ in $x$ given $\lambda$ (similar to Proposition \[lemma:monotone\]). Hence, an $\Omega_{\alpha}$-optimal policy is threshold-type.
Thus far, we have successfully identify the threshold structure of an $\Omega$-optimal policy. The MDP $\Omega$ then can be reduced to a two-dimensional discrete-time Markov chain (DTMC) by applying a threshold-type policy. To find an optimal threshold for minimizing the average cost, in the next lemma we explicitly derive the average cost for a threshold-type policy.
![The post-action age $\tilde{X}(t)$ under the threshold-type policy forms a DTMC.[]{data-label="fig:post-age"}](post-age.eps){width=".35\textwidth"}
\[lemma:threshold-cost\] Given the threshold-type policy $\mu$ with the threshold $\bar{X}\in \{1, 2, \cdots\}$, then the average cost $J(\mu)$, denoted by $\mathscr{C}(\bar{X})$, under the policy is $$\begin{aligned}
\mathscr{C}(\bar{X})=\frac{\frac{\bar{X}^2}{2}+(\frac{1}{p}-\frac{1}{2})\bar{X}+\frac{1}{p^2}-\frac{1}{p}+c}{\bar{X}+\frac{1-p}{p}}. \label{eq:threshold-cost}\end{aligned}$$
Let $\tilde{X}(t)$ be the age *after* an action in slot $t$; precisely, if $\mathbf{S}(t)=(x,\lambda)$ and $A(t)=a$, then $\tilde{X}(t)=x+1-x\cdot a\cdot \lambda$. Note that $\tilde{X}(t)$, called *post-action age* (similar to the post-decsion state [@AMDP:Powell; @online-engery:borkar]), is different from the *pre-action age* $X(t)$. Then, the post-action age $\tilde{X}(t)$ by the threshold-type policy forms an one-dimensional DTMC in Fig. \[fig:post-age\], with the transition probabilities being $$\begin{aligned}
&P[\tilde{X}(t+1)=i+1|\tilde{X}(t)=i]=1, \text{\,\,for\,\,}i=1, \cdots, \bar{X}-1; \\
&P[\tilde{X}(t+1)=i+1|\tilde{X}(t)=i]=1-p, \text{\,\,for\,\,}i=\bar{X}, \cdots;\\
&P[\tilde{X}(t+1)=1|\tilde{X}(t)=i]=p, \text{\,\,for\,\,}i=\bar{X}, \cdots. \end{aligned}$$
To calculate the average cost of the policy, we associate each state in the DTMC with a cost. The DTMC incurs the cost of $c+1$ in slot $t$ when the post-action age in slot $t$ is $\tilde{X}(t)=1$. That is because the post-action age $\tilde{X}(t)=1$ implies that the BS updates the user. In addition, the DTMC incurs the age cost of $y$ in slot $t$ when the post-action age is $\tilde{X}(t)=y \neq 1$.
The steady-state distribution $\boldsymbol{\xi}=(\xi_1, \xi_2, \cdots)$ of the DTMC can be solved as $$\begin{aligned}
\xi_i=\left\{
\begin{array}{ll}
\frac{1}{\bar{X}+\frac{1-p}{p}} & \text{if $i=1, \cdots, \bar{X}$;}\\
\frac{1}{\bar{X}+\frac{1-p}{p}}(1-p)^{i-\bar{X}} & \text{if $i=\bar{X}+1, \cdots$}.
\end{array}
\right.\end{aligned}$$ Therefore, the average cost of the DTMC is $$\begin{aligned}
(1+c)\xi_1 +\sum_{i=2}^{\infty} i\xi_i=\frac{\frac{\bar{X}^2}{2}+(\frac{1}{p}-\frac{1}{2})\bar{X}+\frac{1}{p^2}-\frac{1}{p}+c}{\bar{X}+\frac{1-p}{p}}.\end{aligned}$$
We remark that the post-action age introduced in the above proof are beneficial in many aspects:
- The post-action age can form an one-dimensional DTMC, instead of the original two-dimensional state $\mathbf{S}(t)$.
- We cannot associate each pre-action age with a fixed cost, since the cost in Eq. (\[eq:cost\]) depends on not only state but also action. Instead, the cost for each post-action age is determined by its age only.
- The post-action age will facilitate the online algorithm design in Section \[section:online\].
Derivation of the Whittle index
-------------------------------
Now, we are ready to define and derive the Whittle index as follows.
We define the Whittle index $I(\mathbf{s})$ by the cost $c$ that makes both actions, to update and to idle, for state $\mathbf{s}$ equally desirable.
In the next theorem, we will obtain a very simple expression for the Whittle index by combining Theorem \[theorem:threshold\] and Lemma \[lemma:threshold-cost\].
\[theorem:whittle\] The Whittle index of the sub-problem for state $(x,\lambda)$ is $$\begin{aligned}
I(x,\lambda)=\left\{
\begin{array}{ll}
0 & \text{if $\lambda=0$;}\\
\frac{x^2}{2}-\frac{x}{2}+\frac{x}{p} & \text{if $\lambda=1$.}
\end{array}
\right.\label{eq:index}\end{aligned}$$
It is obvious that the Whittle index for state $(x,0)$ is $I(x,0)=0$ as both actions result in the same immediate cost and the same age of next slot if $c=0$.
Let $g(x)=\mathscr{C}(x)$ in Eq. (\[eq:threshold-cost\]) for the domain of $\{x\in \mathbb{R}: x\geq 1\}$. Note that $g(x)$ is strictly convex in the domain. Let $x^*$ be the minimizer of $g(x)$. Then, an optimal threshold for minimizing the average cost $\mathscr{C}(\bar{X})$ is either $\lfloor x^* \rfloor$ or $\lceil x^* \rceil$: the optimal threshold is $\bar{X}^*=\lfloor x^* \rfloor$ if $\mathscr{C}(\lfloor x^* \rfloor) < \mathscr{C}(\lceil x^* \rceil)$ and $\bar{X}^*=\lceil x^* \rceil$ if $\mathscr{C}(\lceil x^* \rceil) < \mathscr{C}(\lfloor x^* \rfloor)$. If there is a tie, both choices are optimal, i.e, equally desirable.
Hence, both actions for state $(x,1)$ are equally desirable if and only if the age $x$ satisfies $$\begin{aligned}
\mathscr{C}(x)=\mathscr{C}(x+1), \label{eq:equal-desirable}\end{aligned}$$ i.e., $x= \lfloor x^* \rfloor$ and both thresholds of $x$ and $x+1$ are optimal. By solving Eq. (\[eq:equal-desirable\]), we obtain the cost, as stated in the theorem, to make both actions equally desirable.
According to Theorem \[theorem:whittle\], both actions might have a tie. If there is a tie, we break the tie in favor of idling. Then, we can explicitly express the optimal threshold in the next theorem.
\[lemma:optimal-threshold\] The optimal threshold for minimizing the average cost $\mathscr{C}(\bar{X})$ is $x$ if the cost $c$ satisfies $I(x-1,1) \leq c < I(x,1)$, for all $x=1, 2, \cdots$.
Please see Appendix \[appendix:lemma:optimal-threshold\].
Next, according to [@whittle], we have to demonstrate the *indexability* such that the Whittle index is feasible.
For a given cost $c$, let $\mathbf{S}(c)$ be the set of states such that the optimal actions for the states are to idle. The sub-problem is *indexable* if the set $\mathbf{S}(c)$ monotonically increases from the empty set to the entire state space, as $c$ increases from $-\infty$ to $\infty$.
The sub-problem is indexable.
If $c<0$, the optimal action for every state is to update; as such, $\mathbf{S}(0)=\emptyset$. If $c\geq 0$, then $\mathbf{S}(c)$ is composed of the set $\{\mathbf{s}=(x,0): x=1, 2, \cdots\}$ and a set of $(x,1)$ for some $x$’s. According to Lemma \[lemma:optimal-threshold\], the optimal threshold monotonically increases to infinity as $c$ increases, and hence the set $\mathbf{S}(c)$ monotonically increases to the entire state space.
Index scheduling algorithm {#subsection:index}
--------------------------
Now, we are ready to propose a low-complexity *index scheduling algorithm* based on the Whittle index. For each slot $t$, the BS observes age $X_i(t)$ and arrival indicator $\Lambda_i(t)$ for every user $u_i$; then, updates user $u_i$ with the highest value of the Whittle index $I(X_i(t),\Lambda_i(t))$, i.e., $D(t)={\operatornamewithlimits{arg\,max}}_{i=1, \cdots, N} I(X_i(t),\Lambda_i(t))$. We can think of the index $I(X_i(t),\Lambda_i(t))$ as a *value* of updating user $u_i$. The intuition of the index scheduling algorithm is that the BS intends to send the most valuable packet.
The optimality of the index scheduling algorithm for the relaxed version is known [@gittins2011multi]. Next, we show that the proposed index scheduling algorithm is age-optimal for the original problem (without relaxation), when the packet arrivals for all users are *stochastically identical*.
\[lemma:index-optimal\] If the arrival rates of all information sources are the same, i.e., $p_i=p_j$ for all $i \neq j$, then the index scheduling algorithm is age-optimal.
Note that, for this case, the index scheduling algorithm send an arriving packet with the largest age of information, i.e., $D(t)={\operatornamewithlimits{arg\,max}}_i X_i(t)\Lambda_i(t)$ for each slot $t$. Then, in Appendix \[appendix:lemma:index-optimal\] we show that the policy is $\Delta$-optimal.
In Section \[section:simulation\] we will further validate the index scheduling algorithm for stochastically non-identical arrivals by simulations.
Online scheduling algorithm design {#section:online}
==================================
Thus far, we have developed two scheduling algorithms in Sections \[section:mdp\] and \[section:whittle\]. Both algorithms are offline, as the structural MDP scheduling algorithm and the index scheduling algorithm need the arrival statistics as prior information to pre-compute an optimal action for each virtual state and the Whittle index, respectively. To solve the more challenging case when the arrival statistics are unavailable, in this section we develop online versions for both offline algorithms.
An MDP-based online scheduling algorithm {#subsection:mdp-online}
----------------------------------------
We first develop an online version of the MDP scheduling algorithm by leveraging *stochastic approximation* techniques [@learning-book:borkar]. The intuition is that, instead of updating $V^{(m)}(\mathbf{s})$ for all virtual states in each iteration of Eq. (\[eq:rvia\]), we update $V^{(m)}(\mathbf{s})$ by following a *sample path*, which is a set of outcomes of the arrivals over slots. It turns out that the sample-path updates will converge to the $\Delta$-optimal solution. To that end, we need a *stochastic version* of the RVIA. However, the RVIA in Eq. (\[eq:rvia\]) is not suitable because the expectation is inside the minimization (see [@AMDP:Powell] for details). While minimizing the RHS of Eq. (\[eq:rvia\]) for a given current state, we would need the transition probabilities to calculate the expectation. To tackle this, we design *post-action states* for our problem, similar to the proof of Lemma \[lemma:threshold-cost\].
We define post-action state $\tilde{\mathbf{s}}$ as the ages and the arrivals *after* an action. The state we used before is referred to as the *pre-action* state. If $\mathbf{s}=(x_1, \cdots, x_N, \lambda_1, \cdots, \lambda_N) \in \mathbf{S}^{(m)}$ is a virtual state of the MDP $\Delta^{(m)}$, then the virtual post-action state after action $d$ is $\tilde{\mathbf{s}}=(\tilde{x}_1, \cdots, \tilde{x}_N, \tilde{\lambda}_1, \cdots, \tilde{\lambda_N})$ with $$\begin{aligned}
\tilde{x}_i=\left\{
\begin{array}{ll}
1 & \text{if $i=d$ and $\lambda_i=1$;} \\
\left[x_i+1\right]^+_m & \text{else},
\end{array}
\right.\end{aligned}$$ and $\tilde{\lambda}_i=\lambda_i$ for all $i$.
Let $\tilde{V}^{(m)}(\tilde{\mathbf{s}})$ be the value function based on the post-action states defined by $$\begin{aligned}
\tilde{V}^{(m)}(\tilde{\mathbf{s}})=E[V^{(m)}(\mathbf{s})],\end{aligned}$$ where the expectation is taken over all possible the pre-action states $\mathbf{s}$ reachable from the post-action state. We can then write down the post-action average cost optimality equation [@AMDP:Powell] for the virtual post-action state $\tilde{\mathbf{s}}=(\tilde{x}_1, \cdots, \tilde{x}_N,\tilde{\lambda}_1, \cdots, \tilde{\lambda}_N)$: $$\begin{aligned}
&\tilde{V}^{(m)}(\tilde{\mathbf{s}})+V^{(m)*}\\
=&E\left[\min_{d \in \{0, 1, \cdots, N\}} C\left((\tilde{\mathbf{x}},\tilde{\boldsymbol{\lambda}}'),d\right)\right.\\
&\left.\hspace{1cm} +\tilde{V}^{(m)}([\tilde{\mathbf{x}}+\mathbf{1}-\tilde{\mathbf{x}}_d \tilde{\boldsymbol{\lambda}}'_d]^+_m,\tilde{\boldsymbol{\lambda}}')\right],\end{aligned}$$ where $\tilde{\boldsymbol{\lambda}'}$ is the next arrival vector; $\tilde{\mathbf{x}}_i=(0, \cdots, \tilde{x}_i, \cdots,0)$ denotes the zero vector except for the $i$-th entry being replaced by $\tilde{x}_i$; $\tilde{\boldsymbol{\lambda}}'_i=(0, \cdots, \tilde{\lambda}_i, \cdots,0)$ denotes the zero vector except for the $i$-th entry being replaced by $\tilde{\lambda}_i$; the vector $\mathbf{1}=(1, \cdots, 1)$ is the unit vector. From the above optimality equation, the RVIA is as follows: $$\begin{aligned}
\tilde{V}^{(m)}_{n+1}(\tilde{\mathbf{s}})=&E\left[\min_{d \in \{0, 1, \cdots, N\}} C\left((\tilde{\mathbf{x}},\tilde{\boldsymbol{\lambda}}'),d\right)\right.\nonumber\\
&\left.+\tilde{V}^{(m)}_n([\tilde{\mathbf{x}}+\mathbf{1}-\mathbf{x}_d \tilde{\boldsymbol{\lambda}}'_d]^+_m,\tilde{\boldsymbol{\lambda}}')\right] - \tilde{V}^{(m)}_n(\mathbf{0}). \label{eq:rvia-post}\end{aligned}$$
Subsequently, we propose the *MDP-based online scheduling algorithm* in Alg. \[alg:online\] based on the stochastic version of the RVIA. In Lines \[alg:online-init-start\]-\[alg:online-init-end\], we initialize $\tilde{V}^{(m)}(\tilde{\mathbf{s}})$ of all virtual post-action states and start from the reference point. Moreover, by $v$ we record $\tilde{V}^{(m)}(\tilde{\mathbf{s}})$ of the current virtual post-action state. By observing the current arrivals $\mathbf{\Lambda}(t)$ and plugging in Eq. (\[eq:rvia-post\]), the expectation in Eq. (\[eq:rvia-post\]) can be removed; as such, in Line \[alg:online-optimal-decision\] we optimally update a user by minimizing Eq. (\[eq:optimal-online-decision\]). Then, we update $\tilde{V}^{(m)}(\tilde{\mathbf{s}})$ of the current virtual post-action state in Line \[alg:online-value-update\], where $\gamma(t)$ is a *stochastic step-size* in slot $t$ to strike a balance between the previous $\tilde{V}^{(m)}(\tilde{\mathbf{s}})$ and the updated value $v$. Finally, the next virtual post-action state is updated in Lines \[alg:online-state-update\] and \[alg:online-state-update2\]
$\tilde{V}^{(m)}(\tilde{\mathbf{s}}) \leftarrow 0$ for all states $\tilde{\mathbf{s}}\in \mathbf{S}^{(m)}$;\
\[alg:online-init-start\] $\tilde{\mathbf{s}} \leftarrow \mathbf{0}$;\
$v \leftarrow 0$;\
\[alg:online-init-end\]
Next, we show the optimality of the MDP-based online scheduling algorithm as slot $t$ approaches infinity.
If $\sum_{t=0}^{\infty} \gamma(t) = \infty$ and $\sum_{t=0}^{\infty} \gamma^2(t) < \infty$, then Alg. \[alg:online\] converges to $\Delta^{(m)}$-optimum.
According to [@learning:borkar1; @learning:borkar2], we only need to verify that the truncated MDP is unichain, which has been completed in Appendix \[appendix:theorem:truncation\].
In the above theorem, $\sum_{t=0}^{\infty} \gamma(t) =\infty$ implies that Alg. \[alg:online\] needs an infinite number of iterations to learn the $\Delta$-optimal solution, while the offline Alg. \[alg:offline\] converges to the optimal solution in a finite number of iterations. Moreover, $\sum_{t=0}^{\infty} \gamma^2(t) < \infty$ means that the *noise* from measuring $\tilde{V}^{(m)}(\tilde{\mathbf{s}})$ can be controlled. Finally, we want to emphasize that the proposed Alg. \[alg:online\] is asymptotically $\Delta$-optimal, i.e., it converges to the $\Delta$-optimal solution when both the truncation $m$ and the slot $t$ go to infinity. In Section VI, we will also numerically investigate the algorithm over finite slots.
An index-based online scheduling algorithm {#subsection:index-online}
------------------------------------------
Next, we note that the simple Whittle index $I(x, \lambda)$ in Eq. (\[eq:index\]) depends on its arrival probability only. Thus, if the arrival probability is unknown, for each slot $t$ we revise the index by $$\begin{aligned}
I(x,\lambda,t)=\left\{
\begin{array}{ll}
0 & \text{if $\lambda=0$;}\\
\frac{x^2}{2}-\frac{x}{2}+\frac{x}{p(t)} & \text{if $\lambda=1$,}
\end{array}
\right.\end{aligned}$$ where $$p(t)=\frac{\sum_{\tau=0}^t \Lambda(\tau)}{t+1}=\frac{p(t-1)\cdot t+\Lambda(t)}{t+1}$$ is the running average arrival rate. Then, we propose the *index-based online scheduling algorithm* as follows. For each slot $t$, the BS observes age $X_i(t)$ and arrival indicator $\Lambda_i(t)$ for every user $u_i$; then, calculate $p_i(t)$ and update user $u_i$ with the highest value of the revised Whittle index $I(X_i(t),\Lambda_i(t),t)$.
Simulation results {#section:simulation}
==================
In this section we conduct extensive computer simulations for the proposed four scheduling algorithms. We demonstrate the switch-type structure of Alg. \[alg:offline\] in Section \[subsection:sim-switch\]. In Section \[subsection:sim-study\] we compare the proposed scheduling algorithms, especially to validate the performance of the online algorithms over finite slots. Finally, we study the wireless broadcast network with buffers at the BS in Section \[subsection:sim-buffer\].
Switch-type structure of Alg. \[alg:offline\] {#subsection:sim-switch}
---------------------------------------------
![Switch structure of Alg. \[alg:offline\] for (a) $p_1=p_2=0.9$; (b) $p_1=0.9$, $p_2=0.5$. The dots represent $D(t) = 1$ to update user $u_1$ and the stars mean $D(t) = 2$ to update user $u_2$.[]{data-label="fig:switch"}](switch.eps){width=".52\textwidth"}
Figs. \[fig:switch\]-(a) and \[fig:switch\]-(b) show the switch-type structure of Alg. \[alg:offline\] for two users, when the BS has packets for both users. The experiment setting is as follows. We run Alg. \[alg:offline\] with the boundary $m=10$ over 100,000 slots to search an optimal action for each virtual state. Moreover, we consider two arrival rate vectors, with $(p_1,p_2)$ being $(0.9, 0.9)$ and $(0.9, 0.5)$ in Figs. \[fig:switch\]-(a) and \[fig:switch\]-(b), respectively, where the *dots* represent $D(t)=1$ and the *stars* mean $D(t)=2$ when the BS has both arrivals in the same slot. We observe the switch structure in the figures, while Fig. \[fig:switch\]-(a) is consistent with the index scheduling algorithm in Section \[section:whittle\] by simply comparing the ages of the two users. Moreover, by fixing the arrival rate $p_1=0.9$ for the first user, the BS will give a higher priority to the second user as $p_2$ decreases in Fig. \[fig:switch\]-(b). That is because the second user takes more time to wait for the next arrival and becomes a bottleneck.
Numerical studies of the proposed scheduling algorithms {#subsection:sim-study}
-------------------------------------------------------
![Average age for different arrival rate $p_2$, where we fix $N=2$ and $p_1=0.6$.[]{data-label="fig:2user1"}](2users-06.eps){width=".45\textwidth"}
![Average age for different arrival rate $p_2$, where we fix $N=2$ and $p_1=0.8$.[]{data-label="fig:2user2"}](2users-08.eps){width=".45\textwidth"}
![Average age for different arrival rate $p_1=p_2=p$, where we fix $N=2,3,4$, respectively.[]{data-label="fig:4user"}](many-users.eps){width=".45\textwidth"}
In this section, we examine the proposed four algorithms from various perspectives. First, we show the average age of two users for different $p_2$ in Figs. \[fig:2user1\] and \[fig:2user2\] with fixed $p_1=0.6$ and $p_1=0.8$, respectively. Here, we set the boundary $m=30$ for the structural MDP scheduling algorithm. For the MDP-based online scheduling algorithms, we set the boundary $m=100$; moreover, we consider different step sizes in both figures, i.e., $\gamma(t)=1/t$, $\gamma(t)=0.1/t$, and $\gamma(t)=0.01/t$. All the results are averaged over 100,000 slots. How to choose the best step size with provably performance guarantee is interesting, but is out of scope of this paper. By simulation, we observe that $\gamma(t)=0.01/t$ works perfectly for our problem to achieve the minimum average age. Moreover, comparing with the structural MDP scheduling algorithm, the low-complexity index algorithm almost achieves the minimum average age, with invisible performance loss. Even without the knowledge of the arrival statistics, the MDP-based online scheduling algorithm with $\gamma(t)=0.01/t$ and the index-based online scheduling algorithm are both close to the minimum average age.
Second, we show the average age of more than two users with $N=2,3,4$, respectively, in Fig. \[fig:4user\], where we consider $p_1=p_2=p$. According to Lemma \[lemma:index-optimal\], the index scheduling algorithm is age-optimal; thus, we find that both online scheduling algorithms are almost age-optimal again, where we use $\gamma(t)=0.01/t$ only.
Third, we show that the average age of many users in Fig. \[fig:many-user\]. In this setting, the two MDP-based scheduling algorithms may be unfeasible because the resulting huge state space; thus, we consider the two index-based scheduling algorithms only. We find that the low-complexity index-based online scheduling algorithm again achieves the minimum average age.
By these numerical studies, we would suggest implementing the index-based online scheduling algorithm. It is not only simple to implement practically, but also has good performance.
Networks with buffers {#subsection:sim-buffer}
---------------------
Thus far, we consider the no-buffer network only. Finally, we study the buffers at the BS to store the latest information for each user. Similar to Section \[section:mdp\] we can find an age-optimal scheduling by an MDP. However, we need to redefine the states of the MDP $\Delta$. In addition to the age $X_i(t)$ of information at *user* $u_i$, by $Y_i(t)$ we define the *initial age of the information* at the *buffer* for user $u_i$; precisely, $$\begin{aligned}
Y_i(t)=\left\{
\begin{array}{ll}
0 & \text{if $\Lambda(t)=1$;}\\
Y_i(t-1)+1 & \text{else}.
\end{array}
\right.\end{aligned}$$ Then, we redefine the state by $\mathbf{S}(t)=\{X_1(t), \cdots, X_N(t),$ $Y_1(t), \cdots, Y_N(t)\}$. Moreover, the immediate cost is redefined as $$\begin{aligned}
C(\textbf{S}(t), D(t)=d)
=&\sum_{i=1}^N (X_i(t)+1)-(X_d(t)-Y_d(t)),\end{aligned}$$ where we define $Y_0(t)=0$ for all $t$. Then, similar to Section \[section:mdp\], we can show that
- there exists a deterministic stationary policy that is age-optimal;
- the similar sequence of approximate MDPs converges;
- an age-optimal scheduling algorithm is switch-type: for every user $u_i$, if a $\Delta$-optimal action at state $\mathbf{s}=(x_i, \mathbf{x}_{-i}, \mathbf{y})$ is $d^*_{(x_i, \mathbf{x}_{-i}, \mathbf{y})}=i$, then $d^*_{(x_i+1,\mathbf{x}_{-i},\mathbf{y})}= i$, where $\mathbf{y}=(y_1, \cdots, y_N)$ is the vector of all initial ages.
We then modify Alg. \[alg:offline\] for the network with the buffers, as an age-optimal scheduling algorithm.
To study the effect of the buffers, we consider the truncated MDP with the boundary $m=30$ and generate arrivals with $p_1=p_2=p$. After averaging the age over 100,000 slots, we obtain the average age in Fig. \[fig:buffer\] for various $p$, where the red curve with the triangle markers indicates the no-buffer networks (by employing Alg. \[alg:offline\]) and the blue curve with the star markers indicates the network with the buffers.
In this setting, we see mild improvement of the average age by exploiting the buffers. The buffers reduce the average age by only around $(5.6-5.3)/5.6\approx 5\%$ when $p=0.4$, and even lower when $p$ is higher. Let us discuss the following three cases when *both users have arrivals in some slot*:
- *When both $p_1$ and $p_2$ are high*: That means the user who is not updated currently has a new arrival in the next slot with a high probability; as such, the old packet in the buffer seems not that effective.
- *When both $p_1$ and $p_2$ are low*: Then, the possibility of the two arrivals in the same slot is very low. Hence, this would be a trivial case.
- *When one of $p_1$ and $p_2$ are high and the other is low*: In this case, the BS will give the user with the lower arrival rate a higher update priority, as a packet for the other user will arrive shortly.
According to the above discussions, we observe that the buffers might not be that effective as expected. The no-buffer network is not only simple for practical implementation but also works well.
![Minimum average age for the network with/without the buffers by running the MDP-based scheduling algorithms, where the arrival rates $p_1=p_2=p$.[]{data-label="fig:buffer"}](buffer.eps){width=".45\textwidth"}
Concluding remarks
==================
In this paper, we treated a wireless broadcast network, where many users are interested in different information that should be delivered by a base-station. We studied the age of information by designing and analyzing four scheduling algorithms, i.e., the structural MDP scheduling algorithm, the index scheduling algorithm, the MPD-based online scheduling algorithm, and the index-based online scheduling algorithm. We not only theoretically investigated the optimality of the proposed algorithms, but also validate their performance via the computer simulations. It turns out that the low-complexity index scheduling algorithm and both online scheduling algorithms almost achieve the minimum average age.
Some possible future works are discussed as follows. We focused on the no-buffer network in this paper. It is an issue to study provable effectiveness of the buffers and to characterize the regime under which the no-buffer network works with marginal performance loss. Moreover, it is interesting to investigate structural results like ours for simplifying the calculation of the Whittle index for networks with buffers. Finally, the paper treated a single-hop network only. It is interesting to extend our results to multi-hop networks.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of Yu-Pin Hsu is supported by Ministry of Science and Technology, Taiwan (Project No. 107-2221-E-305-007-MY3).
Proof of Proposition \[lemma:optimality-eq\] {#appendix:lemma:optimality-eq}
=============================================
According to [@stationary-policy:Sennott], it suffices to show that $V_{\alpha}(\mathbf{s})< \infty$ for every initial state $\mathbf{s}$ and discount factor $\alpha$. Let $f$ be the deterministic stationary policy of the MDP $\Delta$ that chooses $D(t)=0$ for all $t$. Note that, for initial state $\mathbf{s}=(x_1, \cdots, x_N, \lambda_1, \cdots, \lambda_N)$, we have $$\begin{aligned}
V_{\alpha}(\mathbf{s};f)=& \lim_{T \rightarrow \infty}E_{f}\Bigl[\sum_{t=0}^{T} \alpha^t C(\mathbf{S}(t), D(t)) | \mathbf{S}(0)=\mathbf{s}\Bigr] \\
=&\sum_{t=0}^{\infty} \alpha^t \left[(x_1+t)+\cdots+(x_N+t)\right]\\
=&\frac{x_1+\cdots+x_N}{1-\alpha}+ \frac{\alpha N}{(1-\alpha)^2} < \infty.\end{aligned}$$ By definition of the optimality, we conclude $V_{\alpha}(\mathbf{s}) < \infty$ since $V_{\alpha}(\mathbf{s}; f) < \infty$.
Proof of Proposition \[lemma:monotone\] {#appendix:lemma:monotone}
========================================
The proof is based on induction on $n$ of the value iteration in Eq (\[eq:discount-itr\]). The result clearly holds for $V_{\alpha, 0}(\mathbf{s})$. Suppose that $V_{\alpha, n}(\mathbf{s})$ is non-decreasing in $x_i$. First, note that the immediate cost $C(\mathbf{s},d)=\sum_{i=1}^N(x_i+1)-x_d \lambda_d$ is a non-decreasing function in $x_i$. Second, $E[V_{\alpha,n}(\mathbf{s}')]$ is also a non-decreasing function in $x_i$ according to the induction hypothesis. Since the minimum operator (in Eq. (\[eq:discount-itr\])) holds the non-decreasing property, we conclude that $V_{\alpha, n+1}(\mathbf{s})$ is a non-decreasing function as well.
Proof of Lemma \[thm:stationary\] {#appendix:thm:stationary}
==================================
We divide state space $\mathbf{S}$ into two disjoint sets $\mathbf{S}_1$ and $\mathbf{S}_2$, where the age vectors in the set $\mathbf{S}_1$ belongs to $\mathbf{X}$. Then, set $\mathbf{S}_2$ is a transient set for any scheduling algorithm that will update each user for at lease once. We consider two cases as follow.
First, we focus on the MDP $\Delta$ with the restricted state space $\mathbf{S}_1$, and show the lemma holds. According to [@stationary-policy:Sennott], we need to verify that the following two conditions are satisfied.
1. *There exists a deterministic stationary policy $f$ of the MDP $\Delta$ such that the resulting discrete-time Markov chain (DTMC) by the policy is irreducible, aperiodic, and the average cost $V(f)$ is finite*: We consider the deterministic stationary policy $f$ as the one that updates a user with an arrival and the largest age. It is obvious that the resulting DTMC is irreducible and aperiodic. Next, we transform the age of information into an *age-queueing network* in Fig. \[fig:age-queueing\] consisting of $N$ age-queues $q_1, \cdots, q_N$, age-packet arrivals to each queue, and a server. In each slot an age-packet arrives at the system, since the age increases by one for each slot. In each slot a channel associated with queue $q_i$ is ON with probability $p_i$. For each slot the server can serve a queue with an *infinite* number of age-packets if its channel is ON. Then, the long-run average total age-queue size is the average cost. Note that the arrival rate is interior of the *capacity region* [@neely:book] of the age-queueing network. Moreover, the policy $f$ is the *maximum weight scheduling* algorithm [@neely:book] that is shown to be throughput-optimal; as such, the average age-queue size is finite. Thus, the average cost $V(f)$ is finite as well.
![Age-queueing network.[]{data-label="fig:age-queueing"}](age-queue.eps){width=".35\textwidth"}
2. *There exists a nonnegative $L$ such that the relative cost function $h_{\alpha}(\mathbf{s}) \geq -L$ for all $\mathbf{s}$ and $\alpha$*: Let $C_{\mathbf{s},\mathbf{s}'}(\pi)$ be the expected cost of the first passage from state $\mathbf{s}$ to state $\mathbf{s}'$ under policy $\pi$. Then, using the deterministic stationary policy $f$ in the first condition, we have $C_{\mathbf{s},\mathbf{s}'}(f) < \infty$ (see [@stationary-policy:Sennott Proposition 4]) and $ h_{\alpha}(\mathbf{s}) \geq - C_{\mathbf{0},\mathbf{s}}$ (see [@stationary-policy:Sennott proof of Proposition 5]). Moreover, as $V_{\alpha}(\mathbf{s})$ is a non-decreasing function in $x_i$ (see our Proposition \[lemma:monotone\]), only state $\mathbf{s}$ with $x_i \leq N$ for all $i$ can probably result in a lower value of $V_{\alpha}(\mathbf{s})$ than $V_{\alpha}(\mathbf{0})$. We hence can choose $L=\max_{\mathbf{s}\in \mathbf{S}: x_i \leq N, \forall i} C_{\mathbf{0},\mathbf{s}}$.
Thus, according to [@stationary-policy:Sennott], there exists a deterministic stationary policy that is $\Delta$-optimal and minimum average cost is the constant $V^*$, independent of the initial state.
Second, we note that, if the initial state belongs to $\mathbf{S}_2$, then a $\Delta$-optimal policy will update each user for at least once (e.g, using the above deterministic scheduling algorithm $f$); otherwise, the average cost is infinite. In other words, state $\mathbf{S}(t)$ will enter $\mathbf{S}_1$ in *finite* time, and always stay in the set $\mathbf{S}_1$ onwards. Thus, the average cost *until the state enters $\mathbf{S}_1$* approaches zero as slots go to infinity, and the minimum average cost is still the constant $V^*$ as in the first case. Moreover, there exists a deterministic stationary policy that is $\Delta$-optimal, e.g., following the deterministic stationary policy $f$ before entering $\mathbf{S}_1$ and then following the $\Delta$-optimal deterministic stationary policy in the first case.
Proof of Theorem \[theorem:finite-approximation\] {#appendix:theorem:finite-approximation}
=================================================
Let $V^{(m)}_{\alpha}(\mathbf{s})$ and $h^{(m)}_{\alpha}(\mathbf{s})$ be the minimum expected total $\alpha$-discounted cost and the relative cost function for the MDP $\Delta^{(m)}$, respectively. According to [@finite-state:Sennott], we need to prove the following two conditions are satisfied.
1. *There exists a nonnegative $L$, a nonnegative finite function $F(.)$ on $\mathbf{S}$ such that $-L \leq h^{(m)}_{\alpha}(\mathbf{s}) \leq F(\mathbf{s})$ for all $\mathbf{s} \in \mathbf{S}^{(m)}$, where $m=N+1, N+2, \cdots$ and $0<\alpha<1$*: We consider a randomized stationary algorithm $f$ that updates each user (with packet arrival) with equal probability for each slot. Similar to Appendix \[appendix:thm:stationary\], let $C_{\mathbf{s}, \mathbf{0}}(f)$ and $C^{(m)}_{\mathbf{s},\mathbf{0}}(f)$ be the expected cost from state $\mathbf{s} \in \mathbf{S}^{(m)}$ to the reference state $\mathbf{0}$ by applying the algorithm $f$ to $\Delta$ and $\Delta^{(m)}$, respectively. Then, $h^{(m)}_{\alpha}(\mathbf{s}) \leq C^{(m)}_{\mathbf{s},\mathbf{0}}(f)$ and $C_{\mathbf{s},\mathbf{0}}(f) < \infty$ similar to Appendix \[appendix:thm:stationary\]. In the following, we will show that $C^{(m)}_{\mathbf{s},\mathbf{0}}(f) \leq C_{\mathbf{s},\mathbf{0}}(f)$ and then we can choose the function $F(\mathbf{s})=C_{\mathbf{s},\mathbf{0}}(f)$.
To that end, we first express $P^{(m)}_{\mathbf{s}, \mathbf{s}'}(d)$ as $$\begin{aligned}
P^{(m)}_{\mathbf{s}, \mathbf{s}'}(d)=P_{\mathbf{s}, \mathbf{s}'}(d)+\sum_{\mathbf{r}(\mathbf{s}') \in \mathbf{S}-\mathbf{S}^{(m)}} P_{\mathbf{s}, \mathbf{r}(\mathbf{s}')}(d), \end{aligned}$$ for some (or no) *excess probabilities* [@finite-state:Sennott] on some state $\mathbf{r}(\mathbf{s}') \in \mathbf{S}-\mathbf{S}^{(m)}$, depending on next state $\mathbf{s}'$. Since the scheduling algorithm $f$ is independent of the age, given arrival vector $\boldsymbol{\lambda}$ we have $C_{(\mathbf{i},\boldsymbol{\lambda}),\mathbf{0}}(f) \leq C_{(\mathbf{j},\boldsymbol{\lambda}),\mathbf{0}}(f)$ for age vector $\mathbf{i} \leq \mathbf{j}$. Then, we obtain $$\begin{aligned}
&\sum_{\mathbf{s}' \in \mathbf{S}^{(m)}} P^{(m)}_{\mathbf{s},\mathbf{s}'}(d)C_{\mathbf{s}',\mathbf{0}}(f) \nonumber\\
=& \sum_{\mathbf{s}' \in \mathbf{S}^{(m)}} \bigl(P_{\mathbf{s},\mathbf{s}'}(d)+\sum_{\mathbf{r}(\mathbf{s}') \in \mathbf{S}-\mathbf{S}^{(m)}} P_{\mathbf{s},\mathbf{r}(\mathbf{s}')}(d)\bigr)C_{\mathbf{s}',\mathbf{0}}(f) \nonumber\\
\leq &\sum_{\mathbf{s}' \in \mathbf{S}^{(m)}} P_{\mathbf{s},\mathbf{s}'}(d)C_{\mathbf{s}',\mathbf{0}}(f)+\sum_{\mathbf{k} \in \mathbf{S}-\mathbf{S}^{(m)}} P_{\mathbf{s},\mathbf{k}}(d)C_{\mathbf{k},\mathbf{0}}(f) \nonumber\\
=&\sum_{\mathbf{s}' \in \mathbf{S}} P_{\mathbf{s},\mathbf{s}'}(d)C_{\mathbf{s}',\mathbf{0}}(f). \label{eq:c-inequ}\end{aligned}$$
Using the above inequality, we then conclude $C^{(m)}_{\mathbf{s},\mathbf{0}}(f) \leq C_{\mathbf{s},\mathbf{0}}(f) $ because $$\begin{aligned}
C^{(m)}_{\mathbf{s},\mathbf{0}}(f)=&E_{f}[C({\mathbf{s}},d)+\sum_{\mathbf{s}' \in \mathbf{S}^{(m)}} P^{(m)}_{\mathbf{s},\mathbf{s}'}(d)C_{\mathbf{s}',\mathbf{0}}(f)]\\
\leq & E_{f}[C(\mathbf{s},d)+\sum_{\mathbf{s}' \in \mathbf{S}} P_{\mathbf{s},\mathbf{s}'}(d) C_{\mathbf{s}',\mathbf{0}}(f)]\\
=&C_{\mathbf{s},\mathbf{0}}(f).\end{aligned}$$ On the other hand, we can choose $L=\max_{\mathbf{s} \in \mathbf{S}: x_i \leq N, \forall i} C_{\mathbf{0},\mathbf{s}}(f)$, since $h^{(m)}_{\alpha}(\mathbf{s}) \geq -C^{(m)}_{\mathbf{0}, \mathbf{s}}(f)$ (see Appendix \[appendix:thm:stationary\]) and $ -C^{(m)}_{\mathbf{0}, \mathbf{s}}(f)\geq -C_{\mathbf{0}, \mathbf{s}}(f)$ similar to above.
2. *The value $V^{(m)*}_{\infty}$ is bounded by $V^*$, i.e., $ V^{(m)*}_{\infty}\leq V^*$*: We claim that $V^{(m)}_{\alpha}(\mathbf{s})\leq V_{\alpha}(\mathbf{s})$ for all $m$, and then the condition holds as $$\begin{aligned}
V^{(m)*}=&\limsup_{\alpha \rightarrow 1} (1-\alpha) V^{(m)}_{\alpha}(\mathbf{s}) \\
\leq& \limsup_{\alpha \rightarrow 1} (1-\alpha)V_{\alpha}(\mathbf{s})=V^*. \end{aligned}$$
To verify this claim, we first note that $V_{\alpha}(\mathbf{s})$ is a non-decreasing function in age (see Proposition \[lemma:monotone\]). Then, similar to Eq. (\[eq:c-inequ\]), we have $$\begin{aligned}
\sum_{\mathbf{s}' \in \mathbf{S}^{(m)}} P^{(m)}_{\mathbf{s},\mathbf{s}'}(d)V_{\alpha}(\mathbf{s}')
\leq \sum_{\mathbf{s}' \in \mathbf{S}} P_{\mathbf{s},\mathbf{s}'}(d)V_{\alpha}(\mathbf{s}'). \label{eq:v-star-ineq}\end{aligned}$$
We now prove the claim by induction on $n$ in Eq. (\[eq:discount-itr\]). It is obvious when $n=0$. Suppose that $V^{(m)}_{\alpha,n}(\mathbf{s}) \leq V_{\alpha,n}(\mathbf{s})$, and then $$\begin{aligned}
&V^{(m)}_{\alpha,n+1}(\mathbf{s})\\
=&\min_{d \in \{0, 1, \cdots N\}} C(\mathbf{s},d)+ \alpha \sum_{\mathbf{s}' \in \mathbf{S}^{(m)}} P^{(m)}_{\mathbf{s},\mathbf{s}'}(d) V^{(m)}_{\alpha,n}(\mathbf{s}')\\
\mathop{\leq}^{(a)} & \min_{d \in \{0, 1, \cdots N\}} C(\mathbf{s},d)+ \alpha \sum_{\mathbf{s}' \in \mathbf{S}^{(m)}} P^{(m)}_{\mathbf{s},\mathbf{s}'}(d) V_{\alpha,n}(\mathbf{s}')\\
\mathop{\leq}^{(b)} & \min_{d \in \{0, 1, \cdots N\}} C(\mathbf{s},d)+ \alpha \sum_{\mathbf{s}' \in \mathbf{S}} P_{\mathbf{s},\mathbf{s}'}(d) V_{\alpha,n}(\mathbf{s}')\\
=&V_{\alpha,n+1}(\mathbf{s}),\end{aligned}$$ where (a) results from the induction hypothesis, and (b) is due to Eq. (\[eq:v-star-ineq\]).
\[remark to truncation\] Here, we want to emphasize that we have chosen $m > N$. If not, the state space $\mathbf{S}$ of the MDP $\Delta$ would have to include more *transient* states such that all ages are no more than $N$, e.g., the age vector of $(N, \cdots, N)$. These additional states are not reachable from state $\mathbf{0}$. Thus, we cannot choose the $L$ as in the proof, since $C_{\mathbf{0}, \mathbf{s}}(f)$ is infinite if the age vector in state $\mathbf{s}$ is $(N, \cdots, N)$.
Proof of Theorem \[theorem:truncation\] {#appendix:theorem:truncation}
========================================
According to [@MDP:Puterman Theorem 8.6.6], the RVIA in Eq. (\[eq:rvia\]) converges to the optimal solution in finite iterations if the truncated MDP is *unichain*, i.e., the Markov chain corresponding to every deterministic stationary policy consists of a single recurrent class plus a possibly empty set of transient states. Note that for every truncated MDP, there is only one recurrent class by [@gallager], since the state $(m,\cdots, m, 0, \cdots, 0)$ is reachable (e.g., there is no arrival in the next $m$ slots) from all other states (where remember that $m$ is the boundary of the truncated MDP). Hence, the truncated MDPs are unichain and the theorem follows immediately.
Proof of Theorem \[theorem:stationary-omgea\] {#appendix:theorem:stationary-omgea}
=============================================
Given initial state $\mathbf{S}(0)=\mathbf{s}$, we define the expected total $\alpha$-discounted cost under policy $\mu$ by $$\begin{aligned}
J_{\alpha}(\mathbf{s};\mu)=\limsup_{T \rightarrow \infty} E_{\mu}\left[ \sum_{t=0}^T \alpha^t C(\mathbf{S}(t), A(t))|\mathbf{s}(0)=\mathbf{S} \right],\end{aligned}$$ Let $J_{\alpha}(\mathbf{s})=\min_{\mu}J_{\alpha}(\mathbf{s};\mu)$ be the minimum expected total $\alpha$-discounted cost. A policy that minimizes $J_{\alpha}(\mathbf{s};\mu)$ is called *$\Omega_{\alpha}$-optimal policy*. Again, we check the two conditions in Appendix \[appendix:thm:stationary\].
1. Let $f$ be the deterministic stationary policy of always choosing action $A(t)=1$ for each slot $t$ if there is an arrival. It is obvious that the resulting DTMC by the policy is irreducible and aperiodic. To calculate the average cost, we note that age $X(t)$ by the policy $f$ is also a DTMC in Fig. \[fig:age-dtmc\]. The steady-state distribution $\boldsymbol{\xi}=(\xi_1, \xi_2, \cdots, )$ of the DTMC is $$\begin{aligned}
\xi_i=p(1-p)^{i-1}\,\,\,\,\text{for all $i=1,2, \cdots$}.\end{aligned}$$ Hence, the average age is $$\begin{aligned}
\sum_{i=1}^{\infty} i \xi_i= \sum_{i=1}^{\infty} i p(1-p)^{i-1} = \frac{1}{p}.\end{aligned}$$ On the other hand, the average updating cost is $c \cdot p$ as the arrival probability is $p$. Hence, the average cost under the policy $f$ is the average age (i.e., $1/p$) plus the average updating cost (i.e., $c\cdot p$), which is finite.
![The age $X(t)$ under the policy $f$ forms a DTMC.[]{data-label="fig:age-dtmc"}](age-dtmc.eps){width=".35\textwidth"}
2. Similar to Proposition \[lemma:monotone\], we can show that $J_{\alpha}(x,\lambda)$ is a non-decreasing function in age $x$ for a given arrival indicator $\lambda$; moreover, $J_{\alpha}(x,\lambda)$ is a non-increasing function in $\lambda$ for a given age $x$. Thus, we can choose $L=0$.
By verifying the two conditions, the theorem immediately follows from [@stationary-policy:Sennott].
Proof of Lemma \[lemma:optimal-threshold\] {#appendix:lemma:optimal-threshold}
==========================================
Since $I(x,1)$ is the updating cost $c$ to make both actions for state $(x,1)$ equally desirable and we break a tie in favor of idling, the optimal threshold is $x+1$ if the cost is $c=I(x,1)$, for all $x$. We claim that the optimal threshold monotonically increases with cost $c$, and then the theorem follows.
To verify the claim, we can focus on the discounted cost case according to the proof of Theorem \[theorem:optimal-switch\]. Suppose that an $\Omega_{\alpha}$-optimal action, associated with a cost $c_1$, for state $(x,1)$ is to idle, i.e., $$\begin{aligned}
x+1+\alpha E[J_{\alpha}(x+1,\lambda')] \leq 1+c_1+\alpha E[J_{\alpha}(1,\lambda')] .\end{aligned}$$ Then, an $\Omega_{\alpha}$-optimal action, associated with a cost $c_2 \geq c_1$, for state $(x,1)$ is to idle as well since $$\begin{aligned}
x+1+\alpha E[J_{\alpha}(x+1,\lambda')] \leq &1+c_1+\alpha E[J_{\alpha}(1,\lambda')] \\
\leq & 1+c_2+\alpha E[J_{\alpha}(1,\lambda')].\end{aligned}$$ Then, the monotonicity is established.
Proof of Lemma \[lemma:index-optimal\] {#appendix:lemma:index-optimal}
======================================
Similar to the proof of Theorem \[theorem:optimal-switch\], we can focus on the discounted cost case. Without loss of generality, we assume that age $x_1 \geq \max(x_2, \cdots x_N)$.
Let $\mathbf{x}_{ij}=(0, \cdots, x_j, \cdots, 0)$ be the zero vector except for the $i$-the entry being replaced by $x_j$. By the symmetry of the users, swap of the initial ages of any two users results in the same expected total $\alpha
$-discounted cost, i.e., $$\begin{aligned}
E[V_{\alpha}(x_1,\mathbf{x}_{-1}, \boldsymbol{\lambda})]=E[V_{\alpha}(x_j, \mathbf{x}_{-1}-\mathbf{x}_j+\mathbf{x}_{j1}, \boldsymbol{\lambda})],\end{aligned}$$ for all $j \neq 1$. Similar to the proof of Theorem \[theorem:optimal-switch\], here we focus on the case when $\lambda_1=1$ and $\lambda_j=1$. The result follows from the non-decreasing function of $V_{\alpha}(x_1,\mathbf{x}_{-1},\boldsymbol{\lambda})$ and $x_1 \geq x_j$ for all $j \neq 1$: $$\begin{aligned}
&\nu_{\alpha}(x_1, \mathbf{x}_{-1},\boldsymbol{\lambda};1)-\nu_{\alpha}(x_1, \mathbf{x}_{-1},\boldsymbol{\lambda};j)\\
=&x_j-x_1 +\alpha E[V_{\alpha}(1, \mathbf{x}_{-1}+\mathbf{1}, \boldsymbol{\lambda}')\\
& -V_{\alpha}(x_1+1, \mathbf{x}_{-1}+\mathbf{1}-\mathbf{x}_j, \boldsymbol{\lambda}')]\\
=&x_j-x_1 +\alpha E[V_{\alpha}(x_j+1, \mathbf{x}_{-1}+\mathbf{1}-\mathbf{x}_j, \boldsymbol{\lambda}')\\
&-V_{\alpha}(x_1+1, \mathbf{x}_{-1}+\mathbf{1}-\mathbf{x}_j, \boldsymbol{\lambda}')] \leq 0.\\\end{aligned}$$
[^1]: This paper was presented in part in the Proc. of IEEE ISIT, 2017 [@hsuage] and 2018 [@hsu2018age].
[^2]: The optimal policy for the truncated MDPs is switch-type as well, according to the same proof as Theorem \[theorem:optimal-switch\].
|
---
author:
- Theofilos Petsios
- Jason Zhao
- 'Angelos D. Keromytis'
- Suman Jana
bibliography:
- 'paper.bib'
title: ': Automated Domain-Independent Detection of Algorithmic Complexity Vulnerabilities'
---
<ccs2012> <concept> <concept\_id>10002978.10003022.10003023</concept\_id> <concept\_desc>Security and privacy Software security engineering</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Acknowledgments
===============
We would like to thank the anonymous reviewers for their valuable feedback. This work is sponsored in part by the Office of Naval Research (ONR) grant N00014-17-1-2010, the National Science Foundation (NSF) grants CNS-13-18415 and CNS-16-17670, and a Google Faculty Fellowship. Any opinions, findings, conclusions, or recommendations expressed herein are those of the authors, and do not necessarily reflect those of the US Government, ONR, NSF, or Google.
|
---
abstract: 'Departures of observables from their thermal equilibrium expectation values are studied under heat flow in steady-state non-equilibrium environments. The relation between the spatial and temperature dependence of these non-equilibrium behaviors and the underlying statistical properties are clarified from general considerations. The predictions are then confirmed in direct numerical simulations within the FPU-$\beta$ model. Non-equilibrium momentum distribution functions are also examined and characterized through their cumulants and the properties of higher order cumulants are discussed.'
author:
- 'Kenichiro Aoki[^1] and Dimitri Kusnezov[^2]'
title: Spatial Distributions of Observables in Systems under Thermal Gradients
---
Introduction
============
In studies of non-equilibrium physics, especially those of steady states, local equilibrium is most often invoked and this assumption simplifies calculations through the use of equilibrium statistical mechanics and thermodynamics[@neq]. The local equilibrium assumption allows the use of the equilibrium distribution function to compute observables. If local equilibrium conditions are not assumed, very little can be computed analytically and even the definition of temperature is no longer unique[@keizer; @nonEqT]. Efforts have been made to quantify the goodness of local equilibrium assumptions or how transport coefficients differ from their linear response values, though only few quantitative studies exist[@fluct; @eos; @hk; @local-eq; @onsager; @dhar; @takesue; @ak-le]. Without the knowledge of the non-equilibrium steady-state distribution, theoretical development becomes quite restrictive. We explore how observables depart from their equilibrium expectation values within a given non-equilibrium steady-state, specifically focusing on the spatial dependence of the non-equilibrium expectation values within a given system and their local temperature dependence. To make this concrete, heat flow in the FPU $\beta-$model is simulated to test the predictions. We further quantitatively examine the relationship between the momentum cumulants and the distribution and find that the lower order cumulants characterize the distribution quite well.
For systems in thermal gradients, it is natural to consider how an observable $\cal O$ in the non-equilibrium steady state departs from its equilibrium value, denoted ${\cal O}_{eq}$. The normalized deviation from equilibrium, when ${\cal
O}_{eq}\not=0$, can be expanded as $$\label{eq:le1}
\delta_{\cal O} \equiv\frac{\delta {\cal O}}{\cal O}=\frac{{\cal
O}-{\cal O}_{eq}}{{\cal O}_{eq}} =
C_{\cal O}\left[\frac{\nabla T}{T}\right]^2 + C_{\cal O}'\left[\frac{\nabla
T}{T}\right]^4 + \cdots$$ When ${\cal O}_{eq}=0$, as is the case for higher order momentum cumulants, one can normalize by an observable which has the same dimensions. When local equilibrium is no longer valid, in general, no unique definition of temperature exists and a choice needs to be made. This definition of non-equilibrium temperature can be thought of as a choice of a coordinate system, on which the physics behavior of the system will not depend. If we assume analyticity in $\nabla T$, the deviations $\delta_{\cal
O}$ can be expanded in even powers as above. We shall see below that this expansion is adequate for describing the properties of the system.
The heat flow, $J$, is the flow of energy and can be unambiguously defined in Hamiltonian systems. Near equilibrium, it satisfies Fourier’s law locally as $J=-\kappa \nabla T(x)$, where $\kappa$ is the thermal conductivity, $T(x)$ is the temperature profile inside, and $x$ is the position inside the system. Fourier’s law can be used in to re-express the local departures from equilibrium in terms of the temperature profile $T(x)$, or equivalently the position $x$ once the coefficients $C$, $C'$ are known since $J$ does [*not*]{} depend on $x$; $$\label{eq:spatial1}
\delta_{\cal O} =
C_{\cal O}\left(J\over\kappa(T) T\right)^{2}+
D_{\cal O}' \left(J\over\kappa(T) T\right)^{4}+ \ldots$$ We note that Fourier’s law itself receives non-equilibrium corrections[@ak-le], which is why the coefficient of ${\cal
O}(J^4)$ term in the expansion differs from that of . In the following, the objectives will be to make the formula more explicit and understand its physical properties under rather general assumptions. This relation, together with $\kappa(T)$ (and consequently $T(x)$) provides the basis for defining how non-equilibrium observables vary inside a finite system both near and far from global thermal equilibrium.
The FPU model and Temperature Profiles
======================================
The results we present here are derived from general considerations and we develop them in conjunction with a model in which they can be explicitly analyzed. We study the FPU $\beta$ Hamiltonian, defined generally in the form $$\label{eq:fpu-dimensionful}
{\tilde H} = \sum_{k=0}^L \left[
\frac{\tilde p_k^2}{2m}+\frac{1}{2}m\omega^2
(\tilde q_{k+1}-\tilde q_k)^2 + \frac{\beta}{4} (\tilde
q_{k+1}-\tilde q_k)^4\right].$$ We use the FPU model since its physical properties are of wide interest ([@km; @lepri; @ak-fpu; @fpuRevs] and references therein). Also as the model is well studied, we can understand the physical properties we find within a larger physics context. Under the rescaling $\tilde p_k=p'_k\omega^2\sqrt{m^3}$, $\tilde q_k=
q'_k\omega\sqrt{m}$, we obtain the conventional form of the FPU $\beta$ model, $$\label{eq:fpu-usual}
H_\beta = \frac{1}{2}\sum_{k=0}^L \left[ p_k^{\prime 2}
+ (q'_{k+1}- q'_k)^2
+ \frac{\beta}{2} (q'_{k+1}- q'_k)^4\right],$$ where $H_\beta=\tilde H/(m^2\omega^4)$. We note that in finite temperature simulations, changing the temperature is equivalent to changing the coupling $\beta$. Under the additional rescaling $p'_k=p_k/\sqrt\beta $, $q'_k=q_k/\sqrt\beta$, one obtains a unique, dimensionless, Hamiltonian $H\equiv H_{\beta=1}=\beta
H_\beta$, which we shall use without any loss of generality. Since $p_k^2=\beta p_k'^2$, the temperatures in the two formulations $H$ and $H_\beta$ are related by $T=\beta T'$.
In this work, we study the non-equilibrium steady state physics of the theory under thermal gradients, making use of non-equilibrium states constructed numerically. (For general discussion, see, for instance, [@neqRevs; @thermoRevs].) The model is thermostatted at the boundaries $k=0,L$ at various temperatures $T_1^0, T_2^0$, using the generalized versions of Nosé–Hoover thermostats as detailed in [@ak-long]. These additional thermostat degrees of freedom are added only at the boundaries and the degrees of freedom inside the system ($0<k<L$) are exclusively those of the Hamiltonian . By numerically integrating the equations of motion of the whole system (including those of the thermostats), we obtain the behavior of physical observables in the non–equilibrium steady state by averaging over time, in the standard manner[@thermoRevs]. The local temperature at site $k$ is defined as $T_k=\langle
p_k^2\rangle$. In this work, we study the physics inside the system, away from the boundaries by much more than the mean free path of the system[@ak-fpu]. The sensitivity of the results to the manner in which we apply the boundary conditions — including both the number of thermostats and the strength of the couplings — have been examined to ensure that physics results below remain independent of their implementation. (The only exceptions are the boundary jumps in temperature which we discuss below.) The numerical integrations were performed using the fourth order Runge-Kutta routines with time steps of $0.005\sim0.02$ for $10^7\sim10^{10}$ time steps. The equilibrium properties have been readily verified with this method[@ak-fpu; @ak-long].
![Some examples of temperature profiles for the FPU model with $L=128$. The thermostat temperatures at the boundaries are $(T_1^0,T_2^0)=(0.88,16.72),(2.4,15.2),(4.4,13.2),(6.6,11.0)$ for the four thermal profiles. The profiles predicted from Eq. are indicated by $\times$ and agree well with the results from the numerical simulations.[]{data-label="fig:p2x"}](T-x1.eps){width="8.5cm"}
In some examples of temperature profiles for the FPU theory are shown. Generically, there are temperature jumps just inside the boundaries with smooth temperature variations within. The boundary jumps become larger as one moves away from global equilibrium. The jumps are dynamical in the sense that they depend on the model, the transport coefficient, heat flow, as well as the type of boundary conditions employed. The temperatures at the boundaries are at the thermostat temperatures to high degree of precision. For instance, in the examples of , the boundary temperatures are equal to the prescribed thermostat temperatures to within few in $10^5$ relatively.
From temperature profiles and heat flow calculations, Fourier’s law can be verified to hold up to corrections of the form , and the thermal conductivity, $\kappa$, can be obtained for a given temperature and system size. In the 1-d FPU model, $\kappa$ depends on the system size $L$ and does not display bulk behavior[@lepri]. $\kappa$ is also dependent on the temperature in a known manner[@ak-fpu]. Generally, in cases where we have a one dimensional temperature gradient, the temperature profiles can be obtained by integrating Fourier’s law as long as we are not too far from equilibrium [@ak-le; @profs; @ak-long]: $$\label{eq:integrateFL}
\int_{T_1}^{T(x)} \kappa(T)\,dT=-Jx,\qquad
J=-p_k\left[(q_{k+1}-q_k)+(q_{k+1}-q_k)^3\right]$$ $x$ is the continuum extrapolation of the discrete lattice index $k$. We note here that $J$ is a constant within the system for a given set of temperature boundary conditions since there are no heat sinks or sources inside. $T_1$ in the integral is the temperature extrapolated to the boundary and is explained below.
In many situations, the temperature dependence of the thermal conductivity, within some temperature range, can be well described by $$\kappa(T) = cT^{-\gamma}.
\label{eq:tc}$$ While this power law may not hold globally in $T$, it is often the case that it is sufficient for the region of interest, which is the case here. In such a situation, the temperature profile can be explicitly computed from to be[@ak-long] $$\label{eq:t-profile}
T(x) = \left\{\begin{array}{ll}
T_1\left[1-\left(1-\left(\frac{T_2}{T_1} \right)^{1-\gamma}
\right)
{\frac{x}{L}}\right]^{{\frac{1}{1-\gamma}}},\qquad &
\gamma\neq1\\
T_1 \left( \frac{T_2}{T_1}\right)^{x/L}, &\gamma=1\quad.
\end{array}\right.$$ Here, $T_{1,2}$ denote the boundary temperatures obtained by extrapolating the temperature profile inside the system and differs from the thermostat temperatures $T_{1,2}^0$ by the boundary temperature jumps. From and , the temperatures $T_{1,2}$ are found to obey a relation $$\label{eq:one}
-{ JL\over c}={ T_2^{1-\gamma} - T_1^{1-\gamma}\over1-\gamma}$$
To understand the temperature profile of the whole system, we further need an understanding of the temperature jumps at the boundaries[@ak-jumps]. Similar boundary slips have been seen in sheared systems and these effects have been known for a long time in real systems. To leading order, the temperature jumps can be described by (with $n$ being the normal to the boundary) $$\begin{aligned}
\left|T_i - T_i^0\right|
\backsimeq \frac{\alpha c}{L(1-\gamma)}\left[
T_2^{1-\gamma}-T_1^{1-\gamma}\right]
\sim \lambda\frac{\partial T}{\partial n},\qquad (i=1,2)
\label{eq:jumps}\end{aligned}$$ Here $\lambda$ is the mean free path of the excitations, which for the FPU lattice model, is essentially the $\kappa(T)$ (up to a constant factor of order one) due to kinetic theory arguments[@ak-fpu]. $\alpha$ reflects the efficacy of the boundary conditions. The last relation is obtained by using Fourier’s Law and . The jumps on the hot and cold side are the same provided the system is reasonably close to equilibrium. The jumps at the boundaries and the temperature profile within describe the temperature profile of the complete system. The predicted values for the temperature profiles are plotted in at a number points inside the systems ($\times$ symbols) away from the boundaries and are seen to be consistent with the simulation results. The thermal conductivity is roughly constant with respect to the temperature in this region so that $\gamma=0$ was used in the profile calculations. This demonstrates that all aspects of the non-equilibrium temperature profile can be quantitatively captured through and , irrespective of whether $\kappa(T)$ is a power law in temperature for all $T$ or not. With this understanding of $T(x)$ we can now turn to the question of general observables.
Spatial Dependence of Cumulants in the Non-equilibrium Steady State
===================================================================
In non-equilibrium steady states, physical observables show deviations from their equilibrium values reflecting the lack of local equilibrium in the system. The behavior of the observables have been seen to be well described by on average, at least in some cases[@ak-le]. Here, we now would like to investigate a more detailed issue — whether these properties can be used to understand the nature of the spatial profiles of these observables in a given non-equilibrium situation. We will assume that within some range of $T$ and $L$ that we can represent the expansion coefficients in as $$\label{eq:Cbehavior}
C_{\cal O}=\mu_{\cal O} T^{s_{\cal O}} L^{\alpha_{\cal O}}$$ The behavior of $C_{\cal O}$ with respect to $T,L$ clearly must depend on the dynamics of the theory and is not expected to be generic.
FPU $\beta-$ Model in $d=1$: $(\mu T^s)$ $\alpha$
------------------------------ ------------- ----------
$T=1$ 29(5) 0.87(4)
$T=8.8$ 13(1) 0.99(1)
$T=88$ 7.4(4) 1.04(2)
: Non-Equilibrium coefficients $C_{4}=(\mu T^s) L^\alpha$ for $\cum{p^4}/T^2$ ([*cf.*]{} Eq. ,). The results are shown for the FPU $\beta$ model and the $\phi^4$ theory in $d=1\sim3$–dimensions. The value of $s$ is extracted from fitting to several temperatures. []{data-label="tab:a"}
$\phi^4$ Theory : $(\mu T^s)$ $\alpha$
------------------- ------------- -------------
$d=1\quad T=1$ 3.3(24) 0.96(15)
$T=5$ 1.6(6) 1.18(9)
$d=2\quad T=1$ 1.9(4) 1.09(5)
$T=5$ 0.4(2) 1.6(2)
$d=3\quad T=1$ 4 (1) 0.96(10)
$T=5$ 0.2(5) 1.6(6)
: Non-Equilibrium coefficients $C_{4}=(\mu T^s) L^\alpha$ for $\cum{p^4}/T^2$ ([*cf.*]{} Eq. ,). The results are shown for the FPU $\beta$ model and the $\phi^4$ theory in $d=1\sim3$–dimensions. The value of $s$ is extracted from fitting to several temperatures. []{data-label="tab:a"}
To study the spatial distribution of physical observables in non-equilibrium, we make use of which describes how the observables should behave in non-equilibrium locally in space, given the thermal conductivity. Using this property and , we obtain to leading order that observables will deviate from their local equilibrium values as $$\label{eq:dist}
\delta_{\cal O} = C_{\cal O}\; \left(\frac{J
T(x)^{\gamma-1}}{c}\right)^2 = a_{\cal O}T(x)^{2(\gamma-1)+s}$$ Here $a_{\cal O}$ is defined through this equation and should be proportional to $J^2$. This implicitly contains the spatial distribution since the temperature profile is known and can be understood as in .
While these arguments apply to any physical observable in the system, we choose to study cumulants of momenta, $p$, mainly for the following reasons; conceptual and practical. There seems to be no universal rigorous definition of local equilibrium, yet the concept in the least seems to include a unique meaning for temperature, which in this case would lead to the Maxwellian distribution for $p$. To put another way, when the momentum distribution is not Maxwellian, we can choose different definitions of the temperature based on the various moments of $p$[@keizer; @nonEqT]. The cumulants of the momentum distribution provide insight into how the physical properties of a non-equilibrium system deviates from those of local equilibrium. The cumulants are well defined local variables and their values in local equilibrium are known precisely. The low order cumulants are defined as $$\cum{p^2}=\vev{p^2},\quad
\cum{p^4}=\vev{p^4}-3\vev{p^2}^2,\quad
\cum{p^6}=\vev{p^6}-15\vev{p^2}\vev{p^4}+30\vev{p^2}^3 ,\ \ldots$$ where, in [*equilibrium*]{}, $$\label{eq:cumValues}
\cum{p^2}_{eq}=T,\qquad\cum{p^n}_{eq}=0\ (n\not=2)$$ This property is also of practical importance. Since the deviations we compute can be small, it is desirable to use observables whose local equilibrium values are known exactly. In this case in thermal equilibrium, ${\cal O}_{eq}=0$, so we use $\delta_{\cal
O} = \cum{p^{2n}}/T^n$. We list the coefficient for the case ${\cal O}=\cum{p^4}$ in Table \[tab:a\] for the FPU $\beta-$model as well as $\phi^4$ theory[@ak-long] for comparison.
Let us investigate how well describes the spatial distribution of $\cum{p^4}/T^2$. We find $${\cum{p^4}\over T^2}
=a_4T^{2(\gamma-1)+s_4}
= a_4\left(T_1\left[1-
\left(1-\left(\frac{T_2}{T_1} \right)^{1-\gamma} \right)
{\frac{x}{L}}\right]^{{\frac{1}{1-\gamma}}}\right)^{2(\gamma-1)+s_4}
\label{eq:fpuT}$$ $s_4$ is the temperature dependence of the coefficient $C_4$ which is reflected in Table \[tab:a\]. To understand the validity of the prediction Eq. , fits were made with just one parameter $a_4$ for the whole profile. We find that this describes the situation quite well, as seen in the examples of , where the predictions are denoted by dashes. In these figures, we have compared the fits with the spatial as well temperature dependence of $\cum{p^4}$ for the four systems shown in . In this temperature range, temperature dependence of the thermal conductivity is weak so we used $\gamma=0$ and $s_4=-0.14$ extracted from the data in Table \[tab:a\]. Similar results were found for different temperature boundary conditions and for different $L$.
![(left) Spatial dependence of the rescaled 4-th momentum cumulant, $\cum{p^4}/T^2$ for the four systems in . Larger cumulant values are seen for larger boundary temperature differences. (right) Temperature dependence of $\cum{p^4}/T^2$ for the same systems. In both panels, the predictions are indicated by the dashes. []{data-label="fig:p4"}](p4-x.eps "fig:"){width="8.5cm"} ![(left) Spatial dependence of the rescaled 4-th momentum cumulant, $\cum{p^4}/T^2$ for the four systems in . Larger cumulant values are seen for larger boundary temperature differences. (right) Temperature dependence of $\cum{p^4}/T^2$ for the same systems. In both panels, the predictions are indicated by the dashes. []{data-label="fig:p4"}](p4-T.eps "fig:"){width="8.5cm"}
To further verify the underlying physics, we study the $J$ dependence of the coefficient $a_4$. The behavior for various systems, including the four systems in , are shown in . Each data point represents a system with a particular size and temperature boundary conditions. The central temperature is around $T=8.8$ and is kept fixed.
![\[fig:a2\] $J$ dependence of the non-equilibrium expansion coefficient, $a_4$, for various boundary conditions, $(T_1^0,T_2^0)$ and system sizes $L$. The dashed line is $3.72\,J^2$ and the $\sim J^2$ behavior of the coefficient can be clearly seen, as predicted from theory. Each data point represents a particular temperature boundary condition for $L=32$ ($\times$), $L=64$ ($\Box$) and $L=128$ ($\bigcirc$) systems.](a2.eps){width="8.5cm"}
The observed behavior is clearly well described by $a_4\sim$ const.$\times J^2$. The coefficient $a_4$ seems $L$-independent and this can roughly be understood since $c^2$ grows in $L$ in a manner similar to $C_{4}$. We have in addition systematically studied the results to see if we can discern the contribution of higher order terms in the expansions , (of order $J^4$ and higher) but have found no consistent evidence for them. In other physical situations, non-analytic behavior seems to have been seen in some cases[@shears; @sasa].
While the logic seems to work for the lowest non-trivial order cumulant, $\cum{p^4}$, we find it instructive to analyze if it works at higher orders. In this direction, we have analyzed the next non-trivial order $\cum{p^6}$ and have found that its behavior is quite consistent with physics of , as was the case of $\cum{p^4}$, in all the systems we have studied. In practice, higher order cumulants are more prone to errors and the computations are more difficult. The results for the same four systems in are shown in .
![Spatial dependence (left) and temperature dependence (right) of $\cum{p^6}/T^3$ for the four systems in . Larger cumulant values are seen for larger boundary temperature differences. Predictions are shown with dashes. []{data-label="fig:p6"}](p6-x.eps "fig:"){width="8.5cm"} ![Spatial dependence (left) and temperature dependence (right) of $\cum{p^6}/T^3$ for the four systems in . Larger cumulant values are seen for larger boundary temperature differences. Predictions are shown with dashes. []{data-label="fig:p6"}](p6-T.eps "fig:"){width="8.5cm"}
As in the $\cum{p^4}$ case, the coefficient $a_6$ shows $J^2$ behavior within error, as it should. $a_6$ shows a weak $L$ dependence, as we would generically expect. A common value of $s_6=-1.6$ was adopted for all the data in and . What is evident is that the spatial behavior of non-equilibrium observables can be explicitly related to transport and other physical properties of the system using rather general considerations. From the cumulants we now consider what can be said about the full momentum distribution function.
![\[fig:p63\] $J$ dependence of the coefficient $a_6$ for various boundary conditions, $(T_1^0,T_2^0)$ and system sizes $L$. The dashed line denotes $156\, J^2$. $\sim J^2$dependence of $a_6$ is evident, in agreement with the predictions. Each data point represents a particular temperature boundary condition for system sizes $L=32$ ($\times$), $L=64$ ($\Box$) and $L=128$ ($\bigcirc$), as in .](a2-p6.eps){width="8.5cm"}
Cumulants and the distribution {#sec:distro}
==============================
The cumulants are quantitative indicators of the non–Maxwellian nature of the momentum distribution or the violations of local equilibrium. All the cumulants are non-zero unless the system is in local equilibrium, in which case only the linear and quadratic cumulants are non-zero. There are very few problems where cumulants can all be computed analytically and it becomes numerically intractable to compute them as we go to higher orders. It is then of interest to see how well the lower order cumulants characterize the distribution. The cumulants are properties of the distribution function, which has an infinite number of degrees of freedom. A priori, there is no reason to assume that the lower order cumulants characterize the distribution. In order to clarify this issue, first note that the distribution function $f(p)$ and the cumulants are related explicitly through the generating function as $$\label{eq:genfn}
\int dp\,e^{iup} f(p)= \vev{e^{iup}}
=\exp\left(\sum_{n=0}^\infty{i^nu^n\over n!}\cum{p^n}\right)
=\exp\left(\sum_{n=0}^\infty{(-u^2)^n\over (2n)!}\cum{p^{2n}}\right)$$ Here, in the last equality, the symmetry under $p\leftrightarrow-p$ was used, which leads to $\cum{p^{2n+1}}=0$. We see from this equation that given all the cumulants (or equivalently, moments), we may recover the distribution function by performing an inverse Fourier transform. However, in practice, not all the cumulants are available.
Intuitively, we expect the lower order cumulants to be the leading order results with higher order cumulants becoming more important as we move further away from equilibrium. In , we plot the [*relative difference*]{} of the measured distribution $f(p)$ to the thermal distribution, $f_0(p)$, for the distribution directly measured in the simulations and the distribution computed from the low order cumulants, $\cum{p^{2,4,6}}$. The comparisons are performed for the four systems in at a point in the middle of the system. From these graphs, we observe the following: (a) The agreement between the distribution computed from lower order cumulants and the distribution is quite good in all cases; (b) the relative deviation from the thermal distribution is larger as we move away from equilibrium (larger $\Delta T/T$), as expected; (c) the small discrepancy between the computed distribution and the measured one seems to be larger for larger $\Delta T/T$; (d) the deviation from the thermal distribution becomes more noisy for smaller $\Delta T/T$, since the deviation itself is smaller and the relative error is larger. We mention here that strictly speaking, the distributions can have different behavior, such as long tails, beyond the region we have investigated. However, these tails would have to be quite small since the distributions decay as $\exp(-p^2/(2T))$ and the agreement is good up to reasonably large $p$, as seen in . We have examined numerous systems for different $T$ and $L$ and found similar good agreement. Therefore, we see that the lower order cumulants provide good physical observables that quantitatively describe the deviations of the systems from local equilibrium, at least in the FPU model.
![The relative deviation of the distribution from the Maxwell distribution for the four systems in . Distribution obtained from the cumulants $\cum{p^4},\cum{p^6}$ (dashed) are compared with the measured distributions (solid). The agreement is excellent. $\Delta T/T$ denotes the boundary temperature difference over the average temperature and is an indication of how far the system is from equilibrium. []{data-label="fig:pDist"}](cumDist1.eps "fig:"){width="8.5cm"} ![The relative deviation of the distribution from the Maxwell distribution for the four systems in . Distribution obtained from the cumulants $\cum{p^4},\cum{p^6}$ (dashed) are compared with the measured distributions (solid). The agreement is excellent. $\Delta T/T$ denotes the boundary temperature difference over the average temperature and is an indication of how far the system is from equilibrium. []{data-label="fig:pDist"}](cumDist2.eps "fig:"){width="8.5cm"} ![The relative deviation of the distribution from the Maxwell distribution for the four systems in . Distribution obtained from the cumulants $\cum{p^4},\cum{p^6}$ (dashed) are compared with the measured distributions (solid). The agreement is excellent. $\Delta T/T$ denotes the boundary temperature difference over the average temperature and is an indication of how far the system is from equilibrium. []{data-label="fig:pDist"}](cumDist3.eps "fig:"){width="8.5cm"} ![The relative deviation of the distribution from the Maxwell distribution for the four systems in . Distribution obtained from the cumulants $\cum{p^4},\cum{p^6}$ (dashed) are compared with the measured distributions (solid). The agreement is excellent. $\Delta T/T$ denotes the boundary temperature difference over the average temperature and is an indication of how far the system is from equilibrium. []{data-label="fig:pDist"}](cumDist4.eps "fig:"){width="8.5cm"}
It is possible to examine the characteristics of the higher order cumulants. It should be noted that unlike the even moments $\vev{p^{2n}}$, even cumulants, $\cum{p^{2n}}$, need not be positive and in general will not be. So to study the general trend of the cumulants for higher order, we examine the magnitude of the cumulants. In (left), we show the behavior of the cumulants up to 20-th order for the same four systems in , specifically for the point at which the momentum distributions in were computed. Only data points with reasonable error are shown and an explanation of the relevant errors is given below. We see an increase in the magnitude with the order is roughly exponential. This growth is far milder than the $(2n)!$ seen in . The behavior of the higher order cumulants is of some import and we briefly explain semi-quantitatively why they are difficult to obtain. The difficulty lies mainly in the statistical error in the simulations. This can be estimated from the number of samples for computing the expectation values as $$\label{eq:error1}
{\Delta\vev{p^{n}}\over\vev{p^{n}}}\sim{n\over\sqrt{N}}$$ where $\Delta$ denotes the error and $N$ is the total number of samples or the number of time steps in the simulation. Note that $\cum{p^{n}}=\vev{p^n}+\ldots$ so that an error estimate for the moment should suffice as the error estimate for the cumulant. An adequate value for the moment can be obtained in equilibrium, $$\label{eq:momEq}
{\vev{p^n}\over T^{n/2}}\sim(n-1)!!$$ Combining these relations, we find the statistical error for the cumulants which increases rapidly for higher order cumulants. $$\label{eq:error2}
\Delta\left(\cum{p^{n}}\over T^{n/2}\right)\sim
{ (n-1)!! \,n\over\sqrt{N}}$$ These estimates for the error also apply to the equilibrium situation. In contrast to the non-equilibrium cumulants, the equilibrium cumulants should vanish, with the exception of $\cum{p^2}$. As the measured values will converge to zero, at any given time-step in the simulation, their values will be generically non-zero. In (right), we compare the [*equilibrium*]{} cumulants, in the middle of the system to the above error estimates. It can be seen that the rough estimate seems to be consistent with the results. As one samples more ($N$ increases), these will tend to zero. However, for a finite sample size, this is found to explain the order of the uncertainty.
With $N=10^9$ time-steps — which we used for the values in — for 8 and 10-th order cumulants, the errors are 0.03 and 0.3. As we can see from (left), this means that we can obtain up to the 8 or 10-th cumulant with reasonable error for the four systems but the higher order cumulants are expected to be unreliable for systems closer to equilibrium. These error estimates are quite consistent with the estimates we obtain from the statistical properties of the simulations. These errors can be overcome with higher statistics which quickly becomes unrealistic for higher order. We have analyzed systems with various other temperature boundary conditions and $L$ and have found the increasing behavior of the cumulants seen in (left) to be quite generic.
![(left) Higher order cumulants, $|\cum{p^n}|/T^{n/2} \ (n\leq20)$ for the four systems in , $(T_1^0,T_2^0)=(0.88,16.72), \
(\times)$, $(2.4,15.2),\ (\Box)$, $(4.4,13.2),\ (\bigcirc)$ and $(6.6,11.0),\ (\triangle)$. Only points with reasonably small error are shown. The dashed line is $5.0\times 10^{-6}\exp(1.5n)$ drawn for comparison. (right) The equilibrium cumulants for $L=16$ ($\times$), $L=32$ ($\Box$), $L=64$ ($\triangle$) and $L=128$ ($\bigcirc$) compared to the rough estimate, Eq. (dashes). The cumulants $\cum{p^2}$ were measured at the middle of the system with number of samples $N=10^9$ at $T=8.8$.[]{data-label="fig:cumHigh"}](cum20.eps "fig:"){width="8.5cm"} ![(left) Higher order cumulants, $|\cum{p^n}|/T^{n/2} \ (n\leq20)$ for the four systems in , $(T_1^0,T_2^0)=(0.88,16.72), \
(\times)$, $(2.4,15.2),\ (\Box)$, $(4.4,13.2),\ (\bigcirc)$ and $(6.6,11.0),\ (\triangle)$. Only points with reasonably small error are shown. The dashed line is $5.0\times 10^{-6}\exp(1.5n)$ drawn for comparison. (right) The equilibrium cumulants for $L=16$ ($\times$), $L=32$ ($\Box$), $L=64$ ($\triangle$) and $L=128$ ($\bigcirc$) compared to the rough estimate, Eq. (dashes). The cumulants $\cum{p^2}$ were measured at the middle of the system with number of samples $N=10^9$ at $T=8.8$.[]{data-label="fig:cumHigh"}](cum20Eq.eps "fig:"){width="8.5cm"}
Summary and discussions {#sec:summary}
=======================
The spatial distribution of cumulants in non-equilibrium steady states under thermal gradients were predicted from general considerations and tested in the the FPU model. The understanding of the temperature profile for a given non-equilibrium steady state, combined with the deviations of physical observables from their equilibrium values, can be used to develop a consistent description of the spatial distribution of observables. In principle, the behavior of observables probably have higher order corrections in the non-equilibrium nature of the system, which in this case is $\nabla
T$, but higher order effects could not be separated within the current numerical simulation results.
We quantitatively analyzed the relation between the momentum cumulants and the distribution in the non-equilibrium steady state. It was found that the lower order cumulants characterize the difference of the non-equilibrium distribution from the one in local equilibrium quite well. Understanding and characterizing the properties of the distribution is of manifest importance since the distribution function for physical variables allows us to compute [*any*]{} observable constructed from these variables. To understand the properties of any local variable in the non-equilibrium state, the physical properties of the coordinate variables also need to to be clarified.
A comment is perhaps in order: lack of local equilibrium behavior can in some cases be attributed to the lack of coarse graining[@fluct]. Heuristically speaking, if one does not have a large number of degrees of freedom, one cannot see the equilibrium behavior. This is a [*different*]{} phenomenon from the case at hand, since the effective number of degrees of freedom is the number of samples in the ensemble average which is taken in the time averaging procedure. This number is huge. In fact, as is well known, in these types of ensembles, it makes perfect sense to talk even about the statistical mechanics of one spin degrees of freedom. This is also quite clear from our results; the deviations from local equilibrium seen in , , Eq. are of definite sign and no amount of averaging over space will make it zero. So coarse graining will [*not*]{} average out the violations of local equilibrium seen above. Also, the non-local equilibrium properties found in this paper pertain to systems in the non-equilibrium steady state and therefore are not transient.
We have also performed similar analyses of spatial distributions on the $\phi^4$ model. The physical properties of the model are different from those of FPU model and we intend to report on this in the near future.
[99]{} See, [*e.g.*]{}, S.R. de Groot, P. Mazur, [*“Non-equilibrium Thermodynamics”*]{}, North-Holland, Amsterdam (1962); D. Jou, G. Lebon, J. Casas-Vazques, [*“Extended Irreversible Thermodynamics”*]{} (Springer, Berlin, 1996). J. Keizer, [ *“Statistical Thermodynamics of Nonequilibrium Processes”*]{} (Springer, New York, 1987). J. Casas-Vazquez, D.Jou, [*Rep. Prog. Phys.*]{}, (2003) [**66**]{}, 1937 and references therein. A. Tenenbaum, G. Ciccotti, R. Gallico, [ *Phys. Rev.* ]{}[**A25**]{} (1982) 2778. G. Ciccotti, A. Tenenbaum, [*J. Stat. Phys.*]{} [**23**]{} (1980) 767; C. Trozzi, G. Ciccotti, [*Phys. Rev.*]{} [**A29**]{} (1984) 916. B. Hafskjold, S.K. Ratkje, (1995) 463 W. Loose, G. Ciccotti, [*Phys. Rev.* ]{}[**A45**]{} (1992) 3859; M. Mareschal, E. Kestemont, F. Baras, E. Clementi, G. Nicolis, [*Phys. Rev.* ]{}[**A35**]{} (1987) 3883; A. Tenenbaum, [*Phys. Rev.* ]{}[**A28**]{} (1983) 3132. R. M. Valesco, L.S. Garcia-Colin, [ *J. Noneq. Thermo.*]{} [**18**]{} (1993) 157. A. Dhar, D. Dhar, (1999) 480 S. Takesue, [*Phys. Rev. Lett.* ]{}[**64**]{} (1990) 252 K. Aoki, D. Kusnezov, [*Phys. Lett.* ]{}[**A309**]{} (2003) 377 H. Kaburaki, M. Machida,[*Phys. Lett.*]{} [**A181**]{} (1993) 85; A. Maeda, T. Munakata, (1995) 234. S. Lepri, R. Livi, A. Politi, [*Phys. Rev. Lett.* ]{}[**78**]{} (1997) 1896; [*Europhys. Lett.*]{} [**43**]{} (1998) 271 K. Aoki, D. Kusnezov, [*Phys. Rev. Lett.* ]{} [**86**]{} (2001) 4029 J. Ford, [*Phys. Rep.*]{} [**213**]{} (1992) 271; S. Lepri, R. Livi, A. Politi, [*Phys. Rep.* ]{} 377 (2003) 1 P. Gaspard, [*Chaos, Scattering and Statistical Mechanics*]{}, (Cambridge, New York, 1998); J. R. Dorfman, [ *An Introduction to Chaos in Nonequilibrium Statistical Mechanics*]{} (Cambridge Univ. Press, Cambridge, 1999). D. J. Evans and G. P. Morriss, [*Statistical Mechanics of Nonequilibrium Liquids*]{} (Academic Press, New York, 1990); W. G. Hoover, *Computational Statistical Mechanics (Elsevier, Amsterdam, 1991), *Time Reversibility, Computer Simulation, and Chaos (World Scientific, Singapore, 1999). [*Ann.Phys.* ]{} [**295**]{} (2002) 50 K. Aoki, D. Kusnezov, [*Ann. Phys.* ]{} [**295**]{} (2002) 50; [*Phys. Lett.* ]{}[**B477**]{} (2000) 348 C.S. Kim, J.W. Dufty, [*Phys. Rev.* ]{}[ **A40**]{} (1989) 6723; N. Nishiguchi, Y. Kawada, T. Sakuma, [ *J. Phys. Cond. Matt.* ]{}[**4**]{} (1992) 10227; K. Aoki, D. Kusnezov, (2000) 250 G. Marcelli, B.D.Todd, R. Sadus, [ *Phys. Rev.* ]{}[**E63**]{} (2001) 021204; J.P.Ryckaert, A.Bellemans, G.Ciccotti, G.V.Paolini, [*Phys. Rev. Lett.* ]{}[**60**]{} (1988) 128; S. Rastogi, N. Wagner, S. Lustig, [*J. Chem. Phys.*]{}[**104**]{} (1996) 9234; D.Evans, H.J.M.Hanley,[*Phys. Lett.* ]{}[**80A**]{} (1980) 175. K. Kawasaki, J.D. Gunton, [*Phys. Rev.*]{} [**A8**]{} (1973) 2048; H. Wada, S. Sasa, [cond-mat/0211304]{}**
[^1]: E–mail: [ken@phys-h.keio.ac.jp]{}. Supported in part by the Grant–in–Aid from the Ministry of Education, Science, Sports and Culture.
[^2]: E–mail: [ dimitri@mirage.physics.yale.edu]{}
|
---
abstract: 'We study logamediate inflation in the context of $f(T)$ teleparallel gravity. $f(T)$-gravity is a generalization of the teleparallel gravity which is formulated on the Weitzenböck spacetime, characterized by the vanishing curvature tensor (absolute parallelism) and the non-vanishing torsion tensor. We consider an $f(T)$-gravity model which is sourced by a canonical scalar field. Assuming a power-law $f(T)$ function in the action, we investigate an inflationary universe with a logamediate scale factor. Our results show that, although logamediate inflation is completely ruled out by observational data in the standard inflationary scenario based on Einstein gravity, it can be compatible with the 68% confidence limit joint region of Planck 2015 TT,TE,EE+lowP data in the framework of $f(T)$-gravity.'
author:
- Kazem Rezazadeh
- Asrin Abdolmaleki
- Kayoomars Karami
title: 'Logamediate Inflation in $f(T)$ Teleparallel Gravity'
---
Introduction {#section:introduction}
============
Inflation is accepted as a paradigm to solve some problems of hot Big Bang cosmology, such as the flatness, horizon, and unwanted relics problems [@Guth1981; @Albrecht1982; @Linde1982; @Linde1983]. Furthermore, growth of the perturbations seeded during inflation can provide a convincing explanation for the large-scale structure (LSS) formation in the universe and also for the anisotropies of the cosmic microwave background (CMB) radiation [@Mukhanov1981; @Guth1982; @Hawking1982; @Starobinsky1982].
In the standard inflationary scenario, a canonical scalar field, known as an “inflaton”, is considered in the framework of Einstein’s general relativity (GR) to explain the accelerated expansion of the universe during the inflationary era. Various inflation models with specific potentials or scale factors have been extensively investigated in the setting of the standard inflationary scenario in the light of observational data [@Hossain2014; @Martin2014; @Martin2014a; @Geng2015; @Huang2016].
Impressive observational data have been released by the Planck 2015 collaboration [@Planck2015] following study of the anisotropies in both the temperature and polarization of the CMB radiation. Applying these observational data, we can obtain useful information about the primordial stages of our universe. Furthermore, the observational data from CMB, LSS and other sources can be employed to probe the theory of gravity on astrophysical and cosmological scales [@Baker2015].
Since the early stages of our universe occurred in the regime of high-energy physics, quantum modifications to gravity may play a key role in the inflationary dynamics. Motivated by this concept, numerous inflationary models have been suggested on the basis of extended theories of gravity [@Cerioni2010; @Tsujikawa2013; @Artymowski2014; @Bamba2014; @Artymowski2015; @Myrzakulov2015; @Kumar2016; @Sharif2016; @Tahmasebzadeh2016]. One interesting class of inflationary models is based on teleparallel gravity (TG) and its generalization, $f(T)$-gravity. TG was employed by Einstein in 1928 to attempt to unify gravity and electromagnetism [@Einstein1930; @Unzicker2005]. Although TG and GR are equivalent, they are conceptually completely different theories. In TG, the dynamical object is not the metric but instead is a set of vierbein (or tetrad) fields which forms an orthogonal basis for the tangent space at each point of spacetime. The vierbein fields are transferred parallel in all of the manifolds, which is why TG is sometimes called teleparallelism. Also, in TG, the covariant derivative is defined using the curvature-free Weitzenböck connection rather than the torsionless Levi-Civita version in GR. Furthermore, in TG, the trajectory of motion is determined by the force equations as opposed to the geodesic equations in GR [@Andrade1997].
By a formal analogy with $f(R)$-gravity, the theory of $f(T)$-gravity theory was established by extending the Lagrangian of TG to an $f(T)$ function of a torsion scalar $T$ [@Ferraro2007; @Ferraro2008]. Cosmological implications of $f(T)$-gravity have been extensively studied in the literature [@Bengochea2009; @Linder2010; @Bamba2011; @Bamba2011a; @Bengochea2011; @Miao2011; @Yang2011; @Yang2011a; @Karami2012; @Karami2013; @Karami2013a]. The theory of cosmological perturbations in this scenario has been studied by @Dent2011 [@Chen2011; @Zheng2011; @Izumi2013; @Cai2011], and @Rezazadeh2016. Recently, several inflationary models have been investigated in the framework of $f(T)$-gravity [@Nashed2014; @Jamil2014; @Hanafy2015; @Hanafy2016; @Bamba2016; @Rezazadeh2016; @Wu2016]. For a comprehensive review of $f(T)$-gravity and its cosmological implications, see @Cai2016 and references therein.
In this work, we study logamediate inflation in the framework of $f(T)$-gravity with a minimally coupled canonical scalar field. Logamediate inflation is specified by a scale factor of the form $a(t)\propto\exp\left[A(\ln t)^{\lambda}\right]$ where $A>0$ and $\lambda \ge 1$ [@Barrow2007]. Logamediate inflation can be regarded as a class of possible indefinite cosmological solutions resulting from imposing weak general conditions on cosmological models. @Barrow1996 proposed that there are eight possible asymptotic solutions for cosmological dynamics, of which three lead to non-inflationary expansions. Three others give rise to the power-law ($a(t)\propto t^{q}$ where $q>1$), de Sitter ($a(t)\propto e^{Ht}$ where $H$ is constant) and intermediate ($a(t)\propto\exp\left[At^{\lambda}\right]$ where $A>0$ and $0<\lambda<1$) inflationary expansions. The remaining two inflationary solutions have asymptotic expansions in logamediate form. It is worth mentioning that logamediate inflation arises naturally in some scalar–tensor theories [@Barrow1995].
To date, the power-law and intermediate inflationary models have been investigated in the $f(T)$-gravity scenario, and it has been shown that, using this setting, we can resurrect these models in light of observational data [@Rezazadeh2016]. This motivates us to consider logamediate inflation in the framework of $f(T)$-gravity. Logamediate inflation has already been studied within the standard inflationary scenario [@Barrow2007], and it seems that its predictions are not compatible with the current constraints from the Planck 2015 data [@Planck2015].
The structure of this paper is as follows. In Section \[section:f(T)\] we review the dynamics of the background cosmology in $f(T)$-gravity. We also explore the relations governing the power spectra of the scalar and tensor perturbations in this model. In Section \[section:logamediate\], we consider a power-law form for the $f(T)$ function and study logamediate inflation in this setting. We estimate the inflationary observables in our model and check their viability in light of the Planck 2015 data [@Planck2015]. Finally, in Section \[section:conclusions\], we present our concluding remarks.
The $f(T)$ Theory of Gravity {#section:f(T)}
============================
In the context of $f(T)$-gravity, the action of modified TG can be written as [@Ferraro2007; @Ferraro2008] $$\label{I}
I=\frac{1}{2}\int d^{4}x~e\left[f(T)+L_{\phi}\right],$$ where $e\equiv{\rm det}(e^i_{\mu})=\sqrt{-g}$. Also, $T$ and $L_\phi$ are the torsion scalar and the Lagrangian of the scalar field $\phi$, respectively. It should be noted that throughout of this paper we set the reduced Planck mass to be unity, $M_{P}\equiv1/\sqrt{8\pi G}=1$, for the sake of convenience.
In our notation, $e^i_{\mu}$ is the vierbein or tetrad field which is used as a dynamical object in TG, and satisfies the orthonormality relations $$\label{e^mu_i}
e^{\mu}_{i}e^{i}_{\nu}=\delta^{\mu}_{\nu}, \quad e^{\mu}_{i}e^{j}_{\mu}=\delta^{j}_{i}.$$ Here, Latin and Greek indices label tangent space and spacetime coordinates, respectively. All indices take values from $0$ to $3$. With the help of a dual vierbein, one can obtain the metric tensor as $$\label{g_{mu.nu}}
g_{\mu\nu}(x)=\eta_{ij}e^i_{\mu}(x)e^j_{\nu}(x),$$ where $\eta_{ij}={\rm diag}(-1,1,1,1)$ is the Minkowski metric induced on the tangent space.
In GR, the Levi-Civita connection is $$\label{Gamma,GR}
\Gamma^{\lambda}_{\mu\nu}=\Gamma^{\lambda}_{\nu\mu}\equiv\frac{1}{2}g^{\lambda\rho}(g_{\rho\mu,\nu}+g_{\rho\nu,\mu}-g_{\mu\nu,\rho}),$$ where the comma denotes the partial derivative. The Levi-Civita connection (\[Gamma,GR\]) leads to nonzero spacetime curvature but zero torsion. In contrast, in TG, we have the Weitzenböck connection $$\label{Gamma}
\widetilde{\Gamma}^{\lambda}_{~~\mu\nu}\equiv e^{\lambda}_{i}\partial_{\nu}e^{i}_{\mu}=-e^{i}_{\mu}\partial_{\nu}e^{\lambda}_{i},$$ which yields zero curvature but nonzero torsion.
In GR, the curvature plays the role of gravitational force, and the trajectory of motion is determined by the geodesic equations as $$\label{d.u^lambda,GR}
\frac{du^{\lambda}}{ds}+\Gamma^{\lambda}_{\mu\nu}u^{\mu}u^{\nu}=0,$$ where $u^{\lambda}$ is the four-velocity of the particle. In contrast, in TG, the torsion acts as a force and gravitational interaction is given by the force equations [@Hayashi1979] $$\label{d.u^lambda}
\frac{du^{\lambda}}{ds}+\widetilde{\Gamma}^{\lambda}_{~~\mu\nu}u^{\mu}u^{\nu}=T^{\lambda}_{~~\mu\nu}u^{\mu}u^{\nu},$$ where $T^{\lambda}_{~~\mu\nu}$ is the torsion tensor, expressed as $$\label{T^lambda_{mu.nu}}
T^{\lambda}_{~~\mu\nu}\equiv\widetilde{\Gamma}^{\lambda}_{~~\nu\mu}-\widetilde{\Gamma}^{\lambda}_{~~\mu\nu}=e^{\lambda}_{i}(\partial_{\mu}e^{i}_{\nu}-\partial_{\nu}e^{i}_{\mu}).$$ The difference between the Levi-Civita and Weitzenböck connections gives the contorsion tensor $$\label{K^lambda_{mu.nu}}
K^{\lambda}_{{~~\mu\nu}}\equiv\widetilde{\Gamma}^{\lambda}_{~~\mu\nu}-\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}(T^{\lambda}_{\mu\nu}+T^{\lambda}_{\nu\mu}-T^{\lambda}_{~~\mu\nu}).$$ The torsion scalar $T$ is defined as $$\label{T}
T\equiv S^{~~\mu\nu}_{\lambda}T^{\lambda}_{~~\mu\nu},$$ where $S_{\lambda}^{~~\mu\nu}$ is the superpotential tensor given by $$\label{S_lambda^{mu.nu}}
S_{\lambda}^{~~\mu\nu}\equiv\frac{1}{2}(K^{\mu\nu}_{~~~\lambda}+\delta^{\mu}_{\lambda}T^{\rho\nu}_{~~~\rho}-\delta^{\nu}_{\lambda} T^{\rho\mu}_{~~~\rho}).$$ Using the Weitzenböck connection (\[Gamma\]), the teleparallel covariant derivative, $\widetilde{\nabla}_{\mu}$, of the vierbein fields vanishes, i.e., $$\label{widetilde{nabla}_{mu}}
\widetilde{\nabla}_{\mu}e^i_{\nu}\equiv \partial_{\mu}e^{i}_{\nu}-\widetilde{\Gamma}^{\lambda}_{~~\nu\mu} e^i_{\lambda}=0.$$ This reflects the concept of absolute parallelism or teleparallelism in TG. In GR, the metric covariant derivative, $\nabla_{\lambda}$, of the metric is zero $$\label{nabla_lambda}
\nabla_{\lambda}g_{\mu\nu}\equiv\partial_{\lambda}g_{\mu\nu}-\Gamma^{\rho}_{\lambda\mu}g_{\rho\nu}-\Gamma^{\rho}_{\lambda\nu}g_{\rho\mu}=0.$$
Variation of the action (\[I\]) with respect to the vierbein (tetrad) $e^{i}_{\lambda}$ leads to the field equations in $f(T)$-gravity as [@Li2011b; @Li2011a] $$\begin{aligned}
\nonumber
&& \left[e^{-1}\partial_{\mu}\left(e~e_{i}^{\rho}S_{\rho}^{~~\lambda\mu}\right)-e_{i}^{\sigma}S_{\mu}^{~~\nu\lambda}T_{~~\nu\sigma}^{\mu}\right]f_{,T}+\frac{1}{2}e_{i}^{\lambda}f(T)+ \\
&& e_{i}^{\rho}S_{\rho}^{~~\lambda\sigma}\left(\partial_{\sigma}T\right)f_{,TT}=8\pi G~\Theta_{i}^{\lambda},
\label{f(T),fe}\end{aligned}$$ where we define $f_{,T}\equiv df/dT$ and $f_{,TT}\equiv d^2f/dT^2$. Also $\Theta_{i}^{\lambda}\equiv e^{-1}\delta L_{\phi}/\delta e_{\lambda}^{i}$ and the usual energy-momentum tensor is given in terms of $\Theta^{\lambda}_{i}$ as $$\label{Theta^{mu.nu}}
\Theta^{\mu\nu}=\eta^{ij}\Theta^{\nu}_{i}e^{\mu}_{j}.$$ Note that the set of field equations (\[f(T),fe\]) are second order, and are considerably simpler than the fourth-order equations of $f(R)$ theory [@Wu2010; @Wu2011; @Wei2011].
Contracting with $e^i_{\nu}$, Equation (\[f(T),fe\]) can be rewritten into the form [@Li2011b; @Li2011a; @Li2011] $$\begin{aligned}
\nonumber
&& R_{\mu\nu}f_{,T}-\frac{1}{2}g_{\mu\nu}[(1+T)f_{,T}-f(T)]+S_{\nu\mu}^{~~~\lambda}(\nabla_{\lambda}T)f_{,TT}\\
&& =8\pi G~\Theta_{\mu\nu},
\label{R_{mu.nu}}\end{aligned}$$ which shows that for $f(T)=T$, the field equations coincide completely with those of GR. This is why in the literature, TG is called the teleparallel equivalent of GR (TEGR). This can also be understood in another way. @Li2011b have shown that $T$ and $R$ differ only by a total divergence, i.e., $R=-T-2\nabla^{\mu}(T^{\nu}_{~~\mu\nu})$. Since the total divergence can be neglected inside an integral, the TG Lagrangian density is completely equivalent to the Einstein–Hilbert density.
Now, we consider a spatially flat universe described by the Friedmann-Robertson-Walker metric $$\label{g_{mu.nu},FRW}
g_{\mu\nu}={\rm diag}\left(-1,a^{2}(t),a^{2}(t),a^{2}(t)\right),$$ where $a$ is the scale factor of the universe. Using this together with Equation (\[g\_[mu.nu]{}\]), we get $$\label{e^mu_i,FRW}
e_{\mu}^{i}={\rm diag}\left(1,a(t),a(t),a(t)\right).$$ Substituting the vierbein (\[e\^mu\_i,FRW\]) into (\[T\]) yields $$\label{T,FRW}
T=-6H^2,$$ where $H \equiv \dot{a}/a$ is the Hubble parameter.
Taking $\Theta^\mu_{~\nu}={\rm diag}(-\rho_\phi,p_\phi,p_\phi,p_\phi)$ for the energy–momentum tensor of the scalar field in the perfect fluid form and using the vierbein (\[e\^mu\_i,FRW\]), the field equations (\[f(T),fe\]) yields the Friedmann equations in $f(T)$-gravity as [@Ferraro2007; @Ferraro2008] $$\begin{aligned}
\label{f(T),eq1}
2\rho_{\phi} &=& 12H^{2}f_{,T}+f(T), \\
\label{f(T),eq2}
2p_{\phi} &=& 48H^{2}\dot{H}f_{,TT}-\big(12H^{2}+4\dot{H}\big)f_{,T}-f(T).\end{aligned}$$ Here, $\rho_\phi$ and $p_\phi$ are the energy density and pressure of the scalar field, respectively, and satisfy the conservation equation $$\label{dot{rho}_phi}
\dot{\rho}_\phi+3H\left(\rho_\phi+p_\phi\right)=0.$$ One can rewrite Equations (\[f(T),eq1\]) and (\[f(T),eq2\]) in the standard form of the Friedmann equations as $$\begin{aligned}
\label{Fri1}
H^{2} &=& \frac{1}{3}\left(\rho_{T}+\rho_{\phi}\right),\\
\label{Fri2}
\dot{H}+\frac{3}{2}H^{2} &=&-\frac{1}{2}\left(p_{T}+p_{\phi}\right),\end{aligned}$$ where $$\begin{aligned}
\label{rho_T}
\rho_{T} &\equiv& \frac{1}{2}\left(2Tf_{,T}-f-T\right),\\
\nonumber
p_{T} &\equiv& -\frac{1}{2}\left[-8\dot{H}Tf_{,TT}+\left(2T-4\dot{H}\right)f_{,T}-f+4\dot{H}-T\right],\\
\label{p_T}\end{aligned}$$ are the torsion contribution to the energy density and pressure which satisfy the energy conservation law $$\label{dot{rho}_T}
\dot{\rho}_T+3H(\rho_T+p_T)=0.$$ In the case of $f(T)=T$, from Equations (\[rho\_T\]) and (\[p\_T\]) we have $\rho_T=0$ and $p_T=0$. Therefore, Equations (\[Fri1\]) and (\[Fri2\]) are transformed to the usual Friedmann equations in GR. In the following, we assume the universe to be filled with a canonical scalar field which has energy density and pressure as follows: $$\begin{aligned}
\label{rho_phi}
\rho_{\phi} &=& \frac{1}{2}\dot{\phi}^{2}+V(\phi),\\
\label{p_phi}
p_{\phi} &=& \frac{1}{2}\dot{\phi}^{2}-V(\phi).\end{aligned}$$ Substitution of Equations (\[rho\_phi\]) and (\[p\_phi\]) into the conservation equation (\[dot[rho]{}\_phi\]) yields the evolution equation governing the scalar field as $$\label{ddot{phi}}
\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=0,$$ where $V_{,\phi}\equiv dV/d\phi$.
In order to study inflation in $f(T)$-gravity, it is useful to define the Hubble slow-roll parameters as follows: $$\begin{aligned}
\label{varepsilon_1}
\varepsilon_{1} &\equiv& -\frac{\dot{H}}{H^{2}},\\
\label{varepsilon_{i+1}}
\varepsilon_{i+1} &\equiv& \frac{\dot{\varepsilon}_{i}}{H\varepsilon_{i}}.\end{aligned}$$ Due to having an inflationary epoch ($\ddot{a} > 0$), according to Equation (\[varepsilon\_1\]) we must have $\varepsilon_1 < 1$. It should be noted that the condition $\varepsilon_1 = 1$ can determine the initial (or final) time of inflation if the first Hubble slow-roll parameter $\varepsilon_1$ is a decreasing (or increasing) function of time [@Martin2014; @Zhang2014; @Rezazadeh2015; @Rezazadeh2016].
During inflation, the scalar field $\phi$ and the Hubble parameter $H$ change very slowly. This enables us to use the slow-roll conditions given by $\dot{\phi}^{2}\ll V(\phi)$ and $\big|\ddot{\phi}\big|\ll\big|3H\dot{\phi}\big|,\,\big|V_{,\phi}\big|$. Applying the slow-roll approximation to Equations (\[Fri1\]) and (\[ddot[phi]{}\]), one can find $$\begin{aligned}
\label{V}
V &=& \frac{1}{2}\left(f-2Tf_{,T}\right),\\
\label{dot{phi}}
\dot{\phi}^{2} &=& -2\dot{H}\left(f_{,T}+2Tf_{,TT}\right).\end{aligned}$$ With the help of the above equations, one can obtain the evolutionary behaviors of the potential $V(t)$ and scalar field $\phi(t)$, if the functional form of $f(T)$-gravity and the scale factor $a(t)$ are known. Combining the results of $V(t)$ and $\phi(t)$ to eliminate $t$ between them, one may get $V(\phi)$ determining the shape of the inflationary potential with respect to the inflaton.
In the study of inflation, we usually express the extent of the universe expansion in terms of the $e$-fold number, defined as $$\label{N}
N\equiv\ln\frac{a_{e}}{a},$$ where $a_e$ denotes the scale factor of the universe at the end of inflation. The above definition is equivalent to $$\label{d.N}
dN = - H dt.$$ It is believed that the anisotropies observed in the CMB radiation and in the LSS of the universe are related to the perturbations which exit the Hubble radius around the $e$-fold number $N_* \approx 50-60$ before the end of inflation [@Dodelson2003; @Liddle2003]. Those perturbations remain outside the horizon until a time close to the present time and this enables us to relate the late-time observations to the primordial power spectra of the perturbations produced during inflation.
In the following, we review briefly the basic results of the theory of cosmological perturbations in the $f(T)$-gravity scenario when a canonical scalar field is assumed to be the matter-energy content of the universe (for more details, see @Cai2011 [@Rezazadeh2016]). The primordial power spectrum of the scalar perturbations in the $f(T)$-gravity is given by [@Cai2011; @Rezazadeh2016] $$\label{P_s}
{\cal P}_{s}=\left.\frac{H^{2}}{8\pi^{2}c_{s}^{3}\varepsilon_{1}}\right|_{c_{s}k=aH},$$ which should be evaluated at the sound horizon exit for which $c_s k=a H$. Here, $c_s$ is the sound speed defined as $$\label{c_s,def}
c_{s}^{2}=\frac{f_{,T}}{f_{,T}-12H^{2}f_{,TT}}.$$ It is evident that in the case of TEGR (i.e., $f(T)=T$), we have $c_s=1$ from Equation (\[c\_s,def\]), and then Equation (\[P\_s\]) reduces to the expected relation for the standard inflationary scenario [@Baumann2009].
The scale-dependence of the scalar power spectrum is measured by the scalar spectral index $$\label{n_s,def}
n_{s}-1\equiv\frac{d\ln{\cal P}_{s}}{d\ln k}.$$ In the slow-roll approximation, it is assumed that the Hubble parameter $H$ and the sound speed $c_s$ are slowly varying [@Garriga1999]. Therefore, the relation $c_s k=a H$ leads to $$\label{d.ln.k}
d\ln k \approx Hdt=-dN,$$ which is valid around the sound horizon exit. Now, using Equations (\[varepsilon\_1\]), (\[varepsilon\_[i+1]{}\]), (\[P\_s\]), (\[n\_s,def\]) and (\[d.ln.k\]), we can obtain the scalar spectral index in $f(T)$-gravity scenario as $$\label{n_s}
n_{s}=1-2\varepsilon_{1}-\varepsilon_{2}-3\varepsilon_{s1},$$ where we have defined the sound speed slow-roll parameters as follows: $$\begin{aligned}
\label{varepsilon_{s1}}
\varepsilon_{s1}\equiv\frac{\dot{c}_{s}}{Hc_{s}},\\
\label{varepsilon_{s(i+1)}}
\varepsilon_{s(i+1)}\equiv\frac{\dot{\varepsilon}_{si}}{H\varepsilon_{si}}.\end{aligned}$$ We further can use Equations (\[varepsilon\_[i+1]{}\]), (\[d.ln.k\]), (\[n\_s\]), (\[varepsilon\_[s1]{}\]), and (\[varepsilon\_[s(i+1)]{}\]) to obtain the running of the scalar spectral index as $$\label{d.n_s}
\frac{dn_{s}}{d\ln k}=-2\varepsilon_{1}\varepsilon_{2}-\varepsilon_{2}\varepsilon_{3}-3\varepsilon_{s1}\varepsilon_{s2}.$$
We now focus on the tensor perturbations in the framework of $f(T)$-gravity. Following @Rezazadeh2016, we define the parameters $\gamma$ and $\delta$ as follows: $$\begin{aligned}
\label{gamma}
\gamma &\equiv& \left(\frac{f_{,TT}}{f_{,T}}\right)\dot{T},\\
\label{delta}
\delta &\equiv& \frac{{\left| \gamma \right|}}{{2H}}.\end{aligned}$$ @Rezazadeh2016 proposed that if the $\delta$ parameter is much less than unity ($\delta \ll 1$), then the tensor power spectrum of the tensor perturbations in $f(T)$-gravity reduces to the one for the standard inflationary model, which is given by $$\label{P_t}
{\cal P}_{t}=\left.\frac{2H^{2}}{\pi^{2}}\right|_{k=aH}.$$ It should be noted that the tensor power spectrum must be calculated at the time of horizon crossing specified by $k=a H$. This time is not exactly the same as the time of sound horizon crossing for which $c_s k=a H$, but to lowest order in the slow-roll parameters the difference is negligible [@Garriga1999].
The scale-dependence of the tensor power spectrum is specified by the tensor spectral index $$\label{n_t,def}
n_{t}\equiv\frac{d\ln\mathcal{P}_{t}}{d\ln k}.$$ Using Equations (\[varepsilon\_1\]), (\[d.ln.k\]), (\[P\_t\]), and (\[n\_t,def\]), we obtain this observable for the inflationary model based on the $f(T)$-gravity scenario as $$\label{n_t}
n_t=-2\varepsilon_1.$$ Current experimental devices are not accurate enough to measure this observable, and we may be able to determine it with more sensitive measurements in the future [@Simard2015].
An important inflationary observable which can be applied to discriminate between inflationary models is the tensor-to-scalar ratio $$\label{r,def}
r\equiv\frac{\mathcal{P}_{t}}{\mathcal{P}_{s}}.$$ In $f(T)$-gravity setting, using Equations (\[P\_s\]) and (\[P\_t\]) in Equation (\[r,def\]), it is easy to see that this observable is given by $$\label{r}
r=16c_{s}^{3}\varepsilon_{1}.$$
From Equations (\[n\_t\]) and (\[r\]), the consistency relation in $f(T)$-gravity takes the form $$\label{r,n_t}
r=-8c_{s}^{3}n_{t}.$$ It is obvious that for $c_s=1$, this equation reduces the well-known consistency relation $r=-8n_{t}$ in the standard inflationary scenario [@Baumann2009].
Logamediate Inflation in $f(T)$ Teleparallel Gravity {#section:logamediate}
====================================================
@Barrow2007 investigated logamediate inflation in the framework of the standard inflationary scenario based on Einstein gravity. From their results, it seems that logamediate inflation within the standard inflationary setting is ruled out by current observational data from the Planck 2015 collaboration [@Planck2015]. This motivates us to examine the observational viability of logamediate inflation in $f(T)$ teleparallel gravity.
We consider an $f(T)$-gravity setting in which the $f(T)$ function in action (\[I\]) has the power-law form [@Linder2010; @Wu2010; @Rezazadeh2016] $$\label{f(T)}
f\left( T \right) = T_0\left(\frac{T}{T_0}\right)^n,$$ where $T_0$ and $n$ are constant parameters of the model. For the case $n=1$, Equation (\[f(T)\]) transforms to TEGR, i.e., $f(T)= T$. From the definition of sound speed in Equation (\[c\_s,def\]), we see that the $f(T)$ model (\[f(T)\]) gives rise to a constant sound speed as $$\label{c_s}
c_s^2 = \frac{1}{2n - 1}.$$ The above equation leads to the requirement $n \geq 1$ required to have a physical speed for propagation of the scalar perturbations, i.e., $0<c_s^2\leq 1$ [@Franche2010]. Furthermore, in our $f(T)$-gravity model (\[f(T)\]), the sound speed slow-roll parameters (\[varepsilon\_[s1]{}\]) and (\[varepsilon\_[s(i+1)]{}\]) vanish because of the constant sound speed (\[c\_s\]).
Now, we consider the logamediate scale factor which has the following form [@Barrow2007]: $$\label{a(t)}
a(t)=a_{0}\exp\left[A\big(\ln t\big)^{\lambda}\right],$$ where $a_0>0$, $A>0$ and $\lambda \geq 1$ are constant parameters. For $\lambda=1$, the logamediate scale factor (\[a(t)\]), reduces to the power-law scale factor $a(t)=a_{0}t^{q}$, where $q=A$. With the above scale factor, the Hubble parameter reads $$\label{H}
H=\frac{A\lambda\left(\ln t\right)^{\lambda-1}}{t}.$$ Furthermore, we see from Equation (\[varepsilon\_1\]) that the first slow-roll parameter takes the form $$\label{varepsilon_1,t}
\varepsilon_{1}=\frac{\ln t-\lambda+1}{A\lambda\left(\ln t\right)^{\lambda}}.$$ The above equation shows that at late times, $t \gg 1$, the first slow-roll parameter becomes a decreasing function during inflation, and hence it cannot reach unity at the end of inflation. This demonstrates that, in our model, inflation cannot end with slow-roll violation [@Martin2014; @Rezazadeh2016; @Zhang2014; @Rezazadeh2015].
To obtain the evolution of the inflationary potential, we use Equations (\[T,FRW\]), (\[V\]), (\[f(T)\]), and (\[H\]), and obtain $$\label{V,t}
V(t)=3^{n}\left(2n-1\right)\left[\frac{2}{\left(-T_{0}\right)}\right]^{n-1}\left[\frac{A\lambda\left(\ln t\right)^{\lambda-1}}{t}\right]^{2n}.$$ We further can use Equation (\[dot[phi]{}\]) to find $$\begin{aligned}
\nonumber
\dot{\phi}= && \left[2^{n}n\left(2n-1\right)\left(A\lambda\right)^{2n-1}\left(\frac{3}{\left(-T_{0}\right)}\right)^{n-1}\right]^{1/2}\\
&& \times\left[\left(\ln t\right)^{2n(\lambda-1)-\lambda}\Big(\ln t-(\lambda-1)\Big)\right]^{1/2}t^{-n} .
\label{dot{phi},t}\end{aligned}$$ In general, it is too difficult to solve the above equation for a given value of $n$. Therefore, we cannot combine Equations (\[V,t\]) and (\[dot[phi]{},t\]) and find the shape of the inflationary potential $V(\phi)$ for a general $n$. However, we can check the validity of our results for the simplest case $n=1$ corresponding to TEGR, and we expect that it leads to the same results for logamediate inflation in the standard inflation scenario. For the case of $n=1$, Equations (\[V,t\]) and (\[dot[phi]{},t\]) yield $$\begin{aligned}
\label{V,t,n=1}
V(t)&=&3\left[\frac{A\lambda\left(\ln t\right)^{\lambda-1}}{t}\right]^{2},\\
\label{phi,t,n=1}
\phi(t)&=&\frac{2\sqrt{2A\lambda}}{\lambda+1}\left(\ln t\right)^{(\lambda+1)/2}.\end{aligned}$$ In the derivation of Equation (\[phi,t,n=1\]) we have followed the logic of @Barrow2007, and considered the late time limit which allows us to ignore $(\lambda- 1)$ versus $\ln t$. If we combine the above two equations to eliminate $t$, we find the inflationary potential as $$\label{V(phi),n=1}
V(\phi)=V_{0}\phi^{\alpha}\exp\left(-2B\phi^{\beta}\right),$$ where $$\begin{aligned}
\label{V_0}
V_{0}&\equiv&3\left(A\lambda B^{\lambda-1}\right)^{2},\\
\label{B}
B&\equiv&\left(\frac{\lambda+1}{2\sqrt{2A\lambda}}\right)^{2/(\lambda+1)},\\
\label{alpha}
\alpha &\equiv& \frac{4\left(\lambda-1\right)}{\lambda+1},\\
\label{beta}
\beta &\equiv& \frac{2}{\lambda+1}.\end{aligned}$$ The result in Equation (\[V(phi),n=1\]) is the potential responsible for logamediate inflation in the standard inflationary scenario, and this result is in agreement with that found by @Barrow2007.
Here, we are interested in showing that our result is also in agreement with the analysis performed by @Barrow1995a, who presented asymptotic solutions of the potential $$\label{V(phi),l,m}
V(\phi)=V_{0}\phi^{l}\exp\left(-\kappa\phi^{m}\right),$$ in the slow-roll approximation. In the above equation, $V_0$, $l$, $\kappa$ and $m$ are positive constant parameters. @Barrow1995a found that for the case $m=1$ and $l=0$, and provided that $\kappa^{2}<2$, the potential (\[V(phi),l,m\]) leads to an inflationary expansion in the power-law form $$\label{a(t),m=1,l=0}
a(t)\propto t^{2/\kappa^{2}}.$$ This is just the well-known fact that in the standard inflationary scenario, the exponential potential gives rise to power-law inflation [@Martin2014; @Martin2014a]. In addition, @Barrow1995a obtained that for $0<m<1$ and in the limit of $t \to \infty$, the potential (\[V(phi),l,m\]) provides an inflationary scale factor in the form of $$\label{a(t),0<m<1}
a(t)\propto\exp\left[\frac{1}{\kappa m(2-m)}\left(\frac{2}{\kappa}\right)^{\frac{2-m}{m}}\left(\ln t\right)^{\frac{2-m}{m}}\right].$$
By comparing the potentials (\[V(phi),n=1\]) and (\[V(phi),l,m\]), we find $$\begin{aligned}
\label{alpha,l}
\alpha &=& l,\\
\label{beta,m}
\beta &=& m,\\
\label{B,kappa}
2B &=& \kappa.\end{aligned}$$ First, we focus on the case $m=1$ and $l=0$. In this case, Equations (\[alpha,l\]) and (\[beta,m\]) give $\alpha=0$ and $\beta=1$. As a result, from Equation (\[alpha\]) we get $\lambda=1$. Using this, we see that in Equation (\[a(t)\]) the logamediate scale factor reduces to the power-law one $$\label{a(t),A,m=1,l=0}
a(t)\propto t^A.$$ In addition, Equation (\[B\]) gives $B=1/\sqrt{2A}$. This together with Equation (\[B,kappa\]) gives $A=2/\kappa^{2}$. Finally, using this result in Equation (\[a(t),A,m=1,l=0\]), we reach Equation (\[a(t),m=1,l=0\]) obtained by @Barrow1995a.
Second, we proceed to examine the case $0<m<1$. By use of Equations (\[beta\]) and (\[beta,m\]), we obtain $$\label{lambda,m}
\lambda=\frac{2-m}{m}.$$ Applying this together with Equations (\[B\]) and (\[B,kappa\]), we find $$\label{A,m,kappa}
A=\frac{1}{\kappa m(2-m)}\left(\frac{2}{\kappa}\right)^{\frac{2-m}{m}}.$$ Now, it is obvious that substitution of Equations (\[lambda,m\]) and (\[A,m,kappa\]) into (\[a(t)\]) yields Equation (\[a(t),0<m<1\]) obtained by @Barrow1995a. Therefore, we showed that the results (\[a(t),m=1,l=0\]) and (\[a(t),0<m<1\]) given by @Barrow1995a are recovered in our model.
For the $f(T)$ function given in Equation (\[f(T)\]) with a general $n$, and considering the logamediate scale factor (\[a(t)\]), the scalar power spectrum (\[P\_s\]) becomes $$\label{P_s,t}
\mathcal{P}_{s}=\frac{\left(A\lambda\right)^{3}\left(2n-1\right)^{3/2}\left(\ln t\right)^{3\lambda-2}}{8\pi^{2}t^{2}\left(\ln t-\lambda+1\right)}.$$ We can also obtain the scalar spectral index from Equation (\[n\_s\]) as $$\begin{aligned}
\nonumber
n_{s}= && \left[A\lambda\left(\ln t\right)^{\lambda}\left(\ln t-\lambda+1\right)\right]^{-1} \\
\nonumber
&& \times\left[A\lambda\left(\ln t\right)^{\lambda+1}-A\lambda(\lambda-1)\left(\ln t\right)^{\lambda}-2\left(\ln t\right)^{2}\right. \\
&& \left.+5(\lambda-1)\ln t-3\lambda^{2}+5\lambda-2\right].
\label{n_s,t}\end{aligned}$$ In addition, the running of the scalar spectral index follows from Equation (\[d.n\_s\]) as $$\begin{aligned}
\nonumber
\frac{dn_{s}}{d\ln k}= && (\lambda-1)\left[A\lambda\left(\ln t\right)^{\lambda}\left(\ln t-\lambda+1\right)\right]^{-2}\\
\nonumber
&& \times\left[2\left(\ln t\right)^{3}-(7\lambda-4)\left(\ln t\right)^{2}+\left(8\lambda^{2}-9\lambda+3\right)\ln t\right. \\
&& \left. -\lambda\left(3\lambda^{2}-5\lambda+2\right)\right].
\label{d.n_s,t}\end{aligned}$$
In order to find the expression of the tensor power spectrum for the model under consideration, we note that for the $f(T)$ function (\[f(T)\]), the $\delta$ parameter can be simplified as $$\label{delta,varepsilon_1}
\delta=\left(n-1\right)\varepsilon_{1}.$$ In this paper, we are only dealing with values of $n$ of order unity. Hence, the $\delta$ parameter takes the order of the first slow-roll parameter and therefore it becomes much less than unity in the slow-roll regime. This allows us to use Equation (\[P\_t\]) for the tensor power spectrum to obtain $$\label{P_t,t}
\mathcal{P}_{t}=2\left[\frac{A\lambda\left(\ln t\right)^{\lambda-1}}{\pi t}\right]^{2}.$$ Then, using this together with Equation (\[n\_t\]), the tensor spectral index is obtained as $$\label{n_t,t}
n_{t}=-\frac{2\left(\ln t-\lambda+1\right)}{A\lambda\left(\ln t\right)^{\lambda}}.$$ If we use Equation (\[r\]), we can easily show that the tensor-to-scalar ratio becomes $$\label{r,t}
r=\frac{16\left(\ln t-\lambda+1\right)}{A\lambda(2n-1)^{3/2}\left(\ln t\right)^{\lambda}}.$$
It is interesting to find simplified forms of the equations for the inflationary observables in the case of $\lambda=1$ for which logamediate inflation reduces to power-law inflation $a(t)\propto t^{q}$, where $q=A$. For $\lambda=1$, Equations (\[n\_s,t\]), (\[d.n\_s,t\]), (\[n\_t,t\]), and (\[r,t\]) reduce to $$\begin{aligned}
\label{n_s,lambda=1}
&& n_{s} = 1-\frac{2}{A},\\
\label{d.n_s,lambda=1}
&& \frac{dn_{s}}{d\ln k} = 0,\\
\label{n_t,lambda=1}
&& n_{t} = -\frac{2}{A},\\
\label{r,lambda=1}
&& r = \frac{16}{(2n-1)^{3/2}A}.\end{aligned}$$ These are in agreement with the results of @Rezazadeh2016, where the authors investigated power-law inflation in the $f(T)$-gravity setup (\[f(T)\]). The obtained results for the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ in Equations (\[n\_s,lambda=1\]) and (\[r,lambda=1\]) are independent of the dynamical quantities such as $t$, $N$ or $\phi$. This behavior is familiar for power-law inflation in other inflationary scenarios, for instance, the standard inflationary setting [@Tsujikawa2013], Brans-Dicke inflation [@Tahmasebzadeh2016], tachyon inflation [@Rezazadeh2017], and non-canonical power-law inflation [@Unnikrishnan2013]. Consequently, we can combine Equations (\[n\_s,lambda=1\]) and (\[r,lambda=1\]) to eliminate $A$ between them, and obtain $$\label{r,n_s,lambda=1}
r=\frac{8}{(2n-1)^{3/2}}\left(1-n_{s}\right),$$ implying a linear relation between $r$ and $n_s$. For $n=1$, i.e., $f(T)=T$, the above equation reduces to $r=8\left(1-n_{s}\right)$ which is the well-known result obtained for power-law inflation in the standard inflationary scenario [@Tsujikawa2013].
\[1\][]{}
\[1\][]{}
$\quad \lambda \quad$ $A$ $\frac{dn_{s}}{d\ln k}$
----------------------- -------------------------------------------------------------- -------- -----------------------------------------------------------------------------------
$1$ $52 \lesssim A \lesssim 78$ 68% CL $0$
$2$ $1.64 \lesssim A \lesssim 2.34$ 68% CL $2.18 \times 10^{-5} \lesssim \frac{dn_{s}}{d\ln k} \lesssim 4.95 \times 10^{-5}$
$3$ $0.069 \lesssim A \lesssim 0.093$ 68% CL $5.02 \times 10^{-5} \lesssim \frac{dn_{s}}{d\ln k} \lesssim 1.09 \times 10^{-4}$
$4$ $0.0033 \lesssim A \lesssim 0.0042$ 68% CL $8.78 \times 10^{-5} \lesssim \frac{dn_{s}}{d\ln k} \lesssim 1.79 \times 10^{-4}$
$5$ $1.69 \times 10^{-4}\lesssim A \lesssim 2.06 \times 10^{-4}$ 68% CL $1.35 \times 10^{-4} \lesssim \frac{dn_{s}}{d\ln k} \lesssim 2.66 \times 10^{-4}$
$6$ $9.10 \times 10^{-6}\lesssim A \lesssim 1.04 \times 10^{-5}$ 68% CL $2.15 \times 10^{-4} \lesssim \frac{dn_{s}}{d\ln k} \lesssim 3.72 \times 10^{-4}$
$7$ $4.63 \times 10^{-7}\lesssim A \lesssim 6.55 \times 10^{-7}$ 95% CL $1.42 \times 10^{-4} \lesssim \frac{dn_{s}}{d\ln k} \lesssim 8.29 \times 10^{-4}$
$8$ $2.73 \times 10^{-8}\lesssim A \lesssim 3.46 \times 10^{-8}$ 95% CL $2.20 \times 10^{-4} \lesssim \frac{dn_{s}}{d\ln k} \lesssim 1.07 \times 10^{-3}$
$\lambda \gtrsim 9$ — — —
\[table:d.n\_s\]
We come back to our investigation with general $n$. So far, we have found the inflationary observables in terms of time. In order to estimate these observables, it is necessary to evaluate them at the time of horizon exit which has a specified $e$-fold number. Consequently, we should obtain the relation between time and $e$-fold number in our model. To this end, we solve the differential Equation (\[d.N\]) for the logamediate scale factor (\[a(t)\]) and obtain $$\label{t,N}
t=\exp\left[\left(\left(\ln t_{e}\right)^{\lambda}-\frac{N}{A}\right)^{1/\lambda}\right],$$ where $t_e$ refers to the end time of inflation. To get the above result, we have applied the initial condition $N_{e}\equiv N(t=t_{e})=0$ which is a direct implication of definition (\[N\]) for the $e$-fold number. Here, it is essential to note that we cannot determine $t_e$ in our model by setting $\varepsilon_1=1$, because, as we have mentioned before, inflation in our model cannot end with slow-roll violation [@Martin2014; @Rezazadeh2016; @Zhang2014; @Rezazadeh2015]. To overcome this problem, we follow the approach of @Martin2014 and @Rezazadeh2016, and retain $t_e$ as an extra parameter. In the following, we determine it by fixing the amplitude of the scalar perturbations from the observational results
Inserting Equation (\[t,N\]) into (\[P\_s,t\]), we obtain the scalar power spectrum at the horizon exit as $$\begin{aligned}
\nonumber
\mathcal{P}_{s}\left(N_{*}\right)= && \frac{\left(A\lambda\right)^{3}\left(2n-1\right)^{3/2}\left(\left(\ln t_{e}\right)^{\lambda}-\frac{N_{*}}{A}\right)^{(3\lambda-2)/\lambda}}{8\pi^{2}\left[\left(\left(\ln t_{e}\right)^{\lambda}-\frac{N_{*}}{A}\right)^{1/\lambda}-\lambda+1\right]}\\
&& \times\exp\left[-2\left(\left(\ln t_{e}\right)^{\lambda}-\frac{N_{*}}{A}\right)^{1/\lambda}\right].
\label{P_s,N_*}\end{aligned}$$ The Planck 2015 data provided an estimation for the amplitude of the scalar perturbations as $\ln\left[10^{10}\mathcal{P}_{s}\left(N_{*}\right)\right]=3.094\pm0.034$ (68% CL, Planck 2015 TT,TE,EE+lowP) [@Planck2015]. We use this constraint in the above equation to fix the amplitude of the scalar power spectrum in our model and determine the parameter $t_e$ in terms of the other parameters for a given horizon crossing $e$-fold number $N_*$. Since we cannot determine $t_e$ analytically, we use a numerical approach. Inserting the result of the numerical solution for $t_e$ in Equation (\[t,N\]), we can obtain the time of horizon exit $t_*$ for given parameters $n$, $A$, $\lambda$, and $N_*$. Surprisingly, our computations show that $t_*$ does not depend on $N_*$ at all. To explain this unexpected result, we take the partial derivative of both sides of Equation (\[P\_s,N\_\*\]) with respect to $N_*$, and, keeping in mind that $\partial\mathcal{P}_{s}\left(N_{*}\right)/\partial N_{*}=0$, we obtain $$\label{d.t_e}
\frac{\partial t_{e}}{\partial N_{*}}=\frac{t_{e}}{A\lambda\left(\ln t_{e}\right)^{\lambda-1}}.$$ On the other hand, if we evaluate Equation (\[t,N\]) at the horizon exit with the $e$-fold number $N_*$, and calculate the partial derivative of the result with respect to $N_*$, then we will have $$\begin{aligned}
\nonumber
\frac{\partial t_{*}}{\partial N_{*}}= && \frac{\exp\left[\left(\left(\ln t_{e}\right)^{\lambda}-\frac{N_{*}}{A}\right)^{1/\lambda}\right]}{\left(\left(\ln t_{e}\right)^{\lambda}-\frac{N_{*}}{A}\right)^{(\lambda-1)/\lambda}}\\
&& \times\left(\frac{\left(\ln t_{e}\right)^{\lambda-1}}{t_{e}}\frac{\partial t_{e}}{\partial N_{*}}-\frac{1}{A\lambda}\right).
\label{d.t_*}\end{aligned}$$ It is obvious that substitution of $\partial t_{e}/\partial N_{*}$ from Equation (\[d.t\_e\]) into Equation (\[d.t\_\*\]) leads to $\partial t_{*}/\partial N_{*}=0$. Therefore, in our model, and after fixing the amplitude of the scalar perturbations from the observational data, the time of horizon exit $t_*$ is independent of its $e$-fold number $N_*$. As an important result, we conclude that the inflationary observables (\[n\_s,t\]), (\[d.n\_s,t\]), (\[n\_t,t\]), and (\[r,t\]) evaluated at $t_*$ are independent of $N_*$.
Now, we can estimate the inflationary observables in our model and check their consistency versus the cosmological data. To do so, first we solve Equation (\[P\_s,N\_\*\]) numerically to find $t_e$ for given parameters $n$, $A$, and $\lambda$. Then, we use the obtained value for $t_e$ in Equation (\[t,N\]) and find $t_*$. Subsequently, we evaluate the inflationary observables (\[n\_s,t\]), (\[d.n\_s,t\]), and (\[r,t\]) at the time of horizon exit $t_*$.
In order to check the viability of logamediate inflation (\[a(t)\]) in our $f(T)$-gravity model (\[f(T)\]), we use Equation (\[n\_s,t\]) and (\[r,t\]) and plot the prediction of our model in $r-n_s$ plane as shown in Figure \[figure:r,n\_s\]. In this figure, the marginalized joint 68% CL and 95% CL regions of the Planck 2015 data [@Planck2015] have been specified. We have represented the results of our model with $n=1$ and $n=2$ in the figure as black and orange lines, respectively. Each line is related to a specific value for the parameter $\lambda$, while the parameter $A$ varies. In each case, as $A$ increases, $n_s$ approaches $1$ and $r$ approaches $0$. The case $n=1$ corresponds to TEGR, which provides the same results of the standard inflationary scenario. It is obvious in the figure that for $n=1$, logamediate inflation is completely ruled out by Planck 2015 TT,TE,EE+lowP data [@Planck2015]. But, our study indicates that if we take the parameter $n$ greater than $1$, then logamediate inflation (\[a(t)\]) in our $f(T)$-gravity model (\[f(T)\]) can be compatible with the Planck 2015 results. For instance, as we see in Figure \[figure:r,n\_s\], for $n=2$, logamediate inflation is consistent with the joint 68% CL region of Planck 2015 TT,TE,EE+lowP data [@Planck2015]. In Figure \[figure:lambda,A\], we have specified the parameter space of $A$ and $\lambda$ for which our model with $n=2$ is compatible with the 68% CL or 95% CL regions of Planck 2015 TT,TE,EE+lowP data [@Planck2015]. From the figure, we conclude that for $\lambda \lesssim 6$ ($\lambda \lesssim 8$), our model is compatible with the joint 68% CL (95% CL) region of Planck 2015 TT,TE,EE+lowP data [@Planck2015]. In Table \[table:d.n\_s\], we present the ranges of the parameter $A$ for which our model with $n=2$ and with some typical values of $\lambda$ is consistent with the Planck 2015 observational data [@Planck2015]. In Table \[table:d.n\_s\], we also present the predicted values for the running of the scalar spectral index $dn_s/d\ln k$ obtained using Equation (\[d.n\_s,t\]). The predicted values for $dn_s/d\ln k$ in our model are compatible with the 95% CL constraint provided by Planck 2015 TT,TE,EE+lowP data [@Planck2015].
At the end of this section, it is useful to provide some explicit estimations for the inflationary observables in our model. We choose $n=2$, $A=1.8$, and $\lambda=2$. Using Equation (\[n\_s,t\]), (\[d.n\_s,t\]), and (\[r,t\]), we obtain $n_{s}=0.9657$, $dn_{s}/d\ln k=3.99\times10^{-5}$, and $r=0.0546$, respectively, and these are in good agreement with Planck 2015 TT,TE,EE+lowP data [@Planck2015]. Using Equation (\[n\_t,t\]), our model predicts the tensor spectral index to be $n_{t}=-0.0354$, and this value may be verified by more precise measurements in the future. Within our model, we can also provide some predictions for other parameters, including the time of horizon exit $t_*$ and the end time of inflation $t_e$. With the chosen values for $n$, $A$, $\lambda$, and taking the horizon exit $e$-fold number as $N_*=60$, we obtain the end time of inflation from the numerical solution of Equation (\[P\_s,N\_\*\]) as $t_{e}=6.47\times10^{6}M_{P}^{-1}=1.75\times10^{-36}\,\mathrm{sec}$. Applying this in Equation (\[t,N\]) gives the time of horizon crossing as $t_{*}=2.15\times10^{6}M_{P}^{-1}=5.82\times10^{-37}\,\mathrm{sec}$. Here, we recall that in our logamediate inflationary model, although the value of $t_e$ depends on $N_*$, the value of $t_*$ is completely independent of it.
Conclusions {#section:conclusions}
===========
We have investigated logamediate inflation in the framework of $f(T)$-gravity which is sourced by a canonical and minimally coupled scalar field. For this purpose, we first briefly reviewed the basic equations governing the cosmological background evolution in $f(T)$-gravity and presented the relations of the scalar and tensor power spectra in this scenario. Then, we considered a setting in which the $f(T)$ function in the action has the power-law form $f(T)=T_0\left(T/T_{0}\right)^{n}$, where $n$ and $T_0$ are constant parameters. For $n=1$, this reduces to $f(T)=T$ which provides the same results as for Einstein GR. In addition, in our work we considered the logamediate scale factor $a(t)=a_{0}\exp\left[A\big(\ln t\big)^{\lambda}\right]$, where $a_0>0$, $A>0$ and $\lambda \ge 1$ are constant parameters. For $\lambda=1$, the logamediate scale factor turns into the power-law scale factor $a(t)=a_{0}t^{q}$, where $q=A$.
Our investigation implies that, although logamediate inflation is not consistent with the the Planck 2015 data [@Planck2015] in the standard framework based on Einstein gravity, we can make it compatible with the observational data in our $f(T)$-gravity model, if we take the parameter $n$ greater than $1$. For instance, we showed that for $n=2$, the result of the logamediate inflation in $r-n_s$ plane can lie inside the 68% CL region favored by Planck 2015 TT,TE,EE+lowP data [@Planck2015]. Using the $r-n_s$ test, we determined the parameter space for $A$ and $\lambda$ in our model with $n=2$, and showed that for $\lambda \lesssim 6$ ($\lambda \lesssim 8$), our model is consistent with the joint 68% CL (95% CL) region of Planck 2015 TT,TE,EE+lowP data [@Planck2015]. We further estimated the running of the scalar spectral index $dn_s/d\ln k$ in our model, and concluded that it satisfies the 95% CL bound from Planck 2015 TT,TE,EE+lowP data [@Planck2015].
The authors thank the referee for his/her valuable comments. The work of A.A. was supported financially by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project No. 1/4717-106.
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---
abstract: 'By modeling the interaction of an open quantum system with its environment through a natural generalization of the classical concept of continuous time random walk, we derive and characterize a class of non-Markovian master equations whose solution is a completely positive map. The structure of these master equations is associated with a random renewal process where each event consist in the application of a superoperator over a density matrix. Strong non-exponential decay arise by choosing different statistics of the renewal process. As examples we analyze the stochastic and averaged dynamics of simple systems that admit an analytical solution. The problem of positivity in quantum master equations induced by memory effects $\mbox{[}$S.M. Barnett and S. Stenholm, Phys. Rev. A [**64**]{}, 033808 (2001)$\mbox{]}$ is clarified in this context.'
author:
- 'Adrián A. Budini'
title: 'Stochastic Representation of a Class of Non-Markovian Completely Positive Evolutions'
---
Introduction
============
From the beginning of quantum mechanics there existed alternative formalisms to describe the dynamics of open quantum system. Besides the microscopic derivation of quantum master equations, the theory of quantum dynamical semigroups [@alicki] introduced a strong constraint for the possible structure of a given Markovian master equation. As is well know, the more general structure is given by the so called Kossakowski-Lindblad generator $$\frac{d\rho \left( t\right) }{dt}=-i\left[ H,\rho \left( t\right)
\right] +
\frac{1}{\tau_{0}}{\cal L}_{0}\left[ \rho \left( t\right) \right]. \label{1-1}$$ Here, $\rho(t)$ is the system density matrix, $H$ is the system Hamiltonian, $\tau_{0}$ is the characteristic time scale of the irreversible dynamics and $${\cal L}_{0}\left[ \bullet \right] =\sum_{\beta }
([V_{\beta },\bullet V_{\beta }^{\dagger }]+[V_{\beta }\bullet ,V_{\beta
}^{\dagger }]), \label{zorra}$$ where $\{V_{\beta}\}$ is a set of arbitrary operators. This structure arise after demanding the Markovian property and the completely positive condition (CPC). This last requisite is stronger than positivity. It guarantees the right behavior of the solution map $\rho(0) \rightarrow
\rho(t)$ after extending, with an identity, the original evolution to an ancillary and arbitrary Hilbert space [@alicki; @nielsen].
As a consequence of the Markovian or semigroup condition, the evolution Eq. (\[1-1\]) is local in time. This fact, in general, implies that the dynamics of the density matrix elements is characterized through an exponential decay behavior. Nevertheless, there exist many physical situations that must be described in a quantum regime and whose characteristic decay behaviors are different from an exponential decay.
Some relevant examples arise in atomic and molecular systems subject to the influence of environments with a highly structured spectral density, where the theoretical modeling can be given in terms of a few-modes spin-boson model [@gruebele] and in terms of random-matrix theory[@wong]. In these situations, the characteristic decay of the system dynamics present stretched exponential and power law behaviors. Other examples are one dimensional quasiperiodic systems [@zhong] that develop a non-Gaussian diffusion front, anomalous photon counting statistics for blinking quantum dots [@barkai_dot], many-spin systems [@dobro], fractional derivative master equations [@kuznezov], and structured reservoirs [@dalton].
In all these physical situations the validity of the approximations that allow a Markovian description break down. Therefore, its dynamical description is outside of a Markovian Lindblad evolution. Thus, there seems to be a gap between completely positive evolutions and those with an anomalous decay behavior.
The main purpose of this paper is to establish the possibility of constructing a class of evolution equations for the density matrix that satisfies the CPC and that also lead to strong non-exponential decay. Our basic idea for the derivation of these equations consists in to model the interaction of an open quantum system with its environment as a series of random scattering events represented through the action of a superoperator over the system density matrix, where the elapsed time between the successive events corresponds to an arbitrary random renewal process [@feller]. This stochastic dynamics can be seen as a natural generalization of the classical method of continuous time random walk [@montroll; @metzler], where a particle at random times jumps instantaneously between the sites of a regular lattice. In consequence we will name our starting stochastic dynamics a continuous time quantum random walk (CTQRW).
We remark that the concept of quantum random walks is nowadays used in the context of quantum information and quantum computation [@julia]. Our paper deals a different problem since here we are concerned with a phenomenological description of anomalous irreversible processes in the context of completely positive evolutions.
The dynamics that result from a CTQRW is non-Markovian and can be written as a memory integral over a Lindblad superoperator \[see Eq. (\[master\])\]. This kind of evolution was previously analyzed in Ref. [@barnett] by Barnett and Stenholm, where was raised up the possibility of obtaining non physical solutions from this non-Markovian evolution. Contrarily to their final conclusion, here we will show that, as in a classical context [@sokolov; @barkai], it is possible to use this kind of equation as a phenomenological tool in the description of open systems. Even more, we will see that the correct behavior of this equation is related with the possibility of associating to it a CTQRW.
The paper is organized as follows. In Section II we introduce the stochastic dynamics and the corresponding evolution for the averaged density matrix. The CPC and the relaxation to a stationary state are characterized. In Section III we study some non trivial kernels that leads to a telegraphic and a fractional equation. The dynamics induced by these evolutions are analyzed through simples systems, as a two level system and a quantum harmonic oscillator. The relation with the formalism of intrinsic decoherence is also established. In section IV we give the conclusions.
Continuous Time Quantum Random Walk
===================================
The stochastic dynamics that define a CTQRW involve two central ingredients. First, a completely positive superoperator ${\cal E}[\bullet]$ which represent an instantaneous disruptive intervention of the environment over the system of interest. We will assume that it can be written in a sum representation [@nielsen] as $${\cal E}[\rho ]=\sum_{i} C_{i}\rho C_{i}^{\dagger },\label{super}$$ where the operators $C_{i}$ satisfies the closure condition $$\sum_{i} C_{i}^{\dagger } C_{i}=\text{I}. \label{uno}$$ The second ingredient is a set of random time $t_{1}<t_{2}\cdots< t_{n}$ that define when the disruptive action occurs. We will assume that this set is stationary and defined as a random renewal process, i.e., it can be characterized through a waiting time distribution $w(\tau)$ which gives the probability density for the elapsed time interval $\tau_{i}=t_{i}-t_{i-1}$ between two consecutive disruptive events.
We will work in an interaction representation with respect to the system Hamiltonian and also assume that the unitary evolution commutates with the superoperator ${\cal E}[\bullet]$. Thus, the average evolution of the density matrix over the realizations of the random times can be written in the following way $$\rho (t) =\sum_{n=0}^{\infty }P_{n}(t)\;{\cal E}^{n}[\rho(0)].\label{sol}$$ Here, $P_{n}(t)$ defines the probability that $n$ applications of the superoperator ${\cal E}[\rho]$ have occurred up to time $t$. This set of probabilities is normalized as $$\sum_{n=0}^{\infty }P_{n}(t)=1,$$ and is defined through the expressions $$P_{0}(t)=1-\int_{0}^{t}d\tau w(\tau),$$ and $$P_{n}(t)=\int_{0}^{t}d\tau w(t-\tau )P_{n-1}(\tau ) \label{Pn}.$$ Note that $P_{0}(t)$ defines the survival probability, i.e., the probability of having not any superoperator action up to time $t$. Using recursively Eq. (\[Pn\]), from Eq. (\[sol\]) it is possible to express the average density matrix as $$\rho (t)=P_{0}(t)\rho (0)+\int_{0}^{t}d\tau w(t-\tau ){\cal E}[\rho (\tau )].
\label{integral}$$ In order to obtain a differential equation for the evolution of $\rho(t)$ we follow the calculation in the Laplace domain. Denoting $\tilde{f}(u)=\int_{0}^{\infty} dt
\exp[-ut] f(t)$, from Eq. (\[integral\]), we get $$\tilde{\rho}(u)=\frac{1-\tilde{w}(u)}{u}\left\{ \frac{1}{\text{I}-\tilde{w}%
(u){\cal E}[\bullet ]}\right\} \rho (0).\label{laplace}$$ where we have used $\tilde{P}_{0}(u)=[1-\tilde{w}(u)]/u$. Eq. (\[laplace\]) allows us to express $\rho(0)$ in terms of $\tilde{\rho}(u)$. Thus, it is straightforward to get $$u\tilde{\rho}(u)-\rho (0)=\tilde{K}(u){\cal L}[\tilde{\rho}(u)]\label{iraci}$$ where we have defined $$\tilde{K}(u)=\frac{u\tilde{w}(u)}{1-\tilde{w}(u)}, \label{kernel}$$ and the superoperator $${\cal L}[\bullet ]={\cal E}[\bullet ]-\text{I}.\label{lindblad}$$ Then, the time evolution of the average density matrix reads $$\frac{d\rho (t)}{dt}=\int_{0}^{t}d\tau K(t-\tau ){\cal L}[\rho (\tau
)],\label{master}$$ where the kernel $K(t)$ is defined through its Laplace transform Eq. (\[kernel\]). This evolution, in general, is non-Markovian, and by construction it is a completely positive one. On the other hand, using the sum representation Eq. (\[super\]) and the normalization condition Eq. (\[uno\]) it is possible to write the superoperator Eq. (\[lindblad\]) in a Lindblad form $${\cal L}[\bullet ]=\frac{1}{2}\sum_{i}\{[C_{i},\bullet C_{i}^{\dagger
}]+[C_{i}\bullet
,C_{i}^{\dagger }]\}.$$
[*Random Superoperators*]{}: The previous results can be easily extended to the case in which the scattering superoperator, in each event, is chosen over a set $\{{\cal E}_{a}[\bullet]\}$ with probability $P(a)da$. Assuming that this random selection is statistically independent of the set of random times, the evolution is the same as in Eq. (\[master\]) with $${\cal L}[\bullet ]=\int_{-\infty }^{+\infty }daP(a){\cal E}_{a}[\bullet
]-\text{I}.\label{LA}$$
[ *Infinitesimal Transformations*]{}: At this point, it is important to remark that in general an arbitrary Lindblad structure, Eq. (\[zorra\]), can not be associated with a completely positive superoperator ${\cal E}[\bullet]$ as in Eq. (\[lindblad\]). This fact does not imply any limitation in our approach. In fact, an arbitrary Lindblad term ${\cal L}_{0}[\bullet]$ can be always associated to a completely positive superoperator of the form $${\cal E}_{0} [\rho]=\{\text{I}+[e^{\kappa {\cal L}_{0}}-\text{I}] \} \rho,
\label{infinitesimal}$$ where $\kappa$ must be intended as a control parameter. Then, an arbitrary Lindblad term can be introduced in Eq. (\[master\]) in the limit in which simultaneously $\kappa \rightarrow 0$ and the number of events by unit of time go to infinite, the last limit being controlled by the waiting time distribution $w(t)$. We will exemplify this procedure along the next section.
Completely Positive Condition
-----------------------------
As was mentioned previously, by construction the non-Markov evolution Eq. (\[master\]) is a completely positive one. Nevertheless, from a phenomenological point of view [@barnett] one is also interested to know which kind of arbitrary kernel $K_{d}(t)$ guarantee this condition.
The CPC is clearly satisfied if it is possible to associate to the kernel $K_{d}(t)$ a well defined waiting distribution. Given an arbitrary kernel, from the definition Eq (\[kernel\]), the associated waiting time distribution is $$\tilde{w_{d}}(u)=\frac{\tilde{K_{d}}(u)}{u+\tilde{K_{d}}(u)}=\frac{1}{u/\tilde{K_{d}}(u)+1}.
\label{waiting}$$ This equation defines a positive waiting time distribution if and only if $\tilde{w_{d}}(u)$ is a completely monotone (CM) function [@feller], i.e. $\tilde{w_{d}}(0)>0$ and $(-1)^{n}\tilde{w_{d}}^{(n)}(u)\geqslant 0$, where $\tilde{w_{d}}^{(n)}(u)$ denote the n-derivative. After using that $1/(u+1)$ is a completely monotone function, and that a function of the type $f(g(u))$ is CM, if $f(s)$ is CM and if the function $g(u)$ is positive and possesses a CM derivative [@feller], the Laplace transform of the kernel $K_{d}(u)$ must satisfy $${\displaystyle{u \over \tilde{K_{d}}\left( u\right) }}%
\geqslant 0\text{ and }
{\displaystyle{d[u/\tilde{K_{d}}\left( u\right) ] \over du}}%
\text{ a CM function.} \label{condiciones}$$ As in the classical case, these conditions allow us to classify the kernels in safe and dangerous ones[@sokolov]. The secure ones, independently of the particular structure of the superoperator ${\cal E}[\bullet]$, always admit a stochastic interpretation in terms of a CTQRW. Therefore, they induce a completely positive dynamics. The dangerous ones do not admit a stochastic interpretation and in consequence the CPC is not guaranteed. As we will see in the next examples, in this last case the CPC depends on the particular structure of the superoperator ${\cal E}[\bullet]$.
Integral Solution-Subordination Processes
-----------------------------------------
The solution of the evolution Eq. (\[master\]) can be written in an integral form over the solution of a corresponding Markovian problem. In order to demonstrate this affirmation, first we write Eq. (\[iraci\]) as $$\tilde{\rho}(u)= \frac{1}{\text{u}-\tilde{K}%
(u){\cal L}[\bullet ]} \rho (0) . \label{hebe}$$ Using the expression $$\frac{1}{\text{u}-\tilde{K} (u){\cal L}[\bullet ]} =
\int_{0}^{\infty} d\tau^{'}
e^{ -\{ u-\tilde{K}(u) {\cal L}[\bullet] \} \tau^{'}},$$ and after the change of variable $\tau= \tilde{K}(u) \tau^{'}$ it is possible to write $$\frac{1}{\text{u}-\tilde{K} (u){\cal L}[\bullet ]} =
\int_{0}^{\infty} d\tau \tilde{P} (u,\tau)
e^{ {\cal L}[\bullet] \tau}, \label{eterna}$$ where the function ${\tilde P}(u,\tau)$ is defined by $$\tilde{P}\left( u,\tau \right) =\frac{1}{\tilde{K}\left( u\right) }\exp
[-\tau \frac{u}{\tilde{K}\left( u\right) }]. \label{mirna}$$ Note that from this expression, after a Laplace transform in the second variable $ \tau \to s$, it is possible to obtain ${\tilde P}(u,s)=1/[u+s
{\tilde K}(u)]$, which implies the equivalent definition $$\frac{\partial P(t,\tau)}{\partial t}=-\int_{0}^{t}dt^{'} K(t-t^{'})
\frac{\partial
P(t^{'},\tau)}{\partial \tau}.$$ Inserting Eq. (\[eterna\]) in Eq. (\[hebe\]), the integral solution for the density matrix reads $$\rho \left( t\right) =\int_{0}^{\infty }d\tau P\left( t,\tau \right) \rho
^{(M)}\left( \tau \right), \label{morena}$$ where the density operator $\rho ^{(M)}(\tau) $ is the solution of the Markovian evolution $$\frac{d\rho ^{(M)}(\tau) }{d\tau}={\cal L}[\rho ^{(M)}(\tau)],$$ subject to the initial condition $\rho(0)$, i.e., $\rho ^{(M)}(\tau) =\exp[{\cal L} \tau] [\rho( 0)]$.
When the set of conditions Eq. (\[condiciones\]) is satisfied, from Eq. (\[mirna\]) it is simple to demonstrate that the function $P(t,\tau)$ defines a probability distribution for the $\tau-$variable [@aclaracion], i.e., $$P\left( t,\tau \right) \geqslant 0\text{\ \ \ and\ \ \ }\int_{0}^{\infty
}d\tau P\left( t,\tau \right) =1, \label{claudia}$$ where the normalization of $P\left( t,\tau \right) $ follows from $\int_{0}^{\infty}d\tau P\left( u,\tau \right) =1/u$. This result, joint with Eq. (\[morena\]), allows us to interpret the stochastic evolution as a subordination process [@feller; @sokolov], where the translation between the “internal time" $\tau$ and the physical time $t$ is given by the function $P(t,\tau)$. On the other hand, note that the positivity of this probability function is equivalent to the CPC of the solution map.
Relaxation to the stationary state
----------------------------------
Here we will analyze the relaxation of the density matrix to a stationary state. With the aid of the integral solution Eq. (\[morena\]), the characterization of this process is similar to that of classical Fokker-Planck equations [@barkai]. First, we note that the Markovian evolution of $\rho ^{(M)}(\tau)$ can be always solved in a damping basis[@briegel] as $$\rho ^{(M)}(\tau) =\sum_{\lambda }\check{c}_{\lambda }e^{-\lambda
\tau}P_{\lambda } , \label{relax}$$ where $P_{\lambda}$ are the eigen-operators of the Lindblad term, ${\cal L}[P_{\lambda }]=\lambda P_{\lambda }$, and the expansion coefficients are defined by $\check{c}_{\lambda }=$Tr$[%
\check{P}_{\lambda }\rho ^{(M)}\left( 0\right) ]$. The dual operators $%
\check{P}_{\lambda }$ satisfy the closure condition Tr$[\check{P}%
_{\lambda }P_{\lambda ^{%
%TCIMACRO{\UNICODE[m]{0xb4}}%
%BeginExpansion
{\acute{}}%
%EndExpansion
}}]=\lambda \delta _{\lambda \lambda ^{%
%TCIMACRO{\UNICODE[m]{0xb4}}%
%BeginExpansion
{\acute{}}%
%EndExpansion
}}$ and are defined through ${\check{{\cal L}}}[\check{P}_{\lambda
}]=\lambda \check{P}_{\lambda }$, where ${\check{{\cal L}}}[\bullet ]$ is the dual superoperator of ${\cal L}[\bullet ]$ defined by Tr$\{A{\cal L}%
[\rho ]\}=$Tr$\{\rho {\check{{\cal L}}}[A]\}$[@alicki]. The expansion Eq. (\[relax\]) allows us to write the solution of the non-Markov evolution Eq. (\[master\]) in the form $$\rho \left( t\right) =\sum_{\lambda }\check{c}_{\lambda }h_{\lambda
}(t)P_{\lambda } \label{Relax2}$$ where the functions $h_{\lambda }(t)$ are defined by $$h_{\lambda }(t)=\int_{0}^{\infty }d\tau P\left( t,\tau \right) e^{-\lambda
\tau }. \label{Relax3}$$ In the Laplace domain this definition is equivalent to $$\tilde{h}_{\lambda }\left( u\right) =\frac{1}{u+\lambda \tilde{K}\left(
u\right) } ,\label{Relax4}$$ which also imply $$\frac{d h_{\lambda }(t)}{dt} =-\lambda \int_{0}^{t} d\tau K(t-\tau)
h_{\lambda} (\tau).\label{Relax5}$$ From these expressions it is simple to realize that if the Markovian solution Eq. (\[relax\]) involves a null eigenvalue, the corresponding stationary state maintains this status in the non-Markovian evolution. Furthermore, the typical exponential decay of a Lindblad evolution is translated to that of the characteristic functions $h_{\lambda}(t)$. On the other hand, due to the structure of the solution Eq. (\[Relax2\]), it is clear that any set of relations between the relaxation rates of the Markovian problem [@alicki; @kimura] will be also present in the non Markov solution \[see Eqs. (\[rateexponential\])-(\[ratefractional\])\].
Examples
========
In this section we will analyze different possible dynamics that arise after choosing different memory kernels. Furthermore, we will work out some exact solutions in simple systems.
Markovian Dynamics
------------------
By assuming an exponential waiting time distribution $$w(t)=A_{1}e^{-A_{1}t},$$ from Eq. (\[kernel\]) it is immediate to obtain $$K(t)=A_{1} \delta(t).$$ Thus, the evolution Eq. (\[master\]) reduces to a Markovian one. In this case, it is also possible to obtain all the hierarchy of probabilities $P_{n}(t)$, which read $$P_{n}(t)=\frac{(A_{1}t)^{n}}{n!}e^{-A_{1} t}.$$ This results imply that a Markovian Lindblad evolution can be associated with a Poissonian statistics of the environment action. This stochastic interpretation is also valid for arbitrary Lindblad terms Eq. (\[zorra\]). In this case, the associated superoperator is given by Eq. (\[infinitesimal\]) and it is necessary to take the limit $\kappa
\rightarrow 0$, $A_{1} \rightarrow \infty$ with $\kappa A_{1}=A_{1}^{'}$. Note that this limit is well defined in the sense that the waiting time distribution remains positive and normalized, i.e., $\int_{0}^{\infty} d\tau w(\tau)=1$.
Exponential Kernel
------------------
Now we will analyze the case of an exponential kernel $$K\left( t\right) =A_{\epsilon} \exp [-\gamma t], \label{exponencial}$$ where the units of $A_{\epsilon}$ are $sec^{-2}$. By demanding the conditions Eq. (\[condiciones\]) it is possible to show that this kernel is not a secure one, i.e., in general it is not possible to associate a stochastic dynamics, and in consequence the CPC of the solution map is not guaranteed. Nevertheless, note that in the double limit, $
\gamma \rightarrow \infty $, $ A_{\epsilon} \rightarrow \infty
$, with $A_{\epsilon}/\gamma=A_{1}$ this kernel reduce to the previous case, indicating a possible region of parameters values where the kernel can be a secure one. In order to see this fact, from Eq. (\[waiting\]), after Laplace transform, we get $$w(t)=2 A_{\epsilon} e^{-\gamma t/2} \frac{\sinh [\frac{1}{2} t
\sqrt{\gamma^{2}-4A_{\epsilon}}]}
{\sqrt{\gamma^{2}-4A_{\epsilon}}}. \label{exponential}$$ This function, for $\gamma^{2}> 4A_{\epsilon}$ is a well defined waiting time distribution which delimits the region of parameter values where the evolution is a secure one.
After differentiation of Eq. (\[master\]), the evolution of the density matrix can be written as $$\frac{d^{2}\rho \left( t\right) }{dt^{2}}+\gamma \frac{d\rho \left( t\right)
}{dt}=A_{\epsilon} {\cal L}[\rho \left( t\right) ]. \label{telegrafo}$$ which is a kind of a telegraphic equation [@morse]. This equation must be solved with the initial values $\rho \left( t\right) \mid
_{t=0}=\rho _{0}$ and $d\rho \left( t\right) /dt\mid _{t=0}=0.$ Then, under the condition $\gamma^{2}> 4A_{\epsilon}$ this equation provides an evolution whose solution is a completely positive map. In this case, the characteristic decay functions $h_{\lambda}(t)$, Eq. (\[Relax3\]), results as $$h\left( t,\Phi_{\lambda} \right) =e^{-\gamma t/2}\{\cosh
[\frac{t}{2}\Phi_{\lambda}
]+\frac{\gamma }{\Phi_{\lambda} }\sinh [\frac{t}{2}\Phi_{\lambda} ]\},
\label{dicotomico}$$ where $\Phi_{\lambda}=\sqrt{\gamma^{2}-4 \lambda A_{\epsilon}}$.
We remark that the introduction of an arbitrary Lindblad term ${\cal L}_{0}[\bullet]$ in Eq. (\[telegrafo\]) modifies drastically the previous positivity conditions. In fact, this change requires the use of the superoperator Eq. (\[infinitesimal\]) and the double limit $\kappa \rightarrow 0$, $A_{\epsilon} \rightarrow \infty$, with $\kappa A_{\epsilon}=A_{\epsilon}^{'}$. Nevertheless, from Eq. (\[exponential\]), we note that the limit $A_{\epsilon}
\rightarrow \infty$ leads to a waiting time distribution that always takes negative values. The positivity of $w(t)$ can only be recuperated in the limit $\gamma
\rightarrow \infty$. Nevertheless, as we have commented previously, this extra requirement implies that the final dynamics converge to a Markovian ones. Therefore, for infinitesimal superoperators there is no region of parameter values where the exponential kernel admits a stochastic interpretation. In consequence, the CPC of the solution map is unpredictable and must be checked for each particular case. This result characterizes and generalizes the results obtained in Ref.[@barnett].
Fractional Evolution
--------------------
Now we analyze a case of a sure kernel [@sokolov]. We assume $$\tilde{K}\left( u\right) =A_{\alpha }u^{1-\alpha },\;\;\;\;\;0<\alpha \le 1,
\label{kernelfrac}$$ where the units of $A_{\alpha}$ are $1/sec^{\alpha}$. As is well known, this kind of kernel can be related to a fractional derivative operator [@metzler]. Thus, the density matrix evolution reads $$\frac{d\rho \left( t\right) }{dt}=A_{\alpha \;0}D_{t}^{1-\alpha }{\cal L}%
\left[ \rho \left( t\right) \right] . \label{fraccionaria}$$ The Riemann-Liouville fractional operator is defined by $$_{0}D_{t}^{1-\alpha }f(t)=\frac{1}{\Gamma (\alpha )}
\frac{d}{dt}\int_{0}^{t}dt^{\prime}
\frac{f(t^{\prime })}{(t-t^{\prime })^{1-\alpha }},$$ where $\Gamma \left( x\right) $ is the Gamma function. By using Eq. (\[waiting\]), the Laplace transform of the waiting time distribution reads $$\tilde{w}(u)=\frac{A_{\alpha }}{A_{\alpha}+u^{\alpha }}.$$ Note that for $\alpha=1$ this expression reduces to the Laplace transform of an exponential function. Furthermore, the condition $0<\alpha \le 1$ corresponds to the values of $\alpha$ where ${\tilde w}(u)$ is a CM function, guaranteeing a well defined waiting time distribution. In the time domain it reads $$w(t)=\frac{A_{\alpha }}{t^{1-\alpha }}\sum_{n=0}^{\infty }\frac{(-A_{\alpha
}t^{\alpha })^{n}}{\Gamma \lbrack \alpha (n+1)]}. \label{serie}$$ Thus, the case of fractional derivative provides a well defined evolution, Eq. (\[fraccionaria\]), whose solution is a completely positive map that admits a stochastic interpretation in terms of the waiting time distribution Eq. (\[serie\]). We remark that in this case, the average time between successive applications, $\langle \tau \rangle=\int_{0}^{\infty}
\tau w(\tau) d\tau$, is not defined. As in the classical domain [@metzler], this fact implies the absence of a characteristic time scale and statistically it enables the presence of time intervals of any magnitude. On the other hand, we note that an arbitrary Lindblad superoperator can be always introduced in Eq. (\[fraccionaria\]) in a secure way. In fact, the waiting time distribution Eq. (\[serie\]) is well defined in the limit $\kappa \rightarrow 0$, $A_{\alpha} \rightarrow \infty$ with $\kappa
A_{\alpha}=A_{\alpha}^{'}$.
From Eqs. (\[Relax4\])-(\[kernelfrac\]), the characteristic decay functions $h_{\lambda}(t)$ read $$h_{\lambda}(t)=E_{\alpha}[-\lambda A_{\alpha} t^{\alpha}].$$ Here we have introduced the Mittag-Leffler function $E_{\alpha}(t)$ which is defined through the series [@metzler] $$E_{\alpha }\left[-A_{\alpha} t^{\alpha}\right] =\sum_{k=0}^{\infty
}\frac{(-A_{\alpha} t^{\alpha})^{k}}{\Gamma
\left(\alpha k+1\right) }.$$ The short time regime of this function is governed by an stretched exponential decay $$\lim_{t\rightarrow 0}E_{\alpha }[- A_{\alpha} t^{\alpha }]\approx
e^{- A_{\alpha} t^{\alpha }},$$ while the long time regime converges to a power law decay $$\lim_{t\rightarrow \infty }E_{\alpha }[- A_{\alpha} t^{\alpha
}]\approx \frac{1}{A_{\alpha} t^{\alpha} }.$$ In this way, the fractional kernel allows us to introduce these anomalous behaviors that clearly differ from the typical exponential decay of a standard Lindblad equation. Furthermore, this dynamics can be always associated with a CTQRW characterized through the waiting time distribution Eq. (\[serie\]).
Short Time Regime
-----------------
An important aspect in the theory of open quantum systems is the characterization of the irreversible dynamics at short times [@lu; @budini]. Here we will analyze this regime through the linear entropy $\delta(t)=1-Tr[\rho^{2}(t)]$. For simplicity, we will assume that at the initial time the system is in a pure state, $\rho(0)=|\Psi\rangle\langle
\Psi|$. Defining the average $$\langle\langle {\cal E} \rangle\rangle = \sum_{i} \langle C_{i}^{\dagger}
C_{i}\rangle-\langle
C_{i}^{\dagger}\rangle \langle C_{i} \rangle.$$ where $\langle C \rangle = \langle\Psi|C|\Psi \rangle$, from Eq. (\[morena\]), for the fractional case we get $$\delta(t)\approx \frac{2 A_{\alpha} t^{\alpha}}{\Gamma (1+\alpha)}
\langle\langle {\cal E}
\rangle\rangle.$$ while for the exponential case we get $$\delta(t)\approx A_{\epsilon} t^{2}\langle\langle {\cal E} \rangle\rangle.$$ We note that for the Markovian case $(\alpha=1)$ the increase of entropy is linear in time, while the exponential case present a slower quadratic behavior. On the other hand, the fractional case gives rise to the faster increase, whose rate is not defined, i.e., it is infinite. Nevertheless, as we will show in the next examples, in the long time regime the fractional case induces the slower dynamical behavior.
Two-Level System {#seccion}
----------------
Here we will analyze the non-Markovian dynamics of a two level system driven by different superoperators and memory kernels.
### Depolarizing Reservoir
First we will analyze the case of a depolarizing environment [@nielsen]. Thus, we define the operators that appear in the sum representation Eq. (\[super\]) as $$C_{1}=\sqrt{p_{x}}\; \sigma _{x}\;\;\;\;C_{2}=\sqrt{p_{y}}\;\sigma
_{y}, \label{xy}$$ where $p_{x}+p_{y}=1$, and $\sigma_{x}$, $\sigma_{y}$ are the $x-y$ Pauli matrixes. In order to simplify the final equations, from now on we will assume $p_{x}=p_{y}=1/2$. In this case, the Lindblad superoperator ${\cal L}[\bullet]$ \[Eq. (\[lindblad\])\] reads $${\cal L} \left[ \bullet \right] =\frac{1}{4}([\sigma _{x},\bullet \sigma_{x}
]+[\sigma_{x}\bullet ,\sigma_{x} ]+%
[\sigma_{y} ,\bullet \sigma_{y}]+[\sigma_{y} \bullet ,\sigma_{y
}]).\label{infinito}$$ This Lindblad generator corresponds to the interaction of a two level system with a reservoir at infinite temperature. This fact can be clearly seen by expressing ${\cal L} [ \bullet]$ in terms of the lowering and raising spin operators, $\sigma=(\sigma_{x}- i \sigma_{y})/2$, $\sigma^{\dagger}=(\sigma_{x}+ i \sigma_{y})/2$. We notice that assuming other values of $p_{x}$ and $p_{y}$, extra terms appear in Eq. (\[infinito\]) that do not modify the infinite temperature property of the Lindblad superoperator.
[*Exponential Kernel*]{}: By denoting the density matrix $\rho (t)$ in the basis of the eigenvalues of $\sigma_{z}$ as $$\begin{aligned}
\rho(t) =\left(
\begin{array}{cc}
P_{+}(t) & C_{+}(t) \\
C_{-}(t) & P_{-}(t)
\end{array}
\right),\end{aligned}$$ from Eq. (\[telegrafo\]), the evolution of the upper and lower levels reads $$\frac{d^{2}P_{\pm }(t)}{dt^{2}}+\gamma \frac{dP_{\pm }(t)}{dt}= A_{\epsilon}
\lbrack \pm P_{-}(t)\mp P_{+}(t)],$$ while the coherences evolve as $$\frac{d^{2}C_{\pm }(t)}{dt^{2}}+\gamma \frac{dC_{\pm }(t)}{dt}=- A_{\epsilon}
C_{\pm
}(t).$$ The solution of these equations are $$P_{\pm }(t)=P_{\pm }^{eq}+(P_{\pm }(0)-P_{\pm }^{eq})h(t,\Phi
_{pop}),\label{popular}$$ with $P_{\pm }^{eq}=1/2$, and $$C_{\pm }(t)=C_{\pm }(0)h(t,\Phi _{coh}), \label{coherente}$$ where the function $h(t,\Phi)$ was defined in Eq. (\[dicotomico\]) and $$\Phi _{pop} =\sqrt{\gamma^{2}-8 A_{\epsilon}},\;\;\;\;\;\;\;\;\;\;\;
\Phi_{coh}=\sqrt{\gamma^{2}-4 A_{\epsilon}}.$$ In the Markovian limit $\gamma \rightarrow \infty $, $A_{\epsilon} \rightarrow
\infty$ with $A_{\epsilon}/\gamma=A_{1}$ we get the well know Markovian results $h\left( t,\Phi
\right) =\exp [-\Phi t],$ with $\Phi
_{pop}=2 A_{1}$ and $\Phi _{coh}= A_{1}.$
From our previous results, we know that under the condition $\gamma >4
A_{\epsilon}$ the dynamics must be a completely positive one and that for $\gamma <4 A_{\epsilon}$ this is not guaranteed. Here, we will check these conclusions for this simple model. By using the property $|h(t,\Phi)|\le 1$, from Eq. (\[popular\]) it is possible to conclude that for any value of the parameter $\gamma$ and $A_{\epsilon}$, at all times the populations satisfies $P_{\pm}(t) \ge 0$. On the other hand, the determinant $d(t)$ of $\rho(t)$, for any parameter values, satisfies the inequality $$\begin{aligned}
d(t)&=&\bigg\{\frac{1}{4}+\left(P_{+}(0)-\frac{1}{2}\right)\left(P_{-}(0)-\frac{1}{2}
\right) h^{2}(t,\Phi_{pop}) \nonumber
\\&-& C_{+}(0) C_{-}(0) h^{2}(t,\Phi_{coh}) \bigg \}\ge 0.\end{aligned}$$ In consequence, independently of the values of $\gamma$ and $A_{\epsilon}$, the density matrix is always positive. We remark that this result does not imply that the solution map $\rho(0) \rightarrow \rho(t)$ is a completely positive one. By writing the solution in the sum representation $$\rho(t)=g_{I}(t) \;\rho(0)+\sum_{j=x,y,z} g_{j}(t)\; \sigma_{j} \rho(0)
\sigma_{j},\label{mapa}$$ where $$\begin{aligned}
g_{I}(t)&=&\frac{1}{2} \left[\frac{ 1+h(t,\Phi_{pop})}{2}+h(t,\Phi_{coh})
\right],
\\ g_{x}(t)&=& g_{y}(t)=\frac{ 1-h(t,\Phi_{pop})}{4},
\\ g_{z}(t)&=&\frac{1}{2} \left[\frac{ 1+h(t,\Phi_{pop})}{2}-h(t,\Phi_{coh})
\right],\end{aligned}$$ the CPC is equivalent to the conditions $g_{I}(t) \ge 0$, and $g_{j}(t) \ge 0~\forall j$, for all times. For $\gamma^{2} \ge 4 A_{\epsilon}$ these inequalities are satisfied. On the other hand, for $\gamma^{2} \le 4 A_{\epsilon}$, while the functions $g_{x}(t)$ and $g_{y}(t)$ are still positive, the functions $g_{I}(t)$ and $g_{z}(t)$ take negative values, which imply that the map $\rho(0) \rightarrow \rho(t)$ is not a completely positive ones. Note that in this situation, the map Eq. (\[mapa\]) can be written as a difference of two completely positive maps. This fact agrees with the general results of Ref. [@yu], where it was demonstrated that any positive map can be written as a difference of two completely positive ones.
![Stochastic realizations for the CTQRW defined by the depolarizing operators Eq. (\[xy\]) and the fractional waiting time distribution Eq. (\[serie\]). The graphs correspond to the quantum average of the Pauli matrixes, $M_{j}(t)=Tr[\rho(t) \sigma_{j}]$, $j=x,y,z$. The realization for the normalized average $M_{y}(t)/M_{y}(0)$ is equal to that of the $x$-direction. The parameters were chosen as $p_{x}=p_{y}=0.5$ and $\alpha=0.5$, $A_{\alpha}=1/\sqrt{2}~sec^{-1/2}$, T=$A_{\alpha}^{-1/\alpha}.$[]{data-label="realizations"}](Figure1_Class.eps){width="8.cm"}
[*Fractional kernel*]{}: Now we analyze the dynamics of the the two level system in the case of the fractional kernel Eq. (\[fraccionaria\]). For the evolution of the populations we get $$\frac{dP_{\pm }(t)}{dt}= A_{\alpha \;0}D_{t}^{1-\alpha }
[\pm P_{-}(t)\mp P_{+}(t)],$$ and the evolution of the coherence is $$\frac{dC_{\pm }(t)}{dt}=- A_{\alpha \;0}D_{t}^{1-\alpha } C_{\pm
}(t).$$ The solutions of these equations are $$P_{\pm }(t)=P_{\pm }^{eq}+(P_{\pm }(0)-P_{\pm }^{eq})
E_{\alpha}[-\Phi_{pop}^{(\alpha)}
t^{\alpha}] \label{popfrac},$$ and $$C_{\pm }(t)=C_{\pm }(0)E_{\alpha}[- \Phi_{coh}^{(\alpha)}
t^{\alpha}], \label{coherfrac}$$ where $$\Phi_{pop}^{(\alpha)}=2 A_{\alpha},\;\;\;\;\;\
\Phi_{coh}^{(\alpha)}=A_{\alpha}.$$ These expressions provide a completely positive map that admit a stochastic interpretation in terms of its associated CTQRW. In Fig. (\[realizations\]) we have implemented a numerical simulation of this quantum stochastic process. We show a set of realizations for the quantum averages of the Pauli matrixes, $M_{j}(t)=Tr[\rho(t) \sigma_{j}]$, $j=x,y,z$. After the first application of the depolarizing superoperator, Eq. (\[xy\]), the normalized values of $M_{x}(t)$ and $M_{y}(t)$ go to zero, remaining in this value at all subsequent times. On the other hand, $M_{z}(t)/M_{z}(0)$ oscillates between $\pm 1$ after each scattering event. A notable property of these realizations is the absence of a characteristic time scale both for the first event and for the elapsed time between any successive events. This fact is a consequence of the power law decay of the waiting time distribution $w(t)$, Eq. (\[serie\]). The absence of any time scale can be seen in the realization of $M_{z}(t)/M_{z}(0)$ where it is evident the presence of time intervals of any magnitude.
In Fig. (\[coherenceaverage\]) we show the corresponding average over $10^{4}$ realizations together with the analytical result for $M_{x}(t)$. We have taken $\alpha=1/2$, which allows to use the equivalent expression $E_{1/2}[A_{\alpha}t^{1/2}]= \exp[A_{\alpha}^{2} t]\;
\rm{erfc}[A_{\alpha} t^{1/2}]$ [@metzler]. In the inset we compare the decay behavior induced by the different kernels. Here, the stretched exponential decay at short times and the power law behavior at long times are evident. In order to be able to compare the different time decay scales induced by each kernel, in all figure of the paper we take $\{A_{1}=A_{\alpha}^{1/\alpha}=A_{\epsilon}/\gamma\} \equiv T^{-1}$, which define the dimensionless time scale $t/T$.
![Theoretical result (full line) and average over $10^{4}$ realizations (circles) for $M_{x}(t)$. The inset shows the short time regime together with the theoretical results for the Markovian evolution (dashed line) with $A_{1}=0.5~sec^{-1}$, and the exponential kernel (full line) with $ \gamma=2~sec^{-1}$, $ A_{\epsilon}=1~sec^{-2}$.[]{data-label="coherenceaverage"}](Figure2_Class.eps){width="8.cm"}
[*Linear entropy*]{}: The linear entropy $\delta(t)$ can be used as a probe of the density matrix positivity. In fact, in a two dimensional Hilbert space, the positivity condition $\rho(t) \ge 0$ is equivalent to the inequality $0\leq \delta (t)\leq 1$. This means that if one of the two eigenvalues of $\rho(t)$ is negative, them $\delta(t)<
0$. Furthermore, the dynamical behaviors induced by each kernel can be shown in a transparent way through this object.
In Fig. (\[PositiveEntropy\]) we show the linear entropy for the Markovian, exponential and fractional kernels. As initial condition we have chosen a pure state, an eigenstate of $\sigma_{x}$. In the case of the exponential kernel, consistently, we verify that independently of the parameter values, the linear entropy is always positive.
![Linear entropy for the CTQRW defined by Eq. (\[xy\]) ($p_{x}=p_{y}=1/2$). Long dashed line, Markovian kernel with $A_{1}=0.5~sec^{-1}$. Dashed line, fractional kernel with $\alpha=0.5$, $A_{\alpha}=1/\sqrt{2}~sec^{-1/2}$. Full line, exponential kernel with $\gamma=2~sec^{-1}$, $A_{\epsilon}=1~sec^{-2}$. Dotted line, exponential kernel with $\gamma=0.5~sec^{-1}$, $A_{\epsilon}=0.25~sec^{-2}$.[]{data-label="PositiveEntropy"}](Figure3_Class.eps){width="8.cm"}
### Dephasing Reservoir
Here, we assume that the superoperator ${\cal E}[\bullet]$ is defined through the operator $$C_{1}=\sigma _{z}.$$ The Lindblad superoperator results in ${\cal L}{[\bullet]}={\cal
L}_{d}{[\bullet]}$, where $${\cal L}_{d}\left[ \bullet \right] \equiv\frac{1}{2}([\sigma _{z},\bullet
\sigma _{z}]+[\sigma _{z}\bullet ,\sigma _{z}]).\label{disperso}$$ As is well known, this kind of dispersive contribution destroys coherences without affecting the level occupations.
In the case of the exponential kernel, the matrix elements are given by $$P_{\pm }(t)=P_{\pm }(0),\;\;\;\;\;\;\;\;C_{\pm }(t)=C_{\pm }(0)h(t,\Phi _{d}).$$ where the function $h(t,\Phi )$ was defined in Eq. (\[dicotomico\]) and now $\Phi _{d}=\sqrt{\gamma^{2}-8 A_{\epsilon}}$. It is simple to proof that independently of any parameter value, here the evolution preserves the density matrix positivity. This follows from the inequality $d(t)=P_{+}(0)P_{-}(0)-C_{+}(0)C_{-}(0)(h(t,\Phi
_{d}))^{2}\geqslant 0$, which, added to the preservation of the probability occupations, guarantees the positivity condition. On the other hand, by expressing the density matrix in the sum representation, $\rho(t)=g_{I}(t) \rho(0)+g_{z}(t) \sigma_{z} \rho(0) \sigma_{z}$, with $g_{I}(t)~=~[1+~h(t,\Phi_{d})]/2$ and $g_{z}(t)=[1-h(t,\Phi_{d})]/2$, it is immediate to proof that the dynamics is completely positive for any parameter values. Therefore, for this kind of dispersive superoperator, independently of the possibility of associating to it a stochastic dynamics, the solution map is always completely positive.
In the case of the fractional kernel we get $$P_{\pm }(t)=P_{\pm }(0),\;\;\;\;\;\;\;C_{\pm }(t)=C_{\pm }(0)E_{\alpha
}[-2 A_{\alpha} t^{\alpha }].$$ As in the previous environment model, here the coherence decay displays stretched exponential and power law behaviors.
### Thermal Reservoir
Now we will analyze a dynamics that leads to a thermal equilibrium state. First, we assume $$\begin{aligned}
C_{1} &=&\sqrt{p_{\uparrow}}\left(
\begin{array}{cc}
1 & 0 \\
0 & \sqrt{1-\kappa}
\end{array}
\right), \;\;\;\;\;\;\;\;\;
C_{2} =\sqrt{p_{\uparrow}}\left(
\begin{array}{cc}
0 & \sqrt{\kappa} \\
0 & 0
\end{array}
\right), \\ C_{3}&=&\sqrt{p_{\downarrow}}\left(
\begin{array}{cc}
\sqrt{1-\kappa} & 0 \\
0 & 1
\end{array}
\right), \;\;\;\;\;\;\;\;\; C_{4}=\sqrt{p_{\downarrow}}\left(
\begin{array}{cc}
0 & 0 \\
\sqrt{\kappa} & 0
\end{array}
\right).\end{aligned}$$ where $p_{\uparrow}+p_{\downarrow}=1$ and $0<\kappa \le 1$. These operators correspond to a generalized amplitude damping superoperator [@nielsen]. With these definitions, the Lindblad superoperator Eq. (\[lindblad\]) can be written as $${\cal L}[\bullet]=\kappa {\cal L}_{th}[\bullet]+\tilde{\kappa}{\cal
L}_{d}[\bullet] \label{termico}$$ where ${\cal L}_{d}[\bullet]$ was defined in Eq. (\[disperso\]), and $$\tilde{\kappa}=\frac{1}{2}\left[1-\frac{\kappa}{2}-\sqrt{1-\kappa}\right]
\label{kapita}.$$ On the other hand, the Lindblad term ${\cal L}_{th}[\bullet]$ corresponds to a thermal reservoir $${\cal L}_{th}\left[ \bullet \right] \equiv\frac{ p_{\uparrow}}{2}([\sigma
^{\dagger
},\bullet \sigma ]+[\sigma ^{\dagger }\bullet ,\sigma ])+\frac{
p_{\downarrow} }{2} ([\sigma ,\bullet \sigma ^{\dagger }]+[\sigma \bullet
,\sigma
^{\dagger }]).$$ The temperature is defined by $p_{\uparrow}/p_{\downarrow}=\exp[-\beta
\Delta E]$, where $\Delta E$ is the difference of energy between the two levels.
Before proceeding with the description of this case, we want to remark that a pure thermal evolution can be only introduced through an infinitesimal transformation. In fact, it is possible to demonstrate that the superoperator ${\cal E}_{th}[\bullet]\equiv {\cal
L}_{th}[\bullet]+\text {I}$ is not a completely positive one, i.e., it can not be written in a sum representation Eq. (\[super\]). After noting that the Lindblad superoperator Eq. (\[termico\]) satisfies ${\cal L}[\bullet ]=\kappa {\cal L}_{th}\left[ \bullet \right] + O(\kappa
^{2})$, it is possible to associate $\kappa$ with the control parameter of Eq. (\[infinitesimal\]). Thus, in the limit $\kappa \rightarrow 0$ the dispersive contribution drops out.
The dynamics induced by the Lindblad Eq. (\[termico\]) is similar to those analyzed previously in this section. In fact, the solution for the exponential case can be written as in Eqs. (\[popular\])-(\[coherente\]) with $$\Phi _{pop} =\sqrt{\gamma^{2}-4 \kappa A_{\epsilon} } ,\;\;\;\;
\\ \Phi _{coh} =\sqrt{\gamma^{2}-2 (\kappa+4 {\tilde \kappa})
A_{\epsilon}}.\label{rateexponential}$$ On the other hand, for the fractional kernel, the solutions read as in Eqs. (\[popfrac\])-(\[coherfrac\]) with the definitions $$\Phi_{pop}^{(\alpha)}=\kappa A_{\alpha},\;\;\;\;\;\;
\Phi_{coh}^{(\alpha)}=(\frac{\kappa}{2}+2{\tilde
\kappa})A_{\alpha}. \label{ratefractional}$$
The main difference with the previous solutions are the equilibrium populations which now read $P_{+}^{eq}=p_{\uparrow}$, and $P_{-}^{eq}=p_{\downarrow}$. As a consequence of this fact, it is simple to realize that for $\gamma^{2}
\le A_{\epsilon}$, the exponential kernel produces a mapping that is not completely positive and not even positive. This follows by noting that for $P_{\pm}^{eq}\ne 1/2$, the population solutions Eq. (\[popular\]) can take negative values.
In Fig. (\[NegativeEntropy\]), for each kernel, we show the linear entropy behavior in the case of a zero temperature reservoir. As in the previous figure, as initial condition we use an eigenstate of the $x-$Pauli matrix. In the exponential case, when the stochastic interpretation is not possible the linear entropy takes negative values. Equivalently, this means that $\rho(t)$ is not positive definite.
![Linear entropy for the CTQRW defined by Eq. (\[termico\]) with $p_{\downarrow}=1$, $p_{\uparrow}=0$, and $\kappa=0.75$. Long dashed Line, Markovian kernel with $A_{1}=1~sec^{-1}$. Dashed line, fractional kernel with $\alpha=0.5$, $A_{\alpha}=1~sec^{-1/2}$. Full line, exponential kernel with $\gamma=4~sec^{-1}$, $A_{\epsilon}=4~sec^{-2}$. Dotted line, exponential kernel with $\gamma=1~sec^{-1}$, $A_{\epsilon}=1~sec^{-2}$.[]{data-label="NegativeEntropy"}](Figure4_Class.eps){width="8.cm"}
Dynamics in a Fock Space
------------------------
Here we will analyze the dynamics of a CTQRW in a system provided with a Fock space structure, as for example a quantum harmonic oscillator or a mode of an electromagnetic field. With $a^{\dagger}$ and $a$ we denote the corresponding creation and anhilation operators. This situation will allow us to recover the classical concept of continuous time random walks in the context of completely positive maps.
For the superoperator that defines the CTQRW, we assume the following form $${\cal E}[\rho]=D_{(\beta,\beta^{*})} \rho D_{(\beta,\beta^{*})}^{\dagger},$$ where $D_{(\beta,\beta^{*})}$ is the displacement operator $$D_{(\beta,\beta^{*})}=\exp[\beta a^{\dagger}-\beta^{*} a].$$ Furthermore, we assume that in each application of ${\cal E}[\bullet]$ the complex parameter $\beta$ is chosen with a probability distribution $P_{(\beta,\beta^{*})}$. The induced evolution can be easily analyzed by introducing the Wigner function $$W( \alpha,\alpha^{*},t) =2 Tr [\rho(t) D_{(\alpha,\alpha^{*})}
e^{i \pi a^{\dagger} a}
D_{(\alpha,\alpha^{*})}^{\dagger}],$$ whose evolution from Eqs. (\[master\])-(\[LA\]) then reads $$\begin{aligned}
\frac{d W(\alpha,\alpha^{*},t)}{dt}&=&\int_{0}^{t} d\tau K(t-\tau)
\bigg\{\int_{-\infty}^{\infty}
d\beta d\beta^{*} P_{(\beta,\beta^{*})} \label{eugenia}
\\ &&
W(\alpha-\beta,\alpha^{*}-\beta^{*},\tau)-
W(\alpha,\alpha^{*},\tau)\bigg\}.\nonumber\end{aligned}$$ By construction, the solution of this equation provides a completely positive map. Furthermore, we note that this equation can be interpreted as a “classical” continuous time random walk where the statistic of the “particle jumps” is given by $P_{(\beta,\beta^{*})}$ and the statistics of the elapsed time between the successive jumps is characterized through the waiting time distribution associated to the kernel $K(t)$. Thus, it is evident that this evolution is a classical one [@fischer], which implies that any quantum property can only be introduced through the initial conditions.
When all the moments of the distribution $P_{(\beta,\beta^{*})}$ are finite, i.e., $\langle \beta^{r} \beta^{* s}
\rangle \equiv \int_{-\infty}^{\infty}
d\beta d\beta^{*} P_{(\beta,\beta^{*})} \beta^{r} \beta^{* s}<\infty$ $\forall$ $r$, $s$, the evolution Eq. (\[eugenia\]) can be written in terms of a Kramers-Moyal expansion $$\frac{d W(\alpha,\alpha^{*},t)}{dt}=\int_{0}^{t} d\tau K(t-\tau)
{\cal L} W(\alpha,\alpha^{*},\tau), \label{kramer}$$ where the operator ${\cal L}$ is defined by $${\cal L}=\sum_{n=1}^{\infty} \frac{1}{n !} \int_{-\infty}^{\infty}
d\beta d\beta^{*} P_{(\beta,\beta^{*})} \left(\beta \frac{\partial}{\partial
\alpha} +\beta^{*}
\frac{\partial}{\partial \alpha^{*}}\right)^{n}.$$ These expressions follow after developing in Eq. (\[eugenia\]) the Wigner function $W(\alpha-\beta,\alpha^{*}-\beta^{*},\tau)$ around $W(\alpha,\alpha^{*},\tau)$. In this situation, it is also possible to get a close expression for the average excitation number $n(t)=Tr[\rho(t) a^{\dagger}a]$, which reads $$n(t) =n(0)+\langle |\beta|^{2} \rangle \int_{0}^{t}d\tau K(t-\tau)
\tau.\label{excitacion}$$ Here, we have assumed that the average displacements in the directions $(\beta, \beta^{*})$ are null, i.e., the first moments of the distribution $P_{(\beta,\beta^{*})} $ vanish.
Up to second order, the operator ${\cal L}$ reduces to a Hamiltonian term plus a classical Fokker Planck operator. By truncating the evolution up to this order, the CPC is not broken. This fact can be easily demonstrated by going back to the density matrix representation, where the Lindblad superoperator Eq. (\[LA\]) then reads ${\cal L}[\bullet] \approx {\cal L}_{H}[\bullet]+{\cal L}_{FP}[\bullet]$, with $${\cal L}_{H}\left[ \bullet \right] =[\langle \beta \rangle a^{\dagger}-\langle
\beta^{*} \rangle a,\bullet], \label{shift}$$ and $$\begin{aligned}
{\cal L}_{FP}\left[ \bullet \right] &=& \langle |\beta|^{2} \rangle([a^{\dagger
},\bullet
a]+[a^{\dagger }\bullet ,a]+[a,\bullet a^{\dagger
}]+[a\bullet ,a^{\dagger }]) \nonumber
\\ &+& \langle \beta^{2} \rangle([a^{\dagger
},\bullet
a^{\dagger}]+[a^{\dagger }\bullet ,a^{\dagger}]) \nonumber
\\ &+& \langle \beta^{* 2} \rangle([a,\bullet
a]+[a\bullet ,a]). \label{secondLindblad}\end{aligned}$$ The Lindblad terms proportional to $\langle |\beta|^{2} \rangle$ are equivalent to a reservoir at infinite temperature and the terms proportional to $\langle \beta^{2} \rangle$ and $\langle \beta^{*2} \rangle$ introduce a squeezing effect. On the other hand, it is possible to demonstrate that maintaining only a finite number of higher terms, the evolution for the density matrix can not be written in a Lindblad form and in consequence it is not completely positive. This fact agrees with the predictions of the classical Pawula theorem [@pawula] about Fokker Planck equations.
[*Subdiffusive Processes*]{}: By assuming the fractional kernel Eq. (\[kernelfrac\]), in the limit $A_{\alpha} \rightarrow \infty$, $\langle |\beta|^{2} \rangle \rightarrow 0$, with $A_{\alpha} \langle |\beta|^{2} \rangle= A_{\alpha}^{'}$, the previous second order approximation applies. In this situation, the evolution of the Wigner function is characterized by a subdiffusive process. In fact, the average excitation number reads $$n(t) =n(0)+\frac{2 A_{\alpha }^{'}}{\Gamma (1+\alpha )}
t^{\alpha}.$$ Note that in comparison with a Markovian Lindblad evolution, $\alpha=1$, here the increasing of the average excitations present a slower grow. On the other hand, the evolution of the Wigner function can be written as $$\frac{\partial W\left( x,t\right) }{\partial t}= A_{\alpha}^{'}\;_{0}D_{t}^{1-\alpha }
\frac{\partial ^{2}}{\partial x^{2}}W\left( x,t \right). \label{equis}$$ Here, $x$ is an arbitrary direction in the complex plane, and in order to simplify the expression, we have “traced out” the Wigner function over the perpendicular direction. We remark that this kind of fractional subdiffusive dynamics is allowed in the context of completely positive maps. This equation was extensively analyzed in the literature [@metzler], where it was found that the solution presents a non-Gaussian diffusion front. We notice that the relations between the exponents that characterize this behavior [@front] were found to be universal in the context of quasiperiodic and disordered systems [@zhong].
[*Long Jumps*]{}: When the moments of the distribution $P_{(\beta,\beta^{*})}$ are not defined, the dynamics must be analyzed in the Fourier domain, $(\alpha,\alpha^{*})\rightarrow (k,k^{*})$. Denoting with a hat symbol the Fourier transform, from Eq. (\[eugenia\]), we get $$\frac{d \hat{W}(k,k^{*},t)}{dt}=-\gamma(k,k^{*}) \int_{0}^{t} d\tau K(t-\tau)
\hat{W}(k,k^{*},\tau),$$ where the rates of the Fourier modes is given by $$\gamma(k,k^{*})=1-\hat{P}_{(k,k^{*})}.$$ For example, by assuming a Levy distribution [@metzler] $P_{(k,k^{*})}=exp[-\sigma^{\mu}|k|^{\mu}]$, with $0<\mu \le 2$, the evolution can be written as a series of infinite fractional derivatives with respect to the variables $(\alpha,\alpha^{*})$. Nevertheless, with the present formalism, it is not possible to check the CPC of any truncated evolution.
[*Quantum Random Walks*]{}: Finally we note that the concept of quantum random walks [@julia] used in the context of quantum computation and quantum information can be recovered as a particular case of our approach by using the generalized displacement operator $$D_{(\beta,\beta^{*},\theta,\phi)}=R(\theta,\phi) \;exp[\sigma_{z}(\beta
a^{\dagger}-\beta^{*}
a)],$$ and assuming that $P_{(\beta,\beta^{*})}=\delta_{(\beta-\beta_{0})}\delta_{(\beta^{*}-\beta_{0}^{*})}$, and $w(t)=\delta(t-T_{0})$. Here, $R(\theta,\phi)$ is an arbitrary rotation of an extra spin variable, $(\beta_{0},\beta_{0}^{*})$ is an arbitrary direction in the complex plane and $T_{0}$ is the discreet time step.
Generalized Intrinsic Decoherence Formalism
-------------------------------------------
The intrinsic decoherence formalism [@milburn; @moya] was introduced by Milburn as a phenomenological frame to the description of decoherence phenomema. Here, we will analyze and generalize this formalism by interpreting it as a CTQRW. First, we assume as a superoperator $${\cal E}_{\tau }[\bullet ]=e^{-iH\tau }\bullet e^{iH\tau},$$ where $H$ is an arbitrary Hamiltonian in a given Hilbert space, and $\tau$ is a random variable chosen with a density probability $P(\tau)$. From Eqs. (\[master\])-(\[LA\]), the average density matrix evolves as $$\begin{aligned}
\frac{d\rho (t)}{dt}=\int_{0}^{t} d\tau K(t-\tau)
\bigg\{&&\int_{-\infty }^{+\infty} d\tau^{'} P(\tau^{'} )
\\ && e^{-iH\tau^{'} }\rho (\tau)e^{iH\tau^{'} }-\rho (\tau) \bigg\}.\nonumber\end{aligned}$$ In the basis of eigenstates of the Hamiltonian $H$, $H
|n\rangle=\varepsilon_{n}|n\rangle$, the evolution of the matrix elements $\rho_{nm}=\langle n|\rho | m \rangle$ is given by $$\frac{d\rho _{nm}(t)}{dt}=-\gamma _{nm} \int_{0}^{t} d\tau K(t-\tau) \rho
_{nm}(\tau).$$ Here, the decaying rates $\gamma_{nm}$ read $$\gamma _{nm}=1-\hat{P}(\omega_{nm}),$$ where $\hat{P}(\omega)=\int_{-\infty}^{\infty} d\tau P(\tau)e^{-i \omega\tau}$, is the Fourier transform of the probability and $\omega_{nm}=\varepsilon_n-\varepsilon_m$ are the Bohr frequencies.
The original Milburn proposal is obtained by choosing $$w(t)=(1/\tau _{a})\exp (- t/\tau _{a}),\;\;\;\;\;P(\tau)=\delta (\tau -\tau
_{b}),\label{intrinseco}$$ which implies the density matrix evolution $$\frac{d\rho (t)}{dt}=\frac{1}{\tau _{a}}\left\{ e^{-iH\tau _{b}}\rho
(t)e^{iH\tau _{b}}-\rho (t)\right\}.$$ Thus, our CTQRW provides a natural non-Markovian generalization of this formalism. On the other hand, by choosing the exponential waiting distribution of Eq. (\[intrinseco\]), $P(\tau)=(t/\tau_{b})^{-1}\exp (-t/\tau_{b})$, and using the identity $\ln (1+ix)=\int_{0}^{\infty }ds(e^{-s}/s)(1-e^{isx})$, the rate results $\gamma _{nm}=\ln (1+\omega _{nm}\tau _{b})]/\tau _{a}$. This expression coincides with that obtained in the formalism of Ref. [@bonifaccio].
Summary and Conclusions
=======================
In this paper we have demonstrated that non-Markovian master equations that consist in a memory integral over a Lindblad structure can be considered as a valid tool in the description of open quantum system dynamics.
Our approach for the understanding of this kind of equations consists in a natural generalization of the classical concept of continuous time random walks to a quantum context. We have defined a CTQRW in terms of a set of random renewal events, each one consisting in the action of a superoperator over a density matrix. The selection of different statistics for the elapsed time between the successive applications of the superoperator allowed us to construct different classes of completely positive evolutions that lead to strong non-exponential decay of the density matrix elements. Remarkable examples are the telegraphic master equation, Eq. (\[telegrafo\]), which interpolates between a Gaussian short time dynamics and an asymptotic exponential decay, and the fractional master equation, Eq. (\[fraccionaria\]), which leads to stretched exponential and power law behaviors. On the other hand, in a Fock space the dynamics reduces to a classical one, which allowed us to demonstrate that fractional subdiffusive processes are consistent with a completely positive evolution.
Concerning the possibility of obtaining non-physical solutions from the Non-Markovian master equation Eq. (\[master\]), we have found a set of mathematical conditions on the kernel that guarantee the CPC of the solution map. As in classical Fokker-Planck equations, the set of conditions Eq. (\[condiciones\]) allows us to link each safe kernel with a corresponding waiting time distribution, which in the present case allows to associate to the master equation a CTQRW.
By analyzing the exponential kernel, related to the telegraphic master equation, we have demonstrated that when the kernel can not be associated with a waiting time distribution, the resulting solution map can be either non-physical, only positive, or even completely positive. This case demonstrates that no general conclusions can be obtained outside the regime where a stochastic interpretation is available. Furthermore, we have demonstrated that telegraphic master equations constructed with Lindblad superoperators that can be only introduced through an infinitesimal transformation, Eq. (\[infinitesimal\]), only admit a stochastic interpretation in the Markovian limit. In the case of the fractional kernel we have implemented a numerical simulation that confirms the equivalence between the non-Markovian fractional master equation and the corresponding CTQRW.
Finally we want to remark that from the understating achieved in this work, some interesting open question arise in a natural way, as for example a possible microscopic derivation of these non-Markovian master equations and the finding of alternative stochastic representation based in a continuous measurement theory. In fact, from the examples worked out in this paper, we conclude that the stochastic dynamics of a CTQRW can be thought in a rough way as the continuous measuring action of an environment over an open quantum system, where the scattering superoperator must be associated with the microscopic interaction between the system and the environment, and the statistics of the random times with the spectral properties of the bath.
I am grateful to H. Schomerus and D. Spehner for enlighting discussions.
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abstract: 'We apply the time-delayed Pyragas control scheme to the dissipative Dicke model via a modulation of the atom-field-coupling. The feedback creates an infinite sequence of non-equilibrium phases with fixed points and limit cycles in the primary superradiant regime. We analyse this Hopf bifurcation scenario as a function of delay time and feedback strength, and determine analytical conditions for the phase boundaries.'
author:
- Wassilij Kopylov
- Clive Emary
- Eckehard Schöll
- Tobias Brandes
title: 'Time-delayed feedback control of the Dicke-Hepp-Lieb superradiant quantum phase transition'
---
Interacting quantum systems with time-dependent Hamiltonians offer rich and exciting possibilities to study many-body physics beyond equilibrium conditions. There has be a recent surge in generating correlated non-equilibrium dynamics in a controlled way by changing the interaction parameters as a function of time, for example by periodically modulating the coupling constants, or by abruptly quenching them. Of particular interest then is the fate of coherent quantum dynamics and phase transitions in such scenarios, and indeed intriguing phenomena have been discussed, such as coherent control of tunneling in Bose-Einstein condensates [@Ligetal07], thermalization after quenches [@quenchexperiments], or dynamical and excited state quantum phase transitions [@Ecketal09; @CCI08].
In this Letter, we introduce another and conceptually very different option for driving quantum systems out of equilibrium, i.e., by modulating interaction parameters via a measurement-based feedback loop. The time-delayed Pyragas control scheme [@pyragas1992continuous] that we propose here has been successfully employed in a classical context over the past twenty years, for example, as a tool to stabilize certain orbits in chaotic systems or networks [@schoell_handbook_chaos_control; @stabil_periodorbin-choe; @FLU10b; @SCH13]. Its key idea is to feed back the difference between two signals of the same observable at different times, such that a stabilization occurs when the delay time matches an intrinsic period of the dynamical system.
Our key idea is to generate new non-equilibrium phases via Pyragas control of the interaction between the single bosonic cavity mode and the collection of quantum two-level systems [@Dicke_Modell] in Dicke-Hepp-Lieb superradiance. The superradiant transition without control, which has been observed only recently in cold atoms within a photonic cavity [@Baumann-Dicke_qpt; @Esslinger-Dicke_qpt-1; @Baumann-Dicke_qpt-2; @Nagy-dicke_and_bose_einstein], has an underlying semi-classical bifurcation, which makes it an ideal candidate to study feedback at the boundary between non-linear (classical) dynamics and quantum many-body systems [@Dicke_Chaos_and_qpt].
Control loops of the Dicke model have been studied in the past, for example in the form of periodical modulations of the atom-field-coupling constants [@Dicke-nonequilibrium_qpt-bastidas; @Baumann-Dicke_qpt-2], the level splitting modulation [@Phot_production_from_Vak_to_SR-Vacanti; @Extracavity_radiation_from_single_qbit-Liberato], or as Pyragas-feedback of the cavity mode alone [@Dicke_Rapid_convergence_time_delay-Grimsmo]. In our model, we condition the effective coupling strength between the cavity and the atoms of the Dicke system - in the experiment just proportional to the laser intensity [@Baumann-Dicke_qpt] - on a difference of photon numbers emitted from the cavity at different times. We use a mean field approach and linear stability analysis in order to show that closed loop control dramatically affects the states in the primary superradiant regime, creating a new phase with an infinite sequence of Hopf bifurcations between stable fixed points and limit cycles. We also derive analytical results in the form of a single transcendental equation that determined the boundaries between the different zones in the phase diagrams.
*Open Dicke model with time delayed feedback. —* The Hamiltonian of the Dicke model
$$\label{eq1:Dicke-Hamiltonian}
H = \omega \hat{a}^\dag \hat{a} + \omega_0 \hat{J}_z + \frac{g(t)}{\sqrt{2j}}(\hat{a}^\dag + \hat{a})(\hat{J}_+ + \hat{J}_-)$$
describes the interaction between a single bosonic mode (with frequency $\omega$ and annihilation operator $\hat{a}$) and $N$ two level systems (with level splitting $\omega_0$ and collective angular momentum operators $\hat{J}_{z,\pm}$) with total angular momentum $j=N/2$ [@Dicke_Chaos_and_qpt]. Besides we set the length of the pseudo-spin $j$ to its maximum value. We assume an interaction between the bosonic mode and the collective angular momentum with a time-dependent coupling $g(t)$ that is modulated by a time-delayed feedback loop. Among various models for $g(t)$, the Pyragas form [@pyragas1992continuous] $$\label{eq1:g_modulation_noninvasiv}
g(t) = g_0 + \lambda \left({\left< \hat{a}^\dag \hat{a} \right>}(t - \tau) - {\left< \hat{a}^\dag \hat{a} \right>}(t)\right)$$ with the time-delayed feedback of the boson number at two different times $t$ and $t-\tau$ and feedback strength $\lambda$ turns out to lead to the richest phase diagrams. In the pioneering experiments for the Dicke-Hepp-Lieb phase transition in open photonic cavities [@Baumann-Dicke_qpt], the form Eq. (\[eq1:g\_modulation\_noninvasiv\]) would correspond to measured, average photon fluxes (proportional to the mean cavity photon occupation number [@Oeztop-excitation_of_opticaly_driven_atomic_condensate; @Kopylov_Counting-statistics-Dicke]) coupled back to a pump laser. Apart from the Pyragas delay form, this scheme is in fact close to the original feedback loops used for modulating the photon counting statistics in lasers [@yamamoto_statistic_feedback_and_laser].
We note that by using mean (expectation) values in Eq. (\[eq1:g\_modulation\_noninvasiv\]) instead of operators (and additional noise terms in a stochastic master equation [@Wiseman_Milburn]) for the boson occupations, we already assume a mean field description that we expect to hold for $N\to \infty$ and that we formalize in the following.
![\[figure1\] Phase diagram for fixed time delay $\tau$ and feedback strength $\lambda$. The dashed (orange) line separates the normal from the superradiant phase. The superradiant regime is split by time delayed control into zones with stable fixed points (F) and limit cycles (L) with boundaries (black curves) determined from Eq. . Color encodes the largest real part of the eigenvalue. Parameters: $\tau = 20$ $\mu$s, $\lambda = 5$ MHz and $\omega_0 = 0.05$ MHz, $\kappa = 8.1$ MHz [@Bhaseen_dynamics_of_nonequilibrium_dicke_models; @Baumann-Dicke_qpt]. ](fig1){width="\columnwidth"}
*Semiclassical equations. —* In analogy with semiclassical laser theory, phase transitions in the Dicke model for $N\to \infty$ are well described by mean-field equations for (factorized) operator expectation values [@Bonifacio; @Bhaseen_dynamics_of_nonequilibrium_dicke_models; @Kopylov_Counting-statistics-Dicke] which we denoted by the corresponding symbols without hat. Splitting $a$ and $J_\pm$ into real and imaginary parts, $a = x + i y$ and $J_\pm = J_x \pm i J_y$, these equations read $$\begin{aligned}
\label{eq1:open_dicke_semiclass_eq_alternativ}
\dot{x} &= - \kappa x + \omega y ,\quad
\dot{y} = - \kappa y - \omega x - 2 \frac{g(t)}{\sqrt{2j}} J_x , \\
\dot{J}_x&= - \omega_0 J_y , \quad
\dot{J}_y = \omega_0 J_x - 4 \frac{g(t)}{\sqrt{2j}} \cdot x \cdot J_z ,\notag\\
\dot{J}_z &= 4 \frac{g(t)}{\sqrt{2j}} \cdot x \cdot J_y , \notag\end{aligned}$$ where $\kappa$ is decay rate of the bosonic mode and where the coupling $g(t)$ takes the form $ g(t) = g_0 + \lambda {\left( x_\tau^2 - x^2 + y_\tau^2 - y^2 \right)}$ with the shorthand $f_\tau \equiv f(t - \tau)$. Note that the angular momentum is a conserved quantity even for time dependent $g(t)$, and the time development therefore takes place on the surface of a Bloch sphere with the radius $N/2$. For zero time-delay $\tau = 0$, i.e. without feedback, the phase diagram is well known [@Bhaseen_dynamics_of_nonequilibrium_dicke_models]. For $g < g_c\equiv \sqrt{{\omega_0 (\kappa^2 + \omega^2)}/{4 \omega}}$, a stable normal phase solution corresponds to fixed point $J_x^0 = J_y^0 = x^0=y^0 = 0, J_z^0 = - {N}/{2}$, whereas $ J_x^0 = \pm \sqrt{\frac{N^2}{4} - {J^0_z}^2}$, $J_y^0 = 0$, $J_z^0 = \frac{-N\omega_0 (\kappa^2 + \omega^2)}{8 g_0^2 \omega}$ with $ x^0 = -J_x^0\frac{2 {g_0}\omega}{\sqrt{N}({\kappa^2 + \omega^2 })}$ , $y^0 = \frac{\kappa x_0}{\omega}
$ corresponds to the stable superradiant phase that exists only if $g \geq g_c$.
*Stability and feedback. —* To find out how the time-delayed feedback affects the stability of the system, we linearize Eqs. around the fixed points (these do not depend upon $\tau$ since the feedback Eq. (\[eq1:g\_modulation\_noninvasiv\]) vanishes in the steady state). Using the usual procedure [@HOE05] the linearized equations read $\delta\mathbf{v}\,'(t)= \mathbf{B} \cdot \delta\mathbf{v} (t) + \mathbf{A} \cdot \delta\mathbf{v} (t - \tau)$ with $\delta\mathbf{v} = (\delta J_x, \delta J_y, \delta x,\delta y)^T$ describing the deviation from the fixed point and
$$\begin{aligned}
\mathbf{A} &= \left(
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & -8 J_z^0 {x^0}^2 \lambda & -8 J_z^0 {x^0} {y^0} \lambda \\
0 & 0 & 0 & 0 \\
0 & 0 & -4 J_x^0 {x^0} \lambda & -4 J_x^0 {y^0} \lambda \\
\end{array}
\right) ,
\quad
\mathbf{B} &= \left(
\begin{array}{cccc}
0 & -\omega_0 & 0 & 0 \\
\frac{4 g_0 J_x^0 {x^0}}{J_z^0}+\omega_0 & 0 & 8 J_z^0 {x^0}^2 \lambda -4 g_0 J_z^0 & 8 J_z^0 {x^0} {y^0} \lambda \\
0 & 0 & -\kappa & \omega \\
-2 g_0 & 0 & 4 J_x^0 {x^0} \lambda -\omega & 4 J_x^0 {y^0} \lambda -\kappa \\
\end{array}
\right).\end{aligned}$$
![\[figure2\] Time evolution of mode occupation $a^* a$ and angular momentum component $J_x$ corresponding to stable fixed point P2 (left) and stable limit cycle P1 (right) in the phase diagram Fig. 1. ](fig2a "fig:"){width="0.5\columnwidth"}![\[figure2\] Time evolution of mode occupation $a^* a$ and angular momentum component $J_x$ corresponding to stable fixed point P2 (left) and stable limit cycle P1 (right) in the phase diagram Fig. 1. ](fig2b "fig:"){width="0.55\columnwidth"}
Note, that the $J_z$ component is determined by the conservation of angular momentum. Using the ansatz $\delta\mathbf{v} = \delta\mathbf{v} \, e^{\Lambda t}$, we obtain the characteristic equation $$\label{eq1:stabilitaetsbedingung}
\det{\left( \Lambda \mathbf{1} - \mathbf{B} - \mathbf{A} e^{-\Lambda \tau} \right)} = 0 \,.$$ For $\tau \neq 0$ this transcendental equation has an infinite set of solutions for the eigenvalues $\Lambda \in \mathbb{C}$. The fixed point $\mathbf{v}^{\,0}$ is stable if the real parts of all solutions $\Lambda$ are negative, in which case the fluctuations decay to zero for $t \to \infty$.
*Phase diagrams. —* We obtain the phase diagram of our model in the $\omega$–$g_0$–plane (Fig. 1) from the numerical solution of Eq. for $\Lambda$. First, in the left part of the phase diagram (for $g_0\le g_c$) we recover the usual normal phase, where the boson occupation is zero and as a consequence, the feedback scheme Eq. (\[eq1:g\_modulation\_noninvasiv\]) remains without effect. In contrast, for $g_0> g_c$ and positive $\tau$, the superradiant phase splits up into an infinite sequence of tongue-like areas that alternate between zones with stable, superradiant fixed points (F), and [*limit cycles*]{} (L) with periodically oscillating system observables. We will devote the rest of this Letter to analysing and interpreting this rather surprising effect.
Fig. 2 displays the two markedly different types of time evolution in the superradiant regime: in the fixed point zones (F), the only effect of the Pyragas scheme is to speed up the convergence of the spin-components and the mean boson occupation $a^*a$ towards their fixed point values. This has to be contrasted with the limit-cycle zones (L), where the fixed point is unstable, and the observables end up oscillating with a single frequency.
![\[figure3\] Phase diagram with sequence of stable fixed point (F) and limit cycle (L) zones in the $\lambda$ (coupling strength) vs. $\tau$ (delay time) plane. The black lines represent zone boundaries derived from the single trancendental Eq. . Dashed lines indicate cross-sections shown in Fig. 4. Color represents the largest real part of the eigenvalues. Parameters:$\omega=10$ MHz, $g_0 = 1.5$ MHz, $\omega_0 = 0.05$ MHz, $\kappa = 8.1$ MHz. ](fig3){width="\columnwidth"}
![\[figure4\] Bifurcation scenario for limit cycle amplitudes (upper) and periods $T$ (lower) of $J_z$ as a function of delay time $\tau$ along the cross-sections (dashed lines at fixed $\lambda$) in Fig. 3. The filled symbols in the upper part describe the continuation of the limit cycle for initial value above the unstable limit cycle (unfilled symbols). The inset sketches the appearing saddle-node bifurcation of limit cycles and the dotted arrows show the direction of the phase flow for fixed tau. ](fig4){width="\columnwidth"}
*Analysis of zone boundaries. —* We obtain a simplified transcendental stability equation from Eq. in the limit of very small level splitting $\omega_0\ll \omega,g_0$, which describes the ultra-strong coupling limit of the Dicke model [@ABEB12] and corresponds to a feedback-controlled displaced harmonic oscillator. In this case, the angular momenta deviations $\delta J_{x,y}$ decouple from the field deviations $\delta x,\delta y$ and describe periodic oscillations with the frequency $$\label{eq1:frequency}
\Omega = \frac{4 g_0^2 \omega}{N(\kappa^2+\omega^2)}.$$ Using Eq. we derive an equation for $\delta \ddot{x}$ just by inserting the equations into each other. As a result, we obtain $$\label{eq1:stability_end_condition_w0_is_0}
\tan(\Omega \tau) = \frac{C_2 C_3 \pm C_1 \sqrt{-C_1^2 + C_2^2 + C_3^2}}{C_1^2-C_2^2},$$ with $C_1 = 2 \kappa \Omega + \frac{\lambda \kappa}{g_0 \omega} \Omega^2$ , $C_2 = 4 g_0 \lambda$, $C_3 = \frac{\lambda \kappa}{g_0 \omega} \Omega^2$, which leads to the roots of the Eq. with a vanishing real part i.e., $\Lambda = \pm i \Omega$. Parameter configurations satisfying this equation mark the boundary between stable (F) and unstable (L) fixed points which is included in Fig. 1 and matches the numerically determined boundaries very well.
This analysis also allows us to elucidate the role of the delay time $\tau$ in the control scheme: to obtain real-valued results for the time delay $\tau$, the root in Eq. has to be positive. This condition is only satisfied if the feedback coupling $\lambda$ is larger than some critical value $\lambda_l$, which we determine from the vanishing of the root in Eq. . We corroborate these findings by plotting the largest real part of the eigenvalue numerically determined from Eq. in the $(\lambda,\tau)$-plane for fixed $\omega$ and $g_0$ values, see Fig. 3. We recognize tongue-like zones switching between stable fixed-point and limit cycle (L) zones upon modification of the time delay $\tau$, and furthermore the existence of a critical feedback strength $\lambda_l$ for entering in the (L) zones.
*Limit cycle properties. —* Finally, we discuss the delay time $\tau$ and its role as a control parameter. The alternations between (F) and (L) zones in the superradiant phase in fact constitute an infinite sequence of super- and subcritical Hopf bifurcations of the stationary state generating stable and unstable limit cycles, respectively. Solving the equations of motion for parameter values along the dashed lines in Fig. 3, we find that the amplitude and period $T$ of the limit cycles depend upon tau, as depicted in Fig. 4 for the $J_z$ amplitude. First, we recognize that for initial conditions close to the fixed point both the amplitude and the period show the same Hopf bifurcation scenario (connected lines), with maxima, which mark the end of the L-zone, appearing for certain values of $\tau$. As a particularly striking feature, we observe a drastic collapse of the limit cycle (vertical lines) for values of $\lambda > \lambda_l$ and the birth of an unstable limit cycle (disconnected unfilled symbols, shown only in the upper part of Fig. 4) when the time delay $\tau$ reaches the end of the (L) zone. Our numerics show, however, that this collapse occurs as a jump discontinuity. Furthermore, a stable limit cycle still exists behind the (L) zone (disconnected filled symbols) as a continuation of the previous one but, because of bistability with the stable fixed point, it can only be reached if the initial amplitude lies above the amplitude of the unstable limit cycle, which marks the boundary between the basins of attraction of the fixed point and limit cycle attractors. The branches of the stable and the unstable limit cycles merge in a saddle-node-bifurcation (arrows in Fig. 4). The inset shows schematically this bifurcation, the dotted arrows point to the stable solution (black solid line) the system will take for different initial conditions. As a consequence, the mean number of photons emitted from the system oscillates with a fixed frequency that can be externally controlled. In addition, the theoretical prediction for the oscillating frequency $\Omega$ from the linear stability analysis of the fixed point Eq. matches well with the damped oscillation period in the F-region, see Fig. 4.
We emphasize that the time dependence of $g(t\to \infty)$, Eq. , does not disappear in the L-regions in contrast to the F-regions, leading to the phase diagram discussed above. Our feedback scheme here switches between non-invasive to invasive behavior by crossing the boundaries within the phase diagram. Our results also demonstrate that the Pyragas form in Eq. is essential to create a new stable phase. In contrast, for the direct feedback scheme $ g(t) = g_0 + \lambda {\left< \hat{a}^\dag \hat{a} \right>}(t - \tau)$, depending on parameter values the occupation of the optical mode diverges and the control does not work well, or the time delay does not seriously modify the phase diagram at all (not shown here).
Finally, we also checked that the (experimentally less practical) Pyragas feedback for the angular momentum (instead of the photonic feedback) also leads to the creation of a limit cycle phase in the super radiant regime, but can also not influence the stability of the normal phase.\
We expect that our feedback scheme can be implemented whenever semiclassical equations of motion provide an adequate description for the quantum bifurcation type phase transitions that govern models with collective degrees of freedom, such as the Dicke or the Lipkin-Meshkov-Glick model [@LMG-lipkin1965validity]. An open and challenging problem is the implementation of time-delayed feedback control for quantum critical systems beyond the mean-field level.
*Acknowledgments. —* We thank H. Aoki, J. Lehnert, and N. Tsuji for useful discussions. The authors gratefully acknowledge financial support from the DAAD and DFG Grants BR $1528/7-1$, $1528/8-2$, $1528/9-1$, SFB $910$, and GRK $1558$.
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abstract: |
In this paper we introduce a new mathematical tool to solve fractional equations representing models of fractional systems : The Ultradistributions.\
Ultradistributions permit us to unify the notion of integral and derivative in one only operation. Several examples of application of the results obtained are given.
PACS: 03.65.-w, 03.65.Bz, 03.65.Ca, 03.65.Db.
author:
- |
C.M.Grunfeld and M.C.Rocca\
Departamento de Física, Fac. de Ciencias Exactas,\
Universidad Nacional de La Plata.\
C.C. 67 (1900) La Plata. Argentina.
date: 'March 23, 2009'
title: '[**Modelling and Analysis of Fractional Order Systems using Ultradistributions** ]{} [^1]'
---
Introduction
============
The use of fractional calculus for modelling physical systems has been considered in many works. See for example [@tq1; @tq2; @tq3]. We can find also works dealing with the application of this mathematical tool in control theory [@tq4; @tq5; @tq6; @tq7]..
Moreover, there are many physical systems that can be described by means of a fractional calculus. Some examples are: chaos [@tq8], long electric lines [@tq9], electrochemical process [@tq10] and dielectric polarization [@tq11].
In this paper we want to introduce a new mathematical framework to solve fractional equations representing models of fractional systems which was not treated in none of the previous works: The Ultradistributions.
The paper is organized as follow: in section 2 we introduce definition of fractional derivation and integration. In section 3 we give some examples of application of the formulae of section 2 using the Fourier Transform and the one-side Laplace Transform. In section 3 we present a circuital application. Finally in section 4 we discuss the results obtained in sections 1,2 and 3.
Fractional Calculus
===================
The purpose of this sections is to introduce definition of fractional derivation and integration given in ref. [@tp1]. This definition unifies the notion of integral and derivative in one only operation. Let $\hat{f}(x)$ a distribution of exponential type and $F(\Omega)$ the complex Fourier transformed Tempered Ultradistribution. Then: $$\label{ep2.1}
F(\Omega)=U[\Im(\Omega)]\int\limits_0^{\infty}\hat{f}(x) e^{j\Omega x}\;dx-
U[-\Im(\Omega)]\int\limits_{-\infty}^0\hat{f}(x) e^{j\Omega x}\;dx$$ ($U(x)$ is the Heaviside step function) and $$\label{ep2.2}
\hat{f}(x)=\frac {1} {2\pi}\oint\limits_{\Gamma}F(\Omega)
e^{-j\Omega x}\;d\Omega$$ where the contour $\Gamma$ surround all singularities of $F(\Omega)$ and runs parallel to real axis from $-\infty$ to $\infty$ above the real axis and from $\infty$ to $-\infty$ below the real axis. According to [@tp1] the fractional derivative of $\hat{f}(x)$ is given by $$\label{ep2.3}
\frac {d^{\lambda}\hat{f}(x)} {dx^{\lambda}}=\frac {1} {2\pi}\oint\limits_{\Gamma}
(-j\Omega)^{\lambda}
F(\Omega)
e^{-j\Omega x}\;d\Omega+
\oint\limits_{\Gamma}(-j\Omega)^{\lambda}a(\Omega)
e^{-j\Omega x}\;d\Omega$$ Where $a(\Omega)$ is entire analytic and rapidly decreasing. If $\lambda=-1$, $d^{\lambda}/dx^{\lambda}$ is the inverse of the derivative (an integration). In this case the second term of the right side of (\[ep2.3\]) gives a primitive of $\hat{f}(x)$. Using Cauchy’s theorem the additional term is $$\label{ep2.4}
\oint \frac {a(\Omega)} {\Omega}e^{-j\Omega x} d\Omega=
2\pi a(0)$$ Of course, an integration should give a primitive plus an arbitrary constant. Analogously when $\lambda=-2$ (a double iterated integration) we have $$\label{ep2.5}
\oint \frac {a(\Omega)} {{\Omega}^2}e^{-j\Omega x} d\Omega=
\gamma+\delta x$$ where $\gamma$ and $\delta$ are arbitrary constants. With the change of variables $s=-j\Omega$ formulae (\[ep2.1\]) and (\[ep2.2\]) can be writen as: $$\label{ep2.6}
G(s)=U[\Re(s)]\int\limits_0^{\infty}\hat{f}(x) e^{-sx}\;dx-
U[-\Re(s)]\int\limits_{-\infty}^0\hat{f}(x) e^{-sx}\;dx$$ and $$\label{ep2.7}
\hat{f}(x)=\frac {1} {2\pi i}\oint_{\Gamma} G(s) e^{sx}\;ds$$ where the contour $\Gamma$ surround all singularities of $G(S)$ and runs parallel to imaginary axis from $-j\infty$ to $j\infty$ to the right of the imaginary axis and from $j\infty$ to $-j\infty$ to the left of the imaginary axis. Formula (\[ep2.6\]) represents the two-sided Lapnace Transform. The fractional derivative is now: $$\label{ep2.8}
\frac {d^{\lambda}\hat{f}(x)} {dx^{\lambda}}=\frac {1} {2\pi i}\oint\limits_{\Gamma}
s^{\lambda}G(s) e^{s x}\;ds +
\oint\limits_{\Gamma}s^{\lambda}a(s)
e^{sx}\;ds$$ For the one-side Laplace Transform we have $$\label{ep2.9}
G(s)=U[\Re(s)]\int\limits_0^{\infty}\hat{f}(x) e^{-sx}\;dx$$ $$\label{ep2.10}
\hat{f}(x)=\frac {1} {2\pi j} \int\limits_{a-j\infty}^{a+j\infty}
G(s) e^{sx}\;ds$$ and for the fractional derivative: $$\label{ep2.11}
\frac {d^{\lambda}\hat{f}(x)} {dx^{\lambda}}=
\frac {1} {2\pi j} \int\limits_{a-j\infty}^{a+j\infty}
s^{\lambda}
G(s) e^{sx}\;ds$$
Examples
========
In this section we give some examples of the application of formulae of the precedent section. At first using the Fourier Transform and at second place using the one-side Laplace Transform.
The Fourier Transform {#the-fourier-transform .unnumbered}
---------------------
Let $U(x)$ be the Heaviside step function. $$\label{ep3.1}
\hat{f}(x)=U(x)\;\;\;;\;\;\;
F(\Omega)=U[\Im(\Omega)]\int\limits_0^{\infty}e^{-j\Omega x}\;dx=
\frac {jU[\Im(\Omega)]} {\Omega}$$ The fractional derivative is: $$\frac {d^{\lambda}U(x)} {dx^{\lambda}}=
\frac {je^{-\frac {j\pi\lambda} {2}}} {2\pi}\oint\limits_{\Gamma}
U[\Im(\Omega)]{\Omega}^{\lambda-1}e^{-j\Omega x}\;d\Omega+
\oint\limits_{\Gamma} {\Omega}^{\lambda}
a(\Omega)e^{-j\Omega x}\;d\Omega=$$ $$\label{ep3.2}
\frac {je^{\frac {-j\pi\lambda} {2}}} {2\pi}
\int\limits_{-\infty}^{\infty} (\omega+j0)^{\lambda-1}
e^{-j\omega x}\;d\omega+
\oint\limits_{\Gamma} {\Omega}^{\lambda}
a(\Omega)e^{-j\Omega x}\;d\Omega$$ With the use of the result (see ref.[@tp7]) $$\label{ep3.3}
\int\limits_{-\infty}^{\infty} (\omega+j0)^{\lambda-1}
e^{-j\omega x}\;d\omega=-2\pi j\frac {e^{\frac {i\pi\lambda} {2}}}
{\Gamma(1-\lambda)} x_+^{-\lambda}$$ we obtain: $$\label{ep3.4}
\frac {d^{\lambda}U(x)} {dx^{\lambda}}=
\frac {x_+^{-\lambda}} {\Gamma(1-\lambda)}+
\oint\limits_{\Gamma} {\Omega}^{\lambda}
a(\Omega)e^{-j\Omega x}\;d\Omega$$ When $\lambda=n$ $$\label{ep3.5}
\left.\frac {x_+^{-\lambda}} {\Gamma(1-\lambda)}\right|_{\lambda=n}=
\delta^{(n-1)}(x)$$ $$\label{ep3.6}
\oint\limits_{\Gamma} {\Omega}^n
a(\Omega)e^{-j\Omega x}\;d\Omega=0$$ and we have the ordinary derivative: $$\label{ep3.7}
\frac {d^n U(x)} {dx^n}=\delta^{(n-1)}(x)$$ When $\lambda=-n$ $$\label{ep3.8}
\frac {d^{-n}U(x)} {dx^{-n}}=\frac {x_+^n} {n!}+a_0+a_1x+
a_2x^2+\cdot\cdot\cdot+a_{n-1}x^{n-1}$$ which is a n-times iterated integral.
Let $\delta(x)$ the Dirac’s delta distribution. For it we have: $$\label{ep3.9}
\hat{f}(x)=\delta(x)\;\;\;;\;\;\;
F(\Omega)=\frac {Sgn[\Im(\Omega)]} {2}$$ The fractional derivative is: $$\label{ep3.10}
\frac {d^{\lambda}\delta(x)} {dx^{\lambda}}=
\frac {x_+^{-\lambda-1}} {\Gamma(-\lambda)}+
\oint\limits_{\Gamma} {\Omega}^{\lambda}
a(\Omega)e^{-j\Omega x}\;d\Omega$$ When $\lambda=n$: $$\label{ep3.11}
\frac {d^n \delta(x)} {dx^n}=\delta^{(n)}(x)$$ and when $\lambda=-n$: $$\label{ep3.12}
\frac {d^{-n}\delta(x)} {dx^{-n}}=\frac {x_+^{n-1}} {(n-1)!}+a_0+a_1x+
a_2x^2+\cdot\cdot\cdot+a_{n-1}x^{n-1}$$ Let us consider now the fractional derivative of $e^{jbx}$ $$\label{ep3.13}
\hat{f}(x)=e^{jbx}\;\;\;;\;\;\;F(\Omega)=\frac {j} {\Omega+b}$$ We have: $$\label{ep3.14}
\frac {d^{\lambda}e^{jbx}} {dx^{\lambda}}=
\frac {j} {2\pi}\oint\limits_{\Gamma}
\frac {(-j\Omega)^{\lambda}e^{-j\Omega x}} {\Omega+b}\;d\Omega+
\oint\limits_{\Gamma}{\Omega}^{\lambda}a(\Omega)e^{-j\Omega x}\;
d\Omega=$$ $$\frac {ie^{\frac {-i\pi\lambda} {2}}} {2\pi}
\int\limits_{-\infty}^{\infty}\frac {(\omega+j0)^{\lambda}}
{\omega+b+j0} e^{-j\omega x}d\omega-
\frac {ie^{\frac {-i\pi\lambda} {2}}} {2\pi}
\int\limits_{-\infty}^{\infty}\frac {(\omega-j0)^{\lambda}}
{\omega+b-j0} e^{-j\omega x}d\omega+$$ $$\label{ep3.15}
\oint\limits_{\Gamma}{\Omega}^{\lambda}a(\Omega)e^{-j\Omega x}\;
d\Omega$$ From ref.[@tp10] we obtain: $$\int\limits_{-\infty}^{\infty}\frac {(x+\gamma)^{\lambda}}
{x+\beta}e^{-ipx}dx=$$ $$\label{ep3.16}
2\pi U(p)
\frac {e^{\frac {-j\pi} {2}(1-\lambda)}}
{\Gamma(1-\lambda)}p^{-\lambda}
e^{i\beta p}\phi[-\lambda,1-\lambda,j(\gamma-\beta)p]$$ where $\phi$ is the confluent hypergeometric function. Thus the fractional derivative is: $$\label{ep3.17}
\frac {d^{\lambda}e^{jbx}} {dx^{\lambda}}=
\frac {(x+j0)^{-\lambda}} {\Gamma(1-\lambda)}
\phi(1,1-\lambda,jbx)+
\oint\limits_{\Gamma}{\Omega}^{\lambda}a(\Omega)e^{-j\Omega x}\;
d\Omega$$ With the use of equality: $$\label{ep3.18}
\phi(1,1-\lambda,jbx)=(jbx)^{\lambda}e^{jbx}
\left[\Gamma(1-\lambda)+\lambda\Gamma(-\lambda,jbx)\right]$$ where $\Gamma(z_1,z_2)$ is the incomplete gamma function, (\[ep3.17\]) takes the form: $$\frac {d^{\lambda}e^{jbx}} {dx^{\lambda}}=
(jb)^{\lambda}e^{jbx}\left[1+\frac {\lambda} {\Gamma(1-\lambda)}
\Gamma(-\lambda,jbx)\right]+$$ $$\label{ep3.19}
\oint\limits_{\Gamma}{\Omega}^{\lambda}a(\Omega)e^{-j\Omega x}\;
d\Omega$$ When $\lambda=n$ $$\label{ep3.20}
\frac {d^ne^{jbx}} {dx^n}=(jb)^ne^{jbx}$$ and when $\lambda=-n$: $$\label{ep3.21}
\frac {d^{-n}e^{jbx}} {dx^{-n}}=(jb)^{-n}e^{jbx}+a_0+a_1x+
\cdot\cdot\cdot+a_{n-1}x^{n-1}$$
The Laplace Transform {#the-laplace-transform .unnumbered}
---------------------
If we use the one-side Laplace transform to evaluate the fractional derivative of $U(x)$,then: $$\label{ep3.22}
\hat{f}(x)=U(x)\;\;\;;\;\;\;G(s)=U[\Re(s)]
\int\limits_0^{\infty}e^{-sx}dx=
\frac {U[\Re(s)]} {s}$$ and as a consequence: $$\label{ep3.23}
\frac {d^{\lambda}U(x)} {dx^{\lambda}}=
\frac {1} {2\pi j} \int\limits_{a-j\infty}^{a+j\infty}
U[\Re(s)]s^{\lambda-1}e^{sx}\;ds=$$ $$\label{ep3.24}
\frac {e^{-ax}} {2\pi}\int\limits_{-\infty}^{\infty}
\frac {e^{jsx}} {(a+js)^{1-\lambda}}ds=
\frac {x_+^{-\lambda}} {\Gamma(1-\lambda)}$$ $$\label{ep3.25}
\frac {d^{\lambda}U(x)} {dx^{\lambda}}=
\frac {x_+^{-\lambda}} {\Gamma(1-\lambda)}$$ When $\lambda=n$ we obtain $$\label{ep3.26}
\frac {d^n U(x)} {dx^n}=\delta^{(n-1)}(x)$$ which coincides with (\[ep3.7\]). When $\lambda=-n$ the result is: $$\label{ep3.27}
\frac {d^{-n}U(x)} {dx^{-n}}=\frac {x_+^n} {n!}$$ In a analog way we obtain for Dirac’s delta distribution: $$\label{ep3.28}
\frac {d^{\lambda}\delta(x)} {dx^{\lambda}}=
\frac {x_+^{-\lambda-1}} {\Gamma(-\lambda)}$$ $$\label{ep3.29}
\frac {d^n \delta(x)} {dx^n}=\delta^{(n)}(x)$$ $$\label{ep3.30}
\frac {d^{-n}\delta(x)} {dx^{-n}}=\frac {x_+^{n-1}} {(n-1)!}$$ Finally we consuder the fractional derivative of $e^{jbx}$: $$\label{ep3.31}
\hat{f}(x)=U(x)e^{jbx}\;\;\;;\;\;\;G(s)=\frac {U[\Re(s)]} {s-ib}$$ According to (\[ep2.11\]): $$\label{ep3.32}
\frac {d^{\lambda}U(x)e^{jbx}} {dx^{\lambda}}=\frac {1} {2\pi j}
\int\limits_{a-j\infty}^{a+j\infty}\frac {U[\Re(s)]} {s-jb}
s^{\lambda}e^{sx}ds=$$ $$\label{ep3.33}
-\frac {e^{-\frac {j\pi\lambda} {2}}} {2\pi j}\int\limits_{-\infty}^{\infty}
\frac {(s+j0)^{\lambda}} {s+b+j0}
e^{-jsx} ds$$ And thus: $$\label{ep3.34}
\frac {d^{\lambda}U(x)e^{jbx}} {dx^{\lambda}}=
\frac {U(x)x^{-\lambda}} {\Gamma(1-\lambda)}
\phi(1,1-\lambda,jbx)$$ Using (\[ep3.18\]), (\[ep3.34\]) transforms into: $$\label{ep3.35}
\frac {d^{\lambda}U(x)e^{jbx}} {dx^{\lambda}}=
(jb)^{\lambda}U(x)e^{jbx}\left[1+\frac {\lambda} {\Gamma(1-\lambda)}
\Gamma(-\lambda,jbx)\right]$$ When $\lambda=n$: $$\label{ep3.36}
\frac {d^ne^{jbx}} {dx^n}=(jb)^nU(x)e^{jbx}$$ and when $\lambda=-n$: $$\label{ep3.37}
\frac {d^{-n}e^{jbx}} {dx^{-n}}=(jb)^{-n}U(x)e^{jbx}$$
Circuital Application
=====================
As circuital application we consider a semi-infinite cable with a voltage $V=V_0e^{j\omega t}$ applied at one end. We use first the Fourier transform and then the Laplace transform for see the diferences between both treatments.
The Fourier Transform {#the-fourier-transform-1 .unnumbered}
---------------------
We should solve the system: $$\label{ep4.1}
\begin{cases}
\frac {{\partial}^2f(x,t)} {\partial x^2}-RC\frac {\partial f(x,t)} {\partial t}=0\;\;\;;
\;\;\;x>0\\
f(0,t)=V_0 e^{j\omega t}
\end{cases}$$ where $R$ is the resistance per unit length and $C$ is the capacitance per unit length. Let $V(x,t)$ the voltage along the semi-infinite cable. We use a formalism developed in ref.[@tp2] to solve the system (\[ep4.1\]). It consist in to define: $$\label{ep4.2}
\begin{cases}
V(x,t)=U(x)f(x,t)\\
g(t)=\left.\frac {\partial f(x,t)} {\partial x}\right|_{x=0}
\end{cases}$$ The differential equation in (\[ep4.1\]) transforms into: $$\label{ep4.3}
\frac {{\partial}^2V(x,t)} {{\partial}x^2}-RC
\frac {\partial V(x,t)} {\partial t}=
{\delta}^{'}(x)V_0e^{j\omega t}+
\delta(x)g(t)$$ Taking the Fourier transform of (\[ep4.3\]) we obtain: $$\label{ep4.4}
\hat{V}(\alpha_1,\alpha_2)={\cal F}[V(x,t)]$$ $$\hat{V}(\alpha_1,\alpha_2)=
\pi j V_0\delta(\alpha_1+\omega)\left[
\frac {1} {\alpha_2-\frac {1-j} {\sqrt{2}}
\sqrt{-\alpha_1 RC}}+\right.$$ $$\left.\frac {1} {\alpha_2+\frac {1-j} {\sqrt{2}}
\sqrt{-\alpha_1 RC}}\right]-
\frac {\hat{g}(\alpha_1)} {(1-j)\sqrt{-2\alpha_1 RC}}$$ $$\label{ep4.5}
\left[\frac {1} {\alpha_2-\frac {1-j} {\sqrt{2}}
\sqrt{-\alpha_1 RC}} -
\frac {1} {\alpha_2+\frac {1-j} {\sqrt{2}}
\sqrt{-\alpha_1 RC}}\right]$$ Deprecating the exponential increasing in the solution we obtain: $$\label{ep4.6}
\hat{g}(\alpha_1)=-(1+j)\pi\sqrt{-2\alpha_1RC}\;\delta(\alpha_1+\omega)$$ and then we obtain: $$\label{ep4.7}
V(x,t)=V_0U(x)e^{-\sqrt{\frac {\omega RC} {2}}x}
e^{j(\omega t-\sqrt{\frac {\omega RC} {2}}x)}$$ $$\label{ep4.8}
g(t)=-(1+j)\sqrt{\frac {\omega RC} {2}}\;
V_0e^{j\omega t}$$ The current $i(x,t)$ is: $$\label{ep4.9}
i(x,t)=-\frac {1} {R}\frac {\partial V(x,t)} {\partial x}\;\;\;;\;\;\;x>0$$ As: $$\label{ep4.10}
\frac {\partial V(x,t)} {\partial x}=
(1+j)\sqrt{\frac {\omega RC} {2}}
V_0e^{-\sqrt{\frac {\omega RC} {2}}x}
e^{j(\omega t-\sqrt{\frac {\omega RC} {2}}x)}
\;;\;x>0$$ then: $$\label{ep4.11}
i(x,t)=(1+j)\sqrt{\frac {\omega C} {2R}}
V_0e^{-\sqrt{\frac {\omega RC} {2}}x}
e^{j(\omega t-\sqrt{\frac {\omega RC} {2}}x)}
\;;\;x>0$$ If we take $\lambda=1/2$ in (\[ep3.19\] we obtain: $$\label{ep4.12}
\frac {d^{\frac {1} {2}}e^{j\omega t}} {dt^{\frac {1} {2}}}=
(j\omega)^{\frac {1} {2}}e^{j\omega t}\left[1+\frac {1} {2\sqrt{\pi}}
\Gamma(-\frac {1} {2},j\omega t)\right]+
\oint\limits_{\Gamma}Z^{\frac {1} {2}}a(Z)e^{-jZt}dZ$$ $$\frac {{\partial}^{\frac {1} {2}}V(x,t)} {\partial t^{\frac {1} {2}}}=
(j\omega)^{\frac {1} {2}}\left[1+\frac {1} {2\sqrt{\pi}}
\Gamma(-\frac {1} {2},j\omega t)\right]
e^{-\sqrt{\frac {\omega RC} {2}}x}
e^{j(\omega t-\sqrt{\frac {\omega RC} {2}}x)}+$$ $$\label{ep4.13}
\oint\limits_{\Gamma}Z^{\frac {1} {2}}a(Z,x)e^{-jZt}dZ$$ Thus we have a relation between the current and the time derivative of the voltage: $$i(x,t)=\sqrt{\frac {C} {R}}\left\{\left[
\frac {{\partial}^{\frac {1} {2}}} {\partial t^{\frac {1} {2}}}-
\frac {(j\omega)^{\frac {1} {2}}\Gamma(-\frac {1} {2},j\omega t)} {2\sqrt{\pi}}\right]
V(x,t)\right.-$$ $$\left.\label{ep4.14}
\oint\limits_{\Gamma}Z^{\frac {1} {2}}a(Z,x)e^{-jZt}dZ\right\}$$ If we consider only the first term in the rigth side of (\[ep4.14\]) we obtain the more habitual result: $$\label{ep8.15}
i(x,t)=\sqrt{\frac {C} {R}}\frac {{\partial}^{\frac {1} {2}}V(x,t)}
{\partial t^{\frac {1} {2}}}$$
The Laplace Transform {#the-laplace-transform-1 .unnumbered}
---------------------
If we use the Laplace transform in place of the Fourier transform to evaluate the fractional derivatives, (\[ep4.12\]),(\[ep4.13\]) and (\[ep4.14\]) are replaced by: $$\label{ep4.15}
\frac {d^{\frac {1} {2}}e^{j\omega t}} {dt^{\frac {1} {2}}}=
(j\omega)^{\frac {1} {2}}e^{j\omega t}\left[1+\frac {1} {2\sqrt{\pi}}
\Gamma(-\frac {1} {2},j\omega t)\right]$$ $$\label{ep4.16}
\frac {{\partial}^{\frac {1} {2}}V(x,t)} {\partial t^{\frac {1} {2}}}=
(j\omega)^{\frac {1} {2}}\left[1+\frac {1} {2\sqrt{\pi}}
\Gamma(-\frac {1} {2},j\omega t)\right]
e^{-\sqrt{\frac {\omega RC} {2}}x}
e^{j(\omega t-\sqrt{\frac {\omega RC} {2}}x)}$$ $$\label{ep4.17}
i(x,t)=\sqrt{\frac {C} {R}}\left[
\frac {{\partial}^{\frac {1} {2}}} {\partial t^{\frac {1} {2}}}-
\frac {(j\omega)^{\frac {1} {2}}\Gamma(-\frac {1} {2},j\omega t)} {2\sqrt{\pi}}\right]
V(x,t)$$ Difference between this results and the precedents is the term that contain a contour integral.
Discussion
==========
In this paper we have shown that Ultradistribution Theory is an adequate framework to define a Fractional Caculus and its applications. This definition unifies the notion of integral and derivative in one only operation. Several examples of application of fractional derivative are given, including a circuital application: a semi-infinite cable with a voltage $V=V_0e^{j\omega t}$ applied at one end.
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[^1]: *[This work was partially supported by Consejo Nacional de Investigaciones Científicas Argentina.]{}*
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abstract: 'Traditionally globular clusters and dwarf spheroidal galaxies have been distinguished by using one or more of the following criteria: (1) mass, (2) luminosity, (3) size, (4) mass-to-light ratio and (5) spread in metallicity. However, a few recently discovered objects show some overlap between the domains in parameter space that are occupied by galaxies and clusters. In the present note it is shown that ellipticity can, in some cases, be used to help distinguish between globular clusters and dwarf spheroidal galaxies.'
date: Received
title: Globular Clusters and Dwarf Spheroidal Galaxies
---
\[firstpage\]
(Galaxy:) globular clusters: galaxies: dwarf
INTRODUCTION
============
Recent discoveries of exceedingly faint dwarf spheroidal galaxies have shown quite dramatically that luminosity alone cannot be relied on to distinguish between dwarf spheroidal galaxies and globular clusters. Interesting examples of faint dwarfs are Boötes II \[M$_{v}$ = -3.1\] [@wjw07], Coma \[M$_{v}$ = -3.7\] [@bel07] and Willman 1 \[M$_{v}$ = -2.5\] [@mar07]. All of these galaxies are fainter than the overwhelming majority of globular clusters.
The size of a globular clusters is best described by its half-light radius R$_{h}$, because this parameter remains almost invariant over $\sim$ 10 cluster relaxation times. The vast majority of globular clusters have R$_{h} <$ 10 pc. However, some well-established globular clusters have quite large radii. Examples are NGC 2419 (R$_{h}$ = 18 pc) and Palomar 14 (R$_{h}$ = 25 pc). On the other hand Willman 1 [@mar07], which has a half-light radius of $\sim$ 20 pc, has generally been regarded as a galaxy. This overlap in size and luminosity raises deep questions about the nature of the distinction between these two classes of objects. It should be noted that the distinction between galactic nuclei and globular clusters is also somewhat artificial. Over time tidal forces will, for example, strip away much of the stellar population of the Sagittarius dwarf spheroidal, leaving behind only its nucleus the luminous globular cluster M54. @vdbm04 and @macvan05 have suggested that the position of an object in a plot of M$_{v}$ versus log R$_{h}$ might be used to discriminate between globular clusters and the stripped cores of dwarf galaxies, such as $\omega$ Centauri. However, this proposal now seems less attractive than it once did . In particular @vdb07 has recently noted that the brightest red objects in the halo of the elliptical galaxy NGC 5128 appear to form a continuum in the M$_{v}$ versus log R$_{h}$, [**indicating that globular clusters and dwarf spheroidals may not be clearly separate and distinct types of objects. The same point has recently been made about the brightest objects in NGC5128 by @barmby07 and by @rej07**]{}. Furthermore NGC 2419, which on the basis of its position in the M$_{v}$ versus log R$_{h}$ plane, had been classified as a stripped galaxy core, turns out to have a small metallicity dispersion and lacks a significant population of extra-tidal stars [@bellrip07]. Both of these factors militate against the hypothesis that NGC 2419 is actually a stripped galaxy core. On the other hand the stripped core suspect B514 in the Andromeda galaxy does seem to be embedded in a very low surface density dwarf spheroidal [@fre07]. In summary it appears that the location of a galaxy in the Mv versus log Rh plot is not always a reliable way of separating globular clusters from the stripped cores of dwarf spheroidal galaxies.
The pioneering investigations of @bab39 and @oor40 first demonstrated that dark matter provides a significant contribution to the masses of individual galaxies. More recently @mat98 showed that such dark matter becomes more and more dominant as one proceeds to study ever dimmer galaxies. It has therefore become customary to regard the presence of dark matter as the touchstone that allows one to unambiguously distinguish between galaxies and star clusters. Furthermore, it is often difficult and time consuming to obtain velocity dispersions of faint stars in distant dwarf galaxies. The problems outlined above suggest that it might be useful to have additional criteria to help discriminate between dwarf spheroidal galaxies and globular clusters.
ELLIPTICITY AND CLASSIFICATION
==============================
Table 1 shows the frequency distribution of the ellipticity parameter (a-b)/a , in which a and b are, respectively, the semi-major and semi- minor axes. For Galactic galactic globular clusters the data in the table were taken from the compilation of @har96 that was updated in 2003 at http://physwww.mcmaster.ca/ harris/mwgc.dat. Also given in the table is the distribution of flattening values, drawn from various sources, for those dwarf spheroidal galaxies that have distances $<$ 500 kpc. The individual values of (a-b)/a that were adopted are: CVn II 0.30, Car 0.33, Com 0.50, Dra 0.29, For 0.30, Her 0.67, Leo T 0.0:, Leo I 0.37, Leo II 0.13, Leo IV 0.25, Segue I 0.30, Scl 0.32, Sex 0.35 and UMi 0.66. A comparison between the distribution of globular cluster and dwarf spheroidal galaxy flattenings is listed in Table 1 and plotted in Figure 1. These data show that the the dwarf spheroidal companions to the Galaxy are typically much more flattened than are Galactic globular clusters. A Kolmogorov-Smirnov test shows that the observed difference is significant at $>$99.99%. A K-S test also shows that the distribution of the flattening values of six M31 dwarf spheroidals [@mccirw06] is statistically not distinguishable from that of the Galactic dwarf spheroidals discussed above. The reason for the difference between the the flattening distributions of globular clusters and of dwarf spheroidal galaxies is not yet entirely clear, but might reflect the relative importance of dissipative effects during the evolution of clusters and dwarf galaxies. [**Alternatively the observed flattening of dwarf spheroidal galaxies might reflect the shapes of their original dark matter halos**]{}. Galactic tides might also be important. As @goo97 has pointed out the strength of the tidal field of a parent galaxy may affect the ellipticities of globular clusters. This is so because a strong tidal field might rapidly destroy velocity anisotropies in initially tri-axial rotating globular clusters. However, a possible argument against the importance of tidal effects is that the (a-b)/a values of Galactic globular clusters are not correlated with either their half-light radii or with their concentration indexes. Furthermore tidal fields might not account for all distortions of dwarf spheroidal galaxies. As @coljon07 point out the Hercules dwarf spheroidal, which is located at a distance of 130 kpc, would need to have had a periGalactic distance of $\sim$ 8 kpc to account for its present three-to-one axial ratio.
The large systematic difference in average flattening between dwarf spheroidal galaxies and globular clusters seems to provide a useful way of distinguishing between galaxies and clusters. All Local Group objects with (a-b)/a $>$ 0.3 appear to be galaxies. Unfortunately projection effects only allow one to draw statistical inferences about the nature of any individual object with (a-b)/a $<$ 0.3. Adopting the criterion that objects with (a-b)/a $ >$ 0.3 are dwarf spheroidal galaxies one finds that four out of 23 of the most luminous “globular clusters” surrounding the giant elliptical galaxy NGC 5128 [@rej07] might actually be dwarf spheroidal galaxies. It is noted parenthetically that the faint pair of NGC 5128 clusters C141 and C144, have (a-b)/a values of 0.60 and 0.68 respectively [@har06]. It would clearly be of considerable interest to investigate this close pair (separation 2.’3) of unusually flattened objects in more detail.
Clusters that have been suspected of being the stripped cores of now defunct dwarf spheroidal galaxies have flattening values that are intermediate between those of typical globular clusters and dwarf spheroidal galaxies. For such objects (a-b)/a = 0.17 in NGC 5139 ($\omega$ Cen), 0.06 for NGC 6715 (M54), $\sim$ 0.20 for G1 (Mayall II), $\sim$ 0.17 for 037 (B327) and $\sim$ 0.20 for B514.
CONCLUSIONS
===========
Recent discoveries have shown that there is some overlap in parameter space between the regions occupied by globular clusters and dwarf spheroidal galaxies. It is shown that cluster flattening may be used as an additional parameter to help to distinguish between these two classes of objects. There are, of course, many cases \[such as the Fornax dwarf and its globular clusters\] where it is quite obvious which object is the dwarf spheroidal and which ones are its companion globular clusters. However, there are other cases where this distinction is not quite so obvious [@lee07]. For example, Boo I and Boo II appear to have similar distances and are located at a projected separation of only 1.8 kpc @wal07. Are these two objects companion galaxies, or is Boo II (M$_{v}$ = -3.1, R$_{h}$ =72 pc) a globular cluster companion to Boo I (M$_{v}$ = -5.7, R$_{h}$ = 227 pc)? Inspection of the outer isophotes of Boo II published by @wal07 suggests that 0.1 $\la$ (a-b)/a $\la$ 0.2, which is consistent with its being either a galaxy or a cluster.
In summary it is concluded that dwarf spheroidal galaxies are, on average, significantly flatter than globular clusters. In some cases this may help to distinguish these two classes of objects. Globular clusters that are probably the remnant cores of dwarf spheroidal galaxies appear to have present day flattening values that are, on average, intermediate between these two classes of objects.
ACKNOWLEDGEMENTS
================
I am indebted to Bob Abraham, Merla Geha, Bill Harris, Mario Mateo and Shane Walsh for helpful exchanges of e-mail and [**to a particular helpful referee**]{}. I also thank Brenda Parrish and Jason Shrivell for technical support.
(a-b)/a n(glob) n(dSph)
------------- --------- ---------
0.00 - 0.04 41 1
0.05 - 0.09 26 0
0.10 - 0.14 16 1
0.15 - 0.19 10 0
0.20 - 0.24 5 0
0.25 - 0.29 2 2
0.30 - 0.34 0 5
0.35 - 0.39 0 2
0.40 - 0.44 0 0
0.45 - 0.49 0 0
0.50 - 0.54 0 1
0.55 - 0.59 0 1
0.60 - 0.64 0 0
0.65 - 0.69 0 1
0.70 - 0.74 0 0
: Flattening values of Galactic globular clusters and of dwarf nearby spheroidal galaxies
{height="10.5cm"}
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abstract: 'The compositional scheme of a Bronze Age sword, found near the town of Giurgiu in Romania has been determined by the method of particle-induced X-ray emission (PIXE), at the Tandem accelerator of National Institute for Physics and Nuclear Engineering from Bucharest, Magurele, Romania. The results of the analyses and the comparison with the composition of other swords from the same geographic area, the Danubian plane from Bulgaria and Transylvania regions, show that the sword from Giurgiu could be relatively associated with the swords from Bulgaria, having also the same stylistic, temporal and geographical similitude.'
author:
- |
Agata Olariu$^{\diamond}$, Emilian Alexandrescu$^{\bullet}$, Alexandru Avram$^{\bullet}$, Teodor Badica$^{\diamond}$\
[*$^{\diamond}$Institute for Physics and Nuclear Engineering,*]{}\
[*PO Box MG-6, 76900 Magurele, Bucharest, Romania*]{}\
[*$^{\bullet}$Institute of Archaeology of the Romanian Academy,*]{}\
[*Henri Coandă Street, no. 11, Bucharest, Romania*]{}\
title: 'Compositional analyses of a Reutlingen Bronze Age sword discovered at Giurgiu, Romania'
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Introduction
============
The compositional scheme of a Bronze Age sword, Fig. 1, recently discovered near Giurgiu, a town situated in the south of Romania on the Danube, has been studied using the method of particle-induced X ray emission (PIXE), at the Tandem accelerator of National Institute for Physics and Nuclear Engineering from Bucharest, Magurele, Romania. In order to have a comparative study of composition we have considered also the composition of 6 swords of the same type and from the same period from the south of Danube in Bulgaria, and some different copper-based alloy objects from Şpălnaca deposit, in Transylvania, dated also in the Bronze Age.
Archaeological considerations
=============================
Site of discovery of the sword
------------------------------
The sword has been discovered by Ion Cercel in 1981 in the Mihai Eminescu street of the city of Giurgiu, Romania, in the course of diggings related to the installation of underground electric cables. Since then the sword has been kept by Ion Cercel, who presently is retired, in his house from the Oinacu commune, district of Giurgiu, in southern Romania. In 1999, by a happy concurrence of events, this interesting artifact became available to us, and we take this opportunity to thank Ion Cercel who has been kind to render us the sword for study and analyses.
According to the author of the finding, the artifact has been found at a depth of approximately 1.1 m, no other bronze objects existing in its vicinity. Foundations of buildings from the late medieval period were known to exist in the area, but it seems that no such archaeological remains existed in the specific place where the sword has been found. It seems that the site where the sword has been discovered lies outside the medieval city, buildings being erected there only in the period after 1821.
Description of the sword
------------------------
The sword is in a good state of conservation, being preserved almost entirely, and has the following dimensions: total length 39.5 cm, width at base 5.3 cm, width at the tip 2.9 cm, thickness at base 0.9 cm, thickness toward the tip 0.6 cm, length of the hilt with missing terminal part 9 cm, length of the blade 30.5 cm. On the blade of the sword one can observe, on both faces, channels situated at approximately 0.4 cm from the two edges, as shown in Fig. 1. The blade has a biconvex profile. In the zone of the hilt there are five orifices for the rivets of binding of the hilt and also three rivets still left in the orifices. It is possible that in the missing part of the hilt three more orifices should have existed for the binding of the hilt. The alloy is of a very good quality having a green-dark grey patina.
The sword belongs to the Reutlingen type defined by P. Schauer [@1] and described, with a special view to artifacts attested on Romanian territory, by T. Bader. [@2] Choosing as a criterion of classification in the first place the number of fixation orifices from the blade and the hilt, but also the shape of the nervure of the blade, the latter author distinguished several variants. [@3] Due to the fact that the sword under consideration presents a large median nervure and which is slightly rounded, we think that it is most resembling to two fragmentary pieces belonging to the deposit of Drajna de Jos, district of Prahova, Romania, catalogued by T. Bader with numbers 188 and 189 [@4] and included in the Guşteriţa variant of the Reutlingen type. Moreover, the Giurgiu sword has much smaller dimensions, so that it could be rather considered a “short sword” (Kurzschwert). [@5]
Artifacts of the Reutlingen type have been discovered over a very large area from the south of Scandinavia to Peloponesos and from the Rhine basin to the Black Sea, [@6] and recently discoveries have been reported even in Anatolia. [@7] However, the spreading is not uniform, existing some regions of concentration and others represented by much fewer discoveries. Among the latter, one counts the extra-Carpathian zone in Romania and the territory of Bulgaria and Greece. [@8]
It is however interesting that, in the Balkan area, the discoveries are concentrated mainly in the southern part of Romania, Wallachia, and in the north of Bulgaria, some of them just on the Danube line. In addition to the two pieces from Drajna de Jos, on the Romanian territory one finds specimens belonging to some variants in the ensemble of the Reutlingen type: Bălceşti and Mateeşti (district of V\^ alcea), Techirghiol (district of Constanţa). On the territory of Bulgaria there are 10 discoveries of swords with tongues at the hilt, of which 7 to the north and 3 to the south of the Balkans. [@9] The 7 specimens discovered on the territory between the Danube and the Balkans arise from Orjahovo (Orehovo), [@10] Vărbica (deposit II), [@11], Bajkal, [@12] Kruševo, [@13] Balkanski [@14] and Vasil Levski, [@15] to which one adds the specimen of smaller dimensions from the Razgrad Museum (inventory No. 117), discovered in the neighborhood. [@16] Among these specimens the first two belong to the Reutlingen type.
The artifact from Giurgiu has very close analogues (except, of course, for the dimensions) just in the sword from Orjahovo and in the fragmentary artifact from Vărbica, both cited as belonging to the Guşteriţa variant by T. Bader. [@17] These two specimens have been ascribed in the early horizon of the culture of the fields of urns (von Brunn stages I-III) [@18] by B. H" ansel, [@19] respectively in the subgroup I defined by I. Panayotov (the second horizon of deposits from Bulgaria: XIII$^{th}$ century B.C.). [@20] On the other hand, T. Bader dates, as a function of the synchronisms revealed by the various deposits, the great majority of the specimens belonging to the Reutlingen type discovered on the territory of Romania in the Cincu-Suseni period (HaA1, circa XII$^{th}$ century B.C.), but ascribes three or four deposits (among which is also the one from Drajna de Jos) for the slightly earlier period Uriu-Domăneşti (Bronze D, circa XIII$^{th}$ century B.C.) [@21] Consequently, taking into account the analogies proposed by us with the specimens from Drajna de Jos, Orjahovo and Vărbica II, we favor a dating of the short sword from Giurgiu in the XIII$^{th}$ century B.C., probably towards the end of the century; a date around 1200 B.C. is very likely.
Experimental
============
3 samples from the body of the sword: 1 sample from the tip of the sword and 2 samples from the hilt have been flatted and irradiated with protons of 3 MeV, in a irradiation chamber at the FN Tandem accelerator of National Institute for Physics and Nuclear Engineering from Bucharest, Magurele.
The beam current was kept below 10 nA to maintain a count rate of about 250 counts/s, which implies negligible dead-time and pile-up corrections. X-rays were detected with a HPGe (100 mm$^2\time10$mm) detector with 160 eV energy resolution at 5.9 keV. The X rays spectra have been recorded on a PC with a MCA interface. In the frame of the experimental conditions the following elements have been observed: As, Co, Cr, Cu, Fe, Ni, Pb, Sn and Zn.
The X ray spectra have been processed off line and then the concentrations of the elements have been calculated.
Results and Discussions
=======================
The results of PIXE analysis on the samples from the sword from Giurgiu are shown in the Table 1. The values of the concentrations are given in %. The instrumental errors are generally less than 15 %. We made corrections of the elemental concentrations so that the total value in the sample to be 100 %.
Sample As Co Cu Fe Ni Sn Zn
------------------ -------- -------- ------ -------- -------- ------ --------
Sword tip 0.3530 0.0440 88.2 0.0838 0.3090 10.4 0.6173
Sword big hilt 0.0855 0.0171 85.5 0.3850 0.3250 13.7 0.0470
Sword small hilt 0.2860 0.0224 89.5 0.4740 0.3400 9.35 0.0313
: Composition of the sword from Giurgiu, by PIXE
The composition of the 3 samples form the Giurgiu has been compared with the composition from similar 6 swords from Danubian regions from Bulgaria [@22] and some different archaeological objects from the Bronze Age Şpălnaca deposit, Transylvania [@23].
We present further, in the Table 2 the results of the analyses published by E. N. Černyh, for several of the swords with tongue at the hilt from Bulgaria. [@22]\
1: Vărbica II (10945), category X\
2: Orjahovo (9431), category X\
3: Pavelsko (9220), category X\
4: Bajkal (9432, analysis of the hilt; 9433, analysis of the blade), category X\
5: Kričim (9210), category X\
6: Vasil Levski (10892), category XI\
For all specimens included in Table 2, the copper is the dominant element.
Sn Pb Zn Bi Ag Sb As Fe Ni Co Mn Au
--- ---- ------ ------- -------- -------- ------- ------ ------- ------- ------- --------- ----------------
1 10 0.2 0.01 0.05 0.06 0.06 0.07 0.007 0.05 0.02 - $<$0.001
2 10 0.14 0.01 0.01 0.06 0.04 0.6 0.05 0.4 0.04 - $\approx$0.01
3 12 0.3 ? 0.005 0.06 0.25 0.3 0.003 0.25 0.03 - $\approx$0.003
4 7 0.12 0.006 0.003 0.05 0.3 0.8 0.01 0.3 0.012 - $<$0.01
10 0.3 ? 0.005 0.03 0.3 0.9 ? 0.35 0.02 - $>$0.001
5 7 0.05 - 0.0015 0.01 0.04 0.25 0.005 0.05 0.015 $<0.01$ $>$0.003
6 5 0.09 - - 0.0001 0.015 0.1 0.012 0.035 0.003 - -
: Composition of swords from Bulgaria, % [@22]
------------ ---------- ---------- ---------- ---------- ---------- ----------
Sample As/Cu Co/Cu Fe/Cu Ni/Cu Sn/Cu Zn/Cu
x 10$^6$ x 10$^6$ x 10$^6$ x 10$^6$ x 10$^6$ x 10$^6$
Giurgiu1 4000 500 950 3500 118000 7000
Giurgiu2 1000 200 4500 3800 160000 550
Giurgiu3 3200 250 5300 3800 104500 350
Bulgaria1 779 223 78 556 111305 111
Bulgaria2 6750 450 562 4500 112500 112
Bulgaria3 3430 343 34.3 2860 137300 0
Bulgaria4 8710 131 109 3265 76190 65
Bulgaria5 10140 225 0 3945 112700 0
Bulgaria6 2700 162 53.9 539 75530 0
Bulgaria7 1055 32 1270 369.4 52780 0
Şpălnaca1 6848 0 0 0 188600 0
Şpălnaca2 1193 0 79700 0 3250 0
Şpălnaca3 16600 0 44400 0 0 0
Şpălnaca4 12300 0 50900 0 0 0
Şpălnaca5 13100 0 0 0 81600 0
Şpălnaca6 22100 0 0 0 0 0
Şpălnaca7 15900 0 135000 0 0 0
Şpălnaca8 23000 0 32900 0 0 0
Şpălnaca9 67400 0 98100 0 0 0
Şpălnaca10 2090 0 0 0 0 0
Şpălnaca11 7900 0 21600 0 0 0
Şpălnaca12 7180 0 0 0 203100 0
Şpălnaca13 15600 0 12600 0 1770 0
Şpălnaca14 10360 0 334000 0 0 0
Şpălnaca15 2408 0 0 0 0 0
Şpălnaca16 13170 0 7970 0 0 0
Şpălnaca17 7460 0 49700 0 0 0
Şpălnaca18 2900 0 0 0 253400 0
Şpălnaca19 3400 0 0 0 0 0
Şpălnaca20 7160 0 0 0 0 0
Şpălnaca21 44900 0 0 0 0 0
Şpălnaca22 19200 0 0 0 0 0
Şpălnaca23 32600 0 19800 0 4200 0
Şpălnaca24 8900 0 0 0 0 0
------------ ---------- ---------- ---------- ---------- ---------- ----------
: Ratios of concentrations, in bronze objects of the same type: Giurgiu sword samples, by PIXE, the Bulgarian swords, by atomic spectroscopy, bronze objects from Splanaca, Transylvania, by neutron activation
In Table 3 are shown the elemental composition for all considered objects: the Giurgiu sword, the swords from Bulgaria and different bronze objects from Transylvanian deposit at Şpălnaca. Ratios of concentrations are considered for interpretation of the results to avoid the errors in the absolute calculations of the concentrations. It has been reported value zero in the cases the value of concentrations has been under the limit of detection.
Fig. 2 presents the diagram of ratios of concentrations: Sn/Cu versus As/Sn for the analyzed samples in the present study, and also for Bulgarian and Transylvanian objects, analyzed by atomic spectroscopy and respectively neutron activation analysis.
One could remark that the sword from Giurgiu has a relative closer composition to the Bulgarian ones, especially for the elements: As, Cu, and Sn. The objects from Transylvania are situated relatively outside the cluster formed by the objects from Giurgiu and Bulgaria.
Conclusions
===========
We could express the idea of an association of the sword from Giurgiu with the Bulgarian swords, having a close composition and also similitude in typology, geographic area and dating. Taking into account the analogies proposed by us with the Bulgarian specimens, Especially those of Drajna de Jos, Orjahovo and Vărbica II, we favor a dating of the short sword from Giurgiu in the XIII$^{th}$ century B.C., probably towards the end of that century, around 1200 B.C.
[20]{} P. Schauer, [*Die Schwerter in Süddeutschland, Österreich und in der Schweiz*]{}, Stuttgart, 1971 (PBF, IV, 2), p. 132 and following, who distinguished this type in the frame of the “normal type”, Naue II, Sprockhoff II a, Nenzingen. T. Bader, [*Die Schwerter in Rümanien*]{}, Stuttgart, 1991 (PBF, IV, 8), p. 86. It is difficult to realize a correspondence between the previous classification proposed by Alexandrina D. Alexandrescu, Dacia N.S. s10, 1966, pp. 117-189 (especially p. 133 and following) and that of T. Bader, because the Reutlingen type has not been identified in the frame of the ensemle of the “normal type” at the moment of the publication of the 1966 study. No. 188: I. Andrieşescu, [*Nouvelle contribution sur l’âge du bronze en Roumanie. Le dépôt de bronzes de Drajna de Jos et l’' ep' ee de Bucium*]{}, Dacia, 2, 1925, pp. 349-350 and plate I/1, Alexandrescu, op. cit., p. 178, Cat. No. 104 and fig. XXII/3; No. 189: Andrieşescu, loc. cit. and plate I/3 (fig. 2), Alexandrescu, op. cit., p. 177, Cat. No. 100 and fig. XV/1. Out of both artifacts, only about half (or less than half) of the blades has been conserved, the tongue, and for the piece No. 188 almost the entire hilt. Alexandrescu assumed (“vermutlich”) that No. 101 of his catalogue belongs to the same artifact with the fragment No. 100, an assumption not retained by Bader. As regards the dimensions, but not entirely as type, our sword can be compared to the item discovered around Razgrad (Museum of Razgrad, inventory No. 117), published by I. Panayotov, T. Ivanov, [*Dve bronzovi orăžija ot Razgradki okrăg*]{}, Arheologija, 1979, 1, pp. 29-33, No. 2 and figs. 1/b, 2/b. The conserved length of this sword (of the Nenzingen type, Nane II, but more evolved), out of which only the hilt is missing, is 48.3 cm. T. Bader, op. cit., p. 100. A. M" uller-Karpe, [*Anatolische Bronzenscwerter und S" udosteuropa*]{}, in C. Dobiat, editor, [*Festschrift f" ur Otto-Herman Frey zum 65. Geburtstag*]{}, Marburger Studien zur Vor- und Fr" uhgeschichte 16, 1994, pp. 440-444. Among the three artifacts, one (fig. 2/4) arises from the region of Bodrum (Caria) and shows analogies to the Bucium variant (Bader, Cat. No. 240), another, from the museum of Burdur, is only mentioned as being closest to the first, and a third (fig. 5/1) has been discovered in the Bolu region (at half distance between Istanbul and Ankara), and is likened by the author, among other, to Bader, Cat. No. 239 (Mateeşti variant). T. Bader, op. cit., p. 100: “Vereinzeilt sind die Exemplare aus der Dobrudscha (Techirghiol), aus Muntenien (Drajna de Jos), Oltenien (Bălceşti, Mateeşti) und aus der Moldau (Ilişeni, Bucium) bekannt. Selten und nicht bedeutend sind die Funde des Schwerttypus Reutlingen aus Bulgarien (Orechovo, Vărbica, Smirov dol) und aus dem " ag" aischen Raum (Mykene-Akropolis)”. The provenance of the sword from T\^ argovişte-Valea Voievozilor (Bader, Cat. No. 166, close to the Ighiu variant) is uncertain; cf. Al. Vulpe, Dacia N.S. 22, 1978, p. 372. I. Panayotov, Thracia, 5, 1980, p. 183 and map 2 at p. 180. We have added to the two pieces from the south of Bulgaria (Pavelsko, okr. Smoljan and Kričim, okr. Plovdiv) the item from Smirnov dol (okr. Pernic): M. Čohad' zev, Studia Praehistorica 5-6, 1981, p. 145 and following and fig. 2; V. Ljubenova, in [*Dritter Internationaler Thrakologischer Kongress Wien-Sofia II*]{}, Sofia, 1984, p. 150, fig. 2. The latter item, published after the study of Panayotov, is ascribed by Bader, op. cit. p. 96, also to the variant Guşteriţa. B. H" ansel, PZ 45, 1970, pp. 33-34 and fig. 2/2; E. N. Černyh, [*Gornoe delo i metalurgija v drevnejšej Bolgarii*]{}, Sofia, 1978, p. 237 and figs. 64/5; Panayotov, op. cit. p. 181 and fig. 3/2. B. H" ansel, op. cit., pp. 35-36; E. N. Černih, op. cit., p. 237 and fig. 65/2; I. Panayotov, op. cit., pp. 181-182 and fig. 3/4. B. H" ansel, op. cit., pp. 36-37 and fig. 2/3; E. N. Černih, op. cit., p. 237 and fig. 65/4; I. Panayotov, op. cit., p. 183 and fig. 4/1. A. Milčev and N. Kovačev, [*Neonarodvani pametnici ot Sevlievko*]{}, Arheologija, 1967, p. 40, fig. 1; I. Panayotov, op. cit., p. 183. D. Ivanov, [*Novi materiali ot bronzovata i željaznata epoha, săhrianavani v Rusenskija Muzej*]{}, Godišnik na muzeite ot Severna Bălgarija 4, 1978, pp. 5-9 and figs. 2/a-b; I. Panayotov, op. cit., p. 183 and fig. 4/3. B. H" ansel, op. cit., pp. 37-38 and fig. 2/4; I. Panayotov, op. cit., p. 183 and fig. 4/2. I. Panayotov, T. Ivanov, op. cit.,pp. 29-30 and figs. 1/b, 2/b. T. Bader, op. cit.,p. 96. W. A. von Brunn, [*Mitteldeutsche Hortfunde der jüngeren Bronzezeit*]{}, Berlin, 1968. B. H" ansel, op. cit., p. 34 and note 19, respectively p. 35, where the fragment of sword from Vărbica is dated in conjunction with other constituent pieces of the deposit which present analogies with types from Guşteriţa: “die " ubrigen Gegenst" ande des Hortfundes legen seine Datierung in die fr" uhe Urnenfelderzeit, d.h. in die v. Brunnschen Stufen I und II mit hinl" anglicher Sicherheit fest”. Cf. p. 36: “beide \[the items from Orjahovo and Vărbica (authors’ note)\] vertreten den gleichen Horizont bzw. die $<<$Typenfront$>>$, die in Griechenland mit dem Ende der mykenischen Zivilisation im sp" aten 13. Jahrhundert verkn" upft ist”. I. Panayotov, op. cit., pp. 182 and 185. The typology of the author is somewhat unclear. Cf. p. 181 (the items of swords with tongues at their hilt from the Bulgarian territory “can be related to two subgroups with no sharp differences between them”), pp. 181-182 ( where included in the first subgroup are the pieces from Pavelsko, Orjahovo, Kričim and Vărbica II), p. 182 (“the link between the first and the second subgroup is achieved by the sword from the village of Bajkal”, close to which are the items from Kruševo, Gradinite-Vasil Levski, Balkanski and the museum of Razgrad), p. 185 (“in a purely formal respect we differentiate three subtypes. The first one we mark as Pavelsko type. \[...\] This subtype is synchronic to the Orjahovo subtype to which we also relate the fragments from the blade and a trapezum-like plate on the hilt. Here we also include the swords from the village of Balkanski, district of Razgrad and the one from the village of Kruševo, whose definition is not absolutely certain because of bad state of its upper part”). We add that E. N. Černyh, op. cit., p. 237, includes in the M-6 group (Nenzingen II), alongside the items from Orjahovo and Vărbica, also the sword from Vasil Levski. T. Bader, op. cit., pp. 99-100. E. N. Černyh, op. cit., pp. 357 and following; table with the results of spectrographic analyses. For the definition of the chemical categories X-XII see pp. 178-179. A. Olariu, [*Studies of Archaeometry by Atomic and Nuclear Methods*]{}, Ph. D. Thesis, University of Bucharest, 1998.
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abstract: 'For the two dimensional Schrödinger equation in a bounded domain, we prove uniqueness of determination of potentials in $W^1_p(\Omega),\,\, p>2$ in the case where we apply all possible Neumann data supported on an arbitrarily non-empty open set $\widetilde\Gamma$ of the boundary and observe the corresponding Dirichlet data on $\widetilde{\Gamma}$. An immediate consequence is that one can uniquely determine a conductivity in $W^3_p(\Omega)$ with $p>2$ by measuring the voltage on an open subset of the boundary corresponding to current supported in the same set.'
author:
- 'O. Yu. Imanuvilov, [^1] Gunther Uhlmann,[^2] M. Yamamoto[^3]'
title: 'Inverse Boundary Value Problem by Partial data for the Neumann-to-Dirichlet-map in two dimensions'
---
**Introduction**
================
Let $\Omega\subset \Bbb R^2$ be a bounded domain with smooth boundary $\partial\Omega$ and let $\nu=(\nu_1, \nu_2)$ be the unit outer normal to $\partial\Omega$ and let $\frac{\partial}{\partial\nu} = \nabla\cdot\nu$.
In this domain we consider the Schr[ö]{}dinger equation with a potential $q$: $$L_q(x,D)u=(\Delta +q)u=0\quad\mbox{in}\,\,\Omega.$$
Let $\widetilde \Gamma$ be a non-empty arbitrary fixed relatively open subset of $\partial\Omega.$ Consider the Neumann-to-Dirichlet map $N_{q,\widetilde\Gamma}$ with partial data on $\tilde \Gamma$ defined by $$\label{popo}
N_{q,\widetilde\Gamma}: f\rightarrow u\vert_{\widetilde\Gamma},$$ where $$\label{lopo}
(\Delta+q)u=0\quad\mbox{in}\,\,\Omega,\,\, \frac{\partial
u}{\partial \nu}\vert_{\partial\Omega\setminus \widetilde\Gamma}=0,
\,\, \frac{\partial u}{\partial \nu}\vert_{\widetilde\Gamma}=f$$ with domain $D(N_{q,\widetilde\Gamma})\subset L^2(\widetilde \Gamma).$ Without loss of generality we may assume that $\partial\Omega\setminus\widetilde \Gamma$ contains a non-empty open set. By uniqueness of the Cauchy problem for the Schödinger equation the operator $N_{q,\widetilde\Gamma}$ is well defined since the problem (\[lopo\]) has at most one solution for each $f\in L^2(\widetilde \Gamma).$ Thanks to the Fredholm alternative, we see that $D(N_{q,\widetilde\Gamma})=\overline{D(N_{q,\widetilde\Gamma})}$ and $L^2(\widetilde{\Gamma}) \setminus D(N_{q,\widetilde\Gamma})$ is finite dimensional for any potential $q$ in $W^1_2(\Omega).$
The goal of this article is to prove uniqueness of the determination of the potential $q$ from the Neumann-to-Dirichlet map $N_{q,\widetilde\Gamma}$ given by (\[popo\]) for arbitrary subboundary $\widetilde \Gamma$. More precisely, we consider all Neumann data supported on an arbitrarily fixed subboundary $\widetilde{\Gamma}$ as input and we observe the Dirichlet data only on the same subboundary $\widetilde\Gamma$. This map arises in electrical impedance tomography (EIT) where one attempts to determine the electrical conductivity of a medium by inputting voltages and measuring current at the boundary. After transforming (1) to the conductivity equation, we can interpret $u\vert_{\widetilde\Gamma}$ and $\frac{\partial u}{\partial\nu}_{\widetilde\Gamma}$ respectively as the voltage and the multiple of the current by values of the surface conductivity. In practice, we can realize such inputs and outputs by applying current to electrodes on the boundary and observing the corresponding voltages. The current inputs are modeled by the Neumann boundary data $\frac{\partial u}{\partial \nu}$ and the observation data is modeled by Dirichlet data. See e.g., Cheney, Issacson and Newell [@CIN] for applications to medical imaging of EIT. Moreover it is very desirable to restrict the supports of the current inputs as small as possible. To the authors’ best knowledge there are few works on the uniqueness by such a ”Neumann-to-Dirichlet map" with partial data. In Astala, Päivärinta and Lassas [@APL], the authors consider both the Dirichlet-to-Neumann map and the Neumann-to-Dirichlet map on an arbitrarily subboundary to establish the uniqueness of an anisotropic conductivity modulo the group of diffeomorphisms which is the identity on the boundary where the measurements take place.
The case where the measurements are given by the Dirichlet-to-Neumann map has been extensively studied in the literature. This map is defined in the case of partial data by $$\Lambda_{q,\widetilde \Gamma}: g\rightarrow \frac{\partial
u}{\partial\nu}\vert_{\widetilde\Gamma};\quad
%\begin{equation}\label{popo}
%\mathcal C_q
(\Delta+q)u=0\quad\mbox{in}\,\,\Omega,\,\,
u\vert_{\partial\Omega\setminus \widetilde\Gamma}=0 ,\quad
u\vert_{\widetilde\Gamma}=g.$$ We give some references but the list is not at all complete. In the case of full data $\widetilde{\Gamma} =
\partial\Omega$, this inverse problem was formulated by Calderón [@C]. In the two dimensional case, given a Dirichlet-to-Neumann map $\Lambda_{q,\widetilde \Gamma}$ on an arbitrary subbondary $\widetilde \Gamma$, uniqueness is proved under the assumption $q\in
C^{2+\alpha}(\overline\Omega)$ by Imanuvilov, Uhlmann and Yamamoto [@IUY] and for the uniqueness for potentials $q\in
W^1_p(\Omega), p>2$ see Imanuvilov and Yamamoto [@IY2]. For other uniqueness results by the Dirichlet-to-Neumann map on an arbitrary subboundary $\widetilde{\Gamma}$, we can refer also to Imanuvilov, Uhlmann and Yamamoto [@IUY2], [@IUY1]. Also see Imanuvilov and Yamamoto [@IY1] for uniqueness results for elliptic systems. In Guillarmou and Tzou [@GZ], the result of [@IUY] was extended on Riemmannian surfaces. In particular, for uniqueness in determining a two-dimensional potential with full data: $\Lambda_{q,\partial\Omega}$, we refer to Blasten [@EB], Bukhgeim [@Bu] and, Sun and Uhlmann [@SuU], and for systems in Albin, Guillarmou, Tzou and Uhlmann [@AGTU] and Novikov and Santacesaria [@Nov-San1]. For the case of full data, in [@EB] and [@IY2], it was shown that $\Lambda_{q,\partial\Omega}$ uniquely determines $q$ in the class piecewise $W^1_p(\Omega)$ with $p>2$ and $C^\alpha(\overline\Omega),\alpha>0$, respectively. As for the related problem of recovery of the conductivity in EIT, Astala and Päivärinta [@AP] proved uniqueness for conductivities in $L^\infty(\Omega),$ improving the results of Nachman [@N] and Brown and Uhlmann [@BrU]. Moreover for the case of dimensions $n
\ge 3$ with the full data Sylvester and Uhlmann [@SU] proved the uniqueness of recovery of a conductivity in $C^2(\overline\Omega)$, and later the regularity assumption was improved (see, e.g., Brown and Torres [@BT], Päivärinta, Panchenko and Uhlmann [@PPU] and Haberman and Tataru [@HT]). The case when voltages are applied and current is measured on different subsets was studied in dimensions greater than three in Bukhgeim and Uhlmann [@BuU], Kenig, Sjöstrand and Uhlmann [@KSU] and in Imanuvilov, Uhlmann and Yamamoto [@IUY3] for the two-dimensional case.
Our main result is as follows
\[001\] Let $q_1,q_2\in W^1_p(\Omega)$ for some $p>2.$ If $
D(N_{q_1,\widetilde\Gamma})\subset D(N_{q_2,\widetilde\Gamma})$ and $N_{q_1,\widetilde \Gamma}(f)= N_{q_2,\widetilde\Gamma}(f)$ for each $f$ from $ D(N_{q_1,\widetilde\Gamma})$, then $q_1=q_2$ in $\Omega$.
Notice that Theorem \[001\] does not assume that $\Omega$ is simply connected. An interesting inverse problem is whether one can determine the potential in a domain with holes by measuring $N_{q,\widetilde\Gamma}$ only on some open set $\widetilde{\Gamma}$ in the outer subboundary.
Let $\Omega,G$ be bounded domains in $\Bbb R^2$ with smooth boundaries such that $\overline G\subset \Omega.$ Let $\widetilde{\Gamma} \subset \partial\Omega$ be an open set and $q\in
W^1_p(\Omega\setminus \overline{G})$ with some $p > 2$. Consider the following Neumann-to-Dirichlet map: $$\nonumber
\widetilde N_{q,\widetilde\Gamma}: f\rightarrow u\vert_{\widetilde\Gamma},$$ where $$\thinspace u \in H^1(\Omega\setminus \overline{G}), \quad
(\Delta+q)u=0\,\,\mbox{in}\,\,\Omega\setminus\overline G, \quad
\frac{\partial u}{\partial\nu}\vert_{\partial G\cup
(\partial\Omega\setminus \widetilde\Gamma)}=0, \quad \frac{\partial
u}{\partial\nu}\vert_{ \widetilde\Gamma}=f. \nonumber$$ Then we can directly derive the following from Theorem 1.
\[coro2\] Let $q_1, q_2 \in W^1_p(\Omega\setminus\overline{G})$ with some $p >
2$. If $ D(\widetilde N_{q_1,\widetilde\Gamma})\subset D(\widetilde
N_{q_2,\widetilde\Gamma})$ and $\widetilde N_{q_1,\widetilde
\Gamma}(f)= \widetilde N_{q_2,\widetilde\Gamma}(f)$ for each $f$ from $ D(\widetilde N_{q_1,\widetilde\Gamma})$, then $q_1=q_2$ in $\Omega\setminus \overline{G}$.
For the case of EIT, if the conductivities are known on $\widetilde\Gamma$, then we can apply our theorem to prove uniqueness of the determination of conductivities in $W^3_p(\Omega), p>2$ from the Neumann-to-Dirichlet map.\
\
The remainder of the paper is devoted to the proof of the theorem \[001\]. The main technique is the construction of complex geometrical optics solutions whose Neumann data vanish on the complement of $\widetilde \Gamma$.
Throughout the article, we use the following notations.
[**Notations.**]{} We set $\Gamma_0=\partial\Omega\setminus\overline{\widetilde\Gamma},$ $i=\sqrt{-1}$, $x_1, x_2 \in {\Bbb R}^1$, $z=x_1+ix_2$, $\overline{z}$ denotes the complex conjugate of $z \in \Bbb C$. We identify $x = (x_1,x_2) \in {\Bbb R}^2$ with $z = x_1 +ix_2 \in
{\Bbb C}$ and $\xi=(\xi_1,\xi_2)$ with $\zeta=\xi_1+i\xi_2$. $\partial_z = \frac 12(\partial_{x_1}-i\partial_{x_2})$, $\partial_{\overline z}= \frac12(\partial_{x_1}+i\partial_{x_2})$, $D = \left( \frac{1}{i}\partial_{x_1},
\frac{1}{i}\partial_{x_2}\right),$ $\partial_\zeta=\frac
12(\partial_{\xi_1}-i\partial_{\xi_2}), \partial_{\overline \zeta}=
\frac12(\partial_{\xi_1}+i\partial_{\xi_2}).$ Denote by $B(x,\delta)$ a ball centered at $x$ of radius $\delta.$ For a normed space $X$, by $o_X(\frac{1}{\tau^\kappa})$ we denote a function $f(\tau,\cdot)$ such that $ \Vert
f(\tau,\cdot)\Vert_X=o(\frac{1}{\tau^\kappa})\quad \mbox{as}
\,\,\vert \tau\vert\rightarrow +\infty$. The tangential derivative on the boundary is given by $\partial_{\vec\tau}=\nu_2\frac{\partial}{\partial x_1}
-\nu_1\frac{\partial}{\partial x_2}$, where $\nu=(\nu_1, \nu_2)$ is the unit outer normal to $\partial\Omega$. The operators $\partial_{z}^{-1}$ and $\partial_{\overline z}^{-1}$ are given by $$\partial_{\overline z}^{-1}g=-\frac1\pi\int_\Omega
\frac{g(\zeta,\overline\zeta)}{\zeta-z} d\xi_2d\xi_1,\quad
\partial_{ z}^{-1}g
= \overline{\partial^{-1}_{\overline{z}}\overline{g}}.$$ We call $b(z)$ antiholomorphic if $b(\overline{z})$ is holomorphic. In the Sobolev space $W_2^1(\Omega)$ we introduce the following norm $$\Vert u\Vert_{W_2^{1,\tau}(\Omega)} = (\Vert
u\Vert_{W_2^1(\Omega)}^2 + \vert \tau\vert^2\Vert
u\Vert^2_{L^2(\Omega)})^{\frac{1}{2}}.$$
Proof of Theorem 1
==================
Let $\Phi=\varphi+i\psi$ be a holomorphic function in $\Omega$ such that $\varphi, \psi$ are real-valued and $$\label{1} \Phi\in C^2(\overline\Omega),\quad
\mbox{Im}\, \Phi\vert_{\Gamma_0^*}=0, \quad \Gamma_0\subset\subset
\Gamma_0^*,$$ where $\Gamma_0^*$ is some open set in $\partial\Omega.$ Denote by $\mathcal H$ the set of the critical points of the function $\Phi.$ Assume that $$\label{22}
\mathcal H\ne \emptyset,\quad\partial^2_z\Phi(z)\ne 0,\quad \forall
z\in \mathcal H,\quad \mathcal
H\cap\overline{\widetilde\Gamma}=\emptyset$$ and $$\label{kk}
\int_{\mathcal J}1d\sigma =0,\quad \mathcal J=\{x; \thinspace
\partial_{\vec \tau}\psi(x)=0,x\in \partial\Omega\setminus\Gamma_0^*\}.$$
Let $\Omega_1$ be a bounded domain in $\Bbb R^2$ such that $\Omega\subset\subset \Omega_1$ and $\mathcal C$ be some smooth complex-valued function in $\Omega$ such that $$\label{nono}
2\frac{\partial \mathcal C}{\partial z} =C_1(x)+iC_2(x)\quad
\mbox{in}\,\,\Omega_1,$$ where $C_1,C_2$ are smooth real-valued functions in $\Omega$ such that $$\label{zopaW}
\frac{\partial C_1}{\partial x_1}+\frac{\partial C_2}{\partial
x_2}=1\quad \mbox{in}\quad\Omega_1.$$
The following proposition is proved as Proposition 2.5 in [@IUY1].
\[Theorem 2.1\] Suppose that $q\in L^\infty(\Omega)$, the function $\Phi$ satisfies (\[1\]), (\[22\]), and the function $\mathcal C$ satisfies (\[nono\]), (\[zopaW\]) and $\widetilde v\in W_1^2(\Omega)$ . Then there exist $\tau_0$ and $C(N)$ independent of $\widetilde v$ and $\tau$ such that
$$\begin{aligned}
\label{xxx} \frac{N}{2}\Vert 2\partial_{\overline z}
\widetilde v e^{\tau\varphi+N \mathcal C}\Vert^2 _{L^2(\Omega)}
+\tau\Vert \widetilde v e^{\tau\varphi+N\mathcal
C}\Vert^2_{L^2(\Omega)} +\Vert \widetilde v e^{\tau\varphi+N\mathcal
C}\Vert^2
_{W_2^1(\Omega)}+\tau^2\Vert\vert\frac{\partial\Phi}{\partial z}
\vert \widetilde v e^{\tau\varphi+N\mathcal C}\Vert^2_{L^2(\Omega)}
\nonumber\\
\le \Vert L_q(x,D)\widetilde v e^{\tau\varphi+N \mathcal C}\Vert^2
_{L^2(\Omega)} + C(N)\tau\Vert (\widetilde v
e^{\tau\varphi+N\mathcal C} ,\frac{\partial\widetilde
v}{\partial\nu}e^{\tau\varphi+N\mathcal C}
)\Vert^2_{W_2^{1,\tau}(\partial\Omega)\times L^2(\partial\Omega)}\end{aligned}$$
for all $\tau > \tau_0(N)$ and all positive $N\ge 1.$
Let $\widetilde v\in W_2^2(\Omega)$ satisfy $$L_q(x,D)\widetilde v=f\quad \mbox{in}\,\,\Omega,\quad \frac{\partial
\widetilde v}{\partial\nu}\vert_{\Gamma^*_0}=0.$$ Using Proposition \[Theorem 2.1\], we can show the following.
\[Theorem 2.2\] Suppose that $\Phi$ satisfies (\[1\]), (\[22\]) and $q\in
L^\infty(\Omega) .$ Then there exist $\tau_0$ and $C$ independent of $\widetilde v$ and $\tau$ such that $$\begin{aligned}
\label{Xxxx1}
\tau\Vert \widetilde v e^{\tau\varphi}\Vert^2_{L^2(\Omega)} +\Vert
\widetilde v e^{\tau\varphi}\Vert^2
_{W_2^1(\Omega)}+\tau^2\Vert\vert\frac{\partial\Phi}{\partial z}
\vert
\widetilde v e^{\tau\varphi}\Vert^2_{L^2(\Omega)} \nonumber\\
\le C\left(\Vert L_q(x,D)\widetilde v e^{\tau\varphi}\Vert^2
_{L^2(\Omega)} + \tau\Vert (\widetilde v
e^{\tau\varphi},\frac{\partial \widetilde v
e^{\tau\varphi}}{\partial\nu})\Vert^2_{W_2^{1,\tau}(\widetilde
\Gamma)\times L^2(\widetilde\Gamma)}\right)\end{aligned}$$ for all $\tau > \tau_0$ and for all $\widetilde v\in H^2(\Omega).$
[**Proof.**]{} Let $\{e_j\}_{j=1}^M$ be a partition of unity such that $e_j\in C^\infty_0(B(x_j,\delta))$ where $x_j$ are some points in $\Omega,$ $$\sum_{j=1}^Me_j(x)=1\quad \mbox{on}\,\,\Omega,\quad \frac{\partial
e_j}{\partial\nu}\vert_{\Gamma_0^*}=0\quad\forall j\in\{1,\dots, M\},$$ and $\delta$ be a small positive number such that $B(x_j,\delta)\cap
\Gamma_0\ne\emptyset$ implies $B(x_j,\delta)\cap
\partial\Omega\subset \Gamma_0^*$. Denote $w_j=e_j\widetilde v.$ Let $\mbox{supp}\, w_j\cap (\partial\Omega\setminus\widetilde
\Gamma)=\emptyset$. Then Proposition \[Theorem 2.1\] implies that there exists $\tau_0$ such that for all $\tau\ge\tau_0$ $$\begin{aligned}
\label{bp}
\frac{N}{2}\Vert 2\partial_{\overline z}w_j e^{\tau\varphi}
e^{N \mathcal C}\Vert^2 _{L^2(\Omega)}\nonumber\\
+ \tau\Vert w_j e^{\tau\varphi+N \mathcal C}\Vert^2_{L^2(\Omega)}
+\Vert w_j e^{\tau\varphi+N \mathcal C}\Vert^2
_{H^1(\Omega)}+\tau^2\Vert\vert\frac{\partial\Phi}{\partial z} \vert
w_j e^{\tau\varphi+N \mathcal C}\Vert^2_{L^2(\Omega)} \nonumber\\
\le C\Vert L_q(x,D)w_j e^{\tau\varphi+N \mathcal
C}\Vert^2_{L^2(\Omega)}+C(N)\Vert ( w_j
e^{\tau\varphi},\frac{\partial w_j
e^{\tau\varphi}}{\partial\nu})\Vert^2_{W_2^{1,\tau}(\widetilde
\Gamma)\times L^2(\widetilde \Gamma)}.\end{aligned}$$ Next let $\mbox{supp}\, w_j\cap
(\partial\Omega\setminus\widetilde\Gamma) \ne\emptyset$. We can not apply directly the Carleman estimate (\[Xxxx1\]) in this case, since the function $w_j$ may not satisfy the zero Dirichlet boundary condition. To overcame this difficulty we construct an extension. Without loss of generality, using if necessary a conformal transformation, we can assume that $\mbox{supp}\, w_j\cap
\Omega\subset \{x_2>0\}$ and $\mbox{supp}\, w_j\cap
\partial\Omega\subset \{x_2=0\}.$ Then using the extension $w_j(x_1,x_2)=w_j(x_1,-x_2),
q(x_1,x_2)=q(x_1,-x_2)$ and $\varphi(x_1,x_2)=\varphi(x_1,-x_2)$, we apply Proposition \[Theorem 2.1\] to the operator $L_q(x,D)$ in $\mathcal O=\mbox{supp}\, e_j\cup \{x\vert (x_1,-x_2)\in
\mbox{supp}\, e_j\}.$ We have the same estimate (\[bp\]). Therefore for all $\tau\ge \tau_0$ $$\begin{aligned}
\label{xxx1}
\Vert\widetilde v\Vert^2_* := \tau\Vert \widetilde v
e^{\tau\varphi+N\mathcal C}\Vert^2_{L^2(\Omega)}
+\Vert \widetilde v e^{\tau\varphi+N\mathcal C}\Vert^2
_{H^1(\Omega)}+\tau^2\Vert\vert\frac{\partial\Phi}{\partial z} \vert
\widetilde v e^{\tau\varphi+N\mathcal C}\Vert^2_{L^2(\Omega)} \nonumber\\
\le \sum_{j=1}^M\Vert\widetilde ve_j\Vert^2_* \le
C\sum_{j=1}^M\Vert L_q(x,D)w_j e^{\tau\varphi+N \mathcal
C}\Vert^2_{L^2(\Omega)}+C(N)\Vert (w_j
e^{\tau\varphi},\frac{\partial w_j
e^{\tau\varphi}}{\partial\nu})\Vert^2_{W_2^{1,\tau}(\widetilde\Gamma)\times
L^2(\widetilde\Gamma)}\nonumber\\
\le C \sum_{j=1}^M(\Vert \Delta e_j\widetilde v e^{\tau\varphi+N
\mathcal C}\Vert^2_{L^2(\Omega)}+\Vert2\partial_z e_j
\partial_{\overline z}\widetilde v
e^{\tau\varphi+N \mathcal C}\Vert^2_{L^2(\Omega)}-
N\Vert
\partial_{\overline z}(\widetilde v e_j)e^{\tau\varphi+N \mathcal C}
\Vert^2_{L^2(\Omega)}\nonumber\\
+ \Vert L_q(x,D)\widetilde v e^{\tau\varphi+N \mathcal
C}\Vert^2_{L^2(\Omega)})+C(N)\Vert (\widetilde v
e^{\tau\varphi},\frac{\partial \widetilde
ve^{\tau\varphi}}{\partial\nu})\Vert^2_{W_2^{1,\tau}(\widetilde\Gamma)\times
L^2(\widetilde\Gamma)}.\end{aligned}$$ Fixing the parameter $N$ sufficiently large, we obtain from (\[xxx1\]) $$\begin{aligned}
\label{xxx2} \Vert\widetilde v\Vert^2_*
\le C(N)\left(\Vert \widetilde v e^{\tau\varphi+N \mathcal
C}\Vert^2_{L^2(\Omega)}+ \Vert L_q(x,D)\widetilde v e^{\tau\varphi+N
\mathcal C}\Vert^2_{L^2(\Omega)}+\Vert (\widetilde v
e^{\tau\varphi},\frac{\partial \widetilde
ve^{\tau\varphi}}{\partial\nu})\Vert^2_{W_2^{1,\tau}(\widetilde\Gamma)\times
L^2(\widetilde\Gamma)}\right).\end{aligned}$$ The first term on the right-hand side of (\[xxx2\]) can be absorbed into the left-hand side for all sufficiently large $\tau.$ Since $N$ and $\mathcal C$ are independent of $\tau$, the proof of the proposition is finished.$\square$
The Carleman estimate (\[Xxxx1\]) implies the existence of solutions to the following boundary value problem.
\[Theorem 2.3\] There exists a constant $\tau_0$ such that for $\vert \tau\vert\ge
\tau_0$ and any $f\in L^2(\Omega),r\in W_2^\frac 12 (\Gamma_0^*)$, there exists a solution to the boundary value problem $$\label{lola}
L_q(x,D)u =fe^{\tau\varphi}\quad\mbox{in}\,\,\Omega, \quad
\frac{\partial u}{\partial \nu}\vert_{\Gamma_0}=re^{\tau\varphi}$$ such that $$\label{2}
\Vert u\Vert_{W_2^{1,\tau}(\Omega)}/\root\of{\vert\tau\vert}
\le C(\Vert f\Vert_{L^2(\Omega)}+\vert\tau\vert^\frac 14 \Vert
r\Vert_{L^2 (\Gamma_0)}+\Vert r\Vert_{W_2^{\frac 12}
(\Gamma^*_0)}).$$ The constant $C$ is independent of $\tau.$
The proof of this proposition uses standard duality arguments, see e.g., [@IUY].\
We define two other operators: $$\label{anna}
\mathcal R_{\tau}g = \frac 12e^{\tau(\Phi - \overline{\Phi})}
\partial_{\overline z}^{-1}(g
e^{\tau(\overline{\Phi}-\Phi)}),\,\, \widetilde {\mathcal R}_{\tau}g
= \frac 12 e^{\tau(\overline {\Phi}-{\Phi})}
\partial_{ z}^{-1}(ge^{\tau( {\Phi}
-\overline {\Phi})}).$$ Observe that $$\label{begemot}
2\frac{\partial}{\partial z}(e^{\tau\Phi}\widetilde {\mathcal
R}_{\tau}g)=ge^{\tau\Phi},\quad 2\frac{\partial}{\partial \overline
z}(e^{\tau\overline\Phi} {\mathcal
R}_{\tau}g)=ge^{\tau\overline\Phi} \quad \forall g\in L^2(\Omega).$$ Let $a\in C^6(\overline\Omega)$ be some holomorphic function on $\Omega$ such that $$\label{LL}
\mbox{Im}\, a\vert_{\Gamma^*_0}=0 ,\quad
\lim_{z\rightarrow \hat z}a(z)/\vert z-\hat z\vert^{100}=0, \quad
\forall \hat z\in \mathcal H\cap\Gamma^*_0.$$
Moreover, for some $\widetilde x\in \mathcal H$, we assume that $$\label{begemot}
a(\widetilde x) \ne 0\quad \mbox{and}\quad a(x) = 0,\quad\forall x
\in \mathcal H\setminus \{\widetilde x\}.$$
The existence of such a function is proved in Proposition 9 of [@IY1]. Let polynomials $M_{1}(z)$ and $M_{3}(\overline z)$ satisfy $$\label{begemot2}
(\partial^{-1}_{\overline z}q_{1} -M_{1})(\widetilde x)=0, \quad \quad
(\partial^{-1}_{z}q_{1} - M_{3})(\widetilde x)= 0.$$
The holomorphic function $a_1$ and the antiholomorphic function $b_1$ are defined by formulae $a_1(z)=a_{1,1}(z)+a_{1,2}(z)+a_{1,3}(z)$ and $b_1(\overline
z)=b_{1,1}(\overline z)+b_{1,2}(\overline z) +b_{1,3}(\overline z)$ where $a_{1,1},b_{1,1}\in C^1(\overline \Omega)$ and $$\label{monica}
i\frac{\partial\psi}{\partial\nu}
a_{1,1}(z)-i\frac{\partial\psi}{\partial\nu} b_{1,1}(\overline
z)=-\frac{\partial (a+\overline
a)}{\partial\nu}+i\frac{\partial\psi}{\partial\nu}
\frac{a(\partial^{-1}_{\overline z}
q_{1}-M_{1})}{4\partial_z\Phi}-i\frac{\partial\psi}{\partial\nu}
\frac{\overline{a}(\partial^{-1}_{z} q_{1}-M_{3})}
{4\partial_{\overline z}\overline\Phi}
\quad\mbox{on}\,\,\Gamma_0^*$$ and $a_{1,2}(z,\tau),b_{1,2}(\overline z,\tau)\in C^1(\overline \Omega)$ for each $\tau$ are holomorphic and antiholomorphic functions such that $$b_{1,2}(\overline z,\tau)=-\frac{1}{8\pi}\int_{\partial\Omega}
\frac{(\nu_1-i\nu_2) a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})e^{\tau(\Phi-\overline\Phi)}}{(\overline\zeta-\overline
z)\partial_\zeta\Phi} d\sigma$$ and $$a_{1,2}(z,\tau)=-\frac{1}{8\pi}\int_{\partial\Omega}
\frac{(\nu_1+i\nu_2)\overline{a}(\partial_\zeta^{-1}q_{1}-M_{3})
e^{\tau(\overline\Phi-\Phi)}}{(\zeta-z)\partial
_{\overline\zeta}\overline\Phi} d\sigma.$$ Here the denominators of the integrands vanish in ${\cal H} \cap
\Gamma^*_0$, but thanks to the second condition in (\[LL\]) integrability is guaranteed. We represent the functions $a_{1,2}(z,\tau),b_{1,2}(\overline z,\tau)$ in the form $$a_{1,2}(z,\tau)=a_{1,2,1}(z)+a_{1,2,2}(z,\tau),\quad
b_{1,2}(\overline z,\tau)=b_{1,2,1}(\overline z)
+ b_{1,2,2}(\overline z,\tau),$$ where $$b_{1,2,1}(\overline z)=-\frac{1}{8\pi}\int_{\Gamma_0^*}
\frac{(\nu_1-i\nu_2) a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{(\overline\zeta-\overline z)\partial_\zeta\Phi}
d\sigma ,\quad a_{1,2,1}(z)=-\frac{1}{8\pi}\int_{\Gamma_0^*}
\frac{(\nu_1+i\nu_2)\overline{a}(\partial_\zeta^{-1}q_{1}-M_{3})
}{(\zeta-z)\partial _{\overline\zeta}\overline\Phi} d\sigma.$$
By (\[LL\]), the functions $b_{1,2,1},a_{1,2,1}$ belong to $C^1(\overline \Omega).$ By (\[kk\]) we have $$\Vert b_{1,2,1}(\cdot,\tau)\Vert_{L^2(\Omega)}+\Vert
a_{1,2,1}(\cdot,\tau)\Vert_{L^2(\Omega)}\rightarrow 0\quad\mbox{as}
\,\,\tau\rightarrow +\infty .$$
Finally $a_{1,3}(z,\tau),b_{1,3}(\overline z,\tau)\in W^1_2(
\Omega)$ for each $\tau$ are holomorphic and antiholomorphic functions respectively such that $$\begin{aligned}
\label{dom}
i\frac{\partial\psi}{\partial\nu}
a_{1,3}(z,\tau)-i\frac{\partial\psi}{\partial\nu} b_{1,3}(\overline
z,\tau)=\frac{i}{2\pi}\frac{\partial\psi}{\partial\nu}\int_{\Omega}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{(\overline\zeta-\overline z)}
d\xi_2d\xi_1\nonumber\\-\frac{i}{2\pi}
\frac{\partial\psi}{\partial\nu}\int_{\Omega}
\partial_{\overline\zeta}\left (\frac{\overline{a}(\partial^{-1}_{\overline
\zeta}q_{1}-M_{3})}{\partial_{\overline\zeta}\overline\Phi}\right)\frac{e^{\tau(\overline\Phi-\Phi)}}{(\zeta-z)}
d\xi_2d\xi_1\quad\mbox{on}\,\,\Gamma_0^*\end{aligned}$$ and $$\label{leopard}
\Vert a_{1,3}(\cdot,\tau)\Vert_{L^2(\Omega)}+\Vert
b_{1,3}(\cdot,\tau)\Vert_{L^2(\Omega)}=o(1)
\quad\mbox{as}\,\,\tau\rightarrow +\infty.$$
The inequality (\[leopard\]) follows from the asymptotic formula $$\begin{aligned}
\label{zopa}
\left\Vert \int_{\Omega}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}
d\xi_2d\xi_1\right\Vert_{W^{\frac 12}_2(\Gamma_0^*)} \nonumber\\
+ \left\Vert \int_{\Omega}
\partial_{\overline\zeta}\left (\frac{\overline{a}(\partial^{-1}_{\overline
\zeta}q_{1}-M_{3})}{\partial_{\overline\zeta}\overline\Phi}\right)
\frac{e^{\tau(\overline\Phi-\Phi)}}{\zeta-z}
d\xi_2d\xi_1\right\Vert_{W^{\frac 12}_2(\Gamma_0^*)}
=o(1)\quad\mbox{as}\,\,\tau\rightarrow +\infty.\end{aligned}$$
In order to prove (\[zopa\]) consider the function $e\in
C^\infty_0(\Omega)$ such that $e\equiv 1$ in some neighborhood of the set $\mathcal H\setminus\Gamma_0^*.$ The family of functions $\int_{\Omega} e
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}
d\xi_2d\xi_1\in C^\infty(\partial\Omega),$ are uniformly bounded in $\tau$ in $C^2(\partial\Omega)$ and by Proposition 2.4 of [@IUY] this function converges pointwisely to zero. Therefore $$\label{zopa1}
\left\Vert\int_{\Omega}e
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}
d\xi_2d\xi_1\right\Vert_{W^1_2(\partial\Omega)}
=o(1)\quad\mbox{as}\,\,\tau\rightarrow +\infty.$$
Integrating by parts we obtain $$\int_{\Omega}(1-e)
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}
d\xi_2d\xi_1 =
\frac{(1-e)}{\partial_z\Phi}
\partial_z\left (\frac{ a(\partial^{-1}_{\overline z}q_{1}-M_{1})}
{\tau\partial_z\Phi}\right)e^{\tau(\Phi-\overline\Phi)}$$ $$-\frac{1}{\tau}\int_{\Omega}\partial_{\zeta}\left(\frac{(1-e)}
{\partial_\zeta\Phi}\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right )\right)
\frac{e^{\tau(\Phi-\overline\Phi)}}
{\overline\zeta-\overline z}d\xi_2d\xi_1 .$$
Thanks to (\[1\]) and (\[LL\]), we have $$\label{zopa2}
\left\Vert \frac{1-e}{\partial_z\Phi}
\partial_z\left (\frac{ a(\partial^{-1}_{\overline z}q_{1}-M_{1})}
{\tau\partial_z\Phi}\right
)e^{\tau(\Phi-\overline\Phi)}\right\Vert_{W^{\frac
12}_2(\Gamma_0^*)}=o(1)\quad\mbox{as}\,\,\tau\rightarrow +\infty.$$
The functions $\partial_{\zeta}\left(\frac{1-e}{\partial_\zeta\Phi}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\right)e^{\tau(\Phi-\overline\Phi)}$ are bounded in $L^p(\Omega)$ uniformly in $\tau.$ Therefore by Proposition 2.2 of [@IUY], the functions $\int_{\Omega}\partial_{\zeta}\left(\frac{1-e}{\partial_\zeta\Phi}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}\right)
d\xi_2d\xi_1$ are uniformly bounded in $W^1_p(\Omega).$ The trace theorem yields $$\label{zopa3}
\left\Vert\frac{1}{\tau}\int_{\Omega}\partial_{\zeta}\left(
\frac{1-e}{\partial_\zeta\Phi}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}
{\overline\zeta-\overline z}\right) d\xi_2d\xi_1\right\Vert
_{W^{\frac 12}_2(\Gamma_0^*)}=o(1)\quad
\mbox{as}\,\,\tau\rightarrow +\infty.$$ By (\[zopa1\])-(\[zopa3\]) we obtain (\[zopa\]).
We note that by (\[LL\]) the function $\frac{a}{\partial_z\Phi}\in
C^2(\partial\Omega).$ We define the function $U_1$ by the formula $$\begin{aligned}
\label{mozilaal}
U_1(x)=e^{\tau{\Phi}}{(a+a_{1}/\tau)} +e^{\tau\overline{\Phi}}
{(\overline a+b_{1}/\tau)}
%- \bigg( e^{\tau\Phi}\frac{e_2(\partial^{-1}_{\overline z}
%(aq_{1})-M_{1})}{4\tau\partial_z\Phi} +
% e^{\tau\overline\Phi}\frac{e_2(\partial^{-1}_{z}
%(\overline{a}q_{1})-M_{3})}{4\tau\overline{\partial_z\Phi}} \bigg
%)\nonumber\\
- \frac 12e^{\tau\Phi}\widetilde {\mathcal
R}_\tau\{{a(\partial^{-1}_{\overline z} q_{1}-M_{1})}\} - \frac 12
e^{\tau\overline\Phi}{\mathcal
R}_\tau\{\overline{a}(\partial^{-1}_{z} q_{1}-M_{3})\}.\end{aligned}$$ Integrating by parts, we obtain the following: $$\label{01}
e^{\tau\Phi}\widetilde {\mathcal R}_\tau
\{{a(\partial^{-1}_{\overline z} q_{1}-M_{1})}\}
= \frac{1}{\tau}\Biggl(2b_{1,2}e^{\tau\overline\Phi}
+ \frac{e^{\tau\Phi}a(\partial^{-1}_{\overline z}
q_{1}-M_{1})}{2\partial_z\Phi}$$ $$+ \frac{e^{\tau\overline\Phi}}{2\pi}\int_{\Omega}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}
d\xi_2d\xi_1\Biggr)$$ and $$\label{022}
e^{\tau\overline\Phi}{\mathcal R}_\tau\{\overline{a}(\partial^{-1}_{z}
q_{1}-M_{3})\}
= \frac{1}{\tau}\Biggl(2a_{1,2}e^{\tau\Phi}+\frac
{e^{\tau\overline\Phi}\overline{a}(\partial^{-1}_{z}
q_{1}-M_{3})}{2\partial_{\overline
z}\overline\Phi}$$ $$+ \frac{e^{\tau\Phi}}{2\pi}\int_{\Omega}
\partial_{\overline\zeta}\left (\frac{\overline{a}(\partial^{-1}_{\overline
\zeta}q_{1}-M_{3})}{\partial_{\overline\zeta}\overline\Phi}\right)
\frac{e^{\tau(\overline\Phi-\Phi)}}{\zeta-z} d\xi_2d\xi_1\Biggr).$$
We claim that $$\begin{aligned}
\label{PPPP}
\left\Vert\frac{e^{-i\tau\psi}}{2\pi}\int_{\Omega}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}
d\xi_2d\xi_1\right\Vert_{L^2(\Omega)}\nonumber\\
+ \left\Vert
\frac{e^{i\tau\psi}}{2\pi}\int_{\partial\Omega}
\partial_{\overline\zeta}\left (\frac{\overline{a}(\partial^{-1}_{\overline
\zeta}q_{1}-M_{3})}{\partial_{\overline\zeta}\overline\Phi}
\right)\frac{e^{\tau(\overline\Phi-\Phi)}}{\zeta-z}
d\xi_2d\xi_1\right\Vert_{L^2(\Omega)} \rightarrow 0\quad
\mbox{as}\,\,\tau\rightarrow + \infty.\end{aligned}$$ We prove the asymptotic formula (\[PPPP\]) for the first term. The proof of the asymptotic for the second term is the same. Denote $r_\tau(\xi)=\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)e^{\tau(\Phi-\overline\Phi)}.$ By (\[22\]), (\[begemot\]) and (\[begemot2\]), the family of these functions is bounded in $L^p(\Omega)$ for any $p<2.$ Hence by Proposition 2.2 of [@IUY] there exists a constant $C$ independent of $\tau$ such that $$\label{mk}
\left\Vert\frac{e^{-i\tau\psi}}{2\pi}\int_{\Omega}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}
d\xi_2d\xi_1\right\Vert_{L^4(\Omega)}\le C.$$
By (\[22\]), (\[begemot\]) and (\[begemot2\]), for any $z\ne
\widetilde x_1+i\widetilde x_2$, the function $r_\tau(\xi)/(\bar
\zeta-\bar z)$ belongs to $L^1(\Omega).$ Therefore by Proposition 2.4 of [@IUY], we have $$\label{mk1}
\frac{e^{-i\tau\psi}}{2\pi}\int_{\Omega}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}
d\xi_2d\xi_1\rightarrow 0\quad \mbox{a.e. in }\quad \Omega.$$
From (\[mk\]), (\[mk1\]) and Egorov’s theorem, the asymptotic for the first term in (\[PPPP\]) follows immediately.
We set $$g_\tau=q_1(e^{i\tau\psi}{a_{1}/\tau} +e^{-i\tau\psi} {b_{1}/\tau}-
\frac{e^{i\tau\psi}}{2}\widetilde {\mathcal
R}_\tau\{{a(\partial^{-1}_{\overline z} q_{1}-M_{1})}\} -
\frac{e^{-i\tau\psi}}{2}{\mathcal
R}_\tau\{\overline{a}(\partial^{-1}_{z} q_{1}-M_{3})\}).$$ By (\[01\])-(\[PPPP\]) we have $$\label{inka}\Vert g_\tau\Vert_{L^2(\Omega)}=O(\frac 1\tau
)\quad\mbox{as}\,\,\tau\rightarrow +\infty.$$ Short computations give $$\label{lob}
L_1(x,D)U_1=e^{\tau\varphi}g_\tau\quad \mbox{in}\,\,\Omega,\quad
\frac{\partial
U_1}{\partial\nu}\vert_{\Gamma_0}=e^{\tau\varphi}O_{W_2^\frac
12(\overline{\Gamma^*_0})}(\frac
1\tau)\quad\mbox{as}\,\,\tau\rightarrow +\infty.$$ Indeed, the first equation in (\[lob\]) follows from (\[mozilaal\]), (\[begemot\]) and the factorization of the Laplace operator in the form $\Delta=4\partial_{\overline z}\partial_z.$ In order to prove the second equation in (\[lob\]) we set $\frac{\partial U_1}{\partial\nu}=I_1+I_2$ where $$\begin{aligned}
\label{luba}
I_1=\frac{\partial}{\partial\nu}((a+a_{1,1}/\tau)e^{\tau\Phi}+(\overline
a+b_{1,1}/\tau)e^{\tau\overline\Phi}) \nonumber\\
= e^{\tau\varphi}\left(i\tau\frac{\partial\psi}{\partial\nu}(a+a_{1,1}/\tau)
-i\tau\frac{\partial\psi}{\partial\nu}(\overline
a+b_{1,1}/\tau)+\frac{\partial}{\partial\nu}(a+a_{1,1}/\tau)
+ \frac{\partial}{\partial\nu}(\overline a+b_{1,1}/\tau)\right)
\nonumber\\
= \left(i\frac{\partial\psi}{\partial\nu}
\frac{a(\partial^{-1}_{\overline z}
q_{1}-M_{1})}{4\partial_z\Phi}-i\frac{\partial\psi}{\partial\nu}\frac
{\overline{a}(\partial^{-1}_{z} q_{1}-M_{3})}{4\partial_{\overline
z}\overline\Phi}\right)e^{\tau\varphi}
+ e^{\tau\varphi}O_{C^1(\overline\Gamma^*_0)}(\frac 1\tau).\end{aligned}$$ In order to obtain the last equality, we used (\[LL\]) and (\[monica\]). Then $$\begin{aligned}
\label{luba1}
I_2 = \frac{\partial}{\partial\nu}((a_{1,2}+a_{1,3})e^{\tau\Phi}
+(b_{1,2}+b_{1,3})e^{\tau\overline\Phi})
- \frac 12
\frac{\partial}{\partial\nu}(e^{\tau\Phi}\widetilde {\mathcal
R}_\tau\{{(a(\partial^{-1}_{\overline z}
q_{1}-M_{1})}\} \nonumber\\
+ e^{\tau\overline\Phi}{\mathcal
R}_\tau\{\overline{a}(\partial^{-1}_{z} q_{1}-M_{3})\})\nonumber\\
= -\frac 12\frac{\partial}{\partial\nu}\Biggl(
\frac{e^{\tau\Phi}a(\partial^{-1}_{\overline z}
q_{1}-M_{1})}{2\partial_z\Phi}+\frac{e^{\tau\overline\Phi}}{2\pi}
\int_{\Omega}
\partial_\zeta\left (\frac{ a(\partial^{-1}_{\overline
\zeta}q_{1}-M_{1})}{\partial_\zeta\Phi}\right
)\frac{e^{\tau(\Phi-\overline\Phi)}}{\overline\zeta-\overline z}
d\xi_2d\xi_1 \nonumber\\
+ \frac{e^{\tau\overline\Phi}\overline{a}(\partial^{-1}_{z}
q_{1}-M_{3})}{2\partial_{\overline
z}\overline\Phi}+\frac{e^{\tau\Phi}}{2\pi}\int_{\Omega}
\partial_{\overline\zeta}\left (\frac{\overline{a}(\partial^{-1}_{\overline
\zeta}q_{1}-M_{3})}{\partial_{\overline\zeta}\overline\Phi}\right)
\frac{e^{\tau(\overline\Phi-\Phi)}}{\zeta-z} d\xi_2d\xi_1\Biggr)
\nonumber\\
= -i\frac{\partial\psi}{\partial\nu}\left(\frac
{{a}(\partial^{-1}_{\overline z} q_{1}-M_{1})}{4\partial_{
z}\Phi}-\frac {e^{\tau\overline\Phi}\overline{a}(\partial^{-1}_{z}
q_{1}-M_{3})}{4\partial_{\overline z}\overline\Phi}\right) +
O_{W_2^{\frac 12}(\Gamma^*_0)}(\frac 1\tau).\end{aligned}$$ From (\[luba\]) and (\[luba1\]), we obtain the second equation in (\[lob\]).
Finally we construct the last term of the complex geometric optics solution $e^{\tau\varphi}w_\tau.$ Consider the boundary value problem $$\label{lena}
L_{q_1}(x,D)(w_\tau e^{\tau\varphi})=-g_\tau
e^{\tau\varphi}\quad\mbox{in}\,\,\Omega,\quad \frac{\partial (w_\tau
e^{\tau\varphi})}{\partial\nu}\vert_{\Gamma_0}=-\frac{\partial
U_1}{\partial\nu}.$$
By (\[inka\]) and Proposition \[Theorem 2.3\], there exists a solution to problem (\[lena\]) such that $$\label{ioio}
\Vert w_\tau\Vert_{L^2(\Omega)}=o(\frac 1\tau) \quad
\mbox{as}\,\tau\rightarrow +\infty.$$ Finally we set $$\label{ioioio}
u_1=U_1+e^{\tau\varphi} w_\tau.$$ By (\[ioio\]), (\[ioioio\]), (\[leopard\]) and (\[mozilaal\])-(\[022\]), we can represent the complex geometric optics solution $u_1$ in the form $$\begin{aligned}
\label{mozilaall}
u_1(x)=e^{\tau{\Phi}}{(a+(a_{1,1}+a_{1,2,1})/\tau)}
+e^{\tau\overline{\Phi}} {(\overline a+(b_{1,1}+b_{1,2,1})/\tau)}\nonumber\\
- \bigg( e^{\tau\Phi}\frac{a(\partial^{-1}_{\overline z}
q_{1}-M_{1})}{4\tau\partial_z\Phi} +
e^{\tau\overline\Phi}\frac{\overline{a}(\partial^{-1}_{z}
q_{1}-M_{3})}{4\tau\overline{\partial_z\Phi}} \bigg
)+e^{\tau\varphi}o_{L^2(\Omega)}(\frac 1\tau)\quad
\mbox{as}\,\tau\rightarrow +\infty.\end{aligned}$$
Since the Cauchy data (\[popo\]) for the potentials $ q_1$ and $q_2$ are equal, there exists a solution $u_2$ to the Schrödinger equation with potential $q_2$ such that $\frac{\partial
u_1}{\partial \nu}=\frac{\partial u_2}{\partial\nu}$ on $\partial\Omega$ and $ u_1= u_2$ on $\widetilde \Gamma$. Setting $u=u_1-u_2$, we obtain $$\label{pp}
(\Delta+q_2)u=(q_2-q_1)u_1\quad \mbox{in}\,\,\Omega, \quad
u\vert_{\widetilde\Gamma}=\frac{\partial u}{\partial
\nu}\vert_{\partial\Omega}=0.$$ In a similar way to the construction of $u_1$, we construct a complex geometrical optics solution $v$ for the Schrödinger equation with potential $q_2.$ The construction of $v$ repeats the corresponding steps of the construction of $u_1.$ The only difference is that instead of $q_{1}$ and $\tau$, we use $q_{2}$ and $-\tau,$ respectively. We skip the details of the construction and point out that similarly to (\[mozilaall\]) it can be represented in the form $$\begin{aligned}
\label{mozilaa}
v(x)=e^{-\tau{\Phi}}{(a+(\widetilde a_{1,1}+\widetilde
a_{1,2,1})/\tau)}
+e^{-\tau\overline{\Phi}} {(\overline a+(\widetilde b_{1,1}
+\widetilde b_{1,2,1})/\tau)}\nonumber\\
+ \left (e^{-\tau\Phi}\frac{a(\partial^{-1} _{\overline z}
q_{2}-M_{2})}{4\tau\partial_z\Phi} +e^{-\tau\overline\Phi}
\frac{\overline{a}(\partial^{-1}_{z} q_{2}-M_{4})}
{4\tau\overline{\partial_z\Phi}}\right
)+e^{-\tau\varphi}o_{L^2(\Omega)}(\frac 1\tau) \quad
\mbox{as}\,\tau\rightarrow +\infty,\quad \frac{\partial
v}{\partial\nu}\vert_{\Gamma_0}=0,\end{aligned}$$ where $M_{2}(z)$ and $M_{4}(\overline z)$ satisfy $$(\partial^{-1}_{\overline z}q_{2} -M_{2})(\widetilde x)=0, \quad \quad
(\partial^{-1}_{z} q_{2} - M_{4})(\widetilde x)= 0.$$ The functions $\widetilde a_1(z)=\widetilde a_{1,1}(z)+\widetilde
a_{1,2}(z)$ and $\widetilde b_1(z)=\widetilde b_{1,1}(z)+\widetilde
b_{1,2}(z)$ are given by $$-i\frac{\partial\psi}{\partial\nu} \widetilde a_{1,1}(z)
+i\frac{\partial\psi}{\partial\nu}
\widetilde b_{1,1}(\overline z)=-\frac{\partial (a+\overline a)}
{\partial\nu}+i\frac{\partial \psi}{\partial\nu}
\frac{a(\partial^{-1} _{\overline z}
q_{2}-M_{2})}{4\tau\partial_z\Phi} -i\frac{\partial
\psi}{\partial\nu} \frac{\overline{a}(\partial^{-1}_{z}
q_{2}-M_{4})} {4\tau\overline{\partial_z\Phi}}
\quad\mbox{on}\,\,\Gamma_0,$$ $$\quad \widetilde a_{1,1}, \widetilde b_{1,1}\in
C^1(\overline\Omega)$$ and $\widetilde a_{1,2,1}(z),
\widetilde b_{1,2,1}(\overline z)\in C^1(\overline \Omega)$ are holomorphic functions such that $$\widetilde b_{1,2,1}(\overline z)=\frac{1}{8\pi}\int_{\Gamma_0^*}
\frac{(\nu_1-i\nu_2) a(\partial^{-1}_{\overline
\zeta}q_{2}-M_{2})e^{\tau(\Phi-\overline\Phi)}}{(\overline\zeta-\overline
z)\partial_\zeta\Phi} d\sigma$$ and $$\widetilde a_{1,2,1}(z)=\frac{1}{8\pi}\int_{\Gamma_0^*}
\frac{(\nu_1+i\nu_2)\overline{a}(\partial^{-1}_\zeta
q_{2}-M_{4})e^{\tau(\overline\Phi-\Phi)}}{(\zeta-z)\partial_{\overline\zeta}
\overline\Phi} d\sigma.$$
Denote $q=q_1-q_2.$ Taking the scalar product of equation (\[pp\]) with the function $v$, we have: $$\int_\Omega q u_1vdx=0.$$ From formulae (\[mozilaall\]) and (\[mozilaa\]) in the construction of complex geometrical optics solutions, we have $$\begin{aligned}
\label{lala}
0=\int_\Omega q u_1vdx = \int_\Omega q(a^2+\overline
a^2)dx\nonumber\\
+ \frac 1\tau\int_\Omega q(
a(a_{1,1}+a_{1,2,1}+b_{1,1}+b_{1,2,1})+\overline{a}(\widetilde
a_{1,1}+\widetilde a_{1,2,1}+\widetilde b_{1,1}+\widetilde
b_{1,2,1}))dx \nonumber\\
+ \int_\Omega q(a\overline a e^{2\tau i\psi}+ a\overline a e^{-2\tau i\psi})dx
\nonumber\\
+ \frac{1}{4\tau}\int_{\Omega} \left( qa^2 \frac{\partial_{\overline z}
^{-1}q_{2}-M_{2}} {\partial_z\Phi} + q\overline{a}^2
\frac{\partial_{z}^{-1}q_{2}
-{M_{4}}}{\overline{\partial_z\Phi}}\right)dx \nonumber\\
- \frac{1}{4\tau}\int_\Omega\left( qa^2\frac{\partial_{\overline z}^{-1}
q_{1}-M_{1}}{\partial_z\Phi} +q\overline
a^2\frac{\partial_{z}^{-1}q_{1}-{
M_{3}}}{\overline{\partial_z\Phi}}\right)dx\nonumber\\
+ o(\frac{1}{\tau})=0\quad\mbox{as}\,\,\tau \rightarrow +\infty.\end{aligned}$$
Since the potentials $q_j$ are not necessarily from $C^2(\overline\Omega)$, we can not directly use the stationary phase argument (e.g., Evans [@E]). Let function $\hat q\in
C^\infty_0(\Omega)$ satisfy $\hat q(\widetilde x)=q(\widetilde x).$ We have $$\label{rono}
\int_\Omega q\mbox{Re}\,(a\overline a e^{2\tau i\psi})dx
=\int_\Omega \hat q\mbox{Re}\,(a\overline a e^{2\tau
i\psi})dx+\int_\Omega (q-\hat q)\mbox{Re}\,(a\overline a e^{2\tau
i\psi})dx.$$ Using the stationary phase argument and (\[begemot\]), similarly to [@IUY], we obtain $$\label{masa}
\int_\Omega \hat q(a\overline a e^{2\tau i\psi} + a\overline a
e^{-2\tau i\psi})dx=\frac{2\pi (q\vert a\vert^2)(\widetilde
x)\mbox{Re}\,e^{2{\tau} i\psi(\widetilde x)}} {{\tau}
\vert(\mbox{det}\thinspace \psi'')(\widetilde x)\vert^\frac
12}+o\left(\frac{1}{{\tau}}\right)\quad\mbox{as}\,\,\tau \rightarrow
+\infty.$$ For the second integral in (\[rono\]) we obtain $$\int_\Omega (q-\hat q)(a\overline a e^{2\tau i\psi}+ a\overline a
e^{-2\tau i\psi})dx=\int_\Omega (q-\hat q)\left (a\overline a \frac
{(\nabla\psi,\nabla)e^{2\tau i\psi}}{2\tau i\vert
\nabla\psi\vert^2}-a\overline a \frac {(\nabla\psi,\nabla)e^{-2\tau
i\psi}}{2\tau i\vert \nabla\psi\vert^2}\right )dx$$ $$= \int_{\partial\Omega} q\left (a\overline a \frac
{(\nabla\psi,\nu)e^{2\tau i\psi}}{2\tau i\vert
\nabla\psi\vert^2}-a\overline a \frac {(\nabla\psi,\nu)e^{-2\tau
i\psi}}{2\tau i\vert \nabla\psi\vert^2}\right )d\sigma$$ $$\label{opl}-\frac{1}{2\tau i}
\int_\Omega\left \{ e^{2\tau i\psi}\mbox{div}\,\left ((q-\hat
q)a\overline a \frac {\nabla\psi}{\vert \nabla\psi\vert^2}\right )
- e^{-2\tau i\psi}\mbox{div}\,\left ( (q-\hat q
)a\overline a \frac{\nabla\psi}{\vert \nabla\psi\vert^2}\right )\right\}dx.$$ Since $\psi\vert_{\Gamma_0}=0$ we have $$\int_{\partial\Omega} qa\overline a \left (\frac
{(\nabla\psi,\nu)e^{2\tau i\psi}}{2\tau i\vert \nabla\psi\vert^2}-
\frac {(\nabla\psi,\nu)e^{-2\tau i\psi}}{2\tau i\vert
\nabla\psi\vert^2}\right )d\sigma=\int_{\widetilde\Gamma}
\frac{qa\overline a}{2\tau i\vert \nabla\psi\vert^2}
(\nabla\psi,\nu)(e^{2\tau i\psi}- e^{-2\tau i\psi})d\sigma.$$ By (\[1\]), (\[kk\]) and Proposition 2.4 in [@IUY] we conclude that $$\nonumber
\int_{\partial\Omega} qa\overline a \left (\frac
{(\nabla\psi,\nu)e^{2\tau i\psi}}{2\tau i\vert \nabla\psi\vert^2}-
\frac {(\nabla\psi,\nu)e^{-2\tau i\psi}}{2\tau i\vert
\nabla\psi\vert^2}\right )d\sigma=o(\frac 1\tau)\quad
\mbox{as}\,\,\tau\rightarrow+\infty.$$ The last integral over $\Omega$ in formula (\[opl\]) is $o(\frac{1}{\tau})$ and therefore $$\label{-3}
\int_\Omega(q-\hat q)(a\overline a e^{2\tau i\psi} + a\overline a
e^{-2\tau i\psi})dx=o(\frac 1\tau)\quad\mbox{as}\,\,\tau \rightarrow
+\infty.$$ Taking into account that $\psi(\widetilde x)\ne 0$ and using (\[masa\]), (\[-3\]) we have from (\[lala\]) that $$\frac{2\pi (q\vert a\vert^2)(\widetilde x)} {
\vert(\mbox{det}\thinspace \psi'')(\widetilde x)\vert^\frac 12}=0.$$ Hence $q(\widetilde x)=0.$ In [@IUY1] it is proved that there exists a holomorphic function $\Phi$ such that (\[1\])-(\[kk\]) are satisfied and a point $\widetilde
x\in \mathcal H$ can be chosen arbitrarily close to any given point in $\Omega$ (see [@IUY]). Hence we have $q\equiv 0.$ The proof of the theorem is completed. $\square$
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[^1]: Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO 80523-1874, USA e-mail: oleg@math.colostate.edu
[^2]: Department mathematics, UC Irvine, Irvine CA 92697 Department of Mathematics, University of Washington, Seattle, WA 98195 USA e-mail: gunther@math.washington.edu The second author partly supported by NSF and a Walker Family Endowed Professorship
[^3]: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153, Japan e-mail: myama@ms.u-tokyo.ac.jp
|
---
author:
- |
L. Martínez Alonso$^{1}$ and E. Medina$^{2}$\
*$^1$ Departamento de Física Teórica II, Universidad Complutense*\
*E28040 Madrid, Spain*\
*$^2$ Departamento de Matemáticas, Universidad de Cádiz*\
*E11510 Puerto Real, Cádiz, Spain*
title: ' A common integrable structure in the hermitian matrix model and Hele-Shaw flows [^1]'
---
Introduction
============
The Toda hierarchy represents a relevant integrable structure which emerges in several random matrix models [@ger]-[@avm]. Thus, the partition functions $$\label{1}
Z_N(\mbox{Hermitian})=\int {\mathrm{d}}H \exp\Big(\mbox{tr}(\sum_{k\geq
1}t_k\,H^k)\Big),$$ $$\label{2}
Z_N(\mbox{Normal})=\int {\mathrm{d}}M\,{\mathrm{d}}M^{\dagger}
\exp\Big(\mbox{tr}\,(M\,M^{\dagger}+\sum_{k\geq
1}(t_k\,M^k+\bar{t}_k\,M^{\dagger\,k}))\Big),$$ of the hermitian ($H=H^{\dagger}$) and the normal matrix models ($[M,M^{\dagger}]=0$) , where $N$ is the matrix dimension, are tau-functions of the 1-Toda and 2-Toda hierarchy, respectively. As a consequence of this connection new facets of the Toda hierarchy have been discovered. Thus the analysis of the large $N$-limit of the Hermitian matrix model lead to introduce an interpolated continuous version of the 2-Toda hierarchy: the *dispersionful* 2-Toda hierarchy (see for instance [@tt]). On the other hand, the leading contribution to the large $N$-limit (planar contribution) motivated the introduction of a *classical* version of the Toda hierarchy [@tt] which is known as the *dispersionless* 2-Toda (d2-Toda) hierarchy.
Laplacian growth processes describe evolutions of two-dimensional domains driven by harmonic fields. It was shown in [@zab1] that the d2-Toda is a relevant integrable structure in Laplacian growth problems and conformal maps dynamics. For example, if a given analytic curve $\gamma\, \,(z=z(p),\, |p|=1)$ is the boundary of a simply-connected bounded domain, then $\gamma$ evolves with respect to its harmonic moments according to a solution of the d2-Toda hierarchy. These solutions are characterized by the string equations $$\label{11}
\bar{z}=m,\quad \overline{m}=-z.$$ Here $(z,m)$ and $(\bar{z},\bar{m})$ denote the two pairs of Lax-Orlov operators of the d2-Toda hierarchy. As it was noticed in [@zab1]-[@wz], this integrable structure also emerges in the planar limit of the normal matrix model and describes the evolution of the support of eigenvalues under a change of the parameters $t_k$ of the potential.
The present paper is motivated by the recent discovery [@lee] of an integrable structure provided by the dispersionless AKNS hierarchy which describes the bubble break-off in Hele-Shaw flows. In this work we prove that this integrable structure is also characterized by the solution of a pair of string equations $$\label{trii}
z=\bar{z},\quad m=\overline{m},$$ of the d2-Toda hierarchy. Since the system describes the planar limit of , it constitutes a common integrable structure arising in the Hermitian matrix model and the theory of Hele-Shaw flows.
Our strategy is inspired by previous results [@mel3]-[@mano2] on solution methods for dispersionless string equations. We also develop some useful standard technology of the theory of Lax equations in the context of the d2-Toda hierarchy.
The paper is organized as follows:
In the next section the basic theory of the d2-Toda hierarchy, the method of string equations and the solution of are discussed. In Section 3 we show how the solution of appears in the planar limit of the Hermitian matrix model and the Hele-Shaw bubble break-off processes studied in [@lee].
The dispersionless Toda hierarchy
==================================
String equations in the d2-Toda hierarchy
-----------------------------------------
The dispersionless d2-Toda hierarchy[@tt] can be formulated in terms of two pairs $(z,m)$ and $(\bar{z},\overline{m})$ of Lax-Orlov functions, where $z$ and $\bar{z}$ are series in a complex variable $p$ of the form $$\label{d0a}
z=p+u+\dfrac{u_1}{p}+\cdots,\quad
\bar{z}=\dfrac{v}{p}+v_0+v_1\,p+\cdots,$$ while $m$ and $\bar{m}$ are series in $z$ and $\bar{z}$ of the form $$\label{act0}
m:=\sum_{j=1}^\infty j\, t_j z^{j-1}+\dfrac{x}{z}+\sum_{j\geq
1}\dfrac{S_{j+1}}{z^{j+1}} ,\quad \overline{m} :=\sum_{j=1}^\infty
j\,\bar{t}_j \bar{z}^{j-1}-\dfrac{x}{\bar{z}}+\sum_{j\geq
1}\dfrac{\bar{S}_{j+1}}{\bar{z}^{j+1}}.$$ The coefficients in the expansions and depend on a complex variable $x$ and two infinite sets of complex variables ${\boldsymbol{\mathrm{t}}}:=(t_1,t_2,\ldots)$ and ${\bar{\boldsymbol{\mathrm{t}}}}:=(\bar{t}_1,\bar{t}_2,\ldots)$. The d2-Toda hierarchy is encoded in the equation $$\label{2.a}
{\mathrm{d}}z\wedge{\mathrm{d}}m={\mathrm{d}}\bar{z}\wedge{\mathrm{d}}\overline{m}= {\mathrm{d}}\Big( \log{p}\,{\mathrm{d}}x+ \sum_{j=1}^\infty\Big( (z^{j})_{+}\,{\mathrm{d}}t_j+(\bar{z}^{j})_{-}\,{\mathrm{d}}\bar{t}_j\Big)\Big).$$ Here the $(\pm)$ parts of $p$-series denote the truncations in the positive and strictly negative power terms, respectively. As a consequence there exist two *action* functions $S$ and $\bar{S}$ verifying $$\begin{aligned}
{\mathrm{d}}S&=m\,{\mathrm{d}}z+\log{p}\,{\mathrm{d}}x+ \sum_{j=1}^\infty\Big(
(z^{j})_{+}\,{\mathrm{d}}t_j+(\bar{z}^{j})_{-}\,{\mathrm{d}}\bar{t}_j\Big), \\
{\mathrm{d}}\bar{S}&=\overline{m}\,{\mathrm{d}}\bar{z}+\log{p}\,{\mathrm{d}}x+\sum_{j=1}^\infty \Big( (z^{j})_{+}\,{\mathrm{d}}t_j+(\bar{z}^{j})_{-}\,{\mathrm{d}}\bar{t}_j\Big),\end{aligned}$$ and such that they admit expansions $$\label{act}
S =\sum_{j=1}^\infty t_j z^{j}+x\,\log{z}-\sum_{j\geq
1}\dfrac{S_{j+1}}{jz^{j}},\quad \bar{S} =\sum_{j=1}^\infty \bar{t}_j
\bar{z}^{j}-x\,\log{\bar{z}}-\bar{S}_0+\sum_{j\geq
1}\dfrac{\bar{S}_{j+1}}{j\bar{z}^{j+1}}.$$ From one derives the d2-Toda hierarchy in Lax form $$\label{d3}
\dfrac{\partial \mathcal{K}}{\partial
t_j}=\{(z^j)_+,\mathcal{K}\},\quad \dfrac{\partial
\mathcal{K}}{\partial \bar{t}_j}=\{(\bar{z}^j)_-,\mathcal{K}\},$$ where $\mathcal{K}=z,\,m,\,\bar{z},\,\overline{m}$, and we are using the Poisson bracket $\{f,g\}:=p\,(f_p\,g_x-f_x\,g_p)$.
The following result was proved by Takasaki and Takebe (see [@tt]):
Let $(P(z,m),Q(z,m))$ and $(\overline{P}(\bar{z},\overline{m}),\overline{Q}
(\bar{z},\overline{m})))$ be functions such that $$\{P,Q\}=\{z,m\} ,\quad
\{\overline{P},\overline{Q}\}=\{\bar{z},\overline{m}\}.$$ If $(z,m)$ and $(\bar{z},\overline{m})$ are functions which can be expanded in the form - and satisfy the pair of constraints $$\label{dstring}
P(z,m)=\overline{P}(\bar{z},\overline{m}),\quad
Q(z,m)=\overline{Q}(\bar{z},\overline{m}) ,$$ then they verify $\{z,m\}=\{\bar{z},\overline{m}\}=1$ and are solutions of the Lax equations for the d2-Toda hierarchy .
Constraints of the form are called *dispersionless string equations*. In this paper we are concerned with the system . The first equation $z=\bar{z}$ of defines the 1-Toda reduction of the d2-Toda hierarchy $$\label{d4}
z=\bar{z}=p+u+\dfrac{v}{p},$$ where $$\label{d6}
u=\partial_x S_2,\quad \log{v}=-\partial_x\bar{S}_0.$$ As a consequence the Lax equations imply that $u$ and $v$ depend on $({\boldsymbol{\mathrm{t}}},{\bar{\boldsymbol{\mathrm{t}}}})$ through the combination ${\boldsymbol{\mathrm{t}}}-{\bar{\boldsymbol{\mathrm{t}}}}$.
Due to there are two branches of $p$ as a function of $z$ $$\begin{aligned}
\label{d5}
\nonumber
&p(z)=\dfrac{1}{2}\Big((z-u)+\sqrt{(z-u)^2-4v}\Big)=z-u-\dfrac{v}{z}+\cdots\\\\
\nonumber
&\bar{p}(z)=\dfrac{1}{2}\Big((z-u)-\sqrt{(z-u)^2-4v}\Big)=\dfrac{v}{z}+\cdots.\end{aligned}$$ To characterize the members of the d1-Toda hierarchy of integrable systems as well as to solve the string equations it is required to determine $(z^j)_{-}(p(z))$ and $(z^j)_{+}(\bar{p}(z))$ in terms of $(u,v)$. By using it is clear that there are functions $\alpha_j,\,\beta_j,\,\bar{\alpha}_j, \, \bar{\beta}_j)$, which depend polynomially, in $z$ such that $$\begin{aligned}
\partial_{t_j} S(z)=(z^j)_+(p(z))&=\alpha_j+\beta_j\,p(z),\quad
\partial_{\bar{t}_j} S(z)=(z^j)_-(p(z))=\bar{\alpha}_j+\bar{\beta}_j\,p(z),\\
\partial_{t_j} \bar{S}(z)=(z^j)_+(\bar{p}(z))&=\alpha_j+\beta_j\,\bar{p}(z),\quad
\partial_{\bar{t}_j} \bar{S}(z)=(z^j)_-(\bar{p}(z))=\bar{\alpha}_j+\bar{\beta}_j\,\bar{p}(z),\end{aligned}$$ and $$\label{d8}
\bar{\alpha}_j=z^j-\alpha_j,\quad \bar{\beta}_j=-\beta_j.$$ Now we have $$\label{d7}
\alpha_j+\beta_j\,p(z)=\partial_{t_j} S(z)=
z^j+\mathcal{O}\Big(\dfrac{1}{z}\Big),\quad
\alpha_j+\beta_j\,\bar{p}(z)=\partial_{t_j}\bar{S}(z)
=-\partial_{t_j} \bar{S}_0+ \mathcal{O}\Big(\dfrac{1}{z}\Big),$$ so that $$\label{d7aa}
\alpha_j=\dfrac{1}{2}\Big(z^j-\partial_{t_j}
\bar{S}_0-(p+\bar{p})\,\beta_j\Big),\quad
\beta_j=\Big(\dfrac{z^j}{p-\bar{p}}\Big)_\oplus,$$ where $(\;)_\oplus$ and $(\;)_\ominus$ stand for the projection of $z$-series on the positive and strictly negative powers, respectively. Thus, by introducing the generating function $$\label{d7b}
R :=\dfrac{z}{p-\bar{p}}=\dfrac{z}{\sqrt{(z-u)^2-4v}}=\sum_{k\geq
0}\dfrac{r_k(u,v)}{z^k},\quad r_0=1.$$ we deduce $$\begin{aligned}
\label{d7c}
\nonumber
(z^j)_+(p(z))&=z^j-\dfrac{1}{2}\,\partial_{t_j} \bar{S}_0-\dfrac{z}{2\,R}\,\Big(z^{j-1}\,R\Big)_\ominus\\
&=z^j-\dfrac{1}{2}(\partial_{t_j}
\bar{S}_0+r_j)-\dfrac{1}{2\,z}\,(r_{j+1}-u\,r_j)+\mathcal{O}\Big(\dfrac{1}{z^2}\Big).\end{aligned}$$ Hence $$\partial_{t_j} \bar{S}_0=-r_j,\quad \partial_{t_j}
S_2=\dfrac{1}{2}\,(r_{j+1}-u\,r_j),$$ so that the equations of the $d1$-Toda hierarchy are given by $$\label{dto1}
\partial_{t_j} u=\dfrac{1}{2}\,\partial_x\,(r_{j+1}-u\,r_j),\quad
\partial_{t_j} v=v\,\partial_x\,r_{j}.$$
Furthermore, we have found $$\label{d7d0}
(z^j)_-(p(z))=
-\dfrac{1}{2}\,r_j+\dfrac{z}{2\,R}\,\Big(z^{j-1}\,R\Big)_\ominus,
\quad (z^j)_+(\bar{p}(z))=r_j+(z^j)_-(p(z)).$$ Hence, the first terms of their asymptotic expansions as $z\rightarrow\infty$ are $$\label{d7d}
(z^j)_-(p(z))=\dfrac{1}{2\,z}\,(r_{j+1}-u\,r_j)+\mathcal{O}\Big(\dfrac{1}{z^2}\Big),
\quad
(z^j)_+(\bar{p}(z))=r_j+\dfrac{1}{2\,z}\,(r_{j+1}-u\,r_j)+\mathcal{O}\Big(\dfrac{1}{z^2}\Big).$$ Notice that since $r_0=1$ and $ r_1=u$, these last equations hold for $j\geq 0$.
Hodograph solutions of the $1$-dToda hierarchy
----------------------------------------------
In the above paragraph we have used the first string equation of . Let us now deal with the second one. To this end we set $$m=\overline{m}= \sum_{j=1}^\infty j\,t_j\,(z^{j-1})_++
\sum_{j=1}^\infty j\,\bar{t}_j\,(z^{j-1})_-,$$ which leads to the following expressions for the Orlov functions $(m,\overline{m})$ $$\begin{aligned}
\label{d8a}
\nonumber &m(z)=\sum_{j=1}^\infty j\,t_j\,z^{j-1}+\sum_{j=1}^\infty
j\,(\bar{t}_j-t_j)\,(z^{j-1})_-(p(z)),\\\\
\nonumber &\overline{m}(z)=\sum_{j=1}^\infty
j\,\bar{t}_j\,z^{j-1}-\sum_{j=1}^\infty
j\,(\bar{t}_j-t_j)\,(z^{j-1})_+(\bar{p}(z)).\end{aligned}$$ In order to apply Theorem 1 we have to determine $u$ and $v$ and ensure that $(m,\overline{m})$ verify the correct asymptotic form -. Both things can be achieved by reducing to the form $$\begin{aligned}
\label{d9}
\nonumber &\dfrac{x}{z}+\sum_{j\geq
2}\dfrac{1}{z^j}S_j=\sum_{j=1}^\infty
j\,(\bar{t}_j-t_j)\,(z^{j-1})_-(p(z)),\\\\
\nonumber &-\dfrac{x}{z}+\sum_{j\geq
2}\dfrac{1}{z^j}\bar{S}_j=-\sum_{j=1}^\infty
j\,(\bar{t}_j-t_j)\,(z^{j-1})_+(\bar{p}(z)),\end{aligned}$$ and equating coefficients of powers of $z$. Indeed, from we see that identifying the coefficients of $z^{-1}$ in both sides of the two equations of yields the same relation. This equation together with the one supplied by identifying the coefficients of the constant terms in the second equation of provides the following system of *hodograph-type* equations to determine $(u,v)$ $$\label{ho}\begin{cases}
\sum_{j=1}^\infty j\,(\bar{t}_j-t_j)
r_{j-1}=0,\\\\
\dfrac{1}{2}\,\sum_{j=1}^\infty j\,(\bar{t}_j-t_j)\Big)\,r_j=x.
\end{cases}$$ It can be rewritten as $$\label{hoin}\everymath{\displaystyle}
\begin{cases}
\oint_{\gamma}\dfrac{dz}{2\pi i}
\dfrac{V_{z}}{\sqrt{(z-u)^2-4v}}\, =0,\\\\
\oint_{\gamma}\dfrac{dz}{2\pi
i}\dfrac{z\,V_{z}}{\sqrt{(z-u)^2-4v}}\, =-2\,x,
\end{cases}$$ where $\gamma$ is a large enough positively oriented closed path and $V_{z}$ denotes the derivative with respect to $z$ of the function $$\label{U}
V(z,{\boldsymbol{\mathrm{t}}}-{\bar{\boldsymbol{\mathrm{t}}}}):=\sum_{j=1}^\infty (t_j-\bar{t}_j)\Big)\,z^j.$$ The remaining equations arising from characterize the functions $S_j^{(0)}$ and $\overline{S}_j^{(0)}$ for $j\geq 1$ in terms of $(u,v)$. Therefore we have characterized a solution $(z,m)$ and $(\bar{z},\overline{m})$ of the system of string equations verifying the conditions of Theorem 1 and, consequently, it solves the d1-Toda hierarchy.
Planar limit of the Hermitian matrix model and bubble break-off in Hele-Shaw flows
==================================================================================
The Hermitian matrix model
--------------------------
If we write the partition function of the Hermitian matrix model in terms of eigenvalues and slow variables ${\boldsymbol{\mathrm{t}}}:=\epsilon\,{\boldsymbol{t}}$, where $\epsilon=1/N$, we get $$\label{mat}
Z_n(N\,{\boldsymbol{\mathrm{t}}})=\int_{\mathbb{R}^n}\prod_{k=1}^{n}\Big(d\,x_k\,e^{N\,V(x_k,{\boldsymbol{\mathrm{t}}})})\Big)(\Delta(x_1,\cdots,x_n))^2,\quad
V(z,{\boldsymbol{\mathrm{t}}}):=\sum_{k\geq 1}t_k\,z^k.$$ The large $N$-limit of the model is determined by the asymptotic expansion of $Z_n(N\,{\boldsymbol{\mathrm{t}}}) $ for $n=N$ as $N\rightarrow \infty$ $$Z_N(N\,{\boldsymbol{\mathrm{t}}})
=\int_{\mathbb{R}^N}\prod_{k=1}^{N}\Big(d\,x_k\,e^{N\,V(x_k,{\boldsymbol{\mathrm{t}}})})\Big)(\Delta(x_1,\cdots,x_N))^2,$$ It is well-known [@avm] that $Z_n({\boldsymbol{t}})$ is a $\tau$-function of the semi-infinite 1-Toda hierarchy , then there exists a $\tau$-function $\tau(\epsilon,x,{\boldsymbol{\mathrm{t}}})$ of the dispersionful 1-Toda hierarchy verifying $$\label{rel}
\tau(\epsilon,\epsilon\,n,{\boldsymbol{\mathrm{t}}})=Z_n(N\,{\boldsymbol{\mathrm{t}}}),$$ and consequently $$\label{rel1}
\tau(\epsilon,1,{\boldsymbol{\mathrm{t}}})=Z_N(N\,{\boldsymbol{\mathrm{t}}}).$$ Hence the large $N$-limit expansion of the partition function $$\label{tau1}
{\mathbb{Z}}_N(N\,{\boldsymbol{\mathrm{t}}})=\exp{\Big(N^2\,\mathbb{F}\Big)},\quad
\mathbb{F}=\sum_{k\geq 0}\dfrac{1}{N^{2k}}\,F^{(2k)},$$ is determined by a solution of the dispersionful 1-Toda hierarchy at $x=1$.
As a consequence of the above analysis one concludes that the leading term (planar limit) $F^{(0)}$ is determined by a solution of the 1-dToda hierarchy at $x=1$. Furthermore, the leading terms of the $N$-expansions of the main objects of the hermitian matrix model can be expressed in terms of quantities of the 1-dToda hierarchy. For example, in the *one-cut* case , the density of eigenvalues $$\rho(z)=M(z)\,\sqrt{(z-a)(z-b)},$$ is supported on a single interval $[a,b]$. These objects are related to the leading term $W^{(0)}$ of the one-point correlator [@gin] $$W(z):=\dfrac{1}{N}\,\sum_{j\geq 0}\dfrac{1}{z^{j+1}}\langle tr
M^j\rangle=\dfrac{1}{z}+\dfrac{1}{N^2}\,\sum_{j\geq
1}\dfrac{1}{z^{j+1}}\, \dfrac{\partial \log\,Z_N(N\,{\boldsymbol{\mathrm{t}}})}{\partial
t_j},$$ in the form $$W^{(0)}=-\dfrac{1}{2}V_z(z)+i\pi\,\rho(z).$$ On the other hand, it can be proved (see for instance [@eyn]) that $$W^{(0)}=\label{m1} m(z,1,{\boldsymbol{\mathrm{t}}})-\sum_{j=1}^\infty j\,t_j\,z^{j-1},$$ so that and yield $$\begin{aligned}
\label{e1}
\nonumber &-\dfrac{1}{2}V_z(z)+i\pi\,\rho(z)=-\sum_{j=1}^\infty
j\,t_j\,(z^{j-1})_-(p(z))\\\\
\nonumber &=\dfrac{1}{2}\sum_{j=1}^\infty
j\,t_j\,r_{j-1}-\dfrac{1}{2}\sum_{j=1}^\infty
j\,t_j\,z^{j-1}+\dfrac{1}{2}(p-\bar{p})\sum_{j=2}^\infty
j\,t_j\,\Big(z^{j-2}\,R\Big)_\oplus,\end{aligned}$$ Since we are setting $\bar{t}_j=0,\,\forall j\geq 1$, according to the first hodograph equation the first term in the last equation vanishes. Therefore the density of eigenvalues and its support $[a,b]$ are characterized by $$\begin{aligned}
\label{den}
\nonumber \rho(z)&:=\dfrac{1}{2\pi i}\Big(\dfrac{V_z}{\sqrt{(z-a)(z-b)}}\Big)_\oplus\,\sqrt{(z-a)(z-b)},\\\\
\nonumber a&:=u-2\,\sqrt{v},\quad b:=u+2\,\sqrt{v},\end{aligned}$$ where we set $x=1$ in all the $x$-dependent functions. Observe that according to $$\label{e2}
i\,\pi\,\rho(z)=\dfrac{1}{2}\,V_z(z)+\dfrac{x}{z}+\mathcal{O}\Big(\dfrac{1}{z^2}\Big),\quad
z\rightarrow\infty,$$ so that the constraint $x=1$ means that the density of eigenvalues is normalized on its support $$\int_a^b\, \rho(z)\,dz=1.$$
Moreover, from we obtain $$\label{ho3}
\oint_{\gamma}\dfrac{dz}{2\pi i}\dfrac{V_{z}}{\sqrt{(z-a)(z-b)}}\,
=0,\quad \oint_{\gamma}\dfrac{dz}{2\pi
i}\dfrac{z\,V_{z}}{\sqrt{(z-a)(z-b)}}\, =-2,$$ with $\gamma$ being a positively oriented closed path encircling the interval $[a,b]$. These are the quations which determine the zero-genus contribution or planar limit to the partition function of the hermitian model [@eyn]-[@mig].
Bubble break-off in Hele-Shaw flows
-----------------------------------
A Hele-Shaw cell is a narrow gap between two plates filled with two fluids: say oil surrounding one or several bubbles of air. Let $D$ denote the domain in the complex plane ${\mathbb{C}}$ of the variable $\lambda$ occupied by the air bubbles. By assuming that $D$ is an *algebraic domain* [@lee], the boundary $\gamma$ of $D$ is characterized by a *Schwarz function* ${\mathbb{S}}={\mathbb{S}}(\lambda)$ such that $$\label{sch}
\lambda^*={\mathbb{S}}(\lambda),\quad \lambda\in\gamma.$$ The geometry of the domain ${\mathbb{C}}-D$ is completely encoded in ${\mathbb{S}}$ and it can be conveniently described in terms of the *Schottky double* [@wz]: a Riemann surface $\mathcal{R}$ resulting from gluing two copies $H_{\pm}$ of ${\mathbb{C}}-D$ trough $\gamma$, adding two points at infinity $(\infty,\overline{\infty})$ and defining the complex coordinates $$\begin{cases}
\lambda_+(\lambda)=\lambda,\quad \lambda\in H_+,\\
\lambda_-(\lambda)=\lambda^*,\quad \lambda\in H_-.
\end{cases}$$ In particular ${\mathbb{S}}\,d\lambda$ can be extended to a unique meromorphic differential $\omega$ on $\mathcal{R}$.
The evolution of $\gamma$ is governed by D’Arcy law: the velocity in the oil domain is proportional to the gradient of the pressure. In the absence of surface tension, pressure is continuous across $\gamma$ and then if the bubbles are assumed to be kept at zero pressure, we are lead to the Dirichlet boundary problem $$\label{dir}
\begin{cases}
\bigtriangleup \mathcal{P}=0,\quad \mbox{on ${\mathbb{C}}-D$},\\
\quad \mathcal{P}=0 \quad \mbox{on $\gamma$},\\
\quad \mathcal{P}\rightarrow -\log|z|,\quad z\rightarrow\infty.
\end{cases}$$ If one assumes D’Arcy law in the form $\vec{v}=-2\,\vec{\nabla}\mathcal{P}$, then by introducing the function $$\label{cpo}
\Phi(\lambda):=\xi(\lambda)+i\,\mathcal{P}(\lambda),$$ where $\xi$ and $\mathcal{P}$ are the *stream function* and the pressure, respectively, D’Arcy law can be rewritten as $$\label{dar}
\partial_t\, {\mathbb{S}}=2\,i\,\partial_{\lambda}\,\Phi,$$ where $t$ stands for the time variable.
In the set-up considered in [@lee] air is drawn out from two fixed points of a simply-connected air bubble making the bubble breaks into two emergent bubbles with highly curved tips. Before the break-off the interface oil-air remains free of cusp-like singularities and develops a smooth neck. As it is shown in [@wz]-[@lee], the condition for bubbles to be at equal pressure implies that the integral $$\Pi:=\dfrac{1}{2}\oint_{\beta}\,\omega,$$ where $\omega$ is the meromorphic extension of ${\mathbb{S}}\,d\lambda$ to $\mathcal{R}$ and $\beta$ is a cycle connecting the bubbles, is a constant of the motion. Since at break-off $\beta$ contracts to a point, it is obvious that a necessary condition for break-off is that $\Pi$ vanishes.
The following pair of complex-valued functions were introduced in [@lee] to describe the bubble break-off near the breaking point $$\label{car}
X(\lambda):=\dfrac{1}{2}\Big(\lambda+{\mathbb{S}}(\lambda)\Big),\quad
Y(\lambda):=\dfrac{1}{2\,i}\Big(\lambda-{\mathbb{S}}(\lambda)\Big).$$ They analytically extends the Cartesian coordinates $(X,Y)$ of the interface $\gamma$ $$\label{car1}
X=Re\, \lambda,\quad Y=Im\, \lambda,\quad \lambda\in\gamma,$$ and allow to write the evolution law in the form $$\label{dar1}
\partial_t\, Y(X)=-\partial_X\,\Phi(X).$$
The analysis of [@lee] concludes that after the break-off the local structure of a small part of the interface containing the tips of the bubbles falls into universal classes characterized by two even integers $(4\,n, 2),\, n\geq 1,$ and a finite number $2n$ of real deformation parameters $t_k$. By assuming symmetry of the curve with respect to the $X$-axis, the general solution for the curve and the potential in the $(4\,n, 2)$ class are $$\label{den}
Y:=\Big(\dfrac{U_X}{\sqrt{(X-a)(X-b)}}\Big)_\oplus\,\sqrt{(X-a)(Y-b)},\quad
\Phi=-\sqrt{(X-a)(Y-b)},$$ where $a$ and $b$ are the positions of the bubbles tips and $$\label{U}
U(X,{\boldsymbol{\mathrm{t}}}):=\sum_{j=1}^{2n} t_{j+1}\,X^{j+1}.$$ Here the subscript $\oplus$ denotes the projection of $X$-series on the positive powers. Due to the physical assumptions of the problem, the function $Y$ inherates two conditions for its expansion as $X\rightarrow\infty$ $$\label{hoh}
Y(X)=\sum_{j=1}^{2n}
(j+1)\,t_{j+1}\,X^{j}+\sum_{j=0}^{\infty}\dfrac{Y_n}{X^n}.$$ which determine the positions $a$, $b$ of the tips. The conditions are
1. From $\Phi\rightarrow -i\,\log \lambda$ as $\lambda \rightarrow \infty$. Hence implies that the constant term $Y_0$ in should be equal to $t$.
2. The coefficient $Y_1$ in front of $X^{-1}$ turns to be equal to $\Pi$, so that it must vanish for a break-off [@lee].
As it was shown in [@lee], imposing these two conditions on leads to a pair of hodograph equations which arise in the dispersionless AKNS hierachy. However, from it is straightforward to see that these equations coincide with the hodograph equations associated with the system of string equations provided one sets $$\begin{aligned}
\label{sett}
\nonumber X&=z,\quad Y=2\,m-V_z,\quad \Phi=z-u-2\,p,\\\\
\nonumber t_j&=0,\quad \forall j\geq 2n+2;\quad t=t_1,\quad
x=\dfrac{\Pi}{2}=0.\end{aligned}$$ For instance, we observe that the evolution law derives in a very natural form from the d1-Toda hierarchy. Indeed, from and we have $$p=\dfrac{1}{2}\,(z-u-\Phi),$$ so that implies $$\partial_t Y=2\,\partial_{t_1}
m(z)-1=2\,\partial_{z}(z)_+-1
=2\,\partial_{z}(p+u)-1=-\partial_{z}\Phi=-\partial_{X}\Phi.$$
In this way the integrable structure associated to the system of string equations of the d2-Toda hierarchy manifests a duality between the planar limit of the Hermitian matrix model and the bubble break-off in Hele-Shaw cells. According to this relationship the density of eigenvalues $\rho$ and the end-points $a,\,b$ of its support in the Hermitian model are identified with the interface function $Y$ and the positions of the bubbles tips, respectively, in the Hele-Show model.
[**Acknowledgments**]{}
The authors wish to thank the Spanish Ministerio de Educacion y Ciencia (research project FIS2005-00319) and the European Science Foundation (MISGAM programme) for their financial support.
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\[lastpage\]
[^1]: Partially supported by MEC project FIS2005-00319 and ESF programme MISGAM
|
---
abstract: 'Classical T Tauri stars (CTTS) accrete material from their discs through their magnetospheres. The geometry of the accretion flow strongly depends on the magnetic obliquity, i.e., the angle between the rotational and magnetic axes. We aim at deriving the distribution of magnetic obliquities in a sample of 10 CTTSs. For this, we monitored the radial velocity variations of the HeI$\lambda$5876Åline in these stars’ spectra along their rotational cycle. He I is produced in the accretion shock, close to the magnetic pole. When the magnetic and rotational axes are not aligned, the radial velocity of this line is modulated by stellar rotation. The amplitude of modulation is related to the star’s projected rotational velocity, $v\sin i$, and the latitude of the hotspot. By deriving $v\sin i$ and HeI$\lambda$5876 radial velocity curves from our spectra we thus obtain an estimate of the magnetic obliquities. We find an average obliquity in our sample of 11.4$^{\circ}$ with an rms dispersion of 5.4$^{\circ}$. The magnetic axis thus seems nearly, but not exactly aligned with the rotational axis in these accreting T Tauri stars, somewhat in disagreement with studies of spectropolarimetry, which have found a significant misalignment ($\gtrsim 20^{\circ}$) for several CTTSs. This could simply be an effect of low number statistics, or it may be due to a selection bias of our sample. We discuss possible biases that our sample may be subject to. We also find tentative evidence that the magnetic obliquity may vary according to the stellar interior and that there may be a significant difference between fully convective and partly radiative stars.'
author:
- |
Pauline McGinnis$^{1,2}$[^1], Jérôme Bouvier$^2$ and Florian Gallet$^2$\
$^1$Dublin Institute for Advanced Studies, School of Cosmic Physics, Astronomy & Astrophysics Section, 31 Fitzwilliam Place, Dublin, Ireland\
$^2$Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France
bibliography:
- 'references.bib'
date: 'Accepted 2020 July 08. Received 2020 July 08; in original form 2020 March 15.'
title: The magnetic obliquity of accreting T Tauri stars
---
\[firstpage\]
Accretion, accretion discs – stars: variables: T Tauri – stars: magnetic field – techniques: spectroscopic
Introduction
============
The accretion of circumstellar disc material onto young, low-mass stars known as T Tauri stars is believed to be strongly mediated by the stellar magnetic field. These actively accreting T Tauri stars are often referred to as classical T Tauri stars (CTTS), as opposed to the non-accreting weak-line T Tauri stars (WTTS). Magnetospheric accretion models predict that the stellar magnetosphere of a CTTS interacts with the inner accretion disc at a few stellar radii, truncating the inner disc [@shu94; @romanova02]. At this region, circumstellar disc material is lifted above the disc mid-plane and falls onto the star following the magnetic field lines, forming what are known as accretion funnel flows or accretion columns [@bessolaz08]. When the material traveling at free-fall velocities collides with the stellar surface, accretion shocks are formed near the stellar surface. These shocks produce an excess continuum flux [@calvet98] and narrow components in emission lines such as He I and the Ca II triplet [e.g. @dodin12], which are observed in T Tauri spectra [@joy45; @appenzeller86; @hamman92]. The excess emission flux often veils a T Tauri star’s spectrum, making the photospheric absorption lines appear shallower [@joy49; @rydgren76; @hartigan89].
Magnetohydrodynamics (MHD) simulations have shown that both the strength of the large-scale stellar magnetic field and the magnetic obliquity, i.e. the tilt between the axis of the stellar magnetic field and the stellar rotation axis, affect the geometry of the accretion flow [see, e.g., @kurosawa13]. The magnetic obliquity therefore influences the star-inner disc interaction and has implications on the formation of inner disc warps [@romanova13], as well as on the formation and migration of planets in the inner disc region (few 0.1 au from the star). To better understand its role in the star-disc interaction, it is important to measure the magnetic obliquity of a number of CTTS. However, to do this normally requires mapping stellar magnetic field geometries using spectropolarimetry and Zeeman Doppler Imaging [ZDI, @donati97], a technique that is time-consuming and requires the use of large telescopes. It has therefore only been performed on a small number of CTTSs, making a statistical analysis impracticable.
If we are interested in studying only the magnetic obliquity, and not the strength or complexity of the stellar magnetic field, there is a more cost-effective way to do this without the need to derive full stellar magnetic field configurations. Several CTTSs show narrow HeI$\lambda5876$Åemission with redshifts of a few km s$^{-1}$, which is believed to originate from the post-shock region, where the gas has been considerably decelerated. This line often shows radial velocity variations which may be periodically modulated by the star’s rotation. In these cases, the amplitude of this variability $\Delta V_{rad}(\mathrm{HeI})$ is directly related to the cosine of the latitude $l$ of the accretion shock through the simple formula [@bouvier07]:
$$\label{eq:cosl}
\Delta V_{rad} (\mathrm{HeI}) = 2 \cdot v \sin i \cdot \cos l ,$$
where $v\sin i$ is the projected stellar rotational velocity, which can be measured directly in an observed spectrum. Observations of CTTS magnetic fields using spectropolarimetry have shown that these accretion shocks form within a few degrees of the magnetic poles [e.g, @donati08; @donati10b; @donati11a]. Therefore by measuring the latitude of these shocks in CTTSs, their magnetic obliquities can be inferred without the need of direct magnetic field measurements. Figure \[fig:sketch\] illustrates this scenario, where a hotspot at latitude $l$ on the surface of a star (observed at an inclination $i$) traces the magnetic obliquity (represented by the angle $\Theta$).
![Illustration of a hotspot at latitude $l$ on the surface of a star viewed at inclination $i$ (inclination between the rotation axis and the observer’s line of sight). The magnetic obliquity is identified as the angle $\Theta$. []{data-label="fig:sketch"}](Sketch_grey2.pdf){width="45.00000%"}
As many other emission lines associated with accretion onto a young star, the HeI$\lambda$5876 line is often composed of more than one component. @beristain01 studied a number of He I profiles in CTTSs and found that they often present a narrow component (NC) and a broad component (BC). It is the NC that is believed to arise in the post-shock region, near the stellar surface, while the BC seems to have a composite origin. @beristain01 found that the BC is sometimes observed blueshifted, in which case they attribute it to emission from a hot wind. When it is redshifted, they attribute it to emission from the accretion columns. Since all of these phenomena tend to be dynamic, these components will generally all be variable in their own way. Therefore in order to study one phenomenon or the other, it is important to either study systems that are dominated by one component, or to properly disentangle them. In the present study we attempt to do both.
We propose here to estimate the magnetic obliquity of a sample of well-known accreting T Tauri stars in the Taurus-Auriga star forming region, using the radial velocity variations of the NC of the HeI$\lambda$5876 line. We describe our observations in Sect. \[sec:obs\] and the method we use to derive hotspot latitudes in Sect. \[sec:analysis\]. The results obtained for each object are detailed in Sect. \[sec:results\]. We discuss our main findings in Sect. \[sec:discuss\] and summarize our conclusions in Sect. \[sec:conc\].
Observations {#sec:obs}
============
---------- ------------ ------------ --------------- ------------------- ------------------ ------------------- ------------------ ---------------- --------------- -------------
Name Dates N$_{spec}$ S/N$_{600nm}$ <$V_{rad}$> $\sigma V_{rad}$ <$v\sin i$> $\sigma v\sin i$ Classification Spectral type Mass
(Nov.2011) (km s$^{-1}$) (km s$^{-1}$) (km s$^{-1}$) (km s$^{-1}$) M$_{\odot}$
DE Tau 21-28 13 26-33 14.66 0.61 8.38 0.54 CTTS M2.3 0.38
DF Tau 20-28 15 16-35 17.42 2.42 7.27 1.77 CTTS M2.7 0.32
DK Tau 21-28 15 21-35 15.82 2.38 12.69 2.18 CTTS K8.5 0.68
DN Tau 20-28 16 23-37 16.79 0.44 8.69 0.30 CTTS M0.3 0.55
GI Tau 21-28 11 24-34 17.06 0.53 9.23 0.51 CTTS M0.4 0.58
GK Tau 21-28 15 24-35 16.56 1.94 19.06 3.36 CTTS K6.5 0.76
GM Aur 20-27 14 21-36 14.95 0.98 12.59 1.02 CTTS K6.0 0.88
IP Tau 21-28 10 19-33 15.98 1.73 9.73 0.86 CTTS M0.6 0.59
IW Tau 20-28 15 27-46 16.03 0.30 8.53 0.30 WTTS M0.9 0.49
T Tau 18-28 26 14-39 19.47 1.28 23.54 1.61 CTTS K0 1.99
V826 Tau 18-22 5 21-47 12.95 14.42 5.59 2.83 WTTS K7 0.74
V836 Tau 23-28 7 30-33 17.73 0.86 10.51 0.77 CTTS M0.8 0.58
---------- ------------ ------------ --------------- ------------------- ------------------ ------------------- ------------------ ---------------- --------------- -------------
**Notes:** Spectral types and masses are from @herczeg14.
Observations were carried out from November 18 to 28, 2011, at Observatoire de Haute-Provence using the SOPHIE spectrograph [@perruchot08] in High-Efficiency mode, which delivers a spectral resolution of $R\sim40,000$. The sample consists of ten accreting T Tauri stars (CTTS) and a control sample of two non-accreting T Tauri stars (WTTS). A total of 162 spectra were obtained, comprising between 10 and 16 spectra for each source in the CTTS sample over eight nights (Nov. 21-28), with up to 26 spectra for T Tauri itself over 11 nights (Nov. 18-28), and 5-7 spectra for each of the two WTTS of the control sample over the same time frame. The Journal of Observations is given in Table \[obs\]. Depending on the source brightness, exposure times varied between 137 s (e.g., T Tau) and 4410 s (e.g., IP Tau). The resulting signal-to-noise ratio (S/N) at the continuum level around 600 nm ranges from 15 to 50, with 2/3 of the spectra having S/N$\geq$30.
The raw spectra were fully reduced at the telescope by the SOPHIE real-time pipeline [@bouchy09]. The data products include a re-sampled 1D spectrum with a constant wavelength step of 0.01 Å, corrected for barycentric radial velocity, an order-by-order estimate of the signal-to-noise ratio, and a measurement of the source radial velocity, $V_{rad}$, and projected rotational velocity, $v\sin i$. These were derived from a cross-correlation analysis between the spectrum and a spectral mask template, either a G2 or a K5 mask, depending on the source’s spectral type [e.g., @melo01; @boisse10]. We list the values of these parameters in Table \[obs\] and use them for the subsequent analysis. The error quoted on $V_{rad}$ and $v\sin i$ in Table \[obs\] is the standard deviation of individual measurements. While the latter are usually accurate to within a fraction of a km s$^{-1}$, the photospheric line profile variability induced by surface spots and/or the accretion flow in young stars yields much larger uncertainties [e.g., @petrov01]. The cross-correlation profiles themselves are presented in the online material and their night-to-night variations discussed there.
{width="\textwidth"}
All 1D spectra were normalized to a continuum level of unity in the spectral region of interest. This was done using an IDL routine that identifies the continuum of a portion of the spectrum and fits a polynomial (of fourth or fifth order, depending on the curvature of the spectrum) through it. The continuum is identified by first manually excluding regions with emission lines, then taking only the points with the 20% highest intensities in order to exclude absorption lines. The portion of the spectrum of interest is then divided by the function that fits the continuum, resulting in normalized spectra. The final normalized spectra are shown in the online supplementary material.
Spectral analysis: line profile Doppler shifts {#sec:analysis}
==============================================
We aim at measuring Doppler shifts on the HeI$\lambda5876$ line as a function of rotational phase. The line profiles for our stellar sample are shown in Fig. \[fig:hei\_profiles\]. With the exception of V826 Tau, which has very weak He I emission that is only measurable in one observation, we see that the He I line is observed in emission on every night for every star in this sample. This may seem unexpected if this emission comes from a rotationally modulated hotspot, since we would expect the spot to be behind the star in at least some epochs. We do note, however, that several stars show a significant decrease in the He I line flux during certain epochs (this can be seen in Fig. \[fig:hei\_profiles\]). This could occur if the spot passes behind the star, but a small portion of it always remains visible, because of the system inclination and spot latitude. Studies of ZDI have shown that it is common for the hotspot to be found at high latitudes [e.g. @donati08; @donati10b; @donati11a], which would indeed lead them to be observable during most of the rotational cycle.
{height="5cm"} {height="5cm"} {height="5cm"}
To measure the Doppler shift on each night, we first define a reference profile for each star, to which the profiles from each night are compared. We initially used as reference the line profile with the best S/N and performed the analysis described in this section to find the shifts between this and all other profiles obtained from the spectral series of that star. We then shifted each of these $N$ line profiles in velocity space so they would coincide as closely as possible with each other. We take the mean of these $N$ shifted profiles as the final reference. This is done so as to obtain a mean line profile without artificially broadening it by combining lines at slightly different radial velocities.
The line profile of each night is intensity scaled by a factor $k$ and Doppler shifted by an amount of $\delta\lambda$ in order to fit the reference profile (see Fig. \[chi2\]). For each spectrum in the series, the best fit parameters, $k$ and $\delta\lambda$, are obtained by minimizing the quantity
$$\label{eq:chi2}
\chi_\nu^2 (k, \delta\lambda) = \sum_{i=1}^{n} w_i^2 \frac{\left[k\cdot (I(\lambda_i + \delta\lambda) - 1) - (\left< I(\lambda_i)\right> -1) \right]^2}{(n-2) \sigma_{cont}^2} ,$$
where $n$ is the number of spectral pixels over the line profile, $w_i$ is the weight applied to each pixel, $k$ is a scaling factor that accounts for line intensity variations, $I(\lambda)$ the intensity of the individual profile, $\delta\lambda$ the Doppler shift, $\left< I(\lambda)\right>$ is the intensity of the reference profile, and $\sigma_{cont}$ is the rms noise of the spectrum at the continuum level next to the line. The summation runs over $\pm$2.5$\sigma$ from the line centre, where $\sigma$ is the standard deviation of a Gaussian fit to the average line profile.
A first guess to the scaling factor $k$ is provided by the ratio between the maximum intensity of the average and individual profiles. The scaling parameter is allowed to vary within 20% of their ratio in 30 incremental steps. Doppler shifts are explored over a range of $\pm$25 km s$^{-1}$, with a spectral pixel step of 0.01 Å, corresponding to $\sim$0.5 km s$^{-1}$. Weights, $w_i$, which are chosen to scale as a Gaussian function (determined via a Gaussian fit to the reference profile), are applied to the $\chi_\nu^2$ calculation of each profile, in order to effectively increase the importance of the line core relative to the wings during the fitting procedure[^2]. For each profile, maps of $\chi_\nu^2 (k, \delta\lambda)$ are computed and the location of the deepest $\chi_\nu^2$ minimum is found. As the two model parameters are independent, the $\chi_\nu^2$ surface is a shallow valley, whose main axis lies along the $k$ dimension (see Fig. \[chi2\]). We therefore proceed to obtain a sub-pixel estimate of the location of the $\chi_\nu^2$ minimum in the $(k, \delta\lambda)$ grid by extracting a cut of the $\chi_\nu^2$ surface perpendicular to the $k$ direction around the deepest minimum. The $\chi_\nu^2 (\delta\lambda)$ curve thus obtained is re-sampled and a sub-pixel estimate of its centroid is derived to provide the final estimate on $\delta V_{rad}$, the Doppler shift between the individual and average He I line profiles (see Fig. \[chi2\]). Finally, the 1$\sigma$ uncertainty on $\delta V_{rad}$ is derived along the $\chi_\nu^2 (\delta\lambda)$ curve at the location where $\chi_\nu^2 = \chi_{\nu,min}^2 + 1$ [@press92]. The whole procedure is illustrated in Fig. \[chi2\].
Care must be taken when using this procedure on the stars whose HeI$\lambda5876$ line profiles show more than one component, which is the case for DE Tau, DF Tau, GM Aur and T Tau. These components are believed to have different origins and therefore should present different variabilities [@beristain01]. The magnetic obliquity is related to the variability of the narrow component (NC), which is believed to arise in accretion shocks close to the stellar surface. Therefore, in the cases where a broad component (BC) is also present, we first fit the individual profiles from each night with two Gaussians in order to identify the two components, then analyse only the variability of the narrowest of the two components. However, the profiles that present only one component show that the NC is not symmetric. Thus, taking the centroid of the narrowest Gaussian component may introduce small uncertainties to the radial velocity measurements. On the other hand, a visual inspection of the profiles studied by @beristain01 [see their Fig. 1*a*] shows that the BC often seems to be nearly symmetric and reasonably well reproduced by a Gaussian (this is not always the case, particularly for the strongest accretors, such as AS 353A, DG Tau, DR Tau and RW Aur, all of which show more complex profiles, but it seems to be the case for the four stars in our sample). Therefore, to isolate the NC, we subtract the broadest Gaussian component from the composite profiles to remove the contribution of the BC. We then perform the cross-correlation analysis described above on the residual profile, which should contain only the contribution from the NC. We note that, in our sample, a typical NC has a full width at half maximum (FWHM) of $\sim 35$ km s$^{-1}$, while a typical BC has FWHM $\sim 120$ km s$^{-1}$. Figure \[fig:gausfit\] shows an example of the Gaussian fit and residual profile of each of these stars.
We find that the radial velocity measurements of the residual profiles agree, within the uncertainties, with the centroid velocities of the narrowest component of the Gaussian decomposition. In the case of DE Tau, DF Tau and GM Aur, the results also agree with the ones obtained from the cross-correlation of the full HeI$\lambda5876$ line profile (without subtracting the BC), though the error bars are larger when the BC is not removed. We can see in Fig. \[fig:gausfit\] that the BC mostly affects the wings of these profiles, while the centre of the line is dominated by the NC. Therefore, by allowing the computation of $\chi_\nu^2$ to run over $\pm2.5\sigma$ of a Gaussian approximation to the centre of the line, the broad wings are ignored and as a result the BC does not strongly interfere with the analysis. For the star T Tau, however, the HeI$\lambda5876$ line profile is clearly dominated by the BC. In this case, the cross-correlation analysis of the full profile yields very different results than when the contribution of the BC is removed. For this star, removing the contribution of the BC is essential to properly identify the latitude of the accretion shocks. The Gaussian decomposition of all observations of the star T Tau can be seen in the online material (except for two observations, which have been excluded due to low S/N).
![*Left panels:* Examples of the Gaussian decomposition performed on the HeI$\lambda5876$ line profiles that presented two components. *Right panels:* residual profiles obtained after subtracting the broad component from the observed spectrum. []{data-label="fig:gausfit"}](DETau_gausex.pdf "fig:"){width="23.00000%"} ![*Left panels:* Examples of the Gaussian decomposition performed on the HeI$\lambda5876$ line profiles that presented two components. *Right panels:* residual profiles obtained after subtracting the broad component from the observed spectrum. []{data-label="fig:gausfit"}](DETau_residprof.pdf "fig:"){width="23.00000%"} ![*Left panels:* Examples of the Gaussian decomposition performed on the HeI$\lambda5876$ line profiles that presented two components. *Right panels:* residual profiles obtained after subtracting the broad component from the observed spectrum. []{data-label="fig:gausfit"}](DFTau_gausex.pdf "fig:"){width="23.00000%"} ![*Left panels:* Examples of the Gaussian decomposition performed on the HeI$\lambda5876$ line profiles that presented two components. *Right panels:* residual profiles obtained after subtracting the broad component from the observed spectrum. []{data-label="fig:gausfit"}](DFTau_residprof.pdf "fig:"){width="23.00000%"} ![*Left panels:* Examples of the Gaussian decomposition performed on the HeI$\lambda5876$ line profiles that presented two components. *Right panels:* residual profiles obtained after subtracting the broad component from the observed spectrum. []{data-label="fig:gausfit"}](GMAur_gausex.pdf "fig:"){width="23.00000%"} ![*Left panels:* Examples of the Gaussian decomposition performed on the HeI$\lambda5876$ line profiles that presented two components. *Right panels:* residual profiles obtained after subtracting the broad component from the observed spectrum. []{data-label="fig:gausfit"}](GMAur_residprof.pdf "fig:"){width="23.00000%"} ![*Left panels:* Examples of the Gaussian decomposition performed on the HeI$\lambda5876$ line profiles that presented two components. *Right panels:* residual profiles obtained after subtracting the broad component from the observed spectrum. []{data-label="fig:gausfit"}](TTau_gausex.pdf "fig:"){width="23.00000%"} ![*Left panels:* Examples of the Gaussian decomposition performed on the HeI$\lambda5876$ line profiles that presented two components. *Right panels:* residual profiles obtained after subtracting the broad component from the observed spectrum. []{data-label="fig:gausfit"}](TTau_residprof.pdf "fig:"){width="23.00000%"}
Results {#sec:results}
=======
Variability of the He I line
----------------------------
We present in this section the results obtained for the radial velocity variations of the HeI$\lambda5876$ line profile of a sample of accreting T Tauri stars. The hotspot latitudes derived from the radial velocity variations of each star and Eq. \[eq:cosl\] are shown in Table \[tab:results\]. The variability of the narrow component of the He I line and the veiling variability (with respect to the mean spectrum) for each star in our sample can be verified in Figs. \[fig:he1veil1\] and \[fig:he1veil2\] of the Appendix[^3]. Veiling was measured in a region near the He I line where several photospheric absorption lines are present. The spectrum of each night was compared with the mean spectrum of the same star[^4], in order to find the amount of excess continuum emission that minimized $\chi^2$. The uncertainties (derived at the corresponding $\chi^2_{1\sigma} = \chi^2_{min} + 1$ value) are large, but some of the trends seen in veiling appear to be confirmed by the variability of the equivalent width (EW) of the LiI$\lambda$6708 line (Fig. \[fig:li\_he\_ew\]). The EW of this line (or any deep photospheric absorption line) gives an indirect measurement of veiling, since the excess emission causes the photospheric absorption lines to appear shallower in the normalized spectrum, reducing their EW.
Our analysis also partly rests upon the pattern of spectral variability observed in other lines, such as H$\alpha$ and H$\beta$ (whose profiles are also shown in Figs. \[fig:he1veil1\] and \[fig:he1veil2\] and in more detail in the online material), as well as LiI$\lambda$6708 (whose radial velocity curves and equivalent width variability can be seen in Fig. \[fig:li\_he\_ew\], alongside the variability of $V_{rad}$ and EW of the He I line), and K2 light curves (shown in Fig. \[fig:k2lcs\]). The Li I line profiles can be seen in the online material. Due to the specific behaviour of line profile variability in each object, we discuss each sample source in turn[^5].
---------- ----------------------- --------------- --------------- ------------------------- -------------------------
Star $\Delta V_{rad}$(HeI) $v \sin i$ $\cos l$ $l$ $\Theta$
(km s$^{-1}$) (km s$^{-1}$) ($^{\circ}$) ($^{\circ}$)
DE Tau 1.8$\pm$3.2 8.4$\pm$0.5 0.11$\pm$0.19 84$\substack{+ 6\\-11}$ 6$\substack{+11\\- 6}$
DF Tau 3.7$\pm$2.2 7.3$\pm$1.8 0.25$\pm$0.16 75$\pm10$ 15$\pm10$
DK Tau 7.8$\pm$2.8 12.7$\pm$2.2 0.31$\pm$0.12 72$\substack{+ 7\\- 8}$ 18$\substack{+ 8\\- 7}$
DN Tau 3.8$\pm$3.0 8.7$\pm$0.3 0.22$\pm$0.17 77$\substack{+10\\-11}$ 13$\substack{+11\\-10}$
GI Tau 3.8$\pm$2.8 9.2$\pm$0.5 0.21$\pm$0.15 78$\pm9$ 12$\pm9$
GK Tau 3.0$\pm$4.6 19.1$\pm$3.4 0.08$\pm$0.12 85$\substack{+ 5\\- 7}$ 5$\substack{+ 7\\- 5}$
GM Aur 5.9$\pm$6.6 12.6$\pm$1.0 0.23$\pm$0.26 77$\substack{+13\\-16}$ 13$\substack{+16\\-13}$
IP Tau 2.2$\pm$3.9 9.7$\pm$0.9 0.11$\pm$0.20 83$\substack{+ 7\\-12}$ 7$\substack{+12\\- 7}$
IW Tau 1.9$\pm$4.1 8.5$\pm$0.3 0.11$\pm$0.24 84$\substack{+ 6\\-14}$ 6$\substack{+14\\- 6}$
T Tau 18.4$\pm$7.9 23.5$\pm$1.6 0.39$\pm$0.17 67$\substack{+10\\-11}$ 23$\substack{+11\\-10}$
V836 Tau 3.7$\pm$4.3 10.5$\pm$0.8 0.17$\pm$0.21 80$\substack{+10\\-12}$ 10$\substack{+12\\-10}$
---------- ----------------------- --------------- --------------- ------------------------- -------------------------
: Hotspot latitudes and magnetic obliquities[]{data-label="tab:results"}
### DE Tau
This star’s spectrum presents veiling that is clearly variable in the time-scale of the observations. The veiling increases steadily to reach a maximum on Nov. 26, then drops again, consistent with the rotation period of 7.6 days taken from the literature [@bouvier93]. The intensity of the He I line reaches its maximum two days before the veiling maximum, on Nov. 24. The H$\alpha$ and H$\beta$ line profiles are more intense than average on both these nights, but show no clear trend with the 7.6 day period.
### DF Tau
The intensity of the He I line profile, as well as its radial velocity variability, appear to be modulated on a time-scale of around 6 days, though the time-sampling of our observations is not enough to determine an accurate rotation period. This value is different from the photometric periods given in the literature, of 8.5 days [@bouvier95] and 7.18 days [@artemenko12]. However, neither of these two periods are found in this star’s K2 light curve (N. Roggero private communication). The He I line shows the lowest intensities just before it is most blueshifted, which is consistent with an origin in a stellar spot modulated by rotation. The veiling variability has too small an amplitude to derive any conclusions from it. However, the H$\alpha$ and H$\beta$ line profiles also show a trend with the $\sim$6 day period, where a blueshifted absorption component appears at phase $\phi \sim 0.6$, just after the accretion shock has passed in front of our line of sight (the He I line profile is more redshifted than average). Since this component is associated with outflows intersecting our line of sight, this points to a spatial association between the accretion shocks and a wind. This could be a stellar wind or a non-axisymmetric disc wind that is launched close to the co-rotation radius, in order to show periodic behaviour on the same time-scale as the stellar rotation.
### DK Tau
The intensity and radial velocity of the He I line, as well as the veiling, show a variability that is modulated on the time-scale of the observations (the variability of veiling is just within the error bars, but the trend is confirmed by the EW of the LiI line in Fig. \[fig:li\_he\_ew\]). As the He I line passes in our line of sight, its radial velocity going from blueshifted to redshifted with an amplitude of $\sim 10$ km s$^{-1}$, the veiling and He I intensity increase to reach a maximum on Nov. 25 and Nov. 27, before decreasing again. However, something occurs on Nov. 26 which results in a less intense, more centred He I line, less veiling and much less intense H$\alpha$ and H$\beta$ line profiles. If not for this event, the behaviour of He I and veiling in this star would be consistent with the rotational modulation of a hotspot on the stellar surface with the reported period of 8.18 days [@percy10; @artemenko12]. It is possible that on this date, circumstellar material passed in front of the line of sight, partially obscuring the hotspot and resulting in the less intense emission lines and veiling. Its K2 light curve appears to be dominated by flux dips (see Fig. \[fig:k2lcs\]), which is consistent with circumstellar material obscuring the star from time to time. The H$\beta$ line shows an inverse P Cygni profile on Nov. 24 and 25, just before this event. This type of profile, characterized by a redshifted absorption, occurs when the accretion funnel flows intersect our line of sight.
### DN Tau
The intensity and radial velocity of the He I and Li I lines are modulated with a period that is consistent with the value of 6.32 days given in the literature for this star’s rotation period [@donati13; @artemenko12]. The radial velocity variability of the Li I line appears to be mirrored with respect to the He I line (Fig. \[fig:li\_he\_ew\]). This is consistent with rotational modulation of an accretion shock. While the He I emission originating in the hotspot will be more blueshifted as this spot comes into view and more redshifted as it recedes, photospheric absorption lines such as Li I appear more redshifted as a hot (or cold) spot comes into view and more blueshifted as it recedes, since the line is deformed due to the presence of spots on the stellar surface [see e.g. @gahm13]. The amplitude of the veiling variability is too small to draw any conclusions from it.
This star’s H$\alpha$ and H$\beta$ line profiles present a redshifted absorption component in some observations, which is clearest on Nov. 27, when the He I line is strongest and more centred, meaning that the accretion shock is in full view. As this redshifted absorption originates in accretion funnel flows crossing our line of sight, this points to a spatial association between the accretion shocks and funnel flows.
### GI Tau
The intensity of the main emission line profiles (H$\alpha$, H$\beta$, He I) is modulated on the time-scale of the observations. The intensity of the residual line profiles steadily decreases from Nov. 22-23 to reach a minimum around Nov. 27, then increases again up to Nov. 29, a behaviour which is consistent with the modulation of the line flux by a bright spot rotating with a period of $\sim$7 or 8 days. This is in agreement with the past reported photometric period of 7.1 days [@percy10; @artemenko12]. The veiling appears to decrease over a similar time period, however the variability amplitude is within the uncertainties and therefore cannot be confirmed.
### GK Tau
The variability of the radial velocity of the He I line is consistent with a period of 4.6 days, found in the literature [@percy10; @artemenko12] and in this star’s K2 light curve [@rebull20]. There is no clear trend in veiling or in the H$\alpha$ and H$\beta$ line profiles with this period, even though the variability of the line profiles is very strong.
### GM Aur
The intensity of the He I line is modulated on the time-scale of the observations, decreasing until Nov. 24, then increasing again. The variability of this line’s radial velocity is consistent with rotational modulation of a spot with the period of 6 days given in the literature [@percy10; @artemenko12]. The veiling variability is too small to draw any conclusions from it. As with DN Tau, the radial velocity variability of the Li I line is mirrored with respect to the He I line, in support of rotational modulation of a hotspot. There is no clear trend in the H$\alpha$ and H$\beta$ line profiles with this period.
### IP Tau
The intensity and radial velocity of the He I and Li I lines, as well as the veiling, all show a variability that is consistent with a period of 5.6 days. Veiling increases along with the intensity of He I, both reaching a maximum on Nov. 25, while the He I line profile went from more blueshifted than average on Nov. 23 and 24, to more redshifted on Nov. 25-28, and the Li I line went from more redshifted to more blueshifted in this same time. This behaviour is consistent with the veiling and He I line being modulated by a hotspot rotating at this period, which is different from the photometric period of 3.25 days found by @bouvier93. The light curve from that study was therefore likely dominated by something other than rotational modulation from spots on the stellar surface.
The H$\alpha$ and H$\beta$ line profiles show a redshifted absorption component on one night (Nov. 29), corresponding to phase $\phi \sim 0.25$, when the He I line is more blueshifted and veiling is beginning to increase, meaning that the accretion shock is appearing in our line of sight. As was the case with DN Tau, this points to a spatial association between the accretion funnel flows and accretion shocks.
### IW Tau
This star was one of the two WTTSs (non-accreting T Tauri stars) in our control sample. It shows a relatively narrow H$\alpha$ line profile, with a width at 10% of the maximum intensity of only 180 km s$^{-1}$, which is much lower than the threshold of 270 km s$^{-1}$ usually used to classify a T Tauri star as actively accreting [@white03]. Therefore it would not be classified as a classical T Tauri star based on its H$\alpha$ line profile. However, this star does present a weak He I emission line with a redshift of 4.3 $\pm$ 0.8 km s$^{-1}$, which does not seem to be variable in our observations. @beristain01 also observed weak HeI$\lambda5876$ emission in the spectra of three non-accreting T Tauri stars, however they were all on average centred at the stellar rest velocity, which led to the conclusion that they could be the result of very active chromospheres. The He I emission we observe in IW Tau is redshifted at above 3$\sigma$, which means it must originate in material falling onto the star.
It is possible that this star is still undergoing accretion at a very reduced rate. In order to test this possibility, we estimated a mass accretion rate for this star from several emission lines in its spectra, following the relations given by @alcala17. We find an average value for the logarithmic mass accretion rate of $\log \dot{M}_{acc} = -9.5 (\pm 0.3)~\mathrm{M}_{\odot}~\mathrm{yr}^{-1}$ (corresponding to $\dot{M}_{acc} = 3 \times 10^{-10}~\mathrm{M}_{\odot}~\mathrm{yr}^{-1}$), where the uncertainty in $\log \dot{M}_{acc}$ represents the standard deviation across the ten emission lines used to calculate accretion luminosity. It is worth noting that this likely represents an upper limit to this star’s mass accretion rate, since at these low levels of emission, these lines may be affected by chromospheric activity in a non-negligible way. At the same time, the strong agreement between the mass accretion rates derived from all these different lines (made clear by the relatively small dispersion in the values found) is a good indication of the reliability of this result. This star does not seem to be surrounded by a circumstellar disk [besides possibly a thin debris-disk, typical of WTTSs in general; @beckwith90; @wolk96; @furlan05].
This star’s Li I line is modulated on a 5.5 day time-scale, which could be caused by the rotational modulation of cold spots on the stellar surface, or by a binary companion [since this star has been reported to be a binary system by @richichi94]. This star’s K2 light curve clearly shows a spot-like behaviour, with two distinct periods, one of which agrees with this period found in the Li I line [see Fig. \[fig:k2lcs\] and @rebull20]. It is likely that one of these periods represents the star’s rotation while the other may be due to a companion.
### T Tau
This is a multiple system, consisting of three stars [known as T Tau N, Sa and Sb; @koresko00; @kohler00], although the secondary component T Tau S(a+b) is not detectable in optical wavelengths, likely because of strong extinction from circumstellar or circumbinary material [@duchene05]. Thus the spectra presented here are entirely dominated by the brightest component, T Tau N.
The radial velocity of this star’s He I NC is very well modulated with a period of 2.7$\pm$0.4 days, while the radial velocity of the Li I line gives a similar period of 2.86 days. These periods coincide well with the photometric period of 2.81 days measured by @artemenko12 and in its K2 light curve [@rebull20]. The top two panels of Fig. \[fig:ttaubc\] show the Li I line and the NC of the He I line folded in phase with this period of $P=2.8$ days. The variability amplitude of the veiling is small, and there are no clear trends in the H$\alpha$ or H$\beta$ lines with the stellar rotation period. We can see that the modulus of the equivalent width of the He I line’s NC is largest when it is most redshifted (on Nov. 25 and Nov. 28), which may be due to a favorable viewing geometry. We have found that the magnetic obliquity of this star is $\theta = (23 \pm 10)^{\circ}$, which agrees, within the uncertainties, with its system inclination of $i = (28 \pm 1)^{\circ}$ [@manara19]. This means that when the accretion shock passes in front of us, we observe the flow of matter onto the star projected almost directly in our line of sight.
![Radial velocity variability of *a)* the Li I line; *b)* the narrow component of the He I line; *c)* and *d)* the broad component of the He I line, for the star T Tau. The plots are shown in full in the *left* panels and in phase in the *right* panels, using the period of $P=2.8$ days (panels *a, b* and *c*) and $P=3.1$ days (panel *d*). Different colors represent different rotation cycles. The median error bar on the radial velocity is shown on the top left corner of each plot. In these plots, we do not show the average measurement in each night, but rather include all observations. []{data-label="fig:ttaubc"}](TTau_LiI_vrad.pdf "fig:"){width="47.00000%"} ![Radial velocity variability of *a)* the Li I line; *b)* the narrow component of the He I line; *c)* and *d)* the broad component of the He I line, for the star T Tau. The plots are shown in full in the *left* panels and in phase in the *right* panels, using the period of $P=2.8$ days (panels *a, b* and *c*) and $P=3.1$ days (panel *d*). Different colors represent different rotation cycles. The median error bar on the radial velocity is shown on the top left corner of each plot. In these plots, we do not show the average measurement in each night, but rather include all observations. []{data-label="fig:ttaubc"}](TTau_HeI_vrad_gaus_nc.pdf "fig:"){width="47.00000%"} ![Radial velocity variability of *a)* the Li I line; *b)* the narrow component of the He I line; *c)* and *d)* the broad component of the He I line, for the star T Tau. The plots are shown in full in the *left* panels and in phase in the *right* panels, using the period of $P=2.8$ days (panels *a, b* and *c*) and $P=3.1$ days (panel *d*). Different colors represent different rotation cycles. The median error bar on the radial velocity is shown on the top left corner of each plot. In these plots, we do not show the average measurement in each night, but rather include all observations. []{data-label="fig:ttaubc"}](TTau_HeI_vrad_gaus_bc_p1.pdf "fig:"){width="47.00000%"} ![Radial velocity variability of *a)* the Li I line; *b)* the narrow component of the He I line; *c)* and *d)* the broad component of the He I line, for the star T Tau. The plots are shown in full in the *left* panels and in phase in the *right* panels, using the period of $P=2.8$ days (panels *a, b* and *c*) and $P=3.1$ days (panel *d*). Different colors represent different rotation cycles. The median error bar on the radial velocity is shown on the top left corner of each plot. In these plots, we do not show the average measurement in each night, but rather include all observations. []{data-label="fig:ttaubc"}](TTau_HeI_vrad_gaus_bc_p2.pdf "fig:"){width="47.00000%"}
Another interesting aspect of this star’s He I line profile is that its BC[^6] also shows periodic radial velocity variability, but with a slightly longer period of 3.1$\pm$0.3 days. The bottom two panels of Fig. \[fig:ttaubc\] show the radial velocity variability of this component folded in phase with the period of the NC ($P=2.8$ days) and with the longer period of $P=3.1$ days. It is clear that this component folds much better in phase with the longer period. In order to confirm whether these two slightly different periods are real and not an artefact of the Gaussian decomposition, we performed a 2-dimensional periodogram analysis on the full line (Fig. \[fig:2dper\]). From this figure it appears that the dominant period is the 3.1 day period of the BC, which is clear on the wings of the profile (especially the redshifted wing, between $\sim$40km s$^{-1}$ and 100km s$^{-1}$), where the BC dominates[^7]. However, near the centre of the line (around 5 - 10km s$^{-1}$, where the NC appears), the periodogram seems to split and shows a small peak at 2.7 days. If the full line originated in the hotspot, then this apparent quasi-periodicity could occur, for instance, if the hotspot were to move on the stellar surface, or to split into more than one spot. However, because the wings of the profile come from the BC, which is believed to have a different origin than the NC, it seems more likely that the two components of the He I line indeed have two slightly different periods[^8]. Since the period from the NC agrees well with the rotation period of 2.8 days found from photometry in the literature and also from the Li I line in this study, this likely represents the stellar rotation, while the slightly larger period of 3.1 days from the BC should be caused by a different phenomenon.
![2-dimensional periodogram of the He I line of T Tau. The abscissa shows the velocity bins (centred at stellar rest velocity) for which periodograms were measured, the ordinate shows the range of periods spanned, and the colour gradient represents the periodograms’ intensities. []{data-label="fig:2dper"}](perHask_HeI2TTau.pdf){width="47.00000%"}
According to @beristain01, the broad component of the HeI$\lambda5876$ line may come from a hot wind or from the accretion funnel flows. Since in T Tau this component is redshifted on average, it must be coming mostly from the accretion funnel flows at polar angles less than 54.7$^{\circ}$ [see Fig. 9 of @beristain01]. If this is indeed the case, then a slightly larger period measured from this component when compared to the stellar rotation period may be an indication that there is differential rotation throughout the accretion funnel flow. This could be expected if a star’s magnetosphere were to truncate the inner disc at a radius larger than the co-rotation radius, resulting in the part of the accretion funnel flow that connects to the inner disc rotating more slowly than the stellar surface. The star would then accrete via the propeller regime, though this could be a temporary situation, not necessarily reflecting the steady state of the system. Its K2 light curve, for instance (which was observed in 2015 - four years after the spectroscopic observations in this paper), is periodic, which seems to indicate that this star was in a stable accretion regime at the time of the K2 observations (see Fig. \[fig:k2lcs\]).
We cannot, however, conclude with certainty that there is differential rotation in the accretion columns of T Tau, since the two periods found in fact agree within their error bars. A more in-depth study, with more accurate periods, would be necessary to confirm this hypothesis. Additionally, current radiative transfer models are incapable of reproducing the main characteristics of the broad component of the HeI$\lambda5876$ line with standard accretion models [see, e.g. @kurosawa11], so we cannot say for certain what is the origin of this broad emission.
### V836 Tau
This star shows some variability in the radial velocity of its He I line, but no clear period can be distinguished in the 6-day observation, which is shorter than the period of 7.0 days given in the literature [@rydgren84]. The He I line intensity and veiling show little variability and no clear trend is seen in the H$\alpha$ or H$\beta$ line profiles.
Velocity of the flow of matter traced by He I
---------------------------------------------
We estimate the velocity ($V_{flow}$) of the accretion flow that is traced by the narrow component of the HeI$\lambda5876$ line for each star in our sample, by taking the median He I radial velocity of each star and de-projecting it from the line of sight. Therefore, , where $\alpha$ is the angle between the line of sight and the direction of the flow of matter onto the star. This angle is related to the system inclination ($i$ - given in Table \[tab:vflow\]) and the latitude of the accretion shock ($l$ - given in Table \[tab:results\]) through the relation $\alpha = |i-l|$ (see Fig. \[fig:sketch\]). Table \[tab:vflow\] shows the derived flow velocities $V_{flow}$, as well as the mean ($<V_{rad}>$), median ($V_{rad,med}$) and standard deviation ($\sigma$) of the He I radial velocities measured in each star (with regard to the star’s rest velocity), the mean error in $V_{rad}$ measurements for each star, and the system inclinations $i$.
---------- --------------- --------------- ----------------- ------------------ -------------- ----- ------------------------------
Star $<V_{rad}>$ $V_{rad,med}$ $<err V_{rad}>$ $\sigma V_{rad}$ Inclination Ref $V_{flow}$
(km s$^{-1}$) (km s$^{-1}$) (km s$^{-1}$) (km s$^{-1}$) ($^{\circ}$) (km s$^{-1}$)
DE Tau 5.84 5.96 1.19 0.81 $70\pm7$ P14 14$\substack{ +38\\-7}$
DF Tau 6.54 6.69 1.01 1.61 $19\pm8$ \* 6.7$\substack{ +1.6\\-1.0}$
DK Tau 9.36 9.54 1.86 3.96 $13\pm3$ M19 9.6$\substack{ +2.3\\-1.9}$
DN Tau 6.68 7.36 0.99 1.40 $35\pm 1$ L18 7.9$\substack{ +2.1\\-1.5}$
GI Tau 6.72 6.32 1.25 1.65 $29\pm9$ \* 7.9$\substack{ +9.4\\-2.8}$
GK Tau 7.21 7.40 1.95 1.18 $71\pm5$ S17 18$\substack{ +27\\-9}$
GM Aur 7.10 7.90 2.72 2.56 $55\pm 1$ T17 10.6$\substack{+10.0\\-4.9}$
IP Tau 5.99 6.14 1.08 0.87 $45\pm1$ L18 7.8$\substack{ +3.7\\-2.2}$
IW Tau 4.33 4.14 0.77 0.77 $33\pm9$ \* 4.6$\substack{ +3.0\\-1.3}$
T Tau 4.20 2.00 5.19 7.55 $28\pm1$ M19 2.0$\substack{ +5.5\\-5.2}$
V836 Tau 6.01 6.22 1.30 1.12 $61\pm10$ T17 10$\substack{ +16\\-4}$
---------- --------------- --------------- ----------------- ------------------ -------------- ----- ------------------------------
**Notes:** The second and third columns show the mean and median radial velocity of He I for each star, column 4 shows the mean error on the radial velocity measurements, columns 5 and 6 give the inclination to the system and its reference, and the final column shows the flow velocity derived for each star. References for inclinations are @pietu14 (P14), @manara19 (M19), @long18 (L18), @simon17 (S17), @tripathi17 (T17), and those marked with an asterisk have no direct measurement of the disc in the literature, so inclinations were estimated using the relation $v\sin i = (2\pi R_* / P_{rot}) \sin i$, where $P_{rot}$ and $v\sin i$ are given in this paper, and $R_*$ was taken from @herczeg14.
--------------------- -------------------- -------------------- -------------------
Average $<V_{rad}>$ $\sigma <V_{rad}>$ Average $V_{flow}$ $\sigma V_{flow}$
(km s$^{-1}$) (km s$^{-1}$) (km s$^{-1}$) (km s$^{-1}$)
6.4 1.4 9.0 4.3
--------------------- -------------------- -------------------- -------------------
Discussion {#sec:discuss}
==========
Comparison with other observational studies
-------------------------------------------
As can be seen in Table \[tab:results\], the magnetic obliquities we find for the stars in our sample are between 5$^{\circ}$ and 23$^{\circ}$, with most (80%) below 15$^{\circ}$. Therefore in our sample the magnetic fields seem to be generally well aligned with the stellar rotation axis, which is somewhat in disagreement with what has been found from direct magnetic field measurements in T Tauri stars, many of which have a misalignment larger than 20$^{\circ}$ [see Table \[tab:magob\_otr\] and, e.g., @johnstone14]. It is possible that our sample may be influenced by a selection bias. The stars in this sample were selected based mainly on two criteria: having shown periodic photometric behaviour in a previous study and presenting a HeI$\lambda5876$ line profile in emission that is dominated by the NC. The only exception to the latter criterion is T Tau, which was observed mainly because of its brightness and is the only star in our sample for which the He I line is dominated by the broad component. It is also the star that shows the largest magnetic obliquity in this study, of 23$^{\circ}$. It is possible that there may be a connection between the processes that originate the BC of the HeI$\lambda5876$ line and large magnetic obliquities. This would lead to an observational bias in our sample, where by excluding stars that present very broad HeI$\lambda5876$ emission, we exclude the stars with large magnetic obliquities. In order to confirm this hypothesis, we would need to study a much larger sample of CTTSs with more diverse He I line profiles.
The other selection bias that our sample could possibly be subject to is with periodicity, since the stars in our sample all have a measured period in the literature. However, it is difficult to find an explanation in which this leads to a bias towards magnetic obliquities smaller than $\sim 15^{\circ}$, since MHD models of @kurosawa13 and @blinova16 predict that larger magnetic obliquities should lead to stars accreting more often in a stable accretion regime, which is believed to produce periodic signatures in a star’s photometry. Therefore, based on these models, choosing a sample of stars with known periodicity should not lead us to exclude stars with large magnetic obliquities.
While studying the spectroscopic variability of the very active star EX Lupi, @sicilia-aguilar15 found a periodic signature in the radial velocity variation of different He I lines (at $\lambda = 4713$Å, 5876Å, and 6678Å), as well as the HeII$\lambda4686$ line, which were consistent with the rotational modulation of a hotspot on the stellar surface. They found, however, that the amplitudes of the radial velocity variations differ for the different lines, with lines with higher excitation potentials having larger amplitudes and a clearer modulation. In general, lines with different excitation potentials may originate in regions with different temperatures, which could be at different latitudes, longitudes (a spot may have the highest temperature at its core and be surrounded by regions of lower temperature), or height above the stellar surface [there may be a temperature structure along the accretion column, above the stellar surface, as has been detected by @dupree12; @sicilia-aguilar17]. This could explain the different amplitudes found by @sicilia-aguilar15 for the different lines.
If there is a difference in the temperature structure on the stellar surface along latitude, then this may help to explain our apparently low magnetic obliquities. By choosing the HeI$\lambda5876$ line, which has a lower excitation potential than HeI$\lambda4713$ and HeII$\lambda4686$, we may be tracing a region that is at a slightly higher latitude than the central part of the accretion shock, which may be nearer to the magnetic pole. If true, this effect would probably bias the estimates of magnetic obliquity using this line to lower values, which may explain not only our sample, but also the fact that the three other stars from the literature whose obliquities were derived using the same method as the one described in our paper also present such low values of $\Theta$ [EX Lup, RU Lup and DR Tau[^9], see Table \[tab:magob\_otr\] and @sicilia-aguilar15; @gahm13; @petrov11]. However, this does not seem to be the case. @petrov01 found the same amplitude (within error bars) for the radial velocity variability of He I and He II lines for the star RW Aur. Also, several studies of spectropolarimetry have found misalignements between the main component of the magnetic field and the stellar rotation axis that are consistent with the HeI$\lambda5876$ variability amplitudes found in the same observations [e.g. BP Tau, AA Tau, LkCa15, and CI Tau; @donati08; @donati10b; @alencar18; @donati20].
We can also verify how the variability of photospheric lines compares with that of the He I lines. @crockett12 studied the radial velocity variability of photospheric lines in optical and near-infrared spectra of several CTTSs, in order to study the effect of stellar spots on these lines. Their sample included four of the same stars as our sample. The amplitudes they found for the radial velocity variability of photospheric lines in optical wavelengths agree, within the uncertainties, with our amplitudes for the HeI$\lambda$5876 line, though the values for the photospheric lines are systematically lower, as expected for large cool spots.
Photometric variability
-----------------------
@blinova16 predict that the location of the magnetospheric radius $R_m$ (the disc’s truncation radius) with respect to the disc’s co-rotation radius $R_{co}$ has a strong influence on the accretion regime, stronger than the magnetic obliquity ($\Theta$). None the less, for stars with magnetospheres of similar size, they predict that an increase of the magnetic obliquity leads to a decrease of the amplitude of the light curve oscillations that are associated with instabilities and an increase of the amplitude of the variability associated with the rotation of the star. According to @blinova16, for small values of $\Theta$ ($\lesssim 5^{\circ}$ for $R_{co} \sim 1 - 1.4 R_{m}$, or up to $\sim 10^{\circ}$ for smaller magnetospheres), the main source of variability should be unstable ordered hotspots and the light curve should be very irregular, though the stellar rotation period may still be found in the light curve’s Fourier spectrum. For slightly larger values of $\Theta$ ($\sim 15^{\circ}$), the stellar rotation period should become more evident, but many short-scale oscillations from instabilities could still be seen in the light curve. When $\Theta$ is around $\sim 20^{\circ} - 30^{\circ}$, the oscillations from instabilities would have smaller amplitudes, leading to more regular light curves, but with the stellar rotation period becoming slightly less well defined in the Fourier spectrum.
Among the 11 stars in our sample with He I in emission, 6 were observed recently by the K2 mission [their light curves are shown in Fig. \[fig:k2lcs\] and a more detailed analysis is explored in @rebull20 and in Roggero et al. in prep.]. Even with such a small number, this sample shows a large diversity of light curve morphologies. IW Tau presents a very regular light curve, consistent with modulation by the rotation of stable spots on the stellar surface [as the spot-like light curves of @alencar10], in combination with a second periodic event (possibly due to its binarity). T Tau shows a periodic but somewhat irregular behaviour (altering between what look like flux bursts and flux dips) reminiscent of the class of light curves attributed to stochastic accretion by @stauffer16. The K2 light curve of DF Tau is not periodic and appears to be dominated by flux bursts [as those described in @stauffer14]. Finally, the remaining 3 light curves (those of DK Tau, GI Tau and GK Tau) seem to be dominated by flux dips [as those described in @mcginnis15]. However, it is important to note that GI Tau and GK Tau consist of a binary system with a separation of 13.2" [@akeson19], so the light curve of GI Tau may be slightly contaminated by that of the primary, GK Tau (this is evident from the fact that a peak at 4.6 days, the rotation period of GK Tau, is detected in a periodogram analysis of GI Tau’s light curve).
We can compare our findings with the predictions of @blinova16 regarding the relationship between the magnetic obliquity and the star’s light curve, stated above. The most regular light curve in our sample belongs to IW Tau. This star’s He I line did not show a detectable variation in radial velocity, therefore its magnetic obliquity must be small. This seems not to be in line with the theoretical prediction that with smaller magnetic obliquities, light curves are more irregular. However this particular star has a very low mass accretion rate (if it is in fact accreting and the He I emission we observe does not come from a very active chromosphere), therefore it should have a large magnetosphere, which is also predicted to lead to stable configurations (so long as the magnetic radius $R_m$ does not exceed the co-rotation radius $R_{co}$). Excluding IW Tau, the light curves of T Tau and GK Tau appear to have the least amount of irregular oscillations among the stars in this sample, with both showing a clear periodic signal. T Tau presents the largest magnetic obliquity in the sample, of $23^{\circ} \pm 10 ^{\circ}$, consistent with the prediction that stars with a magnetic obliquity larger than $\sim 20^{\circ}$ would tend to have more regular light curves. However, the magnetic obliquity found for GK Tau is low, of only $5^{\circ} (\substack{ +7^{\circ}\\-5^{\circ}})$, while the values found for DF Tau, DK Tau and GI Tau (all of which present light curves that appear to be more irregular than GK Tau’s) are larger, between $12^{\circ}$ and $18^{\circ}$. It is clear that with this small sample we cannot distinguish any effects the magnetic obliquity may have on a star’s light curve morphology.
Correlations with stellar properties
------------------------------------
In an attempt to shed some light on the origin of the magnetic obliquity, we joined our sample with other values from the literature and searched for possible correlations with stellar parameters. The magnetic obliquities taken from the literature are given in Table \[tab:magob\_otr\], along with their references. They were derived mostly using ZDI, but a few cases were also found in which they were derived in a similar fashion as in our study (or in which a value of $\Delta V_{rad} (He I)$ and $v \sin i$ are given, allowing us to derive $\Theta$).
We find a tentative correlation between the magnetic obliquity and stellar mass. A Kendal $\tau$ analysis gives a false alarm probability of $<1\%$, but it is important to note that there are few stars in this sample, so this result should be taken with care. It is, however, an interesting indication that the inner structure of the star may have an important role in determining its magnetic obliquity since, for stars of a similar age (as should be more or less the case in this sample), a more massive star is expected to have developed a radiative core while less massive stars remain fully convective.
![HR diagram with the stars in our sample (filled diamonds) and other stars from the literature (filled circles). Colours represent the stars’ magnetic obliquities. Two colours are used when two different values were found at different epochs. Dashed lines represent mass tracks from pre-main sequence evolutionary models of @tognelli11, while the dotted line represents the limit where a radiative core begins to develop, according to these models (stars to the right of this line should be fully convective). []{data-label="fig:hrd"}](hrd_pisa.pdf){width="47.00000%"}
In order to further investigate this hypothesis, we plotted an HR diagram using different colours to represent the magnetic obliquity found for each star (Fig. \[fig:hrd\]). In a few cases, two different observations using spectropolarimetry resulted in different values of the magnetic obliquity (with differences of up to 30$^{\circ}$ between the two epochs). Both values are represented in the figure by a two-coloured symbol. We used different symbols to differentiate the sample from our study (filled diamonds) from those taken from the literature (filled circles). Theoretical mass tracks from @tognelli11 are also plotted for reference. The dotted line represents the evolutionary phase in which a radiative core is expected to begin to develop, according to @tognelli11 [see also @gregory12], meaning that stars to the right of this line should be fully convective. This figure seems to suggest that fully convective stars usually have low magnetic obliquities (though there are a few cases where it is as high as 40$^{\circ}$), while the stars with large magnetic obliquities ($\gtrsim 60^{\circ}$) are those that have developed at least a small radiative core. The average magnetic obliquity among the stars in the fully convective part of the HR diagram is 17$^{\circ}$ (with an rms dispersion of 12$^{\circ}$), while in the partly radiative part of the HR diagram it is slightly larger, 35$^{\circ}$ (with an rms deviation of 22$^{\circ}$). This could be complementary with other studies of magnetic fields in young stars that suggest that, as a radiative core develops, stellar magnetic fields tend to become more complex and less intense than in fully convective stars [@gregory12; @folsom16]. If this is indeed the case, it may explain why our sample shows a tendency towards low magnetic obliquities, as most of our stars are fully convective.
The apparent evolution of the magnetic field configuration across the HR diagram [in terms of the strength of the field and its complexity, as demonstrated by @folsom16; @villebrun19 as well as in terms of the magnetic obliquity, as shown here] points to a dynamic origin of stellar magnetic fields, likely as the consequence of a dynamo effect. This study and previous ones such as @folsom16 also seem to suggest that there are intrinsic differences between magnetic fields of stars that are fully convective and those that have begun to develop a radiative core.
Summary and conclusions {#sec:conc}
=======================
We have analysed the variability of the HeI$\lambda$5876 emission line profile in a sample of 10 CTTSs and 1 WTTS, measured with high-resolution spectroscopy over a period of up to 10 days. Most observations show a simple profile, dominated by a narrow component (NC) that is believed to originate in accretion shocks near the stellar surface. For four stars, the HeI$\lambda$5876 line profile is seen to be composed of a combination of narrow and broad component (BC), the latter being believed to have multiple origins, likely formed in both hot winds and accretion columns [@beristain01]. For these four stars, each individual observation was fitted with a combination of two Gaussians, one broader than the other, in order to separately investigate the origin of the two components. Both appear to be variable in their own way on the time-scale spanned by our observations. With the resolution of our spectra we can clearly see that the NC is not truly Gaussian, but this approximation does seem to be valid when analysing its radial velocity. This can be illustrated by the fact that, even though the star T Tau’s He I line profile is clearly dominated by the BC, its stellar rotation period of 2.8 days is recovered from the radial velocity measurements of the NC, while this period is not clear in a radial velocity analysis of the composite profile. This shows that an analysis of the NC of the He I line is still possible even for stars whose He I line profile is clearly dominated by the BC.
The NC of the HeI$\lambda$5876 emission line is shown to be redshifted by an average of 6.4 km s$^{-1}$ in our sample, with a standard deviation of 1.4 km s$^{-1}$. By taking the median redshifts for each star and de-projecting them from our line of sight, we find that the material responsible for this emission is traveling at an average of 9 km s$^{-1}$ in our sample (with a standard deviation of 4.3 km s$^{-1}$). This velocity is consistent with material tracing the post-shock region of accretion shocks, after considerable deceleration of the free-falling material has occurred.
By measuring the amplitude of the radial velocity variability ($\Delta V_{rad}$) of the NC of the HeI$\lambda$5876 emission line, along with the stars’ projected rotational velocities ($v\sin i$), we were able to estimate these stars’ magnetic obliquities ($\Theta$ - the angle between the axis of their magnetic field and rotation axis). We find an average magnetic obliquity of $11.4^{\circ}$ in our sample, with an rms dispersion of $5.4^{\circ}$. The magnetic axis thus seems close to being aligned with the stellar rotation axis in our sample. This is not entirely in agreement with other studies of magnetic field configurations [e.g., @johnstone14], which find several cases of misalignments larger than 20$^{\circ}$. This difference may simply be due to an issue of low number statistics. However, there is also the possibility that our sample may be subject to a selection bias. With the exception of T Tau, the stars in this sample were chosen on the basis that their He I line profiles are dominated by the NC. The star T Tau, whose He I line profile is clearly dominated by the BC (see Fig. \[fig:hei\_profiles\]), is the star that presents the largest magnetic obliquity in our sample (of $23^{\circ} \pm 10^{\circ}$). If the mechanism that originates the BC of the HeI$\lambda$5876 line is somehow connected to larger magnetic obliquities, then this would result in a selection bias in our sample. However, to confirm this hypothesis, a larger study of magnetic obliquities, including stars with more diverse He I profiles, would be needed.
We find tentative evidence for a trend between the position of a star on the HR diagram and its magnetic obliquity. This result is based on a small sample and should be taken with care, but the possibility that a star’s inner structure can play a strong role in determining its magnetic obliquity would be consistent with other studies that show an apparent evolution of a star’s magnetic field as it evolves across the HR diagram [e.g., @gregory12; @folsom16; @villebrun19]. In particular, there may be a considerable difference between the magnetic field configurations of stars that are fully convective and those that are partially radiative, a possibility that merits further investigation as it is unfortunately still subject to low number statistics. If this is indeed the case, it supports the idea that magnetic fields in T Tauri stars are generated by a dynamo, rather than by fossil fields, and that the existence of a boundary between radiative and convective zones plays an important role in determining the geometry of that magnetic field. It could also be responsible for our sample’s bias towards low magnetic obliquities, since most of the stars in our sample are in the fully convective portion of the HR diagram.
Besides our main results, we also find some interesting aspects of individual sources. A joint analysis of the variability of the He I, H$\alpha$ and H$\beta$ lines show, in at least three sources (DK Tau, DN Tau and IP Tau), evidence for a spatial association between the accretion shocks on the stellar surface and the accretion columns, which would be expected in a scenario in which disc-locking takes place. Meanwhile another source, DF Tau, shows evidence for a variable wind that seems to be spatially associated with the accretion shock. This could be a stellar wind that is launched very close to the accretion shock, or it could result from a non-uniform disc wind being launched from close to the truncation radius, in a part of the inner disc where an accretion column is forming. Since this variability seems to be consistent with having the same period as the stellar rotation, this would also be close to the co-rotation radius.
We find that the star IW Tau, previously reported as a non-accreting, weak-line T Tauri star, in fact presents redshifted He I emission, which means that this emission must come from matter falling onto the star. It seems therefore that this star is still weakly accreting. An analysis of the flux of several emission lines that are known to be linked with accretion leads us to estimate a mass accretion rate of $\dot{M}_{acc} = 3 \times 10^{-10}~\mathrm{M}_{\odot}~\mathrm{yr}^{-1}$ for this star. We should note that this value is likely overestimated, since at such low limits of accretion, these lines are likely contaminated with a strong contribution from chromospheric activity.
Finally, we recover the stellar rotation period of 2.8 days for the star T Tau in both the radial velocity analysis of the NC of the He I line, as well as in a portion of a 2-dimensional periodogram of this line. However we find a slightly larger period, of 3.1 days, in the wings of this line and in the radial velocity analysis of its BC, which is redshifted. Since the BC in this case is believed to originate from the accretion columns, while the NC originates at the accretion shock near the stellar surface, this may be an indication that there is differential rotation throughout the accretion columns of this system. Although, seeing as the two periods are very similar, this is not enough evidence to conclude this with certainty and further investigation would be needed to support this claim.
Acknowledgements {#acknowledgements .unnumbered}
================
This paper was based on observations made at Observatoire de Haute Provence (CNRS), France. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 742095, *SPIDI*: Star-Planets-Inner Disk-Interactions, spidi-eu.org; and grant agreement No 743029, EASY: Ejection-Accretion-Structures-in-YSOs). The work also received support from the CAPES/COFECUB collaboration under grant number 88887.160792/2017-00. The authors thank Noemi Roggero for the K2 light curves and Antonella Natta for very helpful discussions.
Data availability {#data-availability .unnumbered}
=================
The data underlying this article can be accessed through the SOPHIE archive at <http://atlas.obs-hp.fr/sophie/>.
Additional figures
==================
The following pages present figures that are discussed in the text (Figs. \[fig:he1veil1\] through \[fig:k2lcs\]).
{width="24.00000%"} {width="24.00000%"} {width="24.00000%"} {width="24.00000%"}
{width="48.00000%"} {width="48.00000%"}
\[fig:he1veil2\]
{width="48.00000%"} {width="48.00000%"}
{width="48.00000%"} {width="48.00000%"}
{width="47.00000%"}
{width="47.00000%"}
\[fig:li\_he\_ew\]
{width="49.00000%"} {width="49.00000%"} {width="49.00000%"} {width="49.00000%"}
{width="49.00000%"} {width="49.00000%"}
{width="24.50000%"} {width="24.50000%"} {width="24.50000%"} {width="24.50000%"}
{width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"}
Additional tables
=================
Table \[tab:eqw\] shows the equivalent widths of the He I, H$\alpha$ and H$\beta$ lines in this study. In order to demonstrate the variability of these emission lines, the average and standard deviation over all nights are given. Table \[tab:magob\_otr\] shows information on the magnetic obliquities of other T Tauri stars taken from the literature.
--------- ------------ ---------- ------------ ---------- ------------ ----------
Star
<EW> $\sigma$ <EW> $\sigma$ <EW> $\sigma$
DETau -65 7 -30 4 -1.0 0.2
DFTau -64 11 -17 6 -2.5 0.7
DKTau -35 19 -9.5 4.7 -1.1 0.4
DNTau -7.6 1.0 -3.3 0.5 -0.34 0.08
GITau -10.3 3.2 -5.0 2.6 -0.54 0.35
GKTau -21 8 -4.1 2.1 -0.44 0.10
GMAur -106 5 -16 2 -1.1 0.4
IPTau -16 6 -7.9 3.1 -0.62 0.37
IWTau -3.8 0.6 -1.7 0.3 -0.07 0.02
TTau -98 13 -17 4 -1.0 0.3
V826Tau -1.5 0.2 -0.27 0.16 -0.07 0.01
V836Tau -14.2 1.4 -3.4 0.4 -0.22 0.04
--------- ------------ ---------- ------------ ---------- ------------ ----------
: Equivalent widths of H$\alpha$, H$\beta$ and HeI$\lambda$5876 for our sample.[]{data-label="tab:eqw"}
**Notes:** We show the mean equivalent width (<EW>) of each emission line over the several days observations, along with the standard deviation ($\sigma$), in order to illustrate the variability of these lines. All values are given in Å.
Star Spectral type Mass ($M_{\odot}$) $\Theta (^{\circ})$ Source of $\Theta$ References
------------------- --------------- -------------------- --------------------- ------------------------------- ----------------------------
AA Tau M0.6 0.57 10 ZDI [@johnstone14; @donati10b]
BP Tau (Feb 2006) M0.5 0.62 10 ZDI [@johnstone14; @donati08]
BP Tau (Dec 2006) “ & ” 30 ZDI [@johnstone14; @donati08]
CI Tau K5.5 0.90 20 ZDI [@donati20]
CR Cha K2 2.00 70 ZDI [@johnstone14; @hussain09]
CV Cha G8 1.05 60 ZDI [@johnstone14; @hussain09]
DN Tau (2010) M0.3 0.55 30 ZDI [@donati13]
DN Tau (2012) “ & ” 15 ZDI [@donati13]
GQ Lup (2009) K5.0 0.89 30 ZDI [@johnstone14; @donati12]
GQ Lup (2011) “ & ” 30 ZDI [@johnstone14; @donati12]
LkCa 15 K5.5 0.89 20 ZDI [@donati19]
TW Hya (2008) M0.5 0.69 40 ZDI [@johnstone14; @donati11b]
TW Hya (2010) “ & ” 10 ZDI [@johnstone14; @donati11b]
V2129 Oph (2005) K6 1.35 20 ZDI [@johnstone14; @donati07]
V2129 Oph (2009) “ & ” 10 ZDI [@johnstone14; @donati11a]
V2247 Oph M1 0.36 40 ZDI [@johnstone14; @donati10a]
V4046 Sgr A K5 0.95 60 ZDI [@johnstone14; @donati11c]
V4046 Sgr B K5 0.85 80 ZDI [@johnstone14; @donati11c]
DR Tau K6 0.88 14 $\Delta V_r (HeI)$ [@petrov11]
EX Lup M0 0.60 13 $\Delta V_r (HeI)$ [@sicilia-aguilar15]
RU Lup K7 0.65 10 $\Delta V_r (HeI)$ [@gahm13; @alcala17]
RW Aur A K6.5 1.13 45 Modeling spectral variability [@petrov01]
**Notes:** To maintain consistency with this work, for the star EX Lup we took the value of $\Theta$ derived from the amplitude of the radial velocity variability of the HeI$\lambda5876$ line. Masses and spectral types are from @herczeg14, where available. For the stars not included in that study, masses were re-derived using the same mass tracks they used [the PISA tracks, @tognelli11], in order to maintain consistency among the sample. \[lastpage\]
[^1]: E-mail: pmcginnis@cp.dias.ie
[^2]: For all objects considered here, the Doppler shifts derived using Gaussian or uniform weights differed by no more than 1 spectral pixel ($\sim$0.5 km s$^{-1}$) with no systematics.
[^3]: When more than one spectrum of a given star were taken on the same night, we averaged these profiles in order to obtain a better signal to noise ratio. Therefore the plots shown in Figs. \[fig:he1veil1\], \[fig:he1veil2\] and \[fig:li\_he\_ew\] show only one measurement per night.
[^4]: Because the spectra were compared with their own mean spectrum, rather than with the spectrum of a non-accreting star of the same spectral type (as is usually the case when determining veiling in a CTTS spectrum), both positive and negative values were found, representing the variation of veiling, rather than the value of veiling itself.
[^5]: We do not discuss the star V826 Tau, because it is clear from Fig. \[fig:hei\_profiles\] that this is a spectroscopic binary (SB2) which presents almost no He I emission.
[^6]: The Gaussian decomposition of each observation of this star can be verified in the online material.
[^7]: Another peak can also be seen at approximately 1.5 days, but this is simply an artefact, equal to half of the 3.1 day period.
[^8]: Because we are analysing the normalized spectra, variations of the continuum caused, for instance, by variability of veiling, would also affect the periodogram. Nevertheless, one can see in Fig. \[fig:he1veil2\] that the veiling variability in this star is small and not periodic, so any period measured in Fig. \[fig:2dper\] should not have any influence from this effect.
[^9]: These are three very active stars and all present both a NC and BC in their HeI$\lambda$5876 line profiles. In all three cases, the two components were deconvolved using Gaussian fits and the radial velocity of the NC was used to derive the hotspot latitude, in accordance with our study.
|
---
abstract: 'Conventional prior for Variational Auto-Encoder (VAE) is a Gaussian distribution. Recent works demonstrated that choice of prior distribution affects learning capacity of VAE models. We propose a general technique (embedding-reparameterization procedure, or ER) for introducing arbitrary manifold-valued variables in VAE model. We compare our technique with a conventional VAE on a toy benchmark problem. This is work in progress.'
author:
- |
Eugene Golikov\
Neural Networks and Deep Learning Lab\
Moscow Institute of Physics and Technology\
Russia\
[golikov.ea@mipt.ru]{}\
Maksim Kretov\
Neural Networks and Deep Learning Lab\
Moscow Institute of Physics and Technology\
Russia\
[kretov.mk@mipt.ru]{}
title: 'Embedding-reparameterization procedure for manifold-valued latent variables in generative models'
---
Introduction
============
Variational Auto-Encoder (VAE) [@vae] and Generative Adversarial Networks (GAN) [@gan-orig] show good performance in modelling real-world data such as images well. The key idea of both frameworks is to map a simple distribution (typically Gaussian) of lower dimension to a high-dimensional observation space by a complex non-linear function (typically neural network). Most of research efforts are concentrated on the enhancement of training procedure and neural architectures giving rise to a variety of elegant extensions for VAE and GANs [@gan-overview].
We consider prior distribution that is mapped to data distribution $p(x)$ as one of design choices when building generative model. Its importance is highlighted in a number of works [@elbo-surgery; @s-vae18; @homeo-vae; @sphere-nlp]. Although [@s-vae18] provides an extensive overview of usage of $L_2$-normalized latent variables (points lying on a hypersphere); this is clearly just one of the possible design choices for prior distribution in generative model.
Recent works [@s-vae18; @homeo-vae; @sphere-nlp] argued that manifold hypothesis for data [@belkin] provides evidence in favor of using more complicated priors than Gaussian, for which the topology of latent space matches that of the data. The above mentioned works derived analytic formulas for reparameterization of probability density on manifold (hypersphere in [@s-vae18] and Lie group $SO(3)$ in [@homeo-vae]).
A somewhat less rigorous argument in favor of using manifold-valued latent variables is that we can represent generative process for data as having two sources of variation (see Figure \[sym\]): one is uniform sampling from a group of transformations that we consider as compact symmetry groups (for example group of rotations) and another one is all the rest. This favors the choice of such topology of the latent space that would match “real” generative process: choose uniform distribution on some compact symmetry group as a prior distribution for latent variables.
Once a universal procedure for fast prototyping of VAE with different manifold-valued variables is available, such VAE can be used for estimating the likelihood integral $p(x|Model)$ (for example using IWAE estimate [@iwae]) and thus make conclusions about latent symmetries that are present in the data. This was one of the key motivations for the current work.
![Observed data $X$ are generated by *uniform* sampling from compact symmetry group $G$ and other independent factors of variation $V$ (for example, label of the class).[]{data-label="sym"}](sym.png)
All of above brings to the focus the case of continuously differentiable symmetry groups (Lie groups), which is a special case of manifold-valued latent variables.
Manifold-valued latent variables
================================
Let us make the following preliminary assumption:
*Data $x \sim p(x), x \in S \subset \mathbf{R}^n$ are generated as on Figure \[sym\] with Lie group $G$ embedded in $\mathbf{R}^m$ and there is a continuous mapping $G \rightarrow S$.*
When using images as a test bed it implies that images generated by “close” symmetry elements (say two similar rotation angles $\phi_1$ and $\phi_2$) are also close in the pixel space. It justifies using additional tricks such as continuity loss [@homeo-vae] for training VAE with manifold-valued latent variables.
Construction of VAE
-------------------
Recall the optimization problem for VAE [@vae]: $${\mathcal{L}}(\phi, \psi) =
{\mathbb{E}}_{x \sim {\mathcal{D}}} \left[{\mathbb{E}}_{z \sim q_\phi(z | x)}[\log p_\psi(x | z)] - {\mathrm{KL}}(q_\phi(z | x) \| p(z))\right] \to \max_{\phi, \psi},$$ where ${\mathcal{D}}$ denotes the data distribution, $q_\phi(z|x)$ is a posterior distribution on latent space $Z$, $p(z)$ is the corresponding prior, and $p_\psi(x|z)$ is the likelihood of a data point $x$ given $z$. In order to construct a VAE with manifold-valued latent variables, we need the following:
1. An encoder that produces the posterior distribution $q(z|x)$ from a parametric family of distributions on a manifold.
2. An ability to sample from this posterior distribution: $z \sim q(z | x)$.
3. An ability to compute KL-divergence between this posterior and a given prior.
Recent works [@s-vae18; @homeo-vae] proposed approaches to working with manifold-valued latent variables that are similar in spirit to ours: they derive a reparameterization of probability density defined on smooth manifold and use it in VAE. Problem is that such derivation appears to be complicated and needs to be done for all manifolds of interest.
Our approach is the following. First of all, we introduce a hidden latent space $Z_{hid}$, such that $\mathrm{dim} \, Z_{hid} = \mathrm{dim} \, {\mathcal{M}}= n$, where ${\mathcal{M}}$ is our manifold lying in a latent space $Z$ of dimension $m > n$. Let $p(z_{hid})$ be a prior distribution on $Z_{hid}$.
Suppose then, we have an embedding $f:\; Z_{hid} \to Z$, so that $f(Z_{hid}) \subset {\mathcal{M}}$. Being an embedding requires $f$ to be a diffeomorphism with its image, in particular, $f$ should be a differentiable injective map. We also pose an additional constraint on $f$: it should map the prior on $Z_{hid}$ to a prior on the manifold ${\mathcal{M}}$; in other words, if $z_{hid} \sim p(z_{hid})$, then $f(z_{hid}) \sim p(z)$.
Using this embedding $f$, we can construct a VAE with manifold-valued latent variables as depicted on the right part of Figure \[fig:wae\_and\_manifold\_latent\_vae\]. In this case the posterior distribution $q(z_{hid}|x)$ on $Z_{hid}$ together with the embedding $f$ induce a posterior distribution $q(z|x)$ on ${\mathcal{M}}\subset Z$. We then have to compute KL-divergence between this induced posterior and the prior $p(z)$ on the manifold. Despite the fact that in this case the probability mass is concentrated on the manifold ${\mathcal{M}}$ and hence the probability density on $Z$ is degenerate, we can define the manifold probability densities $q_{{\mathcal{M}}}(z|x)$ and $p_{{\mathcal{M}}}(z)$ (see Appendix 5.1 for details). Moreover, the corresponding KL-divergence is equivalent to the KL-divergence between distributions defined on $Z_{hid}$ (Appendix 4.3): $${\mathrm{KL}}(q_{{\mathcal{M}}}(z|x) \| p_{{\mathcal{M}}}(z)) =
{\mathrm{KL}}(q(z_{hid}|x) \| p(z_{hid}))$$
Hence the final optimization problem for model on the right part of Figure \[fig:wae\_and\_manifold\_latent\_vae\] becomes the following: $${\mathcal{L}}(\phi, \psi) =
{\mathbb{E}}_{x \sim {\mathcal{D}}} \left[{\mathbb{E}}_{z_{hid} \sim q_\phi(z_{hid} | x)}[\log p_\psi(x | f(z_{hid}))] - {\mathrm{KL}}(q_\phi(z_{hid} | x) \| p(z_{hid}))\right] \to \max_{\phi,\psi},$$ where $\phi$ are parameters of VAE encoder, which encodes the object $x$ into $Z_{hid}$ space, and $\psi$ are parameters of VAE decoder which maps the manifold ${\mathcal{M}}\subset Z$ to data-manifold in feature space; ${\mathcal{D}}$ is our data distribution.
Thereby working with probability distributions induced on manifold of interest is easy: both terms in VAE loss (reconstruction error and KL-divergence) are easily calculated in the original hidden space $Z_{hid}$ that is further mapped on a manifold.
Learning manifold embedding
---------------------------
To apply the procedure described above, we have to construct an embedding $f$. In order to do this, we propose the following procedure:
1. Sample data from $p(z)$ (distribution on $\mathcal{M}$).
2. Train Wasserstein Auto-Encoder (WAE) [@wae] on the data from $p(z)$ (feature space) and the latent space $Z_{hid}$ with the prior $p(z_{hid})$: see the left part of Figure \[fig:wae\_and\_manifold\_latent\_vae\].
3. Use the decoder of this trained WAE as our embedding function $f$.
Our motivation is the following: since the dimension of latent space $Z_{hid}$ and the dimension of manifold ${\mathcal{M}}$ are the same, the reconstruction term in WAE objective constraints its decoder to be an injective map. Since it is represented with a neural network, it is also differentiable. The objective of WAE learning also forces its decoder to map a prior distribution on a latent space (in our case, $p(z_{hid})$) to a distribution of data to the feature space (in our case, $p(z)$). Hence WAE decoder is an ideal candidate for an embedding $f$.
![**Left:** Scheme diagram of WAE with feature space $Z$ and latent space $Z_{hid}$ that learns the prior distribution $p(z)$ on manifold ${\mathcal{M}}\subset Z$. **Right:** The generative model with manifold-valued latent variables $z$. The squared node is deterministic.[]{data-label="fig:wae_and_manifold_latent_vae"}](wae_and_manifold_latent_vae.pdf){width="80.00000%"}
At first glance the described model leaves quite similar questions as vanilla VAE: we “shifted” the complex task of learning non-homeomorphic manifolds of a different topology (latent space and data space) from the VAE decoder to sub-module of the same VAE but pretrained using WAE. Nevertheless, the procedure ensures better control over mapping to manifold and one can develop corresponding metrics to control the quality of mapping.
Introducing symmetries of latent manifold into encoder
======================================================
Recall that in our scheme an encoder $q(z_{hid}|x)$ together with embedding $f: \; Z_{hid} \to Z$ induce a family of posterior distributions $q(z|x)$ on ${\mathcal{M}}$; let us call this family ${\mathcal{Q}}$.
A natural requirement to ${\mathcal{Q}}$ is to have the same symmetries as ${\mathcal{M}}$ has. Suppose we have a symmetry group $G$ of ${\mathcal{M}}$ acting on $Z$, i.e. $$\forall z \in {\mathcal{M}}, \, \forall g \in G \quad gz \in {\mathcal{M}}.$$ For example, if ${\mathcal{M}}$ is an $n$-dimensional sphere $S^n$ in $Z = {\mathbf{R}}^{n+1}$, $G$ is a group of rotations $SO(n+1)$. We require $G$ to also be a symmetry of ${\mathcal{Q}}$ also: $$\forall q \in {\mathcal{Q}}, \, \forall g \in G \quad \exists q' \in {\mathcal{Q}}: \; \forall z \in {\mathcal{M}}\quad q'(gz) = q(z).$$ This means that if a symmetry $g$ of ${\mathcal{M}}$ acts on samples $z$ from a distribution $q \in {\mathcal{Q}}$, we should get samples from another distribution $q'$ from the same family ${\mathcal{Q}}$. Note that we did not pose this requirement while training $f$, hence it would not generally be satisfied. Therefore we have to symmetrize ${\mathcal{Q}}$ explicitly.
![The generative model with group action-encoder acting on manifold-valued latent variable $z$.[]{data-label="fig:manifold_latent_det_action_vae"}](manifold_latent_det_action_vae.pdf){width="50.00000%"}
In order to do this we introduce a group action encoder $a(x)$, see Figure \[fig:manifold\_latent\_det\_action\_vae\]. This group action encoder produces an element $g = a(x)$ of the symmetry group $G$ of ${\mathcal{M}}$, which further acts on a sample $z = f(z_{hid})$. This effectively enriches the posterior family ${\mathcal{Q}}$ with $q': \; q'(gz) = q(z)$.
This procedure has close connection with homeomorphic VAE [@homeo-vae]. Suppose our manifold ${\mathcal{M}}$ is a compact Lie group. Then it is homeomorphic to its own symmetry group: ${\mathcal{M}}\cong G$. Then our group action-encoder is equivalent to $R_\mu$ of [@homeo-vae].
Experiments and conclusions
===========================
Models ELBO
--------------------------------------------------------------------------------------------- -----------------------------
VAE, $\mathrm{dim}\, Z = 1$ $183.98 \pm 11.66$
Manifold-latent VAE with learned $f$, $\mathrm{dim}\, Z_{hid} = 1$ $197.19 \pm 20.46$
Manifold-latent VAE with $f = f_{proj}$, $\mathrm{dim}\, Z_{hid} = 1$ $193.40 \pm 24.57$
Manifold-latent VAE with learned $f$ and group action encoder, $\mathrm{dim}\, Z_{hid} = 1$ $\mathbf{259.03 \pm 59.14}$
VAE, $\mathrm{dim}\, Z = 2$ $\mathbf{356.53 \pm 22.96}$
: Results on the toy task for different models.[]{data-label="results"}
We followed the same experimental setup as for a toy task in paper [@s-vae18], but without noise. [^1] Sampling of a batch from the dataset consisted of two steps:
1. We generated uniformly distributed points on a 1-dimensional unit sphere embedded in $\textbf{R}^2$.
2. We applied a non-linear fixed transformation ${\mathbf{R}}^2 \rightarrow {\mathbf{R}}^{100}$ implemented as a randomly initialized multilayer perceptron with one hidden layer of size 100 and ReLU nonlinearity. Xavier-uniform initialization scheme was applied to the hidden layer.
All models are VAEs with the posterior distribution $q(z_{hid}|x)$ (Beta on $[0,1]^n$), the prior distribution $p(z)$ (uniform of $[0,1]^n$) and the likelihood $p(x|z)$ (Gaussian on ${\mathbf{R}}^{100}$). As for the reparameterization function $f(z_{hid})$, it was either WAE-MMD or the exact mapping from segment $[0,1]$ into a 1-dimensional circle (“Projection”) in the first layer of decoder: $$f_{proj}(z_{hid}) = \begin{pmatrix}
\cos(2 \pi z_{hid}) & \sin(2 \pi z_{hid}))
\end{pmatrix}^T.$$ The dimensions of latent variables $n$ were either 1 or 2.
In a case when the group action encoder is used, it produces an angle (element of $SO(2)$), which is further used to rotate the sample $z = f(z_{hid}) \in {\mathbf{R}}^2$.
The results are presented in Table \[results\]. All decoder structures that include manifold mapping show better results than a vanilla VAE with 1-dimensional latent Gaussian space.
### Acknowledgments {#acknowledgments .unnumbered}
This work was supported by National Technology Initiative and PAO Sberbank project ID 0000000007417F630002.
Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. , abs/1312.6114, 2013.
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, editors, [*Advances in Neural Information Processing Systems 27*]{}, pages 2672–2680. Curran Associates, Inc., 2014.
Antonia Creswell, Tom White, Vincent Dumoulin, Kai Arulkumaran, Biswa Sengupta, and Anil A. Bharath. Generative adversarial networks: An overview. , abs/1710.07035, 2017.
Matthew D. Hoffman and Matthew J. Johnson. Elbo surgery: yet another way to carve up the variational evidence lower bound, 2016.
Tim R. Davidson, Luca Falorsi, Nicola De Cao, Thomas Kipf, and Jakub M. Tomczak. Hyperspherical variational auto-encoders. , 2018.
Luca Falorsi, Pim de Haan, Tim R. Davidson, Nicola De Cao, Maurice Weiler, Patrick Forr[é]{}, and Taco S. Cohen. Explorations in homeomorphic variational auto-encoding. , abs/1807.04689, 2018.
Jiacheng Xu and Greg Durrett. Spherical latent spaces for stable variational autoencoders, 2018.
Mikhail Belkin, Partha Niyogi, and Vikas Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. , 2006.
Yuri Burda, Roger B. Grosse, and Ruslan Salakhutdinov. Importance weighted autoencoders. , abs/1509.00519, 2015.
Ilya Tolstikhin, Olivier Bousquet, Sylvain Gelly, and Bernhard Scholkopf. Wasserstein auto-encoders, 2017.
Appendix
========
Probability density functions with manifold support
---------------------------------------------------
Suppose we have a probability distribution on $Z=\mathbf{R}^n$ with density $p(z)$ and a diffeomorphism $f: Z \to X$, where $X = \mathbf{R}^n$ as well. Then, $f$ induces a probability distribution on $X$ with the following density: $$p(x) =
p(f^{-1}(x)) |\det J_f(f^{-1}(x))|^{-1} =
p(f^{-1}(x)) |\det J_{f^{-1}}(x)|.$$
Suppose now that $X = R^m$ with $m > n$, and $f: Z \to X$ is a smooth embedding (which requires $f$ to be a diffeomorphism between $Z$ and $f(Z)$). From this follows that $f$ induces degenerate probability distribution on $X$ since all the probability mass in $X$ is concentrated on a manifold $\mathcal{M} = f(Z)$. The corresponding probability measure is trivial: $$P(f(A)) = P(A)$$ for some event $A$ on $Z$. Although we cannot define a valid probability density of $X$, we can define a manifold probability density on $\mathcal{M} = f(Z)$ as follows: $$\begin{split}
p_\mathcal{M}(f(z)) &\coloneqq
\lim_{{\mathrm{Vol}}(A) \to 0 \; s.t. \; z \in A} \frac{P(f(A))}{{\mathrm{Vol}}_\mathcal{M}(f(A))} \\&=
\lim_{{\mathrm{Vol}}(A) \to 0 \; s.t. \; z \in A} \frac{P(A)}{{\mathrm{Vol}}_\mathcal{M}(f(A))} \\&=
\lim_{{\mathrm{Vol}}(A) \to 0 \; s.t. \; z \in A} \frac{\int_A p(z) \, dz_1 \ldots dz_n}{{\mathrm{Vol}}_\mathcal{M}(f(A))} \\&=
p(z) \lim_{{\mathrm{Vol}}(A) \to 0 \; s.t. \; z \in A} \frac{\int_A dz_1 \ldots dz_n}{{\mathrm{Vol}}_\mathcal{M}(f(A))},
\end{split}$$
where by ${\mathrm{Vol}}_\mathcal{M}(f(A))$ we denote an $n$-dimensional volume of $f(A) \subset \mathcal{M}$; let us define this volume. Let $\Omega$ be an open subset of $Z$. Then its image under embedding $f$ is an open subset of a manifold $f(\Omega)$ (open in terms of the topology of $\mathcal{M}$). If $Z$ is a Euclidean space, than the “volume” of $\Omega$ is given simply as: $${\mathrm{Vol}}(\Omega) = \int_{\Omega} dz_1 \ldots dz_n.$$ Since $\mathcal{M}$ is embedded into $X$, and $X$ is a Euclidean space, we can measure an $n$-dimensional “volume” of $f(\Omega) \subset \mathcal{M}$. It is given as: $${\mathrm{Vol}}_\mathcal{M}(f(\Omega)) = \int_{f(\Omega)} \sqrt{|\det G(z)|} \, dz_1 \ldots dz_n,$$ where $G(z)$ is a metric tensor on $Z$, induced by the scalar product $\langle \cdot \rangle$ on $X$ and the embedding $f$: $$G_{ij}(z) = \left\langle \frac{df(z)}{dz^i}, \frac{df(z)}{dz^j} \right\rangle.$$
Returning to our formula for probability density on $\mathcal{M}$, we now have: $$\begin{split}
p_\mathcal{M}(f(z)) &=
p(z) \lim_{{\mathrm{Vol}}(A) \to 0 \; s.t. \; z \in A} \frac{\int_{A} dz_1 \ldots dz_n}{\int_{f(A)} \sqrt{|\det G(z)|} \, dz_1 \ldots dz_n} \\&=
p(z) |\det G(z)|^{-1/2}.
\end{split}$$ Or, $$p_\mathcal{M}(x) =
p(f^{-1}(x)) |\det G(f^{-1}(x))|^{-1/2}.$$
Calculation of KL divergence in the case of normalizing flow
------------------------------------------------------------
$$\begin{split}
{\mathrm{KL}}(q_{{\mathcal{M}}}(z|x) \| p_{{\mathcal{M}}}(z)) &=
{\mathbb{E}}_{z \sim q_{{\mathcal{M}}}(z|x)} (\log q_{{\mathcal{M}}}(z|x) - \log p_{{\mathcal{M}}}(z)) \\&=
{\mathbb{E}}_{z_{hid} \sim q(z_{hid}|x)} (\log q(z_{hid}|x) + \log |\det J_f(z_{hid})|^{-1} \\&- \log p(z_{hid}) - \log |\det J_f(z_{hid})|^{-1}) \\&=
{\mathbb{E}}_{z_{hid} \sim q(z_{hid}|x)} (\log q(z_{hid}|x) - \log p(z_{hid})) \\&=
{\mathrm{KL}}(q(z_{hid}|x) \| p(z_{hid})).
\end{split}$$
where $q(z_{hid}|x)$ is the posterior distribution (i.e. fully-factorized Gauss or Beta) on latent variables of WAE, which we use for manifold embedding, $p(z_{hid})$ is the corresponding prior (i.e. standard Gauss or Uniform), $f$ is the decoder of the WAE, which we use to transform the latent space of WAE into manifold ${\mathcal{M}}$, and $J_f(z_{hid})$ is the Jacobian of this transformation. As we see, log-determinants of Jacobians cancel out, and we are left with the KL-divergence on latent space of WAE.
Calculation of KL divergence in case of embedding map
-----------------------------------------------------
$$\begin{split}
{\mathrm{KL}}(q_{{\mathcal{M}}}(z|x) \| p_{{\mathcal{M}}}(z)) &=
{\mathbb{E}}_{z \sim q_{{\mathcal{M}}}(z|x)} (\log q_{{\mathcal{M}}}(z|x) - \log p_{{\mathcal{M}}}(z)) \\&=
{\mathbb{E}}_{z_{hid} \sim q(z_{hid}|x)} (\log q(z_{hid}|x) + \log |\det G(z_{hid})|^{-1/2} \\&- \log p(z_{hid}) - \log |\det G(z_{hid})|^{-1/2}) \\&=
{\mathbb{E}}_{z_{hid} \sim q(z_{hid}|x)} (\log q(z_{hid}|x) - \log p(z_{hid})) \\&=
{\mathrm{KL}}(q(z_{hid}|x) \| p(z_{hid})),
\end{split}$$
where $G$ denotes the metric tensor of the embedding $f$. As in Appendix 4.2, the corresponding terms cancel out.
[^1]: Our code is available on GitHub: <https://github.com/varenick/manifold_latent_vae>
|
---
abstract: 'We investigate the formation via tunneling of inflating (false-vacuum) bubbles in a true-vacuum background, and the reverse process. Using effective potentials from the junction condition formalism, all true- and false-vacuum bubble solutions with positive interior and exterior cosmological constant, and arbitrary mass are catalogued. We find that tunneling through the same effective potential appears to describe two distinct processes: one in which the initial and final states are separated by a wormhole (the Farhi-Guth-Guven mechanism), and one in which they are either in the same hubble volume or separated by a cosmological horizon. In the zero-mass limit, the first process corresponds to the creation of an inhomogenous universe from nothing, while the second mechanism is equivalent to the nucleation of true- or false-vacuum Coleman-De Luccia bubbles. We compute the probabilities of both mechanisms in the WKB approximation using semi-classical Hamiltonian methods, and find that – assuming both process are allowed – neither mechanism dominates in all regimes.'
author:
- 'Anthony Aguirre & Matthew C. Johnson'
bibliography:
- 'tunneling.bib'
title: Two Tunnels to Inflation
---
Introduction
============
It has long been appreciated that in a field theory with multiple vacua – including some models of cosmological inflation – the nucleation of true-vacuum bubbles in a false-vacuum background can and does occur. The study of such transitions, with and without gravity, was pioneered by Coleman and collaborators [@Coleman:1977py; @Callan:1977pt; @Coleman:1980aw] and has become a large enterprise.
Real understanding of the [*reverse*]{} process, nucleation of false-vacuum (inflating) regions in a background of (non-inflating) true-vacuum, has, however, been somewhat more elusive. It has been proposed that his may occur by the same Coleman-DeLuccia (CDL) instanton responsible for true-vacuum nucleation [@Lee:1987qc; @Banks:2002nm; @Garriga:1993fh], by the tunneling of a small false-vacuum bubble through a wormhole to become an inflating region (the Farhi-Guth-Guven, or ‘FGG’ mechanism) [@Farhi:1989yr; @Fischler:1990pk; @Fischler:1989se], or by thermal activation [@Garriga:2004nm; @Gomberoff:2003zh].
This paper comprises the second in a series studying the general process of the nucleation of inflating regions from non-inflating ones. In the first [@Aguirre:2005sv] we cataloged and interpreted all single-bubble thin-wall solutions with an interior false-vacuum de Sitter (‘dS’) space, and discovered and investigated an instability in such bubbles to non-spherical perturbations. In this paper we attempt to unify the treatment of both false- and true-vacuum bubble nucleations, via the CDL, FGG, and thermal activation mechanisms, in the thin-wall limit. We find that these can all be studied within a single framework based on the junction condition potentials developed by Guth and collaborators [@Farhi:1989yr; @Blau:1986cw] and further generalized by Aurilia et. al. [@Aurilia:1989sb] [^1]. This allows us to both catalog all true- or false-vacuum bubble spacetimes, and to calculate tunneling exponents using the semi-classical Hamiltonian formalism of Fischler et al. [@Fischler:1990pk; @Fischler:1989se].
Understanding the quantum mechanical [^2] genesis of inflating regions is very important in assembling a picture of spacetimes containing fields with multiple false vacua, and in understanding how inflation might have begun in our past. These are related because if inflation can begin from a non-inflating region like our own, then [*our*]{} inflationary past may have nucleated from non-inflation, and this raises troubling questions [@Dyson:2002pf; @Albrecht:2004ke] if spawning inflation is less probable than spawning a large homogeneous big-bang region. This is indeed suggested by singularity theorems showing that inflating false vacuum regions must be larger than the [*true*]{} vacuum horizon size [@Penrose:1964wq; @Vachaspati:1998dy] according to some observers [@Aguirre:2005sv]. The FGG mechanism provides a potential loophole [@Albrecht:2004ke] because according to an observer in the background true vacuum spacetime, only a region the size of the black hole event horizon is removed.
There have, however, been lingering questions about whether the Farhi-Guth-Guven [@Farhi:1989yr] “tunneling" process can actually occur. The oldest objection is the fact that the euclidean tunneling spacetime is not a regular manifold [@Farhi:1989yr]. A more modern objection comes from holography: in the FGG mechanism, an observer in the background spacetime only sees a small back hole, whereas the inflating region “inside" should be described by a huge number of states [@Bousso:2004tv; @Banks:2002nm]. This entropy puzzle was recently considered by Freivogel et. al. [@Freivogel:2005qh], who have used the AdS/CFT correspondence to study thin-walled dS bubbles embedded in a background Schwarzschild-Anti-de Sitter space (Alberghi et. al. have also used the ADS/CFT correspondence to study charged vacuum bubbles [@Alberghi:1999kd]). They find that bubbles containing inflating regions which reside behind a wormhole are represented by mixed states in the boundary field theory. This resolves the entropy puzzle, and also implies that inflating regions hidden behind a wormhole cannot arise from a background spacetime by any unitary process, including tunneling. It does not, however, suggest why semi-classical methods break down, nor how we should interpret the seemingly-allowed tunneling.
The formalism that we outline in this paper indicates that there are two ways to interpret tunneling through the effective potential of the junction conditions. The existing interpretation (the FGG mechanism) requires that the wall of a false-vacuum bubble (and in some cases of true-vacuum bubbles) must tunnel through a wormhole to produce an inflating region. In this paper, we use the global properties of the Schwarzschild-de Sitter spacetime to show that there is another interpretation corresponding to a mechanism that does not require the existence of a wormhole.
In this mechanism, a small bubble of true- or false-vacuum, which would classically collapse, instead tunnels to a large bubble that exists outside of the [*cosmological*]{} horizon of the background spacetime. Consequently, this mechanism exists only in spacetimes with a positive cosmological constant. The zero-mass limit of this mechanism correctly reproduces the tunneling exponent for both true- and false-vacuum CDL bubbles [@Coleman:1980aw; @Lee:1987qc]. In light of the objections to the FGG mechanism, this new process may be an alternative, in which case the formation of inflating false-vacuum regions by tunneling is forbidden in flat spacetime. On the other hand, these may just be two competing processes, and we will directly compare the tunneling exponents under this assumption.
In section \[classy\], we classify the possible thin-wall true and false one-bubble spacetimes using the effective potential formalism. We then introduce the possible tunneling mechanisms and outline the calculation of the tunneling exponents for the various possibilities in section \[tunneling\]. We compare the tunneling rates for the allowed processes in section \[comparison\], interpret our results in section \[bl\], and conclude in section \[conclusions\].
Classical Dynamics of True and False Vacuum Bubbles {#classy}
===================================================
We will model the true- and false-vacuum bubbles as consisting of a dS interior with cosmological constant $\Lambda_{-}>0$ separated by a thin wall of surface energy density $\sigma$ from a Schwarzschild de Sitter (SdS) exterior with cosmological constant $\Lambda_{+}>0$. If $\Lambda_{-} > \Lambda_{+}$ we will refer to the configuration as a false-vacuum bubble, otherwise it will be denoted a true-vacuum bubble. The exterior metric in the static foliation is given by $$\label{gsds}
ds_{+}^2=-a_{\rm sds}dt^2 + a_{\rm sds}^{-1}dR^2 + R^2 d\Omega^2,$$ $$\label{defa}
a_{\rm sds}=1-\frac{2M}{R} - \frac{\Lambda_{+}}{3} R^2.$$ where $M$ is the usual Schwarzschild mass parameter. The interior metric in the static foliation is $$\label{gds}
ds_{-}^2 = -a_{\rm ds} dt^2 + a_{\rm ds}^{-1} dR^2 +R^2 d\Omega^2,$$ $$a_{\rm ds}=1- \frac{\Lambda_{-}}{3}R^2,$$
The classical dynamics of thin-walled vacuum bubbles can be determined from the Israel junction conditions, and the problem has been solved in full generality by Aurilia et. al. [@Aurilia:1989sb], building on the work of Guth et. al. [@Farhi:1989yr; @Blau:1986cw]. Assuming spherical symmetry, the radius of curvature of the bubble is the only dynamical variable, so Einstein’s equations yield just one equation of motion: $$\label{israel}
\beta_{\rm ds} - \beta_{\rm sds} = 4 \pi \sigma R,$$ where $$\label{betadef}
\beta_{\rm ds} \equiv - a_{\rm ds} \frac{dt}{d\tau},\ \ \beta_{\rm sds} \equiv
a_{\rm sds} \frac{dt}{d\tau}.$$ Here, $a$ is the metric coefficient in dS or SdS, and $\tau$ is the proper time of an observer on the bubble wall. The sign of $\beta$ is determined by the trajectory because $dt/d\tau$ could potentially be positive or negative.
Effective potentials
--------------------
A set of dimensionless coordinates can be defined, in which Eq. \[israel\] can be written as the equation of motion of a particle of unit mass in a one dimensional potential. Let: $$\label{ztor}
z=\left(\frac{L^2}{2M}\right)^{\frac{1}{3}}R,\ \ T = \frac{L^2}{2k} \tau,$$ where $M$ is the mass appearing in the SdS metric coefficient, and $$k=4\pi\sigma,$$ $$\label{Lsq}
L^2=\frac{1}{3}\left[\left| \left(\Lambda_{-} + \Lambda_{+} + 3k^2 \right)^2 - 4\Lambda_{+}\Lambda_{-}\right| \right]^{\frac{1}{2}}.$$ With these definitions, Eq. \[israel\] becomes $$\label{juncteom}
\left[\frac{dz}{d T }\right]^2=Q-V(z),$$ where the potential $V(z)$ and energy $Q$ are $$\label{potential}
V(z)=-\left[z^2+\frac{2Y}{z}+\frac{1}{z^4} \right],$$ with $$\label{y}
Y=\frac{1}{3}\frac{\Lambda_{+}-\Lambda_{-}+3k^2}{L^2},$$ and $$\label{Qtom}
Q=-\frac{4k^2}{\left(2M\right)^{\frac{2}{3}}L^{\frac{8}{3}}}.$$ Note that a small negative $Q$ corresponds to a large mass, so that even between $-1 < Q < 0$ the mass can be arbitrarily large. The scale of all quantities of interest is set by some power of the bubble wall surface energy density ($k$) if the interior and exterior cosmological constants are written in terms of $k^2$ as $$\Lambda_{+}=Ak^2, \ \ \Lambda_{-}=Bk^2.$$
From the constant-$Q$ trajectories in the presence of the potential of Eq. \[potential\], one can construct the full one-bubble spacetimes [@Blau:1986cw; @Farhi:1989yr; @Aguirre:2005sv]. Shown in Fig. \[Bgt3Am1\] is an example of two of the possible potential diagrams. In addition to the potential Eq. \[potential\], there are other landmarks in Fig. \[Bgt3Am1\]. Intersections with the dashed line $Q_{\rm ds}$ (which is obtained by solving $a_{\rm ds}=0$ for $Q$) as one moves along a line of constant $Q$ represent a crossing of either the past or future horizon of the interior dS spacetime. Every intersection with the dashed line $Q_{\rm sds}$ represents a horizon crossing in the SdS spacetime (this could represent either the past- or future-black hole [*or*]{} cosmological horizons). It can be shown [@Aurilia:1989sb] that $\beta_{\rm ds}$ and $\beta_{\rm sds}$ are monotonic functions of $z$, which will have zeros where $Q_{\rm ds}$ or $Q_{\rm sds}$ intersect the potential. These points demarcate sign changes in $\beta_{\rm ds}$ or $\beta_{\rm sds}$, and are denoted by the vertical dotted lines in Fig. \[Bgt3Am1\].
{width="17cm"}
For there to be a $\beta_{\rm ds}$ sign change, $Y$ in Eq. \[y\] must be in the range $-1 \leq Y < 0$ [@Aurilia:1989sb], which yields the condition that $B > A+3$ if a sign change is to occur. This inequality shows that $\beta_{\rm ds}$ does not change sign for true vacuum bubbles ($A>B$). For there to be a $\beta_{\rm sds}$ sign change, the function $$\tilde{Y} = \frac{1}{3}\frac{\Lambda_{+}-\Lambda_{-}-3k^2}{L^2}$$ must be in the range $-1 \leq \tilde Y < 0$ [@Aurilia:1989sb], which yields the condition that $B > A-3$ if a $\beta_{\rm sds}$ sign change is to occur. If a $\beta_{\rm sds}$ sign change does exist, it can occur to the left (if $B > 3(A-1)$) or right ($B < 3 (A-1)$) of the maximum in the potential [@Aguirre:2005sv]. Given these conditions, there are a total of seven qualitatively different potential diagrams to consider, examples of which are shown in Figs. \[Bgt3Am1\], \[Blt3Am1\], \[AgtBo3p1\], and \[A\_6B\_5\].
{width="17cm"}
{width="17cm"}
Conformal diagrams
------------------
The one-bubble spacetimes, represented by lines of constant $Q$ on the junction condition potential diagrams, are shown in Figs. \[diags1\], \[diags2\], and \[thermalons\] [^3]. The shaded regions of the conformal diagrams shown in the left column cover the interior of the vacuum bubble. The shaded regions of the diagrams in the right column cover the spacetime outside the bubble. The conformal diagrams in each row are matched along the bubble wall (solid line with an arrow). For solutions with qualitatively similar SdS diagrams, the various options for the dS interior are connected by labeled solid lines.
{width="14cm"}
The conformal diagrams shown in Fig. \[diags1\] are all solutions in which the bubble wall remains to the right of the wormhole of the SdS conformal diagram. The bound solutions, Solutions 1 and 2, exist for both true- and false-vacuum bubbles. For false-vacuum bubbles, they represent a regime in which the inward pressure gradient and bubble wall tension dominate the dynamics, causing the bubble to ultimately contract. In the case of true-vacuum bubbles, this corresponds to cases where the wall tension overwhelms the outward pressure gradient.
In the monotonic Solutions 3-5 of Fig. \[diags1\] the bubble wall has enough kinetic energy to reach curvatures comparable to the exterior horizon size, at which time the bubble cannot collapse. Solutions 3 and 4 represent either true- or false-vacuum bubbles where the wall tension and/or the inward pressure gradient causes the wall to accelerate towards $r=0$, but which are saved from collapse by the expansion of the exterior spacetime. Solution 5 exists only for true-vacuum bubbles, and describes a solution which accelerates away from the origin due to the outward pressure gradient while also being pulled out of the cosmological horizon by the expansion of the exterior spacetime.
The unbound Solution 6 also exists only for true-vacuum bubbles. Here, the bubble expands, all the while accelerating towards the false-vacuum. The zero mass limit ($M \rightarrow 0$, or $Q \rightarrow -\infty$) of this solution is the one-bubble spacetime of the analytically continued true-vacuum Coleman-De Luccia (CDL) instanton [@Coleman:1980aw] in the limit of an infinitely thin wall. This can be seen by considering the limit as the potential (Eq. \[potential\]) goes to $-\infty$, where on the right (unbound) side of the potential hump the $z^2$ term dominates. Solving for $R$ using Eq. \[ztor\], we find the radius at turnaround to be $$\label{r_0inst}
R = 6k \left[\left|\left(\Lambda_{+} + \Lambda_{-} +3k^2 \right)^2 - 4\Lambda_{+}\Lambda_{-} \right| \right]^{-1/2},$$ which is indeed the radius of curvature of the CDL instanton [@Coleman:1980aw] at nucleation.
{width="16cm"}
The solutions shown in Fig. \[diags2\] are all behind the wormhole in the SdS spacetime, save Solutions 12 and 13, which correspond to evolution in a spacetime without horizons. The false-vacuum bubble solutions 7 and 9, and true- or false-vacuum bubble solution 8 are unbound solutions which exist to the left of the worm hole on the SdS conformal diagram. It can be seen that at turnaround, each of these bubbles will be larger than the exterior horizon size. Observers in region III of the SdS conformal diagram will see themselves sandwiched between a black hole and a bubble wall which encroaches in from the cosmological horizon. Observers inside the bubble are also surrounded by a bubble wall, and so we are faced with the rather odd situation that both observers will perceive themselves inside bubbles of opposite phase.
Solutions 7 and 8 have interesting zero mass limits. Since these solutions involve both sides of the wormhole, the zero mass limit corresponds to an exactly dS universe consisting of regions I, II’, III’, and IV’ (encompassed by the vertical dashed lines shown on the right side of the first diagram of Fig. \[diags2\]) of the SdS diagram (in which nothing happens), and a dS universe consisting of regions III, II”, and IV” (encompassed by the other set of vertical dashed lines) which contains a CDL true- or false-vacuum bubble. The radius at the turning point is still given by Eq. \[r\_0inst\], and so the bubble to the left of the wormhole is the analytic continuation of the true- or false-vacuum CDL instanton. However, note that the Lorentzian evolution of the true-vacuum bubbles is very different from the canonical CDL instanton discussed in the previous paragraph. As seen from the outside (region III of the SdS diagram on the right), the bubble wall accelerates towards the true-vacuum (driven by the wall tension); in the absence of the cosmic expansion of the false-vacuum, this solution would be bound.
Because the SdS manifold is non-compact, there are actually many more options. We have so far placed special significance on the singularities in regions II and IV of the SdS diagram. However, there will be other singularities both to the left and right of these regions which can also be viewed as the origin of coordinates. It is perfectly legitimate to construct bubble wall solutions using any origin of coordinates one wishes, and therefore each of the solutions in Fig. \[diags1\] and \[diags2\] represents only one of an infinity of possible solutions. An example of an alternative solution is shown in Fig. \[noworm\], which is identical to the Solution 7 in Fig. \[diags2\] in every way, except different regions of the conformal diagram are physical. This observation is key for the tunneling mechanisms we will describe in the next section.
![Solutions can be to the right of region I instead of behind the wormhole. This solution is identical to Solution 7 of Fig. \[diags2\]. \[noworm\][]{data-label="default"}](fig7.eps){width="8.6cm"}
Moving on to the other solutions in Fig. \[diags2\], Solution 10 (corresponding to either true- or false-vacuum bubble) and Solution 11 (corresponding to a false-vacuum bubble) are massive unbound solutions which lie outside the cosmological horizon of a region III observer. Solution 12 (corresponding to a false-vacuum bubble) and Solution 13 (corresponding to either a true- or false-vacuum bubble) are monotonic solutions with mass greater than the Nariai mass of the SdS spacetime. This can be seen by noting that these constant $Q$ trajectories never cross the $Q_{\rm sds}$ line in the potential diagrams. The false-vacuum bubble Solution 14, and the true- or false-vacuum bubble solution 15 are monotonic solutions which must lie to the left of the wormhole.
There is one more class of solutions, shown in Fig. \[thermalons\], which exist in unstable equilibrium between the bound and unbound solutions of Fig. \[diags1\] and \[diags2\]. Solution 16 corresponds to true- or false-vacuum bubbles with $B < 3 (A-1)$, while Solution 17 corresponds to true- or false-vacuum bubbles with $B > 3 (A-1)$. These solutions can be identified as the spacetimes of the thermal activation mechanism of Garriga and Megevand [@Garriga:2004nm], which we will discuss further in Sec. \[highlow\] and \[comparison\].
{width="12cm"}
Classical trajectories exist on either side of the potential diagrams of Figs. \[Bgt3Am1\], \[Blt3Am1\], \[AgtBo3p1\], and \[A\_6B\_5\], and so one can ask if there is any quantum process that connects two solutions of the same mass through the classically forbidden region under the potential. This would correspond to transitions from the bound spacetimes shown in Fig. \[diags1\] (Solutions 1 and 2) to the unbound spacetimes shown in Figs. \[diags1\] and \[diags2\] (Solutions 6-11). Such processes do seem to occur [@Farhi:1989yr; @Fischler:1989se; @Fischler:1990pk; @Berezin:1988dz; @Khlebnikov:2003ve], at least within the framework of semi-classical quantum gravity, and we now turn to the problem of determining which transitions are allowed and with what probabilities.
Tunneling
=========
The potential diagrams discussed in the previous section nicely summarize the classically allowed one-bubble spacetimes. They also illustrate the possibility that there might exist some process akin to the tunneling of a point particle through a potential barrier. Such a process would correspond to the quantum tunneling between thin-wall bubbles of equal mass, but different turning-point radii. We will find that in SdS, there are actually two different semi-classical tunneling processes which connect equal mass solutions: the FGG mechanism [@Farhi:1989yr] and a process which is only allowed in the presence of an exterior cosmological constant, and which has a zero-mass limit that corresponds to CDL true- or false-vacuum bubble nucleation [@Coleman:1980aw; @Lee:1987qc].
Hamiltonian formalism
---------------------
In a pair of papers, Fischler et. al. (FMP) [@Fischler:1990pk; @Fischler:1989se] presented a calculation of the probability for transitions between various thin-wall false-vacuum bubble solutions. This calculation was done using Hamiltonian methods in the WKB approximation for the case where the exterior cosmological constant is zero. A similar calculation of such tunneling events was performed by Farhi et. al. [@Farhi:1989yr] using a path integral approach. Both methods encounter the difficulty that the interpolating geometry involves a two-to-one mapping to the exterior spacetime, and thus is [*not*]{} a manifold. We will use the Hamiltonian approach, which is the most direct route to a tunneling exponent and temporarily skirts this issue. A discussion of the interpolating geometry will appear in a forthcoming publication [@Aguirre:xi].
Here, we extend the calculation of FMP to include all spacetimes with arbitrary non-negative interior ($\Lambda_{-}$) and exterior ($\Lambda_{+}$) cosmological constants. This formalism, with the catalog of all classically allowed solutions, will allow us to create a complete listing of the possible tunneling events.
Following FMP, we begin by making a coordinate transformation to recast the interior and exterior metrics in Eqs. \[gsds\] and \[gds\] into the form $$\begin{aligned}
\label{metric}
ds^2 &=& - N^{t}\left(t,r\right)^2 dt^2 + L\left(t,r\right)^2 \left[dr + N^{r}\left(t,r\right) dt \right]^2 \nonumber \\ && + R\left(t,r\right)^2 \left(d\theta^2 + \sin^2 \theta d\phi^2 \right),\end{aligned}$$ where $N^{t}\left(t,r\right)$ is the lapse function, $N^{r}\left(t,r\right)$ is the shift, and $L \equiv ds/dr$. The action for a general theory of matter coupled to gravity is then given by $$S = \int dt \ p \ \dot{q} + \int dr \ dt \ \left(\pi_{L} \dot{L} + \pi_{R} \dot{R} - N^{t} H_{t} - N^{r} H_{r} \right)$$ where $\pi_{L}$ is the momentum conjugate to $L$, and $\pi_{R}$ is the momentum conjugate to $R$. This action, with the four constraints
\[constraints\] $$H_{t, r} \left(q, L, R, p, \pi_{L}, \pi_{R} \right)=0,$$ $$\pi_{N^{t}}=\pi_{N^{r}}=0,$$
fully determines the classical evolution of the system. For a thin-walled bubble with an arbitrary surface energy density $k$ and interior and exterior cosmological constant ($\Lambda_{-}$ and $\Lambda_{+}$), the Hamiltonian densities are given by $$\begin{aligned}
\label{Ht}
H_{t} &=& \frac{L \pi_{L}^{2}}{2 R^2} - \frac{\pi_{L} \pi_{R}}{R} \nonumber \\ &&+ \frac{1}{2} \left[ \left[\frac{2RR'}{L}\right]' - \frac{R'^2}{L} - L + \Lambda_{+} L R^2 \right] \nonumber \\ &&+ \Theta\left(r_{w} -r \right) \frac{\left(\Lambda_{-} - \Lambda_{+} \right)}{2} L R^2 \nonumber \\ && + \delta\left(r_{w}-r\right) \left(L^{-2} p_{w}^{2} + k^2 R_{w}^{4} \right)^{1/2},\end{aligned}$$ $$\label{Hr}
H_{r} = R' \pi_{R} - L \pi_{L}' - \delta\left(r_{w} - r \right) p_{w},$$ where a prime denotes a derivative with respect to $r$ and $r_{w}$ is the position of the bubble wall (quantities with the subscript $w$ are evaluated at this position).
A linear combination of the constraints Eq. \[Ht\] and \[Hr\] can be use to eliminate $\pi_{R}$ $$\frac{R'}{L} H_{t} + \frac{\pi_{L}}{RL} H_{r} = 0,$$ which, if we define $$\label{M}
\mathcal{M} \equiv \frac{\pi_{L}^{2}}{2R} + \frac{R}{2} \left[1 - \left[\frac{R'}{L}\right]^2 - \frac{\Lambda_{\pm} R^2}{3} \right],$$ can be written as $$\label{M'}
\mathcal{M}' = \delta\left(r_{w}-r\right) \left(\frac{R'}{L} \left(L^{-2} p_{w}^{2} + k^2 R_{w}^{4} \right)^{1/2} + \frac{\pi_{L}}{RL} p_{w} \right).$$ It can be seen from Eq. \[M’\] that $\mathcal{M}$ is zero for $r < r_{w}$ and independent of $r$ for $r > r_{w}$. We will define $\mathcal{M}(r>r_{w}) \equiv M$, which is the mass enclosed by a surface with $r>r_{w}$. Solving for $\pi_{L}$ at $r=0$ and $r=\infty$ using the conditions on $\mathcal{M}$ yields: $$\label{piLsmall}
\pi_{L}^{2} = - R^2 \left[ 1 - \left[\frac{R'}{L} \right]^2 - \frac{\Lambda_{-}R^2}{3} \right], \ \ \ \ \ r < r_{w}$$ $$\label{piLbig}
\pi_{L}^{2} = - R^2 \left[ 1 - \left[\frac{R'}{L} \right]^2 - \frac{\Lambda_{+}R^2}{3} - \frac{2M}{R} \right], \ \ \ \ \ r > r_{w}.$$ From $H_{r} = 0$, solving for $\pi_{L}'$, and integrating from $r_{w} - \epsilon$ to $r_{w} + \epsilon$, one finds that the discontinuity in $\pi_{L}$ across the wall ($\Delta \pi_{L}$) is $$\Delta \pi_{L} = - \frac{p_{w}}{L_{w}},$$ From $H_{t} = 0$, solving for $R''$, and integrating from $r_{w} - \epsilon$ to $r_{w} + \epsilon$, one finds that the discontinuity in $R'$ across the wall ($\Delta R'$) is $$\Delta R' = - \frac{1}{R_{w}} \left[ p_{w}^2 + k^2 L^2 R_{w}^4 \right].$$ These discontinuity equations are equivalent to the Israel junction conditions, and can be manipulated to reproduce Eq. \[israel\]. There are classically allowed and forbidden regions in the space of $R$, $L$, and $r$, the boundaries between which can be found by looking for where the conjugate momenta are zero. There is, however, only one true degree of freedom, the classically allowed/forbidden region for which is classified by the potential Eq. \[potential\]. The unphysical degrees of freedom will allow for a variety of physically equivalent paths through the the space of $(L, R, r)$.
To quantize the system, we impose the constraints of Eq. \[constraints\] on the wave functional $\Psi$: $$\hat{H}_{t} \Psi=\hat{H}_{r} \Psi=\hat{\pi}_{N^{t}}\Psi=\hat{\pi}_{N^{r}}\Psi=0.$$ The last two constraints restrict the wave functional to depend only upon $L$, $R$, and $r$, which in the WKB approximation is taken to be $$\label{psi}
\Psi\left(L,R,r\right) = \exp \left[ i \Sigma_{0} \left(L,R,r \right) / \hbar + O\left(\hbar \right)\right].$$ We explicitly include $\hbar$ here to emphasize the order of our approximation, but note that we use geometrical units in all other cases. Acting with $\hat{H}_{t}$ and $\hat{H}_{r}$, and keeping terms in the Taylor expansion only to leading order in $\hbar$ (which removes any operator ordering ambiguities) yields the Hamilton-Jacobi equations $$H_{r,t} \left(r,L,R,\frac{\delta \Sigma_{0}}{\delta r}, \frac{\delta \Sigma_{0}}{\delta L},\frac{\delta \Sigma_{0}}{\delta R} \right) = 0.$$ We will integrate $$\label{dSigma}
\delta \Sigma_{0} = \hat{p} \delta \hat{r} + \int_{0}^{\infty} dr \left[ \pi_{L} \delta L + \pi_{R} \delta R \right],$$ to solve for the exponent of the wave functional Eq. \[psi\].
Calculating tunneling rates
---------------------------
The problem that we wish to solve is the tunneling amplitude in the WKB approximation to connect bound solutions with turning point $R_{1}$ to equal-mass unbound solutions with turning point $R_{2}$. An example of this is the FGG mechanism [@Farhi:1989yr], which consists of two steps. First, an expanding region of false/true-vacuum, which would classically collapse into a black hole, is formed and evolves to the classical turning point. Here, there is a chance for the bubble wall to tunnel through the wormhole to one of the unbound solutions, as shown in Fig. \[tunnel\]. The result of this process is a black hole in the region of the old phase, which is connected by a wormhole to a universe containing an expanding bubble of the new phase.
As we saw in Sec. \[classy\], because SdS is non-compact, there are many possible one-bubble spacetimes where region I of the SdS conformal diagram is not physical. We can therefore imagine tunneling from the bound Solution 1 or Solution 2 of Fig. \[diags1\] to the unbound spacetime shown in Fig. \[noworm\]. This process, which can occur only in the presence of a a positive exterior cosmological constant, is depicted in Fig. \[tunnelnoworm\]. For every transition which goes through the wormhole, as in the FGG mechanism, there is another transition which instead goes out the cosmological horizon.
There are many possible transitions to consider, corresponding to the many qualitatively different spacetimes shown in Figs. \[diags1\] and \[diags2\]. In each case, the tunneling probability in the WKB approximation is given by $$\label{probR1toR2}
P \left( R_{1} \rightarrow R_{2} \right) = \left| \frac{\Psi\left( R_{2} \right)}{ \Psi\left( R_{1} \right) } \right|^2 \simeq e^{2i\Sigma_{0}\left[R_{2}-R_{1}\right]},$$ where $\left[R_{2} - R_{1}\right]$ represents evaluation between the two turning points of the classical motion, and $\Sigma_{0}$ is obtained by integrating Eq. \[dSigma\]. The plan of attack is to split the integral into three parts: one over the interior of the bubble, one over the exterior, and one in the neighborhood of the wall. We thus write: $$i \Sigma_{0} = F_{I}\left[R_{2} - R_{1} \right] + F_{O}\left[R_{2} - R_{1} \right] + F_{w} \left[R_{2} - R_{1} \right].$$
The integrals $F_{I}$ and $F_{O}$ are found by holding $r_{w}$ and the geometry in the neighborhood of the wall fixed, while allowing nontrivial variation of $L$ and $R$ in the interior and exterior spacetimes. Following FMP, we will integrate $L$ along a path of constant $R$ to the boundary of the classically allowed/forbidden region, and then integrate along this boundary to the desired configuration of $L(r), R(r)$. The momenta vanish along this second leg, and so the integral will be of $\pi_{L}$ over $L$ $$\begin{aligned}
F_{I} &=& \int_{0}^{\hat{r}} dr \int dL (\pm \pi_{L}) \\
&=& \pm \int_{0}^{\hat{r}} dr \left[ i \pi_{L} - RR' \cos^{-1} \left(\frac{R'}{L a_{\rm ds}^{1/2}}\right) \right]. \nonumber\end{aligned}$$ Note that there is an ambiguity in the sign. This comes from the fact that the constraints (Eq. \[piLsmall\] and \[piLbig\]) are second order in the momenta, and so we must account for both the positive and negative roots. To keep track of this ambiguity, we will define a variable $\eta \equiv \pm1$ with $\sqrt{\pi_{L}^{2}} = \eta \pi_{L}$. We shall have more to say about this issue later.
At the turning point, $\pi_{L}$ vanishes. The integral evaluated between the two turning points is then $$\label{FI}
F_{I} \left[ R_{2}-R_{1} \right] = \eta \int_{R_{1}}^{R_{2}} dR R \cos^{-1} \left(\frac{R'}{L a_{\rm ds}^{1/2}}\right)$$
The integral outside the bubble wall ($r > r_{w}$) is given by $$F_{O} = \eta \int_{r_{w}}^{\infty} dr \left[ i \pi_{L} - RR' \cos^{-1} \left(\frac{R'}{L a_{\rm sds}^{1/2}}\right) \right]$$ which evaluated between the two turning point becomes $$\label{FO}
F_{O}\left[R_{2} - R_{1}\right] = \eta \int_{R_{1}}^{R_{2}} dR R \cos^{-1} \left(\frac{R'}{L a_{\rm sds}^{1/2}}\right)$$
At the turning point, $R'$ inside and outside of $r_{w}$ is given by solving Eqs. \[piLsmall\] and \[piLbig\] for $R'$: $$R'(r_{w} - \epsilon) = \pm L a_{\rm ds}^{1/2}, \ \ \ \ R'(r_{w} + \epsilon) = \pm L a_{\rm sds}^{1/2}.$$ Therefore, the inverse cosine in the integrals of Eq. \[FI\] and \[FO\] are either $0$ when $R'$ is positive or $\pi$ when $R'$ is negative. To perform these integrals, imagine moving the wall along the tunneling hypersurface ($t=0$) between the two turning points (for an example, see Fig. \[tunnel\]). The sign of $\beta$ is positive if the coordinate radius $r$ is increasing in a direction normal to the wall and negative if it is decreasing. Therefore, the sign of $R'$ is equal to the sign of $\beta$ as one moves along the tunneling hypersurface, and the integrals Eq. \[FI\] and \[FO\] will be zero in regions of positive $\beta$ and $\pi$ in regions of negative $\beta$.
Shown in table \[tableworm\] are the values of $F_{O}$ and $F_{I}$ for all of the possible transitions where the unbound solution is to the left, on the conformal diagram, of the bound solution (for example, the process shown in Fig. \[tunnel\]), which in all cases but $B>3 (A-1)$ with $M>M_S$ (the mass at which $\beta_{sds}$ changes sign on the effective potential) occurs through a wormhole (for $B>3 (A-1)$ with $M>M_S$, the most massive bound and unbound solutions can both be behind a worm hole). We will refer to these solutions as L(eft) tunneling geometries. These were the solutions studied by FGG and FMP, but we have seen above that there are actually many other allowed processes due to the non-compact properties of the SdS spacetime. These are tunneling processes where the unbound solution lies to the right of the bound solution on the conformal diagram, which we will refer to as R(ight) tunneling geometries. The values of the integrals $F_{I}$ and $F_{O}$ in this case are shown in table \[tablenoworm\]. In all cases except for $B > 3 (A-1)$ with $M>M_S$, the bubble wall exits the cosmological horizon (whereas the L tunneling geometries went through a wormhole), as in Fig. \[tunnelnoworm\] (for $B>3 (A-1)$ with $M>M_S$, the bubble wall traverses a wormhole and cosmological horizon).
There still is one more integral to evaluate, which allows for the variation of the geometry at the position of the wall
$$\begin{aligned}
\label{Fw}
F_{w} \left[ R_{2} - R_{1} \right] &=& \int_{R_{1}}^{R_{2}} dR_{w} R_{w} \left[ \cos^{-1}\left[\frac{6M + 3 k^2 R_{w}^{3} - R_{w}^{3} \left(\Lambda_{-} - \Lambda_{+} \right) }{6k R_{w}^{2} a_{\rm ds} } \right] \right. \nonumber \\ && \left. - \cos^{-1}\left[\frac{6M - 3 k^2 R_{w}^{3} - R_{w}^{3} \left(\Lambda_{-} - \Lambda_{+} \right) }{6k R_{w}^{2} a_{\rm sds} } \right] \right].\end{aligned}$$
We have been unable to find an analytic expression for this integral, and so have evaluated it numerically.
Putting everything together, we can evaluate the tunneling exponent for the various cases shown in tables \[tableworm\] and \[tablenoworm\]. Shown in Fig. \[nosub\] is an example of $2 i \Sigma_{0}$ for both the L (blue dashed line) and R (red solid line) tunneling geometries with $3 (A-1) < B < A + 3$ ($A=1$, $B=6$), where we have taken $\eta=+1$. The vertical dashed lines represent the mass scales $M_D$ (left) and $M_{S}$. L tunneling geometries with $M < M_S$ correspond to tunneling through a wormhole. The magnitude of these tunneling exponents is fixed by the inverse bubble wall tension squared ($k^{-2}$), which in geometrical units ranges from $k^{-2} \simeq 10^{102}$ for a tension set by the Weak scale to $k^{-2} \simeq 1$ for a tension set by the Planck scale.
High- and low-mass limits {#highlow}
-------------------------
Note that as the mass increases, the width of the potential barrier that must be crossed decreases (see the potential diagrams in Fig. \[Bgt3Am1\], \[Blt3Am1\], \[AgtBo3p1\], and \[A\_6B\_5\]). We therefore expect that the tunneling exponent (for tunneling through the effective potential) goes to zero at the top of the barrier. However, the tunneling exponent is not always zero at the top of the potential, as can be seen from the tunneling exponent for the R tunneling geometry shown in Fig. \[nosub\] (red solid line). To see how this happens, consider a mass slightly below the maximum of the effective potential. The bound solutions are the same for both the L and R tunneling geometries (Solutions 1 or 2), but the unbound solutions to which we are tunneling differ. For a bound Solution 1, we are tunneling to one of the two versions (corresponding to the L or R tunneling geometry) of either Solution 6, 10, or 11 depending on the values of $A$ and $B$ . For a bound Solution 2, we are tunneling to one of the two versions of either Solution 8 or 9.
In the case where $B > 3(A-1)$ (the situation pictured in Fig. \[nosub\]), the most massive L tunneling geometry will have the bound and unbound solutions smoothly merge as the top of the potential barrier is approached. The most massive R tunneling geometry in this case will find the bound and unbound solutions separated by both a black hole and cosmological horizon, and so the tunneling exponent at the top of the potential well will be given by $2 i \Sigma_{0}= \pi \left( R_{S}^{2}-R_{C}^{2} \right)$. This situation is reversed when $B < 3(A-1)$, where the R tunneling geometry will possess the smooth high mass limit, and the most-massive L tunneling geometry will have a non-zero tunneling exponent.
Now consider the other end of the mass spectrum: the zero mass limit of the two different tunneling geometries. In either case, as the mass is taken to zero, the turning point of the bound solution goes to zero, and the turning point of the unbound solution approaches the nucleation radius of a CDL bubble (see Eq. \[r\_0inst\]). Even so, there is a fundamental difference between these two solutions when the background spacetime is considered.
As the mass is taken to zero in the L tunneling geometry (corresponding to the FGG mechanism), the worm hole separating the background of the old phase and the bubble of the new phase disappears. This leaves a background spacetime in which absolutely nothing happens, along with a universe containing a CDL bubble which is created from nothing. At least in the zero-mass limit, this means that we are calculating Vilenkin’s tunneling wave function for an inhomogenous universe [@Vilenkin:1982de; @Vilenkin:1984wp; @Vilenkin:1998dn] with the tunneling exponent equal in magnitude to the CDL instanton action (without the background subtraction term).
This situation is rather strange: if considered one physical system, we have seemingly created new degrees of freedom. It is therefore unclear how we should interpret the tunneling probability; what are we fluctuating out of, and probability per unit what? The massive case seems to create new degrees of freedom as well, since the region to the left of the worm hole (containing large regions of both the old and new phase) in Fig. \[tunnel\] does not exist prior to the tunneling event. It is perhaps not so surprising then that Freivogel et. al. [@Freivogel:2005qh] have found that when a conformal field theory dual to FGG tunneling from AdS is constructed using the AdS/CFT correspondence, it corresponds to a non-unitary process.
The zero mass limit of the R tunneling geometry corresponds to the nucleation, in some background, of a CDL true- or false-vacuum bubble. The CDL tunneling exponent (including the background subtraction) can be written as [@Garriga:2004nm; @Feng:2000if] $$\label{BCDL}
S_{\rm CDL} = \frac{3 \pi}{2} \left[\frac{1}{\Lambda_{+}} \left( 1 - b \alpha_{+} \right) - \frac{1}{\Lambda_{-}} \left( 1 - b \alpha_{-} \right) \right],$$ where $$\alpha_{\pm} = \frac{\Lambda_{+}-\Lambda_{-}}{6k} \mp \frac{k}{2},$$ and $$b = \sqrt{\frac{3}{\Lambda_{-} + 3 \alpha_{-}^{2}}}.$$ The horizontal dotted line in Fig. \[nosub\] is the value of the CDL tunneling exponent for a particular choice of parameters, and it can be seen that the zero mass limit ($Q \longrightarrow -\infty$) of the R tunneling geometry asymptotes to this. Similar results were found in the case of [*true*]{}-vacuum bubbles by Ansoldi et. al. [@Ansoldi:1997hz], who were able to reproduce the CDL tunneling exponent using a Hamiltonian formalism.
It can be seen in Fig. \[nosub\], that the tunneling exponent takes opposite signs for the two tunneling geometries (Fig. \[tunnel\] and Fig. \[tunnelnoworm\]). For both tunneling probabilities to be less than one, $\eta$ must take opposite signs in each case. We have seen that the zero-mass limit of the L tunneling geometry (FGG mechanism) corresponds to creation of an inhomogenous universe from nothing. This perspective suggests that the sign choice we are forced to make is a reflection of some quantum-cosmological boundary conditions, since choosing the sign of $\eta$ is tantamount to choosing the growing or decaying wave function in the region under the well. Taking linear combinations of the growing and decaying wave functionals would yield any one of the three existent sign conventions of Hartle and Hawking [@Hartle:1983ai], Linde [@Linde:1983mx], and Vilenkin [@Vilenkin:1982de]. In contrast, the sign choice is rather straightforward for the R tunneling geometries. This process has a clear-cut interpretation in terms of a fluctuation between true- and false-vacuum regions. Thus, we might physically interpret the low CDL probability as the low probability for a downward entropy fluctuation in the background spacetime to occur [@Banks:2002nm].
If both tunneling geometries are allowed, we have two processes which correspond to tunneling under the same potential well Eq. \[potential\]. It is unclear exactly how one is to interpret this situation, but if it were the case that only one of these two interpretations were valid, there would be a number of important consequences. For example, if the FGG mechanism (L tunneling geometry) is in fact forbidden, then there would be no possible thin-wall false-vacuum bubble nucleation events in Minkowski space. We have also seen above that the bound and unbound solutions will merge into the monotonic solution at the top of the potential for [*either*]{} the L or R tunneling geometry, but never both. Since in the low mass limit only the R tunneling geometry matches the tunneling exponent for CDL bubbles, if one were to choose between the two mechanisms, either the low or the high mass end of the spectrum would be discontinuous for some range of parameters. We hope to explore these points further in future work.
$A$ and $B$ $M$ $F_{I}\left[ R_{2}-R_{1} \right]$ $F_{O}\left[ R_{2}-R_{1} \right]$
----------------------- ---------------------- ----------------------------------------------------- ----------------------------------------------------
$ 3(A-1) < A+3 < B $ $M < M_{D}$ $\frac{\pi}{2}\left(R_{D}^{2} - R_{2}^{2} \right) $ $\frac{\pi}{2}\left(R_{2}^{2} - R_{S}^{2} \right)$
$3(A-1) < A+3 < B $ $M_{D} < M < M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{S}^{2} \right)$
$3(A-1) < A+3 < B $ $M > M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{1}^{2} \right)$
$3(A-1) < B < A+3 $ $M < M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{S}^{2} \right)$
$3(A-1) < B < A+3 $ $M > M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{1}^{2} \right)$
$A+3 < B < 3(A-1)$ $M < M_{D}$ $\frac{\pi}{2}\left(R_{D}^{2} - R_{2}^{2} \right) $ $\frac{\pi}{2}\left(R_{2}^{2} - R_{S}^{2} \right)$
$A+3 < B < 3(A-1)$ $M_{D} < M < M_{SD}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{S}^{2} \right)$
$A+3 < B < 3(A-1)$ $M<M_{SD}$ $0$ $\frac{\pi}{2}\left(R_{C}^{2} - R_{S}^{2} \right)$
$B < A+3 < 3(A-1)$ $M<M_{SD}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{S}^{2} \right)$
$B < A+3 < 3(A-1)$ $M>M_{SD}$ $0$ $\frac{\pi}{2}\left(R_{C}^{2} - R_{S}^{2} \right)$
$A > B+3$ $M < M_{\rm CRIT}$ $0$ $\frac{\pi}{2}\left(R_{C}^{2} - R_{S}^{2} \right)$
$A > \frac{B}{3} + 1$ $M < M_{SD}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{S}^{2} \right)$
$A > \frac{B}{3} + 1$ $M > M_{SD}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{S}^{2} \right)$
$A < \frac{B}{3} + 1$ $M < M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{S}^{2} \right)$
$A < \frac{B}{3} + 1$ $M > M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{1}^{2} \right)$
$A$ and $B$ $M$ $F_{I}\left[ R_{2}-R_{1} \right]$ $F_{O}\left[ R_{2}-R_{1} \right]$
----------------------- ---------------------- ----------------------------------------------------- ----------------------------------------------------------------------------
$ 3(A-1) < A+3 < B $ $M < M_{D}$ $\frac{\pi}{2}\left(R_{D}^{2} - R_{2}^{2} \right) $ $\frac{\pi}{2}\left(R_{2}^{2} - R_{C}^{2} \right)$
$ 3(A-1) < A+3 < B $ $M_{D} < M < M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{C}^{2} \right)$
$ 3(A-1) < A+3 < B $ $M > M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{1}^{2} +R_{S}^{2}-R_{C}^{2} \right)$
$3(A-1) < B < A+3 $ $M < M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{C}^{2} \right)$
$3(A-1) < B < A+3 $ $M > M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{1}^{2} +R_{S}^{2}-R_{C}^{2} \right)$
$A+3 < B < 3(A-1)$ $M < M_{D}$ $\frac{\pi}{2}\left(R_{D}^{2} - R_{2}^{2} \right) $ $\frac{\pi}{2}\left(R_{2}^{2} - R_{C}^{2} \right)$
$A+3 < B < 3(A-1)$ $M_{D} < M < M_{SD}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{C}^{2} \right)$
$A+3 < B < 3(A-1)$ $M>M_{SD}$ $0$ $0$
$B < A+3 < 3(A-1)$ $M<M_{SD}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{C}^{2} \right)$
$B < A+3 < 3(A-1)$ $M>M_{SD}$ $0$ $0$
$A > B+3$ $M < M_{\rm CRIT}$ $0$ $0$
$A > \frac{B}{3} + 1$ $M < M_{SD}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{C}^{2} \right)$
$A > \frac{B}{3} + 1$ $M > M_{SD}$ $0$ $0$
$A < \frac{B}{3} + 1$ $M < M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{C}^{2} \right)$
$A < \frac{B}{3} + 1$ $M > M_{S}$ $0$ $\frac{\pi}{2}\left(R_{2}^{2} - R_{1}^{2} + R_{S}^{2} - R_{C}^{2} \right)$
Having developed the necessary tools to calculate the exponent for tunneling from bound to unbound vacuum bubbles, we now finish the development of a framework which will allow us to compare the relative likelihood for all thin-walled vacuum transitions to occur.
Comparison of the Tunneling Exponents {#comparison}
=====================================
Assuming that both the L and R tunneling geometries exist, and that we can choose the overall tunneling exponent to be negative in either case, we now venture to directly compare the tunneling rates for these two processes. In a cosmological setting, we must fluctuate the bound solution which will expand to its turning point and possibly tunnel to one of the unbound solutions. In the absence of a detailed theory of the nature of these fluctuations, we assume that the probability of fluctuating a solution of a given mass is given by the exponential of the entropy change due to the change in the area of the exterior dS horizon in the presence of a mass [@Gibbons:1977mu; @Albrecht:2004ke] $$P_{\rm seed} =\exp\left[-\pi \left(\frac{3}{\Lambda_{+}} - R_{C}^{2} \right) \right],$$ where $R_{C}$ is the radius of curvature of the cosmological horizon in SdS.
Once the bound solution has been fluctuated, it must survive until it reaches the turning point of the classical motion. The authors have shown [@Aguirre:2005sv] that any solution with a turning point is unstable against non-spherical perturbations. Even quantum fluctuations present on the bubble wall at the time of nucleation will go nonlinear over some range of initial size and mass. Presumably, these asphericities will affect the tunneling mechanism discussed in the previous section, and may be a significant correction to these processes. Seed bubbles can, however, avoid this instability by forming as near-perfect spheres very near the turning point; in the spectrum of possible fluctuations, there will inevitably be some such events.
Assuming that the seed bubble is still reasonably spherically symmetric when it reaches the turning point, the probability to go from empty dS to the spacetime containing an expanding vacuum bubble is given by the product $$\label{P}
P \simeq C P_{\rm seed} e^{2 i \Sigma_{0}} \equiv C e^{-S_{E}},$$ where $C$ is a pre-factor that will be neglected in what follows.
Shown in Fig. \[A1B6sigma\] is $-S_{E}$ as a function of $Q$ for ($A=1$, $B=6$), normalized to $k^{-2}$, for both the L tunneling geometries (blue dashed line) and R tunneling geometries (red solid line). In this case, it can be seen that the L tunneling geometries (which pass through the worm hole) are always more probable than R tunneling geometries (which pass through the cosmological horizon). Also, note that the zero mass ($Q \longrightarrow \infty$) solution is in both cases the most probable, even though the width of the potential barrier is largest in this limit.
We can locate and match the tunneling exponent for thermal activation [@Garriga:2004nm] in Fig. \[A1B6sigma\] as the most massive R tunneling geometry (the solution resting on top of the potential in Fig. \[Blt3Am1\]), which is denoted by the dot at the far right of the red solid curve. These solutions are bubbles which form in unstable equilibrium between expansion and collapse. We find, in agreement with Garriga and Megevand [@Garriga:2004nm], that thermal activation is always sub-dominant to CDL.
We have seen above that the R tunneling geometry possesses a smooth high-mass limit only for $B < 3 (A-1)$. The post-tunneling spacetime for this range of parameters is Solution 16 (see Fig. \[thermalons\]). However, our picture of the spacetime for $B > 3 (A-1)$ is somewhat different than Solution 17 of Fig. \[thermalons\], which is the post-tunneling spacetime found in Ref. [@Garriga:2004nm]. We find instead that the bubble nucleates outside the cosmological horizon (in the process removing a large section of the background de Sitter) as opposed to behind a worm hole (which leaves the background de Sitter space intact).
We have studied examples of the tunneling exponent for all of the possible situations listed in Tables \[tableworm\] and \[tablenoworm\]. The zero mass solution is always the most probable for both the L and R tunneling geometries. Depending on the values of $A$ and $B$, either the L or R tunneling geometries can dominate. Shown in Fig. \[A9B20sigma\] is an example of a true-vacuum bubble with ($A=9$, $B=20$); in this case the R tunneling geometries dominate. We can solve for the regions of parameter space where one geometry or another dominates by looking at the zero mass limit. The zero mass limit of the R tunneling geometry is CDL, and the tunneling exponent is given by Eq. \[BCDL\] (this includes the background subtraction). The zero mass limit of the L tunneling geometry (FGG) corresponds to the creation from nothing of a universe of the old phase containing a CDL bubble. The tunneling exponent in this case is numerically equal to $3\pi/\Lambda_{+} - S_{CDL}$. Taking the difference of the two tunneling exponents, we find that the L tunneling geometries will be dominant when $2 S_{CDL} > 3\pi/\Lambda_{+}$.
Depending on the values of the interior and exterior cosmological constant, the picture of vacuum transitions can be very complicated. For comparable cosmological constants, the situation is the most complicated, with both tunneling geometries and all mass scales having tunneling exponents of the same order of magnitude. While one mechanism will dominate, it may not overwhelm the slightly less probable possibilities. In the case where $\Lambda_{+} \ll \Lambda_{-}$, the zero mass limit of the L tunneling geometry (creation of a universe from nothing containing a CDL bubble) dominates. In the case where $\Lambda_{+} \gg \Lambda_{-}$, the zero mass limit of the R tunneling geometry (CDL true-vacuum bubbles) will dominate.
The bottom line {#bl}
===============
In the context of the junction condition potentials Figs. \[Bgt3Am1\], \[Blt3Am1\], \[AgtBo3p1\], and \[A\_6B\_5\], we now have a very organized picture of the types of vacuum transitions which are allowed. At one extreme, corresponding to $Q \rightarrow - \infty$ ($M \rightarrow 0$), we have [*both*]{} CDL bubble nucleation or the creation of a bubble spacetime from nothing. Moving up the potential in $Q$, we have the L tunneling geometries (FGG mechanism) and/or the R tunneling geometries. These are two-step processes, involving both a thermal fluctuation of the bound solution and a quantum tunneling event through the potential. At the top of the potential, we have the thermal activation mechanism, which is a one step, entirely thermal process. This completes our picture of the possible vacuum transitions, but still leaves unclear which processes actually occur – as mentioned in the introduction, there has been some question in the community as to whether the FGG process, for instance, is really a valid tunneling mechanism.
The comprehensive semi-classical picture that we have assembled raises a number of important questions in this regard. For instance, we have seen in the derivation of the tunneling exponent that the L and R tunneling geometries require different sign conventions to ensure a well-defined transition amplitude. Since the zero-mass limit of the L tunneling geometry describes the creation of a universe from nothing, this sign choice may well be connected with the notorious sign ambiguity in quantum cosmology [@Vilenkin:1998dn]. However, at the current level of treatment of the problem, there does not seem to be any well defined reason to choose one sign convention over the other, or to allow both.
There is also the question of how to reconcile the high- and low mass-limits of the L and R tunneling geometries. We have seen that the zero-mass limit of the R tunneling geometry always describes the nucleation of true- or false-vacuum CDL bubbles. It is therefore tempting to use this as evidence that the L tunneling geometries are not allowed. However, in a number of cases the high-mass limit of the R tunneling geometry is discontinuous in the sense that the pre-tunneling bound solution does not approach the post-tunneling unbound solution as the top of the effective potential is reached. In these same cases, the high-mass limit of the L tunneling geometry [*is*]{} continuous. Thus, even though the low-mass limit of the L tunneling geometry is rather strange (the creation of a universe from nothing), the high-mass limit seems completely reasonable. This complicates any hope of ruling out all L or all R tunneling geometries based on the reasonableness of the high- and low-mass limits of the effective potential.
There is also the problem that the tunneling geometry for some of the processes is not a manifold [@Farhi:1989yr; @Fischler:1990pk], and it is unclear that such metrics should be included in the path integral. The methods of FGG [@Farhi:1989yr] can be straightforwardly extended to show that the L tunneling geometry is never a manifold. That is, the Euclidean interpolating geometry between the pre- and post-tunneling states always has a degenerate metric. Analyzing the R tunneling geometries is much more involved because both the black hole and cosmological horizons come into play (requiring two coordinate patches), so we will defer a complete analysis of the L and R tunneling geometries to a separate follow-up paper [@Aguirre:xi].
The unfortunate bottom line, then, is that while the relation between the various nucleation processes is much clearer, the question of which ones actually occur remains open.
Conclusions
===========
We have catalogued all possible spherically symmetric, thin-wall, one-bubble (true- and false-vacuum) spacetimes with positive cosmological constant and have provided an exhaustive list of the possible quantum transitions between these solutions. Although there are undoubtedly many more possibilities as one relaxes the assumptions of spherical symmetry and a thin wall, this analysis should provide guidance in searching for more realistic processes.
The effective potentials of the junction condition formalism which were used to construct this catalog clearly indicate the existence of a region of classically forbidden radii separating bound solutions from unbound solutions. There are seemingly two processes which correspond to quantum tunneling through this same region, which we refer to as the L and R tunneling geometries. Both processes begin with a bound solution, which might be fluctuated by the background dS spacetime as we have assumed in Section \[comparison\]. This bound solution then evolves to its classical turning point, where it has a chance to tunnel to an unbound solution, which is typically either through a wormhole in the case of the L tunneling geometries (the Farhi-Guth-Guven, or FGG, mechanism) or through a cosmological horizon in the case of the R tunneling geometries.
The R tunneling geometries without a wormhole have a relatively clear interpretation in terms of the transition of a background spacetime to a spacetime of a different cosmological constant. Indeed, the zero-mass limit corresponds exactly to the nucleation of true- and false-vacuum CDL (Coleman-De Luccia) bubbles, correctly reproducing the radius of curvature of the bubble at the time of nucleation, as well as the tunneling exponent.
The L tunneling geometries (FGG mechanism) have a rather perplexing interpretation, which is most clearly seen by studying the zero mass limit. This corresponds to absolutely nothing happening in the background spacetime, while a completely topologically disconnected universe containing a CDL bubble of the new phase is created from nothing. The massive L tunneling geometries also have an element of this creation from nothing. Before the tunneling event, there is no wormhole, but after the tunneling event, there is a wormhole behind which is a large (eventually infinite) region of the old phase surrounded by a bubble of the new phase. It is unclear how we are to interpret this as the transition of a background spacetime to a spacetime of a different cosmological constant, since the background spacetime remains completely unaffected save for the presence of a black hole.
We have found that the sign of the Euclidean action is opposite for the L and R tunneling geometries, and while the second order constraints on the momenta introduce a sign ambiguity, it is unclear how to correctly fix the signs in light of the existence of two seemingly different processes for tunneling in the same direction through the same potential. A complete explanation of these processes may well require the resolution of some very deep problems in quantum cosmology.
If we take the stance that the L and R tunneling geometries are in competition as two real descriptions of a transition between spacetimes with different cosmological constants, then we must directly compare their relative probabilities. We have shown in Section \[comparison\] that the zero-mass solution is always the most probable for either the L or R tunneling geometries, and that the L tunneling geometry will be dominant when $2 B_{CDL} > 3\pi/\Lambda_{+}$. Therefore, if one is considering drastic transitions of the cosmological constant, the zero-mass FGG mechanism will be the dominant mechanism for upward fluctuations and the nucleation of true-vacuum CDL bubbles will be the dominant mechanism for downward fluctuations. This situation upsets the picture of fluctuations in the cosmological constant satisfying some kind of detailed balance [@Lee:1987qc; @Banks:2002nm].
It does, however, help to explain how spawning an inflationary universe from a non-inflating region might be a feasible cosmology [@Albrecht:2004ke]. In the picture that we have presented, both the L and R tunneling geometries are constructed by carving some volume out of the background spacetime and filling it with the new phase. The size of this region is in some sense a measure of how special the initial conditions for inflation are. In the case of the R tunneling geometries, a huge number of the states of the background spacetime must be put into the false vacuum at high cost in terms of the probability of such a fluctuation occurring [@Banks:2002nm]. The L tunneling geometries avoid this cost by fluctuating new states already in the false vacuum (seemingly a non-unitary process as discussed by Frievogel et. al. [@Freivogel:2005qh]), with the result that beginning inflation is no longer prohibitively difficult. The question of how much of the background spacetime must make the transition to the false vacuum is therefore crucial to determining exactly how special the initial conditions for inflation are. Unfortunately, detailed balance and this resolution of the paradoxes associated with the initial conditions for inflation are seemingly incompatible, but hopefully future work will yield further insight into the old but still interesting theory of vacuum transitions.
The authors wish to thank A. Albrecht, T. Banks, M. Dine, S. Gratton, and A. Shomer for their assistance in the development of this work.
[^1]: The study of thin wall junctions is a vast subject [@Berezin:1982ur; @Maeda:1981gw; @Ipser:1983db; @Aurilia:1984cm; @Sato:1986uz; @Berezin:1987bc]. In this paper, we will use the notation of Aurilia et. al. [@Aurilia:1989sb].
[^2]: It can be shown that this process cannot occur classically unless the weak energy condition is violated [@Farhi:1986ty; @Dutta:2005gt; @Aguirre:2005sv].
[^3]: Many of these solutions have appeared in previous work [@Blau:1986cw; @Sato:1986uz; @Berezin:1987bc; @Aurilia:1989sb; @Gomberoff:2003zh; @Garriga:2004nm; @Aguirre:2005sv], but with specific assumptions about the mass and/or the interior and exterior cosmological constants.
|
---
author:
- 'S.-N. X. Medina, S. A. Dzib, M. Tapia, L. F. Rodríguez, and L. Loinard'
date: 'Received 2017; '
title: The richness of compact radio sources in NGC 6334D to F
---
[The presence and properties of compact radio sources embedded in massive star-forming regions can reveal important physical properties about these regions and the processes occurring within them. The NGC 6334 complex, a massive star-forming region, has been studied extensively. Nevertheless, none of these studies has focused in its content in compact radio sources.]{} [Our goal here is to report on a systematic census of the compact radio sources toward [ NGC 6334]{}, and their characteristics. This will be used to try and define their very nature.]{} [We use VLA C band (4–8 GHz) archive data with [ 0.36 (500 AU)]{} of spatial resolution and noise level of 50 $\mu$Jy bm$^{-1}$ to carry out a systematic search for compact radio sources within NGC 6334. We also search for infrared counterparts to provide some constraints on the nature of the detected radio sources.]{} [A total of 83 compact sources and three slightly resolved sources were detected. Most of them are here reported for the first time. We found that 29 of these 86 sources have infrared counterparts and three are highly variable. Region D contains 18 of these sources. The compact source toward the center, in projection, of region E is also detected.]{} [From statistical analyses, we suggest that the 83 reported compact sources are real and most of them are related to NGC 6334 itself. A stellar nature for 27 of them is confirmed by their IR emission. Compared with Orion, region D suffers a deficit of compact radio sources. The infrared nebulosities around two of the slightly resolved sources are suggested to be warm dust, and we argue that the associated radio sources trace free-free emission from ionized material. We confirm the thermal radio emission of the compact source in region E. However, its detection at infrared wavelengths implies that it is located in the foreground of the molecular cloud. Finally, three strongly variable sources are suggested to be magnetically active young stars.]{}
Introduction
============
Massive star-forming regions contain high- and low-mass stars, which often present radio emission from different origins. The high-mass stars may ionize the medium around them, creating an region, which produces free-free emission detectable at radio wavelengths. Depending on the size and density of the ionized region they can be classified as classic, compact (C), ultra-compact (UC), and hyper-compact (HC) regions (Kurtz 2005). Also, massive stars may produce radio emission from their ionized winds (Contreras et al. 1996) and, in massive multiple stellar systems, from their wind collision regions (e.g., Ortiz-León et al. 2011). On the other hand, young low-mass stars may produce free-free radio emission in their jets (Anglada 1996), from their externally ionized disk by the UV photons of an OB star (proplyds; O’dell et al. 1993), and gyrosynchrotron radio emission when they are magnetically active. Even for nearby star forming regions (d $<2$ kpc), the radio emission of HC regions and from low-mass stars occurs on subarcsecond scales (see Rodríguez et al. 2012 for a detailed description on the characteristics of the different types of compact radio emission). Massive star forming regions contain many unresolved, or slightly resolved, radio sources (e.g. Orion and M17; Forbrich et al. 2016, Rodríguez et al. 2012). The compact radio sources have long been known to exist also within extended HII regions such as Orion, and have recently been found to be abundant in other cases. They are important because they might play a role in the time evolution of the HII regions themselves. In this work we present the first radio detection of a large number of radio sources in NGC 6334.
The giant molecular cloud NGC 6334 is a complex with very active spots of massive star formation at different evolutionary stages (see Persi & Tapia 2008 for a detailed review). Located at the relative nearby distance of 1.34$^{+0.15}_{-0.12}$ kpc (Reid et al. 2014) it is an ideal target for the study of the different phases of massive star formation and the detailed analysis of its content in compact radio sources may provide strong clues on the phases to which they belong.
Early radio maps with high resolution ($\sim1''$) of NGC 6334 showed the existence of six strong radio sources which were named as sources NGC 6334A to F (Rodríguez et al. 1982, from now on we will refer to them only with the alphabetic name). Source B is extragalactic (Moran et al. 1990; Bassani et al. 2005), while the others are compact and UC- regions. In this manuscript we will focus in the study of compact sources around the regions D, E and F and their surroundings, including the region called NGC 6334I(N) (hereafter I(N)). These sources are located in the north-east portion of the cloud.
Briefly described, source D is an evolved region with a nearly circular shape of approximate radius of 75$''$ (0.5 pc), centered on what appears to be one of its main ionizing early B type stars, 2MASS J17204800-3549191, reddened by about A$_V$ = 10 (Straw et al. 1989). At its western edge, the expansion of the region seems to be halted by a dense dark cloud, [ where CXOU 172031.76-355111.4 (also known as 2MASSJ17204466-3549168 or NGC6334II-23), a more luminous late O-type ionizing star, lies (Feigelson et al. 2009; Straw & Hyland 1989)]{}. This interaction appears to have triggered a second star formation stage in the region inside the dense dark cloud (Persi & Tapia 2008). Regions E and F, widely studied at several wavelengths, are regions that contain many signs of star formation (Persi & Tapia 2008 and references therein). [The C- region E]{} shows an extended shell-like structure with a compact radio source at its center (Carral et al. 2002). Source F is a younger and complex UC region with a cometary shape and a radio flux of $\sim~3$ Jy. This source and its surroundings are collectively known as NGC 6334I. Finally, the region I(N) is the youngest of all, as it has just started to form massive stars (Rodríguez et al. 2007; Hunter et al. 2006).
Observations
============
We will use an archival observation of NGC 6334 obtained with the Karl G. Jansky Very Large Array (VLA) telescope of the NRAO[^1] in the C-band (4 to 8 GHz) obtained as part of the project 10C-186. The data were taken on 7 July 2011, while the array was in A-configuration, its most extended. Two sub-bands, each 1 GHz wide and centered at 5.0 and 7.1 GHz, respectively, were recorded simultaneously. The average frequency of the whole observation is 6.0 GHz. The quasars J1717$-$3342, J1924$-$2914 and J1331+3030, were used as the gain, bandpass, and flux calibrators, respectively. The total time spent on source was 84 minutes. First results of this observation were reported by Hunter et al. (2014) and Brogan et al. (2016), we point the reader to these papers for further details of the observation.
The data were edited, calibrated and imaged using the software CASA with the help of the VLA Calibration Pipeline. The image at 6.0 GHz was produced by combining the two sub-bands using a multi-frequency deconvolution software (e.g., Rau & Cornwell 2011) and with a pixel size of 0$''$06. The resulting beamsize is 0$''67\times$0$''$19; P.A. = $-2\rlap{.}^\circ$7. The used weighting scheme was intermediate between natural and uniform (robust = 0, Briggs 1995). Similar images were also produced for both sub-bands separately, with pixel sizes of 0$''$06 and 0$''$04 for 5.0 GHz and 7.1 GHz, respectively. The beamsize for the 5.0 GHz image is 0$''91\times$0$''$25; P.A. = $-2\rlap{.}^\circ$3, and 0$''66\times$0$''$17; P.A. = $-2\rlap{.}^\circ$8 for the image at 7.1 GHz. [ All the images are corrected for the primary beam response.]{}
Source extraction
=================
The source extraction was performed using the BLOBCAT software (Hales et al. 2012). This software is a flood fill algorithm that cataloges islands of agglomerated pixels (blobs) within locally varying noise. It is designed for two-dimensional input FITS images of surface brightness (SB). BLOBCAT makes improvements in the morphological assumptions and applies bias corrections to extract blob properties (see Hales et al. 2012 for details). To run BLOBCAT, we also use the rms estimator algorithm implemented within the SExtractor package (Bertin & Arnouts 1996, Holwerda 2005) to make a suitable noise map of the SB image. We use a mesh size of 60$\times$60 pixels and Signal to Noise Ratio (SNR) $=$ 5 as threshold for detecting blobs for the 6.0 GHz image. For the 5.0 and 7.1 GHz images the mesh size was 80$\times$80 pixels and SNR = 3 as threshold, but in this case only sources that were also founded in the 6.0 GHz map were considered as real. The previous calculations were made following the derivations of Hales et al. (2012).
The expected number of false detections is calculated using the complementary cumulative distribution function $\Phi(x)=1- \phi(x)$ where $\phi(x)$ is the cumulative distribution function. Assuming that the noise in our radio maps follows a Gaussian distribution $$\phi(x)=\frac{1}{2} \bigg [ 1 + erf \bigg( \frac{x}{\sqrt{2}} \bigg) \bigg ]$$ where erf is the error function given by $$erf(x)= \frac{1}{\sqrt{\pi}} \int_{-x}^{x} e^{-t^2} dt.$$ $\Phi(x)$ is the probability that a value of a standard normal random variable X will exceed an $x$ level. So, the probability that any independent pixel (synthesized beam) will have a value up to 5$\sigma$ is $\Phi(5) \approx 3\times10^{-7}$. In consequence, we expect a total of 1 source above 5$\sigma$ in our 6 GHz maps, and conclude that essentially all our sources catalogued in the region are real. In consequence, the sources above 5$\sigma$ at 6.0 GHz are trustworthy. [ Even so, we did a visual inspection of all these sources to confirm their detection. We excluded the artifacts located close to the edge of the image.]{}
Results
=======
Radio sources
-------------
We obtain an image of $10'\times10'$, which is displayed in Figure \[fig:full\], combining the two observed sub-bands. The noise level is position dependent and produces the following two effects. The first is that the noise increases at the edges of the image, which is expected from the Gaussian primary beam pattern. The second is due to the imperfect sampling of the UV-space and it mainly affects areas around the extended strong emission. Thus, the noise level is close to 50 $\mu$Jy near the regions, but only about 8 $\mu$Jy in areas free of extended emission and not far from the center of the image. A similar effect is present in the Orion observations of Forbrich et al. (2016).
In Figure \[fig:full\], the UC region F and C region E are immediately appreciated, while only a fraction of the extended emission of source D is recovered. Only a small portion of region C falls inside the primary beam of this observation. Additional to the extended emission of the regions, a total of 83 compact[^2] radio sources were also detected (Table \[tab:RS\]), including those previously reported in the E, I, and I(N) region. The compact sources cannot be easily appreciated in this large field figure and they are represented by small red circles. The astrometric accuracy for the positions is better than 0.1, and we have used the positions of the sources to name them according to the IAU suggestion. Using equation A11 from Anglada et al. (1998) the expected number of background extragalactic sources in the imaged area with flux densities above 50 $\mu$Jy is 7$\pm3$. This previously computed number only reflects an upper limit because the emission of the extended sources affects the noise distribution and this changes the value of the expected background sources. Most of the detected compact radio sources are Galactic objects and most probably related to the NGC 6334 complex.
Region D, presented in Figure \[fig:D\], contains a significant fraction of all of the detected compact radio sources. Also, at the north side of this region there are two cometary shaped radio sources and at its center a double source. All of them, the compact radio sources, the cometary radio sources and the double radio sources, are reported here for the first time.
In order to compute the spectral index ($\alpha$; $S_{\nu}\propto\nu^{\alpha}$), we measured the flux density of the detected sources in the 5.0 GHz and 7.1 GHz maps and calculated the slope between both points. The error was calculated by using the standard error propagation theory. As the field of view of the map at 7.1 GHz is smaller than at 5.0 GHz, we could not calculate the spectral index for some sources that lie outside the edges of the 7.1 GHz image. Additionally, some sources were detected in one band but not in the other, in those cases we just calculate a limit for $\alpha$. The spectral index for each individual source is shown in Table \[tab:RS\].
[ Using the SIMBAD database we searched for counterparts of detected radio sources. Inside a radius of 0$''$5 from the VLA position, we found 17 radio sources with counterparts at different wavelengths and are shown in Table \[tab:RS\]. Additionally, the source SSTU J172053.96-3545.6 is at an angular distance of 0$''$9 from VLA J172054.06-354548.4. This shift is larger than the combined position errors of both telescopes (pointing accuracy for IRAC is $\sim$0$''$5) and indicates that these sources are not the same object, but we cannot discard that they may be related (e.g., a binary system). Finally, source \[S2000e\] SM6 is at an angular distance from source VLA J172053.26-354305.7 of $\sim$1$''$2, this agrees within the position error of 2$''$ for the millimeter source (Sandell 2000). To look for more infrared counterparts we perform a more detailed search, that is described in the next section.]{}
Infrared counterparts
---------------------
We searched for infrared counterparts of the compact radio sources searched in section 4.1. We use the source catalog from the [*Spitzer*]{} space mission as well as published near- and mid-IR photometry data by Willis et al. (2013) and Tapia, Persi & Roth (1996).
We complemented the search with unpublished $JHK_s$ photometric data from M. Tapia (in preparation). Defining reliable criteria for assigning a positive infrared counterpart to a compact radio source is not straightforward, as we are dealing with several infrared sets of infrared photometric measurements, namely from [*Spitzer*]{} images in the mid-IR and from ground-based observations with three different telescopes (and set-ups) in the near-IR. For the present work, we cross-checked the coordinates of the VLA compact sources with those from the [*Spitzer*]{} and also from the ground-based images. We selected those with coordinates coinciding (in all wavelength ranges) within 0$''$9 or less to obtain a list of candidate counterparts. We then examined by eye each source on all available individual images (i.e., each wavelength) and checked for the consistency of the corresponding flux measurements. The mean differences in the radio, mid- and near-IR source positions for the 27 counterparts were -0$''$07 in right ascension and +0$''$07 in declination, with standard deviations of 0$''$44 and 0$''$30, respectively. In all, no systematic coordinate offsets were found in the (small) area covered by this survey.
From the 83 VLA sources in Table 1, only 27 IR counterparts were found. Interestingly, 10 of these infrared sources are inside the dark cloud in region D. The ground-based near-IR and [*Spitzer*]{}/IRAC mid-IR photometry of all these sources are listed in Table \[tab:IR\]. The diagnostic two-color and color-magnitude diagrams are presented in Figs. \[fig:3\], \[fig:4\], \[fig:5\], and \[fig:6\]. The $J-H$ versus $H-K_s$ and the $H-K_s$ versus $K_s$-\[3.6\] diagrams are accurate tools for discriminating reddened photospheres (those lying along the reddening vectors) from stars with disks (i.e. near-IR excesses). The latter stars would lie shifted towards redder colors in these diagrams. The amount of dust extinction (i.e., the value of $A_V$ along the line of sight) for the former set of stars, is determined by the color-excess indices, under a “standard” reddening law, represented by the reddening vectors. References for the intrinsic colors and reddening law assumed for determining these parameters are given in the figure captions.
The color-magnitude plot $K_s$ versus $H-K_s$, on the other hand, is the best tool for estimating approximate spectral types and intra-cloud extinction values for embedded stellar sources located at the distance of the star formation complex ($d = 1.34$ kpc and foreground value of $A_V = 1.0$ in this case). Finally, the IRAC two-colour magnitude \[3.6\]-\[4.5\] versus \[4.5\]-\[5.8\] diagram provides a simple diagnostics for classifying the evolutionary status of YSOs and of emission-line-dominated regions (mainly Polycyclic Aromatic Hydrocarbons, PAHs, or shocked molecular hydrogen). This is described in detail by Ybarra et al. (2014) and references therein. The combined results for these analyses in terms of the derived properties of a number of individual sources are listed in the last column of Table 2, where we also indicate whether it was detected as an X-ray source (within 0$''$6) by Feigelson et al. (2009).
[ccccccccr]{} ID & J & H & K & \[3.6\] & \[4.5\] & \[5.8\] & \[8\] & Notes\
\
3 & – & 17.76 $\pm$ 0.04 & 15.25 $\pm$ 0.02 & 12.33 $\pm$ 0.18 & 12.41 $\pm$ 0.19 & – & – & 2,4,9,X\
4 & 16.75 $\pm$ 0.02 & 15.30 $\pm$ 0.08 & 13.89 $\pm$ 0.07 & – & – & – & – & 2,4,8,X\
5 & ... & – & 15.98 0.10 & 12.65 $\pm$ 0.10 & 11.98 $\pm$ 0.11 & – & – &\
8 & 19.72 $\pm$ 0.11 & 15.47 $\pm$ 0.02 & 13.15 $\pm$ 0.02 & 11.61 $\pm$ 0.16 & 11.32 $\pm$ 0.15 & – & – & 1,8,X\
9 & – & – & 15.55 $\pm$ 0.05 & – & – & 9.78 $\pm$ 0.12 & 7.25 $\pm$ 0.13 & X\
10 & – & 17.86 $\pm$ 0.12 & 16.24 $\pm$ 0.08 & – & – & – & – &\
11 & 15.15 $\pm$ 0.02 & 13.24 $\pm$ 0.02 & 12.36 $\pm$ 0.03 & – & – & – & – & 2,4,8,X\
12 & – & 17.44 $\pm$ 0.12 & 14.27 $\pm$ 0.05 & 12.47 $\pm$ 0.18 & – & – & – & 1,9,X\
13 & 16.36 $\pm$ 0.10 & 12.34 $\pm$ 0.08 & 10.14 $\pm$ 0.09 & 8.53 $\pm$ 0.04 & 7.87 $\pm$ 0.05 & 7.52 $\pm$ 0.04 & 6.85 $\pm$ 0.16 & 1,9,6,X\
14 & 17.48 $\pm$ 0.08 & 15.56 $\pm$ 0.04 & 13.58 $\pm$ 0.03 & 11.37 $\pm$ 0.10 & 10.21 $\pm$ 0.10 & – & – & 2,9,4,5\
15 & 11.34 $\pm$ 0.03 & 9.79 $\pm$ 0.03 & 8.93 $\pm$ 0.03 & 8.06 $\pm$ 0.12 & 7.54 $\pm$ 0.11 & 6.78 $\pm$ 0.07 & 99.99 $\pm$ 9.99 & 1,4,5,6,X\
\
& 11.93 $\pm$ 0.02 & 9.97 $\pm$ 0.03 & 9.18 $\pm$ 0.02 & 8.12 $\pm$ 0.03 & 7.84 $\pm$ 0.01 & 7.37 $\pm$ 0.03 & 6.77 $\pm$ 0.06 & 1,8,4\
[ 27]{} & 16.49 $\pm$ 0.01 & 14.18 $\pm$ 0.02 & 13.00 $\pm$ 0.01 & 11.96 $\pm$ 0.05 & 12.13 $\pm$ 0.05 & 10.61 $\pm$ 0.09 & 8.73 $\pm$ 0.13 & 2,4,X\
[ 35]{} & – & – & 16.60 $\pm$ 9.15 & 13.14 $\pm$ 0.03 & 12.00 $\pm$ 0.03 & – & – & 5\
[ 38]{} & 14.28 $\pm$ 0.02 & 12.78 $\pm$ 0.01 & 11.95 $\pm$ 0.01 & 11.05 $\pm$ 0.07 & 10.50 $\pm$ 0.10 & – & – & 2,8\
[ 39]{} & – & – & – & 14.07 $\pm$ 0.03 & 10.75 $\pm$ 0.03 & 8.89 $\pm$ 0.02 & 7.68 $\pm$ 0.03 & 5,7\
[ 42]{} & – & – & 14.40 $\pm$ 0.02 & 9.49 $\pm$ 0.03 & 7.29 $\pm$ 0.07 & 5.82 $\pm$ 0.06 & 4.46 $\pm$ 0.07 & 5,7,9, CHII-E\
[ 43]{} & – & 16.48 $\pm$ 0.04 & 14.09 $\pm$ 0.04 & 9.25 0.08 & 7.27 $\pm$ 0.06 & 6.08 $\pm$ 0.04 & 5.08 $\pm$ 0.05 & 4,5,7\
[ 44]{} & 15.98 $\pm$ 0.01 & 13.53 $\pm$ 0.01 & 12.40 $\pm$ 0.02 & 10.93 $\pm$ 0.07 & 11.06 $\pm$ 0.05 & 10.09 $\pm$ 0.08 & – & 2,8,5,X\
[ 45]{} & – & – & – & 13.22 $\pm$ 0.08 & 12.02 $\pm$ 0.03 & 10.99 $\pm$ 0.08 & – & 5,7,X\
[ 49]{} & – & – & – & 13.28 $\pm$ 0.03 & 12.28 $\pm$ 0.09 & – & – & X\
[ 51]{} & 16.95 $\pm$ 0.02 & 13.83 $\pm$ 0.03 & 12.20 $\pm$ 0.03 & 10.59 $\pm$ 0.03 & 10.05 $\pm$ 0.04 & 10.88 $\pm$ 0.19 & 8.65 $\pm$ 0.04 & 1,4,5,6,8,X\
[ 52]{} & – & 17.49 $\pm$ 0.04 & 15.88 $\pm$ 0.05 & – & – & – & – &\
[ 53]{} & – & – & 16.09 0.05 & 11.80 $\pm$ 0.02 & 10.17 $\pm$ 0.02 & 9.29 $\pm$ 0.02 & 8.67 $\pm$ 0.02 & 5,7,X\
[ 60]{} & 10.97 $\pm$ 0.03 & 10.45 $\pm$ 0.03 & 10.10 $\pm$ 0.02 & 9.74 $\pm$ 0.07 & 9.68 $\pm$ 0.06 & 9.46 $\pm$ 0.07 & – & 3\
[ 65]{} & – & – & – & 15.59 $\pm$ 0.03 & 12.17 $\pm$ 0.07 & 12.01 $\pm$ 0.02 & 9.12 $\pm$ 0.15 & 5\
[ 75]{} & 19.40 $\pm$ 0.07 & 16.21 $\pm$ 0.02 & 14.67 $\pm$ 0.02 & 13.58 $\pm$ 0.02 & 13.44 $\pm$ 0.02 & – & – & 3,8\
\[tab:IR\]
[ccccccccccr]{} R.A. & Dec. & & & & & & & & &\
(J2000) & (J2000)& J& H & K & \[3.6\]& \[4.5\] & \[5.8\] & \[8\]& [ Notes\*]{} & Other\
\
17:20:41.92 &$-$35:48:04.7 &14.69$\pm$0.01 &13.40$\pm$0.01 &12.37$\pm$0.02 &9.50$\pm$0.15 &9.14$\pm$0.12 &6.46$\pm$0.12 &5.33$\pm$0.12 &2,8,4,5,7 &Mir-1\
17:20:41.93 &$-$35:48:09.7 & - & - & - &10.52$\pm$0.18 &9.39$\pm$0.13 &6.64$\pm$0.13 &5.35$\pm$0.12 &5,7,X &Mir-2\
17:20:42.31 &$-$35:48:18.3 &21.35$\pm$0.13 &17.67$\pm$0.01 &14.40$\pm$0.02 &10.63$\pm$0.17 &9.52$\pm$0.12 &8.21$\pm$0.20 &- &1,9,4,5,7 &Mir-3\
17:20:42.19 &$-$35:48:12.5 &19.95$\pm$0.09 &17.12$\pm$0.03 &15.22$\pm$0.01 &- &- &- &- &2,9,4 &-\
17:20:42.00 &$-$35:48:07.7 &18.78$\pm$0.12 &16.45$\pm$0.04 &15.73$\pm$0.03 &- &- &- &- &3 &-\
17:20:42.02 &$-$35:48:11.0 & - &17.95$\pm$0.05 &15.80$\pm$0.09 &- &- &- &- &2,9 &-\
17:20:42.49 &$-$35:48:09.5 &21.48$\pm$0.12 &19.49$\pm$0.24 &16.56$\pm$0.04 &- &- &- &- &2,4 &-\
17:20:42.49 &$-$35:48:08.5 & - & - &17.2$\pm$0.2 &- &- &- &- &-\
17:20:42.48 &$-$35:48:03.8 & - &- &16.9$\pm$0.2 &- &- &- &- &-&-\
17:20:42.29 &$-$35:48:03.2 &20.01$\pm$0.10 &17.93$\pm$0.11 &16.79$\pm$0.04 &- &- &- &- &3&-\
17:20:42.20 &$-$35:48:02.6 &20.33$\pm$0.17 &17.49$\pm$0.11 &16.52$\pm$0.06 &- &- &- &- &3 &-\
17:20:42.02 &$-$35:49:10.1 & - &17.42$\pm$0.03 &16.70$\pm$0.07 &- &- &- &- &- &-\
17:20:42.15 &$-$35:48:17.8 & - & - &17.0$\pm$0.2 &- &- &- &- &- &-\
17:20:42.11 &$-$35:48:13.2 & - & - &16.05$\pm$0.10 &- &- &- &- &- &-\
17:20:42.10 &$-$35:48:05.2 & - & - &15.8$\pm$0.2 &- &- &- &- &- &-\
\
17:20:41.58 &$-$35:48:36.9 &- & 16.09$\pm$0.15 & 14.85$\pm$0.05 & 11.07$\pm$0.09 & 10.08$\pm$0.13 & 7.14$\pm$0.10 & 5.51$\pm$0.15 & 7,X &Mir-4\
\[tab:IRcome\]
Discussion
==========
To define the nature of the radio emission of the compact sources, we need information on the different characteristics ($\alpha$, polarization, time variability). However we are only able to do a rough measure of the spectral index, which only gives clues on the emission mechanism. While thermal free-free radio emission has spectral indices with values $-0.1\leq\alpha\leq+2.0$, gyrosynchrotron emission has values in the range $-2.0\leq\alpha\leq+2.0$, and optically-thin synchrotron emission from wind collision regions has values $\alpha\sim-0.7$ (see Rodríguez et al. 2012 and references therein).
Region D
--------
### Compact sources
Although the reported compact radio sources are spread over the entire observed area, there is a group of 15 objects, reported here for the first time, that is well concentrated (within a radius of 0.3 pc) in the dark cloud on the western edge of region D. Of these, 73% (11 sources) have unresolved infrared counterparts (see Fig. \[fig:D\]) and all but 3 are X-ray emitters (Feigelson et al. 2009). In the first parts of Tables \[tab:RS\] and \[tab:IR\] we list the observational data of the compact sources in region D that we now focus on.
The clustering of compact radio sources clearly coincides with the densest part of the dark cloud, where one would suspect to have the latest stage of star formation in the region. From the near-IR colors, we can distinguish only four stars that have a late-O or early-B spectral types, while the rest are less massive. Two of these three massive stars have a negative spectral index, and thus their radio emission may originate in wind collision regions. The majority of the remaining sources (eight sources) have negative spectral indices (non-thermal) values and only three are positive. The association of these sources with X-ray emission is compatible with a non-thermal origin for the radio emission, but there are a few cases where the spectral index is positive in spite of the presence of an X-ray source (Dzib et al. 2013). As the sources are compact and likely associated with the star forming region, an attractive interpretation is that they are magnetically active low mass YSOs with gyrosynchrotron radio emission. However, the errors are large and this will have to be confirmed with multi-epoch monitoring to measure their flux variability, as nonthermal low mass stars tend to be very variable on scales of days (André 1996). These future observations will also help to obtain a better estimate of their spectral indices.
It is worth highlighting that in the HII region associated with the Orion core [ (at d$\simeq$400 pc; Menten et al. 2007; Kounkel et al. 2017)]{}, there is a significant population of YSOs that produce radio emission (e.g. Forbrich et al. 2016, Zapata et al. 2004). In order to compare with region D, if the Orion core were located at 1.34 kpc, the distance of the NGC 6334 complex, the flux density of the 556 compact sources detected in Forbrich at al. (2016) would be 11.2 times lower. Considering 50 $\mu$Jy as the detection threshold, only 47 compact sources from Forbrich et al. (2016) could be detected. We are only detecting 14 compact sources, or about one third of the expectation based on the Orion population. This is consistent with other similar comparisons (e.g., Masqué et al. 2017), which may indicate that the Orion core may be richer in radio sources than other similar regions. Also, our images are highly contaminated by the extended emission making it difficult to identify all compact sources. Finally, there may be more sources below our detection threshold but we need deeper radio observations to clarify this point.
### Cometary Nebula North (CNN) = VLA J172041.75-354808.2
Close to the north-western edge of the expanding HII region D, we discovered an extended radio source with a cometary structure. From now on, we will refer to this source as CNN (for Cometary Nebula North). Its flux density at 6.0 GHz is 18.4$\pm$0.1 mJy with a peak flux of 385$\pm12\, \mu$Jy bm$^{-1}$. The position of its peak flux is R.A.=[$17^{\rm h}20^{\rm m}41\mbox{$^{\rm s}\mskip-7.6mu.\,$}75$]{}; Dec.= [$-35^{\circ}48'08\mbox{$''\mskip-7.6mu.\,$}2$]{}. This peak of the extended radio emission reported here is at the center of a mid-IR nebulosity with an ovoid shape of size around $15''$. It is relatively bright in the 4.5, 5.8 and 8 $\mu$m [*Spitzer*]{}/IRAC images, as shown in Fig. \[fig:CN\]–Top. Although their spatial resolution (1.5 to 2.0) is much worse than that obtained with the VLA, a similar morphology of the diffuse emission is evident. The mid-IR structure is complicated by the presence of at least three bright, unresolved mid-IR sources (Mir-1, Mir-2 and Mir-3), most likely of protostellar nature embedded in the nebulosity. Fig. \[fig:CN\]–Bottom shows the same field at 2.2 $\mu$m as observed in excellent seeing conditions ($\sim 0\rlap{$''$}\,.55$) with the 2.5-m DuPont telescope at Las Campanas Observatory (Tapia, Persi & Roth, in preparation). In addition to the mid-IR sources, there are a dozen near-IR stars detected within the nebula. Comparison with the surrounding field shows that at least 60% of them must be embedded in it. IRAC and $JHK$ photometry of these sources are reported in Table \[tab:IRcome\] and their colors are plotted in Figs. 3, 4, 5 and 6.
The nature of the IR sources in CNN can be deduced from the multi-wavelength photometry. From the nebula-subtracted fluxes of sources Mir-1, Mir-2 and Mir-3, we conclude that they are Class I young stellar objects with spectral types earlier than A and each with quite different characteristics. The notes in [ column 9 of]{} Table \[tab:IRcome\] summarize the results. Interestingly, in the cometary nebulae there is another [emission peak]{} evident in the radio maps at the three frequencies (see Fig. \[fig:CNRadio\]) at the position R.A.=[$17^{\rm h}20^{\rm m}41\mbox{$^{\rm s}\mskip-7.6mu.\,$}93$]{}; Dec.= [$-35^{\circ}48'10\mbox{$''\mskip-7.6mu.\,$}5$]{}. This position is consistent, within errors, with the position of source Mir-2 and we suggest that it may be its radio counterpart.
-- -- --
-- -- --
Concerning the diffuse, nebular emission, from the present images, we attempted to measure nebular fluxes at several IR wavelengths. Because of many limitations (poor and very variable resolutions, presence of point-like sources, etc.) we found it impossible to deduce total fluxes reliably. Nevertheless, [ by integrating the near- and mid-IR flux densities that are well-derived from the ground-based and [*Spitzer*]{} observations and extrapolate them into the far-IR, we can estimate a lower limit to the total IR luminosity of the nebulosity to be $\sim 10^3 L_\odot$.]{} Undoubtedly, this arises from thermal emission of warm dust.
Thus we will assume that the CNN source is an region, surrounded by a dusty envelope, and derive its properties under that assumption. [ At a frequency of 6.0 GHz,]{} we derive[^3] a brightness temperature of $44$ K, an electron density of 4.8$\times10^3$ cm$^{-3}$, an emission measure of $0.6\times10^6$ cm$^{-6}$ pc, an ionized mass of 8$\times10^{-3}$ M$_{\odot}$, and a flux of ionizing photons of 1.4$\times10^{46}$ s$^{-1}$ assuming a circular shape with a radius of 2$''$ = 0.013 pc, and an electron temperature of T$_e = 10^4$ K. These values are consistent with an UC region being photoionized by a B0.5 ZAMS star (Kurtz et al. 1994, Panagia 1973). This is consistent with the approximated spectral types of sources Mir-1, Mir-2 and Mir-3.
### Cometary Nebula South (CNS) = VLA J172041.59-354837.2
As can be seen in Fig. \[fig:D\], a second small mid-IR nebula was found about $27''$ south of CNN (hereafter, we will refer to this source as CNS for Cometary Nebula South). Both nebulosities have similar mid- and far-IR colors implying matching dust temperatures, though the integrated Herschel flux at 70 $\mu$m for CNS is four times fainter than that from CNN. Unfortunately, no photometry could be obtained at longer wavelengths and, thus, no reliable dust temperature can be derived. CNS has at its center a compact near- and mid-IR source (Mir-4 in Fig. 9 and Table 3). The IR photometry of this source suggests that this is an intermediate luminosity Class I YSO with $A_V \simeq 13-16$. This object was also detected in X-rays by Feigelson et al. (2009). At the same position, our 6.0 GHz image shows the presence of a small, roundish radio source with a diameter of around $2''$. Its total flux [ density]{} at 6.0 GHz is 4.5$\pm$0.4 mJy. As for CNN, the presence of warm dust also indicates that the radio emission of CNS also traces ionized gas. The derived parameter in this case are a brightness temperature of $39$ K, an electron density of 6.2$\times10^3$ cm$^{-3}$, an emission measure of $0.5\times10^6$ cm$^{-6}$ pc, an ionized mass of 1$\times10^{-3}$ M$_{\odot}$, and a flux of ionizing photons of 3.0$\times10^{45}$ s$^{-1}$, which suggests that CNS is being photoionized by a B1 star (Panagia 1973).
It should be noted that the neighboring bright star that appears “blue” in the mid-IR IRAC image (Fig. \[fig:CNS\]) is, given its IR colors, a foreground star unrelated to CNS.
### Double Nebula Source (DNS) = VLA J172044.4-354917
Most intriguing is the extended double radio source VLA J172044.47-3549017 (hereafter DNS for Double Nebula Source), located almost at the center of the radio H II region D (e.g. the 1.6 GHz map by Brooks & Whiteoak 2001). This source is surrounded by three radio compact sources (VLA 13, VLA 14 and VLA 15) with IR counterparts, see Fig. \[fig:DNS\]. Their IR counterparts indicate that they are massive stars with spectral types from late to early-B.
The nature of the double radio source is unknown. As can be seen in Fig. \[fig:full\] and Fig. \[fig:DNS\], there is a lot of diffuse emission in region D. Thus, we speculate that DNS is gas ionized by the above mentioned three massive stars. To test this hypothesis we analyze additional VLA archive data. The observations were made on 2013 July 25 in L (1 to 2 GHz) band under project 13A-448. The array was in the C configuration. The data were calibrated following the standard CASA procedures. To obtain the best angular resolution possible an image was made using the spectral window with the highest frequency, centered at 1.92 GHz and with a bandwidth of 128 MHz. We additionally used superuniform weighting in the CLEAN task of CASA. The image was also corrected for the response of the primary beam and is shown in blue contours in Fig. \[fig:DNS\] . This image indicates that source D has a shell-like morphology and that the double radio source coincides with the brightest, eastern edge of the shell. Then, the double source may not be a true independent source but simply the brightest part of a larger structure.
The central source in the C-${\mbox{\ion{H}{\small II}}}$ region E
------------------------------------------------------------------
This compact radio source was discovered by Carral et al. (2002) close to the center of the C- region E (see Fig. \[fig:E\]) and it is our source No. 42 (Tables \[tab:RS\] and \[tab:IR\]). It is coincident with IR source 161 of Tapia et al (1996), and was undetected in Feigelson et al.’s (2009) X-ray survey. Carral et al. (2002) measured a spectral index of $\alpha=1.0\pm0.7$ which is compatible with an ionized stellar envelope and, thus, they interpreted this source as the ionizing star of source E. The IR photometry here reported confirms that this is a Class I YSO, with $A_V > 55$ and $L_{\rm IR} > 10$ $L_\odot$. Lacking mid- and far-IR fluxes, we cannot rule out, nor confirm that this star is the ionizing star of the C- region, though the spatial coincidence makes it very probable.
The measured spectral index in our maps is $\alpha=1.1\pm0.3$ which is consistent with the previous results and the thermal nature of this source is corroborated. Using the estimated spectral index to extrapolate our measured flux to 8.4 GHz, we obtain a flux density of 1.2$\pm$0.3 mJy, which is in good agreement with the flux density measured by Carral et al. (2002). This non-variability also supports the thermal nature of the radio source.
Long term variability
---------------------
Most of the previous radio observations on NGC 6334 were focused on regions E, F, and I(N), as they have shown the most recent star formation on this molecular cloud. Furthermore, they reached noise levels that were much poorer than those of the final images reported on this paper. Still, we may use these observations to roughly measure variability on some sources. In the VLA archive we found a VLA observation, in A configuration, obtained on 11 August 1995 at 8.4 GHz, that is part of project AM495. These observations were reported in Carral et al. (2002). The pointing center was RA=[$17^{\rm h}20^{\rm m}53\mbox{$^{\rm s}\mskip-7.6mu.\,$}40$]{}; Dec.=[$-35^{\circ}46'25\mbox{$''\mskip-7.6mu.\,$}0$]{} (i.e. roughly equidistant from regions E, F, and I(N)). The data were edited and calibrated in a standard way. Images were produced with a pixel size of 0.06 and, as it was done by Carral et al. (2002), removing the short spacings below 100 k$\lambda$ (to filter out structures larger than 2$''$). [ The final beamsize of this image was 0$''$48$\times$0$''$19; PA=0.7.]{} We look at the position of sources in Table \[tab:RS\] and that are inside the imaged area. In this case, we used a threshold of three times the noise level to consider a detection. Five sources were detected using this restriction and their fluxes from this image are reported in Table \[tab:95vs2011\]. Additionally, using the calculated spectral index, we extrapolate our measured fluxes at 6.0 GHz and determine their fluxes at 8.4 GHz, these are also listed on Table \[tab:95vs2011\]. Most of the sources are in good agreement with their expected flux in 2011, which suggests that these sources are not strongly variables. The only exception is source VLA J172058.14$-$354934.6 which shows a decrease in its flux from 1995 to 2011 by a factor of $2.4\pm0.3$.
On the other hand, we notice that due to the smaller imaged area and larger noise, most of the sources are not expected to be detected because they fall outside the primary beam or their flux will be below three times the noise level. However, using the extrapolated flux in epoch 2011 and assuming non-variability, three sources should be detected above this threshold in the 1995 image. Interestingly, they are not. In Table \[tab:95vs2011\] we list these sources, an upper limit of three times the noise level (with the noise level of the area as the flux error) and its predicted flux at 8.4 GHz [ for the epoch 2011]{}. Clearly sources VLA J172052.02$-$354938.0 and VLA J172053.65-354548.4 are strong variables [ ($\gtrsim$100%)]{} on scales of years. We could not support a variability for source VLA J172057.98$-$354431.6, since its non-detection may be due only to fluctuation in the flux caused by the noise of the observations.
We also searched for possible additional new sources by using BLOBCAT and with the parameters used for the 6.0 GHz map. We did not find any additional new source.
The three strongly variable sources reported here may be good examples of magnetically active YSOs. With future deep observations we could study variability from the weakest radio sources.
----------------------- ---------------- ----------------
$S_\nu$ (1995) $S_\nu$ (2011)
VLA name ($\mu$Jy) ($\mu$Jy)
J172103.47$-$354618.8 406$\pm$59 453$\pm$52
J172052.02$-$354938.0 $<408\pm$136 1038$\pm$93
J172053.65$-$354548.4 $<171\pm$57 338$\pm$62
J172057.98$-$354431.6 $<216\pm$72 220$\pm$24
J172050.91$-$354605.0 $1380\pm$110 1188$\pm$290
J172055.19$-$354503.8 401$\pm$107 365$\pm$23
J172058.14$-$354934.6 1240$\pm$110 510$\pm$52
J172054.62$-$354508.5 290$\pm$60 192$\pm$25
\[tab:95vs2011\]
----------------------- ---------------- ----------------
: Comparison from radio sources of epoch 1995 and 2011.
Other sources
-------------
We could not determine the spectral classification for most of the remaining radio sources. However, the infrared emission of two sources, 24 and 50 in Table \[tab:RS\], indicate that they are early B stars. Their spectral index has large errors but suggest a flat spectrum, so they are most probably thermal emitters. The radio emission in these cases may originate in the winds of the massive stars.
It is hard to speculate on the radio emission of the remaining sources, because of the scarce information. Future multi-wavelength and multi-epoch observations with better sensitivity may help to reveal their nature.
Conclusions
===========
We have presented a deep radio observation ($\sigma\sim50\,\mu$Jy bm$^{-1}$) with high angular resolution (0.2) of the NE of the NGC 6334 complex (covering the regions D, E, F, I(N), and part of the region C) searching for compact radio sources. We also searched for infrared counterparts of detected compact radio sources and characterized them. Now we list the results and conclusions from our analysis.
- A total of 83 radio compact sources in the NE of NGC 6334 are detected, 15 of them are located inside the region D. Most of these sources are new detections and only around 10 of them were previously reported and are located in regions E, I, and I(N).
- The stellar nature of 27 of the 83 compact radio sources is confirmed by the properties of their infrared emission.
- We computed the spectral index of the sources in order to speculate about the nature of their radio emission. In region D the values tend to be negative, suggesting non-thermal emission. Most of these sources are likely magnetically active low mass YSOs as in the Orion core. However, the IR emission of three of them suggest that they are early B stars and their radio emission may originate in strong shocks of wind collision regions.
- Two interesting cometary radio sources, CNN and CNS, were detected close to region D and are here reported for the first time. They are spatially coincident with more extended mid-IR nebulosities of similar shape. We suggest that they are regions (traced by the radio emission) surrounded by dusty envelopes (traced by the mid-IR). Interestingly we found three stars (Mir-1, Mir-2 and Mir-3) with spectral types earlier than A, which could be the sources of the ionizing photons of CNN. On the other hand, the possible ionizing source of CNS is Mir-4.
- Through the inspection of an additional 1.92 GHz image, we suggest that the double source VLA J1720444.4-354917 (DNS) is part of the diffuse ionized gas from the region D.
- Our observations support the thermal nature of source VLA J172050.91$-$354605.0, which is located near the center of the radio region E.
- By comparing with an observation obtained in 1995, we analyzed the variability in flux of eight sources. Three of them show strong variability [ suggesting]{} that they are magnetically active low mass stars.
Our analysis has provided clues on the nature of several of the detected compact radio sources. However, future observations are necessary to better establish the nature of most of them. Ideally, these observation should be multi-wavelength and multi-epoch.
S.-N.X.M. acknowledges IMPRS for a Ph.D. research scholarship. L.F.R., M. T. and L.L. acknowledge the financial support of DGAPA, UNAM, and CONACyT, México. MT and LL acknowledge support for this work through UNAM-PAPIIT grants IN104316, and IN112417. This paper makes use of archival data obtained with the [*Spitzer Space Telescope*]{}, which is operated by the Jet Propulsion Laboratory, California Institute of Technology (CIT) under National Aeronautics and Space Administration (NASA) contract 1407. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration
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[^1]: The National Radio Astronomy Observatory is operated by Associated Universities Inc. under cooperative agreement with the National Science Foundation.
[^2]: Compact sources are defined here as source whose deconvolved sizes are similar to, or smaller than the synthesized beam.
[^3]: We have used the basic equations of brightness temperature, emission measure, electron density, and ionizing photon rate (e.g., Table 4 in Masqué et al. 2017).
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---
abstract: 'A region crossing change at a region of a spatial-graph diagram is a transformation changing every crossing on the boundary of the region. In this paper, it is shown that every spatial graph consisting of theta-curves can be unknotted by region crossing changes.'
author:
- 'Ayaka Shimizu [^1]'
- 'Rinno Takahashi [^2]'
title: 'Region crossing change on spatial theta-curves'
---
Introduction
============
A [*knot*]{} is an embedding of a circle in $S^3$, and a [*link*]{} is an embedding of some circles in $S^3$. A [*spatial graph*]{} is an embedding of a graph in $S^3$. A [*diagram*]{} of a knot, link or spatial graph $G$ is a projection of $G$ to $S^2$ with over/under information at each crossing, where each crossing is made of two arcs intersecting transversely. A knot, link or spatial graph $G$ is [*unknotted*]{}, or [*trivial*]{}, if $G$ has a diagram which has no crossings. It is well-known that any diagram of a knot or link can be transformed into a diagram of a trivial knot or link by a finite number of [*crossing changes*]{}, where a crossing change is a local transformation shown in Figure \[cc\][^3].
![Crossing change. []{data-label="cc"}](cc.eps){width="30mm"}
A [*region crossing change*]{} at a region of a diagram of a knot, link or spatial graph is a local transformation which yields crossing changes at all the crossings on the boundary of the region. For knots, the following theorem is shown:
Any diagram of any knot can be transformed into a diagram of the trivial knot by a finite number of region crossing changes. \[thm11\]
For links, the following is shown:
Any diagram of a link $L$ can be transformed into a diagram of a trivial link by a finite number of region crossing changes if and only if $L$ is a proper link. \[cheng-thm\]
For knots and links, the unknottability on region crossing change does not depend on the choice of diagram. On the other hand, for spatial graphs, it depends on the choice of diagram. For the two diagrams representing the same spatial graph in Figure \[diagram-dependent\], the left one cannot be unlinked by region crossing changes at any set of regions, whereas the right one gets unlinked by a region crossing change at the shaded region.\
![Unknottability on region crossing change is diagram-dependent for spatial graphs. []{data-label="diagram-dependent"}](diagram-dependent.eps){width="35mm"}
For a spatial graph with a single component, i.e., an embedding of a connected graph, region crossing change is studied in [@HSS]. In this paper, we study region crossing change on a spatial graph with some components, and show the following theorems.
Let $G$ be a two-component spatial graph consisting of a spatial $\theta$-curve[^4] and a knot. There exists a diagram $D$ of $G$ such that $D$ can be unknotted by a finite number of region crossing changes. \[theta-knot\]
Let $n$ be a positive integer. Let $G$ be an $n$-component spatial graph whose components are all spatial $\theta$-curves. There exists a diagram $D$ of $G$ such that $D$ can be unknotted by a finite number of region crossing changes. \[theta-n\]
The rest of the paper is organized as follows: In Section \[spatial-graphs\], we review the study of region crossing change on spatial graphs. In Section \[proofs\], we prove Theorems \[theta-knot\] and \[theta-n\]. In Section \[s-handcuff\], we also consider spatial handcuff graphs. In Section \[incidence-matrices\], we study incidence matrices for spatial-$\theta$-curve diagrams. In Section \[ineffective-sets\], we consider ineffective sets for spatial-$\theta$-curve diagrams.
Spatial-graph diagrams {#spatial-graphs}
======================
In this section, we prepare some terms of spatial-graph diagrams regarding that knots and links are included by spatial graphs, and review some results on region crossing change.\
A [*graph*]{} is a pair of sets of vertices and edges. Each connected graph $g$ which has at least one vertex has a [*maximal tree*]{}, where a maximal tree is a connected subgraph of $g$ which includes no cycles and includes all the vertices of $g$. Since a spatial graph is an embedding of a graph, every connected spatial graph which has a vertex has a maximal tree.\
It is known that two diagrams represent the same spatial graph if and only if they are equivalent up to the [*Riedemeister moves*]{} shown in Figure \[reidemeister\].
![Reidemeister moves. []{data-label="reidemeister"}](reidemeister.eps){width="90mm"}
A crossing $p$ of a spatial-graph diagram $D$ is [*reducible*]{} if one can draw a circle $C$ on $S^2$ such that $C$ intersects only $p$ transversely as shown in Figure \[reducible\]. A diagram $D$ is said to be [*reducible*]{} if $D$ has a reducible crossing. Otherwise, $D$ is said to be [*reduced*]{}, or [*irreducible*]{}.
![A reducible diagram with a reducible crossing $p$. []{data-label="reducible"}](reducible.eps){width="40mm"}
A [*cutting circle*]{} of a diagram $D$ is a circle on $S^2$ intersecting an edge transversely at exactly one point as shown in Figure \[cutting\]. We call such an edge a [*cutting edge*]{}.
![A diagram with a cutting edge. []{data-label="cutting"}](cutting.eps){width="45mm"}
For spatial graphs of one component, the following is shown:
Let $G$ be a spatial graph of one component, and let $D$ be a diagram of $G$ without cutting edges. Any crossing change on $D$ is realized by a finite number of region crossing changes. \[HSSthm\]
A [*$\theta$-curve*]{} is a connected graph consisting of two vertices $v_1$ and $v_2$ and three edges which are adjacent to both $v_1$ and $v_2$. A [*spatial $\theta$-curve*]{} is an embedding of a $\theta$-curve in $S^3$. Since any diagram of a spatial $\theta$-curve does not have a cutting edge, we have the following:
Any crossing change on any diagram of a spatial $\theta$-curve is realized by a finite number of region crossing changes. \[thHSS\]
Theorem \[HSSthm\] is a generalization of the following:
Any crossing change on any knot diagram is realized by a finite number of region crossing changes. \[realize-k\]
Theorem \[thm11\] is obtained from Lemma \[realize-k\] by repeating such region crossing changes. Similarly, we can unknot any diagram of a spatial $\theta$-curve by a finite number of region crossing changes by Lemma \[thHSS\].
Proofs of Theorems \[theta-knot\] and \[theta-n\] {#proofs}
=================================================
In this section we prove Theorems \[theta-knot\] and \[theta-n\].
[*Proof of Theorem \[theta-knot\]*]{}. For a spatial graph $G$ consisting of a spatial $\theta$-curve and a knot, take a maximal tree $T$ of the $\theta$-curve component. Then $G$ has a diagram $D_0$ such that the corresponding part of $T$ has no crossings (see, for example, [@kawauchi]). We call the vertices $v_1$, $v_2$ and edges $e_1$, $e_2$ and $e_3$ of the $\theta$-curve component as indicated on $D_0$ in Figure \[d0\]. We note that the maximal tree $T$ consists of $v_1$, $v_2$ and $e_3$ without crossings, and that $e_1$ and $e_2$ may have crossings.\
![A diagram $D_0$ of $G$ without crossings on $e_3$. []{data-label="d0"}](d0.eps){width="30mm"}
Shrink $e_1$ so that $v_1$ moves to the right side, near $v_2$, and that $e_2$ and $e_3$ follow $e_1$ making a sufficiently narrow wheel track. We call the result $D_1$. Next, similarly shrink $e_2$ so that $v_2$ moves to the left side, and that $e_1$ and $e_3$ follow $e_2$ making a wheel track, and obtain $D_2$. On $D_2$, the $\theta$-curve component and the knot component make pairwise crossings at the wheel tracks.\
We can change any such crossing pair by region crossing changes as follows: Let $p$ be a crossing pair. Let $t_i$ be the wheel track where $p$ belongs, and let $v_i$ be the adjacent vertex of $t_i$. Apply region crossing changes at the regions in the wheel track $t_i$ in order from $v_i$ to $p$. Then only the crossing pair $p$ changes.\
Apply region crossing changes on $D_2$ so that the $\theta$-curve component is over than the knot component at every crossing between them. An example is shown in Figure \[d012\]. We call the result $D_3$.\
![Take a diagram $D_0$ so that $e_3$ has no crossing. Shrink $e_1$ so that $v_1$ moves to the right side, and obtain $D_1$. Shrink $e_2$ so that $v_2$ moves to the left side, and obtain $D_2$. On $D_2$, we can lift up the $\theta$-curve component over than the knot component by region crossing changes at some regions inside the wheel tracks. []{data-label="d012"}](d012.eps){width="100mm"}
Let $D_3^{\theta}$ be the $\theta$-curve component diagram in $D_3$. Recall that $D_3^{\theta}$ has a set of regions $S^{\theta}$ such that $D_3^{\theta}$ is transformed into a diagram of the trivial $\theta$-curve by region crossing changes at $S^{\theta}$ (Lemma \[thHSS\]). Apply region crossing changes on $D_3$ at the regions $S_3^{\theta}$ of $D_3$ corresponding to the regions in $S^{\theta}$. Then, the $\theta$-curve component gets unknotted, while crossings between the components are unchanged because there are disjoint four regions around each crossing between them, and an even number of them belongs to $S_3^{\theta}$.\
Thus we obtain a diagram $D_4$, representing a splittable spatial graph with the $\theta$-curve component unknotted. Let $D_4^k$ be the knot component diagram in $D_4$. Recall that $D_4^k$ has a set of regions $S^k$ such that $D_4^k$ is transformed into a diagram $O_4^k$ of the trivial knot by region crossing changes at $S^k$ (Lemma \[realize-k\]). Apply region crossing changes to $D_4$ at the regions of $D_4$ corresponding to the regions in $S^k$. Then $D_4^k$ gets unknotted, and crossings between the components are unchanged. Remark that some reducible crossings of $D_4^k$ may be different from $O_4^k$ after the region crossing changes while non-reducible crossings are the same. This does not matter since the unknottedness is unchanged even if a reducible crossing is changed. Similarly, the unknottedness of the $\theta$-curve component is also unchanged. Thus, we obtain a diagram of an unknotted spatial graph by a finite number of region crossing changes from a diagram $D_2$ of $G$. $\Box$
Next, we prove Theorem \[theta-n\] in a similar way.
[*Proof of Theorem \[theta-n\]*]{}. Let $G$ be a spatial graph consisting of $n$ spatial $\theta$-curves, and let $D$ be a diagram of $G$. Take a maximal tree for each $\theta$-curve component. Gather the $n$ maximal trees to the same place in the following way. Choose a region $R$ of $D$. Take a small part of an edge of each maximal tree, and move it into $R$ by Reidemeister moves. Then move the adjacent vertices into $R$ along the edge, by Reidemeister moves. We call the result $D_0$, and name the components ${\theta}^k$, vertices $v^k _i$ and edges $e^k _j$ as shown in Figure \[d0n\].\
![Diagram $D_0$. []{data-label="d0n"}](d0n.eps){width="70mm"}
Move $v^i_1$ to another side near $v^i _2$ by shrinking $e^i _1$ for each $i$ with an order, where $e^i _2$ and $e^i _3$ follow $e^i _1$ making a wheel track, and we obtain a diagram $D_1$. Then move $v^i_2$ to another side by shrinking $e^i _2$ for each $i$, where $e^i _1$ and $^i _3$ follow $e^i _2$ making a wheel track. Thus, we obtain a diagram $D_2$ of $G$, and this is the diagram we required.
We can transform $D_2$ into a diagram such that ${\theta}^i$ is over than ${\theta}^j$ ($i < j$) at each crossing between them by region crossing changes using the wheel-track method of the proof of Theorem \[theta-knot\]. We call such a diagram $D_3$. Each diagram of ${\theta}^i$ has a set $S^i$ of regions such that ${\theta}^i$ gets unknotted by the region crossing changes by Lemma \[thHSS\]. Apply region crossing changes at the corresponding regions in order from $S^1$ to $S^n$. Then we obtain a diagram of the unknotted spatial graph. $\Box$
From Theorems \[cheng-thm\], \[theta-knot\] and \[theta-n\], we have the following corollary:
Let $G$ be a spatial graph consisting of some spatial $\theta$-curves and a proper link. There exists a diagram $D$ of $G$ such that $D$ can be unknotted by a finite number of region crossing changes.
Spatial handcuff graphs {#s-handcuff}
=======================
A [*handcuff graph*]{} is a connected graph consisting of two vertices $v_1$, $v_2$ and two loops based on $v_1$ and $v_2$, and an edge connecting $v_1$ and $v_2$. A [*spatial handcuff graph*]{} is an embedding of a handcuff graph in $S^3$. Similarly to spatial $\theta$-curve, we can unknot a spatial graph consisting of a spatial handcuff graph and a knot by region crossing change by taking a suitable diagram as shown in Figure \[handcuff-unknotting\].
![A spatial-graph diagram which can be unlinked by region crossing changes by moving the diagram. []{data-label="handcuff-unknotting"}](handcuff-unknotting.eps){width="90mm"}
We have the following corollary:
Let $G$ be a two-component spatial graph consisting of a spatial handcuff graph and a knot. There exists a diagram $D$ of $G$ such that $D$ can be unknotted by a finite number of region crossing changes.
Let $e_1$, $e_2$ be the loop edge based on $v_1$, $v_2$, respectively, and let $e_3$ be the non-loop edge of the handcuff component. Then $v_1$, $v_2$ and $e_3$ form a maximal tree of the handcuff component. Take a diagram $D$ of $G$ such that $e_3$ has no crossings. If the handcuff component in $D$ has a cutting edge, apply Reidemeister moves to $D$ so that there are no cutting edges, and call the result $D_0$. Move $v_1$ along $e_1$ until it backs to the initial position, where $e_1$ and $e_3$ makes a wheel track, and we call the result $D_1$. Similarly, move $v_2$ along $e_2$ and obtain a diagram $D_2$, and this is the diagram we required. The rest steps are same to the proof of Theorem \[theta-knot\].
We also have the following corollary:
Let $G$ be a spatial graph consisting of some spatial $\theta$-curves, some spatial handcuff graphs and a proper link. There exists a diagram $D$ of $G$ such that $D$ can be unknotted by a finite number of region crossing changes.
Incidence matrices
==================
In this section, we consider incidence matrices, and show the following:
Let $D$ be a diagram of a spatial $\theta$-curve, and let $D'$ be a diagram obtained from $D$ by crossing changes at some crossings. There exist exactly eight sets of regions of $D$ such that $D$ is transformed into $D'$ by region crossing changes at the regions. \[eight\]
We show an example:
For the diagram $D$ in Figure \[t31\], if one wants to change the crossing $c_1$, one should solve the following simultaneous equations (see [@AS] for knots):
![A diagram of a spatial $\theta$-curve. []{data-label="t31"}](t31.eps){width="30mm"}
$$\begin{aligned}
\begin{cases}
c_1: \ x_1+x_3+x_4+x_6 \equiv 1 \pmod 2 \\
c_2: \ x_2+x_3+x_5+x_6 \equiv 0 \pmod 2 \\
c_3: \ x_3+x_4+x_5+x_6 \equiv 0 \pmod 2 \\
\end{cases}\end{aligned}$$
The first equation implies that one should choose an odd number of regions from $R_1$, $R_3$, $R_4$ and $R_6$ to change $c_1$. The second equation implies that one should choose an even number of regions from $R_2$, $R_3$, $R_5$ and $R_6$ not to change $c_2$. For the simultaneous equations, we have eight solutions, and then we have eight sets of regions $\{ R_1 \}$, $\{ R_5 , R_6 \}$, $\{ R_1 , R_2 , R_4 , R_6 \}$, $\{ R_2 , R_4 , R_5 \}$, $\{ R_1 , R_3 , R_6 \}$, $\{ R_3 , R_5 \}$, $\{ R_1 , R_2 , R_3 , R_4 \}$, $\{ R_2 , R_3 , R_4 , R_5 , R_6 \}$ to change only $c_1$ by region crossing changes. \[ex-t31\]
Let $G$ be a spatial graph, and let $D$ be a diagram of $G$. For crossings $c_1, c_2, \dots$ and regions $R_1, R_2, \dots$ of $D$, the [*region choice matrix $M=(a_{ij})$ of $D$*]{} is a matrix defined by the following: $$\begin{aligned}
a_{ij}=
\begin{cases}
1 & \text{if } R_j \text{ is adjacent to } c_i \\
0 & \text{otherwise}
\end{cases}\end{aligned}$$ We note that the region choice matrix for knots is introduced in [@AS] not only for modulo 2. We also note that the region choice matrix of modulo 2 is the transposed matrix of the [*incidence matrix*]{} introduced in [@CG]. The region choice matrix of the diagram $D$ in Figure \[t31\] is $$\begin{aligned}
\begin{pmatrix}
1 & 0 & 1 & 1 & 0 & 1 \\
0 & 1 & 1 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 & 1 & 1
\end{pmatrix}\end{aligned}$$ and this is equivalent to the coefficient matrix of the simultaneous equation in Example \[ex-t31\].\
For knots, the size of a region choice matrix of a knot diagram with $n$ crossings is $n \times (n+2)$, and then it is shown in [@CG] that the rank is $n$ using Lemma \[realize-k\], and the knot version of Proposition \[eight\] is shown in [@kawauchi-ed] and [@hashizume] as the number of sets is four. We show Proposition \[eight\] in the same way. First, we show the following:
Let $D$ be a diagram of a spatial $\theta$-curve with $n$ crossings. The size of a region choice matrix of $D$ is $n \times (n+3)$. \[c3\]
Since crossings correspond to the rows, and regions correspond to the columns, Lemma \[c3\] follows from the following lemma:
Every diagram of a spatial $\theta$-curve has the number of regions three more than the number of crossings.
Let $D$ be a diagram of a spatial $\theta$-curve with $n$ crossings, and let $|D|$ be a graph obtained from $D$ by regarding each crossing to be a vertex. That is, $|D|$ is a graph on $S^2$ with $v=n+2$ vertices. Looking locally at each vertex of $|D|$ which corresponds to a crossing on $D$, there are four edges around it, and looking at each vertex of $|D|$ which corresponds to a vertex on $D$, there are three edges around it. Hence, the number of total endpoints of edges of $|D|$ is $4n+6$. Since each edge has two endpoints, the number $e$ of edges of $|G|$ is $2n+3$. By substituting to the equation $v-e+f=2$ of Euler’s characteristic of $S^2$, we have the number of the regions, $f=n+3$.
Secondary, we show the following lemma:
Let $D$ be a diagram of a spatial $\theta$-curve with $n$ crossings, and let $M$ be a region choice matrix of $D$. Then, the rank of $M$ is $n$, namely, $M$ is full-rank. \[fullrank\]
Let $c_1, c_2, \dots$ and $c_n$ be the crossings of $D$. By Lemma \[thHSS\], we can change only $c_i$ by region crossing changes at some regions. In terms of matrices, we can create the column vector such that the $i$th element is $1$ and the others are $0$ by a linear combination of some columns of $M$, for any $i \in \{ 1, 2, \dots ,n \}$. This means the rank of $M$ is $n$.
Then we prove Proposition \[eight\].
[*Proof of Proposition \[eight\].*]{} Consider a simultaneous equations whose coefficient matrix is $M$. Since the degree of freedom of the solution is obtained by subtracting the rank of $M$ from the number of columns of $M$, in this case the degree is $(n+3)-n=3$ by Lemmas \[c3\] and \[fullrank\]. Since we work on modulo 2, the number of the solutions is $2^3=8$. $\Box$
We remark that Proposition \[eight\] does not hold for spatial-handcuff-graph diagrams. Some crossing changes on a diagram with a cutting edge are not realized by region crossing changes, as shown in Figure 4 in [@HSS].
Ineffective sets
================
In this section we consider the diagramatical implications of Proposition \[eight\]. A set $S$ of regions of a diagram is said to be [*ineffective*]{} when the region crossing changes at all the regions in $S$ do not change the diagram [@IS]. The following is shown for knots:
Let $D$ be a reduced knot diagram with a checkerboard coloring. Then the set $B$ of the black-colored regions of $D$ is ineffective.
For reducible diagrams, we may need a modification to $B$ at some reducible crossings to get ineffective (see Figure 11 in [@shimizu-rcc]). For spatial $\theta$-curves, we have the following:
Let $G$ be a spatial $\theta$-curve consisting of vertices $v_1, v_2$ and edges $e_1, e_2$ and $e_3$. Let $D$ be a reduced diagram of $G$, and let $k^i$ be the knot diagram obtained by removing $e_i$ $(i \in \{ 1,2,3 \} )$. The set of regions of $D$ which are black-colored on a checkerboard coloring on $k^i$ is ineffective. \[k-ine\]
If we choose such black regions, diagonal two regions are chosen around each crossing of $k^i$, four or no regions are chosen at each crossing of $e_i$ because $e_i$ is ignored on $k^i$, and adjoining two regions are chosen around each crossing between $k^i$ and $e_i$ for the same reason. Hence, all the crossings are unchanged by the region crossing changes.
For reducible diagrams, we may need a modification at some reducible crossings (see Figure 12 in [@HSS]). Let $D$ be a reduced diagram of a spatial $\theta$-curve on $\mathbb{R}^2$. Let $k^i$ be the knot diagram as mentioned in Corollary \[k-ine\]. Give checkerboard coloring to each $k^i$ so that the outer region is colored white. We call the set of regions of $D$ which are black-colored (resp. white-colored) $B^i$ (resp. $W^i$) on the checkerboard coloring to $k^i$. We have the following:
The equality $ B^l = \left( B^m \cup B^n \right) \setminus \left( B^m \cap B^n \right)$ holds, where $(l, m, n)$ is a permutation of $(1, 2, 3)$. \[lem-set\]
For each region of $k^l$ with the above checkerboard coloring, give the value $1$ (resp. $0$) if the region is colored black (resp. white), for each $l \in \{ 1,2,3 \}$. And then, for each region $r_i$ of $D$, give the value $f(r_i)$ which is the sum of the values of the corresponding regions for $k^1$, $k^2$ and $k^3$. An example is shown in Figure \[ex02\].\
![Give the checkerboard coloring to each $k^l$ so that the outer region is white, and give the values $1$ and $0$. The labeling $f$ to each region of $D$ is obtained by summing the three values. []{data-label="ex02"}](ex02.eps){width="100mm"}
Now we show that each region $r_i$ of $D$ has $f(r_i)=0$ or $2$. Let $r_i$ and $r_j$ be regions sharing an edge $e^l$. Then $r_i$ and $r_j$ take different values on both $k^m$ and $k^n$ because $e^l$ exists on $k^m$ and $k^n$. And they take the same value on $k^l$ because $r_i$ and $r_j$ belongs to the same region on $k^l$. Hence, for each pair of regions $r_i$ and $r_j$ sharing an edge, the difference between $f(r_i)$ and $f(r_j)$ is $0$ or $2$. Since the outer region $r_0$ has $f(r_0)=0$, and the value of $f$ can be at most $3$, every region $r_i$ has $f(r_i)=0$ or $2$, with the breakdown $0+0+0$ or $0+1+1$. Therefore, $B^l$ is obtained by $ \left( B^m \cup B^n \right) \setminus \left( B^m \cap B^n \right)$.
Let $S$ be a set of regions of a reduced diagram $D$. Let $I$ be an ineffective set for $D$. We can obtain the same result of region crossing changes at the regions in $S$ by retaking the regions to $\left( S \cup I \right) \setminus \left( S \cap I \right)$ (see [@shimizu-rcc] for knots). For a reduced diagram of a spatial $\theta$-curve, by taking $B^1$, $W^1$ and $B^2$ as $I$ and the above retaking, we can obtain eight sets of regions whose effects by region crossing changes are the same. We remark that $B^1$, $W^1$ and $B^2$ are independent; Looking around a vertex, we can see that neither $B^1$, $W^1$ nor $B^2$ can be obtained by a combination of the others. See Figure \[aroundV\].
![Neither $B^1$, $W^1$ nor $B^2$ can be obtained by a combination of the others. []{data-label="aroundV"}](aroundV.eps){width="60mm"}
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank Kota Koashi, Atsushi Oya, Hiroaki Saito, Shunta Saito and Mao Totsuka for valuable discussions in the seminars in Gunma College. They are also very grateful to Yoshiro Yaguchi for valuable discussions and helpful comments.
[99]{} K. Ahara and M. Suzuki, An integral region choice problem on knot projection, J. Knot Theory Ramifications [**21**]{} (2012), 1250119 \[20 pages\]. Z. Cheng, When is region crossing change an unknotting operation?, Mathematical Proceedings of the Cambridge Philosophical Society, [**155**]{} (2013), 257–269. Z. Cheng and H. Gao, On region crossing change and incidence matrix, Science China Mathematics [**55**]{} (2012), 1487–1495. M. Hashizume, On the image and the cokernel of homomorphism induced by region crossing change, JP J. Geom. Topol. [**18**]{} (2015), 133–162. K. Hayano, A. Shimizu and R. Shinjo, Region crossing change on spatial-graph diagrams, J. Knot Theory Ramifications [**24**]{} (2015), 1550045 \[12 pages\]. A. Inoue and R. Shimizu, A subspecies of region crossing change, region freeze crossing change, J. Knot Theory Ramifications [**25**]{} (2016), 1650075. A. Kawauchi, On a trial of early childhood education of mathematics by a knot (in Japanese), in: Chapter one of: Introduction to Mathematical Education on Knots for primary school children, junior high students, and the high school students, No.4 (ed. A. Kawauchi and T. Yanagimoto), 2014. A. Kawauchi, A. Shimizu and Y. Yaguchi, Cross-index of a graph, to appear in Kyungpook Math. J. A. Shimizu, Region crossing change is an unknotting operation, J. Math. Soc. Japan [**66**]{} (2014), 693–708.
[^1]: Department of Mathematics, National Institute of Technology, Gunma College, 580 Toriba-cho, Maebashi-shi, Gunma, 371-8530, Japan. Email: shimizu@nat.gunma-ct.ac.jp
[^2]: Department of Information and Computer Engineering, National Institute of Technology, Gunma College, 580 Toriba-cho, Maebashi-shi, Gunma, 371-8530, Japan.
[^3]: Some spatial graphs cannot be unknotted by crossing changes, such as the complete graph $K_5$.
[^4]: Spatial $\theta$-curve is explained in Section \[spatial-graphs\].
|
---
abstract: |
We compute the quantum string entropy $S_s(m, H)$ from the microscopic string density of states $\rho_s (m,H)$ of mass $m$ in de Sitter space-time. We find for high $m$, (high $Hm \rightarrow c/\alpha' $), a [**new**]{} phase transition at the critical string temperature $ T_{s}= (1/2\pi k_B)L_{c\ell}~c^2/\alpha'$, [**higher**]{} than the flat space (Hagedorn) temperature $t_{s}$. ($L_{c\ell}= c/H$, the Hubble constant $H$ acts at the transition as producing a smaller string constant $\alpha'$ and thus, a higher tension). $ T_s$ is the precise quantum dual of the semiclassical (QFT Hawking-Gibbons) de Sitter temperature $T_{sem}=\hbar c /(2\pi k_B L_{c\ell})$. By precisely identifying the semiclassical and quantum (string) de Sitter regimes, we find a [**new**]{} formula for the full de Sitter entropy $S_{sem} (H)$, as a function of the usual Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$. For $L_{c\ell}\gg \ell_{Planck}$ , ie. for low $H\ll c/\ell_{Planck}$, $S_{sem}^{(0)}(H)$ is the leading term, [**but**]{} for high $H$ near $c/\ell_{Planck}$, a [**new**]{} phase transition operates and the whole entropy $S_{sem} (H)$ is drastically [**different**]{} from the Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$.
We compute the string quantum emission cross section $\sigma_{string}$ by a black hole in de Sitter (or asymptotically de Sitter) space-time (bhdS). For $T_{sem~bhdS}\ll T_{s}$, (early evaporation stage), it shows the QFT Hawking emission with temperature $T_{sem~bhdS}$, (semiclassical regime). For $T_{sem~bhdS}\rightarrow T_{s}$, $\sigma_{string}$ exhibits a phase transition into a string de Sitter state of size $L_s = \ell_s^2/L_{c\ell}$, ($\ell_s= \sqrt{\hbar \alpha'/c}$), and string de Sitter temperature $T_s$. Instead of featuring a single pole singularity in the temperature (Carlitz transition), it features a square root [*branch point*]{} (de Vega-Sanchez transition). [**New**]{} bounds on the black hole radius $r_g$ emerge in the bhdS string regime: it can become $r_g = L_s/2$, or it can reach a more quantum value, $r_g = 0.365 ~\ell_s$.
author:
- 'A. Bouchareb$^{1,2}$, M. Ramón Medrano$^{3,2}$ and N.G. Sánchez$^{2}$'
title: 'Semiclassical (Quantum Field Theory) and Quantum (String) de Sitter Regimes: New Results'
---
Introduction and Results\[sec:intro\]
=====================================
The understanding of semiclassical and quantum gravity de Sitter regimes is particularly important for several reasons:\
(i) the physical (cosmological) existence of de Sitter (or quasi de Sitter) stages, describing inflation at an early stage of the universe (semiclassical or quantum field theory (QFT) regime), and acceleration at the present time (classical regime).\
(ii) the need of describing de Sitter (or quasi de Sitter) quantum regimes. Besides their conceptual interest, these regimes should be relevant in the stage preceeding semiclassical inflation, their asymptotic behaviour should provide consistent initial states for semiclassical inflation, and clarify, for instance, the issue of the dependence of the observable primordial cosmic microwave fluctuations on the initial states of inflation.\
(iii) the lack till now of a full conformal invariant description of de Sitter background in string theory [@14]. This should not be considered as an handicap for de Sitter space-time, but as a motivation for going beyond the current scarce physical understanding of string theory. The flow of consistent cosmological data (cosmic microwave background, large scale structure, and supernovae observations) place de Sitter (and quasi-de Sitter) regimes as a real part of the standard (concordance) cosmological model, [@17]-[@20].\
(iv) the results of classical and quantum string dynamics in de Sitter space-time. Besides its interesting features, solving the classical and quantum string dynamics in conformal and non conformal invariant string backgrounds it shows that the [**physics**]{} is the same in the two class of backgrounds: conformal and non conformal. The mathematics is simpler in conformal invariant backgrounds, the main physics, in particular the string mass spectrum, remains the same, [@11]-[@10],[@14].
In this paper, we describe classical, semiclassical and quantum de Sitter regimes. A clear picture for de Sitter background is emerging, going [**beyond**]{} the current picture, both for its semiclassical and quantum regimes.
A central object is $\rho_s (m,H)$, the microscopic string density of states of mass $m$ in de Sitter background. $\rho_s (m,H)$ is derived from the string density of levels $d(n)$ of level $n$ and from the string mass spectrum $m (n, H)$ in de Sitter background. The mass formula $m (n, H)$ is obtained by solving the quantum string dynamics in de Sitter background [@11]-[@1]. The density of levels $d(n)$ is the same in flat and in curved space-times. As a result, the mass formula $m (n, H)$ and $\rho_s (m, H)$ are [*different*]{} from the respective flat space-time string mass spectrum and flat space mass level density. The formulae $m(n, H)$ and $\rho_s (m, H)$ depend on the two characteristic lengths in the problem: $L_{c\ell} = c/H$ , the de Sitter radius and $\ell_s = \sqrt{\hbar \alpha'/c}$, the fundamental string length, or equivalently on the respective mass scales: $M_{c\ell} = c^2 L_{c\ell}/G$ and $ m_s =\ell_s / \alpha'$; relevant combinations of them emerge in the mass formula $m(n,H)$ as the de Sitter string length $L_s =\hbar/c M_s$ or string de Sitter mass $M_s = L_{c\ell}/\alpha'$. The temperature $T_s = \frac{1}{2\pi k_B}~M_s~c^2 $ emerges as a true critical string temperature for strings in de Sitter background, and moreover as the intrinsic de Sitter temperature in the quantum (string) de Sitter regime, that is in the high $H$ or high curvature de Sitter regime.
From $\rho_s (m, H)$, we get the string entropy $S_s^{(0)}(m, H)$. We find that a phase transition takes place at $m = M_s$, ie $T = T_s$. This is a [**gravitational**]{} like phase transition: the square root [*branch point*]{} behaviour near the transition is analogous to the thermal self-gravitating gas phase transition of point particles [@13]. This is also the same behaviour as for strings in flat space-time but with the spin modes $j$ included in the high $j$ regime, ($j\rightarrow m^2\alpha' c$), ([*extreme*]{} transition) [@5]. As pointed out in [@4], this string behaviour is [*universal*]{}, it holds in any number of dimensions, and is similar to the Jeans’s instability at finite temperature but with a more complex structure.\
The transition occurs at the string de Sitter temperature $ T_{s}= \Big(L_s/\ell_s\Big) t_s $ [*higher*]{} than the flat space (Hagedorn) value $t_s $. This is so, since for high masses, the true critical string temperature in de Sitter background is $T_s$, instead of $t_s$. The flat space (Hagedorn) temperature $t_s$ is the scale temperature in the low $Hm$ regime. $H$, which acts in the sense of the string tension, does appear in the transition temperature as an “effective string tension” $(\alpha^{'}_H) ^{-1} $ : a smaller $\alpha^{'}_H = (\hbar/c)\Big(~H \alpha'/c\Big)^2$, (and thus a [*higher tension*]{}).\
$T_{s}$ is the precise quantum dual of the semiclassical (QFT Hawking-Gibbons) de Sitter temperature $T_{sem}=\hbar c /(2\pi k_B L_{c\ell})$, the two temperatures satisfy: $T_{s} = t_{s}^2~ T_{sem}^{-1}$.
When $m \rightarrow M_s$ the string becomes “[*classical*]{} ” reflecting the classical properties of the background, $m$ becomes $M_{cl}$, (with $\alpha '$ instead of $G/c^2$), thus the string becomes the “ background ”[@4]. Conversely, and interestingly enough, string back reaction supports this fact: $M_s$ is the mass of de Sitter background in its string regime, a de Sitter phase with mass $M_s$ and temperature $T_s$ is sustained by strings [@1]. ($L_s$, $M_s$, $T_s$) are the [*intrinsic*]{} size, mass and temperature [*of*]{} de Sitter background in its string (high curvature) regime.
By precisely identifying the semiclassical and quantum (string) de Sitter regimes, we find a [**new**]{} formula for the de Sitter entropy $S_{sem} (H)$, which is a function of the usual Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$. For $M_{cl}\gg m_{Pl}$, that is $L_{c\ell} \gg \ell_{Pl}$, i.e. low Hubble constant $H\ll c/\ell_{Pl}$, (semiclassical regime), $S_{sem}^{(0)}(H)$ is the leading term of this expression, [**but**]{} for $M \rightarrow m_{Pl}$, that is $L_{c\ell}\rightarrow \ell_{Pl}$, i.e high Hubble constant $H \rightarrow c/\ell_{Pl}$ (quantum gravity regime), $S_{sem}^{(0)}(H)$ is sub-dominant, a gravitational phase transition operates and the whole de Sitter entropy $S_{sem}(H)$ is drastically [**different**]{} from the semiclassical (Bekenstein-Gibbons-Hawking) de Sitter entropy $S_{sem}^{(0)}(H)$. This [**new phase transition**]{} takes place at the Planck mass $m_{Pl}$, or equivalently at the Planck temperature $t_{Pl}$. Its signature is a [*branch point square root*]{} behaviour at the transition temperature in all space-time dimensions.
We also consider the string regimes of a black hole in a de Sitter (or asymptotically) de Sitter background (bhdS). We compute the quantum string emission cross section $\sigma_{string}$ by a Schwarschild black hole in de Sitter background. For $T_{sem~bhdS}\ll T_{s}$, (early evaporation stage), $\sigma_{string}$ shows the Hawking emission with temperature $T_{sem~bhdS}$, (semiclassical regime). For $T_{sem~bhdS}\rightarrow T_{s}$, $\sigma_{string}$ exhibits a phase transition at $T_{sem~bhdS} = T_{s}$: the massive emission condensates into a string de Sitter state of string de Sitter size $L_s$ and string de Sitter temperature $T_s$. Again, this is not like the flat (or asymptotically flat) space string phase transition (of Carlitz type [@12], [@2]), but this is a de Sitter string transition: instead of featuring a single pole singularity at $T_s$, $\sigma_{string}$ features a square root branch point at $T_s$ in any D dimensions, similar to the one exhibited by $\rho_s(m, H)$ and by the partition function of the string gaz in de Sitter space-time . This is the same behavior exhibited by the thermal selfgravitating gas of point particles (de Vega-Sanchez transition, [@13])
[**New**]{} string bounds on the black hole emerge in the bhdS string regime, ie. when $T_{sem~ bhdS} = T_s$. The bhdS space-time allows an intermediate string regime, not present in the Schwarzschild black hole alone (H=0): in the asymtotically flat black hole, the black hole radius becomes $\ell_s$ in the string regime. In the asymptotically de Sitter black hole, the black hole radius $r_g$ becomes the de Sitter string size $L_s$, ($r_g = L_s/2$). If the de Sitter radius $L_{c\ell}$ reaches $L_s$, (which implies $L_s = \ell_s$ as well), then $r_g$ becomes determined by $\ell_s$, ($r_g = 0.365 ~\ell_s$).
This work does not make use of AdS (Anti-de Sitter space), neither of CPP’s (conjectures, proposals, principles, assumed in string theory in the last years).
Semiclassical de Sitter background\[sec:dS\]
============================================
The D-dimentional de Sitter metric can be expressed in terms of the so called static coordinates as $$ds^{2}=-A(r) c^2dt^{2} + A^{-1}(r) dr^2 + r^2 d\Omega_{D-2}
\label{eq:ds}$$ where $$A(r) = 1 - \Big( \frac{r}{L_{c\ell}}\Big)^2 ~~,~~~~ L_{c\ell}= c H^{-1}
\label{eq:Lcl}$$ the horizon being located at $$r = L_{c\ell}
\label{eq:h}$$ $H$ is the Hubble constant and $L_{c\ell}$ the classical de Sitter length.
In the context of Quantum Field Theory (Q.F.T) in curved space time, de Sitter background has a semiclassical (Hawking-Gibbons) temperature [@7] given by $$T_{sem}= \frac{\hbar}{2\pi k_B}~H = \frac{\hbar c}{2\pi k_B} ~\frac{1}{L_{c\ell}}
\label{eq:Tsem}$$ Eq. (\[eq:Tsem\]) holds in any number of space time dimensions $D$. The Hubble constant $H$ and scalar curvature $R$, or cosmological constant $\Lambda$ ($\Lambda > 0$), are related by $$R = D(D-1) \frac {H^2}{c} ~~,~~ H = c ~\sqrt{\frac{2 \Lambda}{(D-1)(D-2)} }
\label{eq:H}$$ de Sitter space time can be viewed as generated by a constant equation of state (with positive energy density $\epsilon$ and negative pressure density $p$), satisfying $$\epsilon + p =0
\label{eq:p}$$ In $D=4$, the following relations hold $$H = \frac{2}{c} ~\sqrt{\frac{2 \pi G \epsilon}{3}} ~~;~~~~\Lambda = \frac{8 \pi G}{c^4}\epsilon
\label{eq:Hla}$$
being $G$ the Newton gravitational constant.
QFT Semiclassical de Sitter Entropy\[sec:Ssem\]
================================================
In the semiclassical or Q.F.T regime, the relation between the semiclassical entropy $S_{sem}(H)$ and the density of states $\rho_{sem}(H)$ of de Sitter background is given by $$\rho_{sem}(H) = e^{\frac{S_{sem}(H)}{k_B}}
\label{eq:ro}$$
The zeroth order semiclassical de Sitter entropy is given by $$S_{sem}^{(0)}(H) = \pi k_B ~ \Bigg( \frac{L_{c\ell}}{\ell_{pl}}\Bigg)^2 =\pi k_B ~ \Bigg(\frac{M_{cl}}{m_{pl}}\Bigg)^2
\label{eq:SH0}$$ where $$\ell_{pl} = \sqrt{\frac{\hbar ~ G}{c^3}}~~;~~~~m_{pl}=\sqrt{\frac{\hbar ~ c}{G}}
\label{eq:lmPl}$$ $\ell_{pl}$ and $m_{pl}$ being the Planck length and Planck mass respectively, and $M_{cl}$ is the (classical) mass scale of de Sitter background $$M_{cl} = \frac{c^2}{G}~L_{c\ell}=\frac{c^3}{G~H}~~~~~~~~(D=4)
\label{eq:Mcl}$$
Eq. (\[eq:SH0\]) is better expressed as $$S_{sem}^{(0)}(H) = \frac{1}{2} ~\frac{M_{cl}~ c^2}{T_{sem}}
\label{eq:S0}$$ where $T_{sem}$ is the semiclassical (Gibbons-Hawking) de Sitter temperature defined by Eq. (\[eq:Tsem\]).
In terms of the classical and the semiclassical masses, $T_{sem}$ is expressed as $$T_{sem} = \frac{c^2}{2\pi k_B} ~\frac{m_{pl}^2}{M_{c\ell}}=\frac{1}{2\pi k_B}~M_{sem}~c^2
\label{eq:TM}$$ where $$M_{sem}=\frac{m_{pl}^2}{M_{c\ell}}
\label{eq:Msem}$$
Eq. (\[eq:S0\]) for the (zero order) gravitational entropy is the [*ordinary*]{} entropy expression for [*any*]{} ordinary system, where $T_{sem}$ is the Hawking temperature. The Hawking temperature $T_{sem}$ is [*just*]{} the Compton length of Sitter space in the units of temperature, that is, the temperature scale of the semiclassical gravity properties for which the [*mass scale*]{} is precisely $M_{sem}$ Eq. (\[eq:Msem\]). This semiclassical or intermediate energy regime interpolates between the classical and the quantum regimes of gravity. We discuss more on these regimes in Section (\[sec:dual\]).\
As we will see in Section (\[sec:dual\]), the Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$ Eq.(\[eq:S0\]), is just one term in a more general expression of the semiclassical entropy $S_{sem}(H)$ which is a function of $S_{sem}^{(0)}(H)$. For low H, that is high mass, $M_{cl} \gg m_{Pl}$, or large de Sitter radius $L_{c\ell} >> \ell_{Pl} $, (semiclassical de Sitter regime), $S_{sem}^{(0)}(H)$ is the leading term. But for high $H$, (that is, $L_{c\ell}$ near $\ell_{Pl} $), a gravitational phase transition operates and the whole de Sitter entropy $S_{sem}(H)$ is very different from $S_{sem}^{(0)}(H)$. The whole de Sitter entropy $S_{sem}(H)$ Eq. (\[eq:ro\]), as a function of the de Sitter Bekenstein-Hawking entropy $S_{sem}^{0}(H)$, will be discussed in Section (\[sec:dual\]).
Quantum String Entropy in de Sitter Background \[sec:qs\]
=========================================================
The entropy of quantum strings in de Sitter background is defined by $$\rho_{s}(m, H) = e^{\frac{S_{s}(m,H)}{k_B}}
\label{eq:rhos}$$ where $\rho_{s}(m, H)$ is the string density of mass levels in de Sitter space time.
In order to derive $\rho_{s}(m, H)$, let us notice that the degeneracy $d_n(n)$ of level $n$ (counting of oscillator states) is the same in flat and in curved space time. The differences, due to the space-time curvature, enter through the relation $m=m (n)$ of the mass spectrum. As is known, asymptotically for high $n$, the degeneracy $d_n(n)$ behaves universally as
$$d_n(n) = n^{-a'}~ e^{b~ \sqrt n}
\label{eq:d}$$
where the constants $a'$ and $b$ depend on the space time dimensions and on the type of the strings; for bosonic strings: $$b=2\pi\sqrt{\frac{D-2}{6}}~,~~~~~a'=\frac{D+1}{4}~~~\text{(open)}~;~~~~~~a'=\frac{D+1}{2}~~~\text{(closed)}
\label{eq:ba}$$
For large $n$, the mass formula for quantum strings in de Sitter background is given by [@8]-[@10], $$\Bigg( \frac{m}{m_s} \Bigg)^2 \simeq 4 ~n \Bigg[1- n \Bigg(\frac{m_s}{M_s} \Bigg)^2 \Bigg] ~~\text{(closed)}
\label{eq:mmsc}$$
$$\Bigg( \frac{m}{m_s} \Bigg)^2 \simeq n \Bigg[1- n \Bigg( \frac{m_s}{M_s} \Bigg)^2 \Bigg] ~~\text{(open)}
\label{eq:mmso}$$
For $H=0$, we recover from the above expressions the flat space-time mass string spectrum.
In Eq.(\[eq:mmsc\]) and Eq.(\[eq:mmso\]), $m_s$ is the fundamental string mass and $M_s$ the characteristic string mass in de Sitter space time [@8]-[@10],[@1],[@4] $$m_s = \sqrt{\frac{\hbar}{\alpha'c}}\equiv~\frac{\ell_s}{\alpha'}~;~~~~M_s = \frac{L_{c\ell}}{\alpha'} = \frac{c}{H ~ \alpha'};~~~~\Bigg(\frac{m_s}{M_s} \Bigg) = \frac{\ell_{s}}{L_{c\ell}}= \frac{H}{c}\ell_{s}
\label{eq:mMs}$$
$\alpha'$ is the fundamental string constant ($\alpha'^{-1}$ is a mass linear density), $\ell_{s}$ is the fundamental string length and $L_{c\ell}$ is given by Eq. (\[eq:Lcl\]). Furthermore, $M_s$ defines the quantum string de Sitter length $L_s$: $$L_s = \frac{\hbar }{M_s ~c}= \frac{\ell_s^2}{L_{c\ell}}=\frac{\hbar\alpha'}{c^2}H~~,
\label{eq:Ls}$$ and the string de Sitter temperature $T_s$: $$T_s = \frac{1}{2\pi k_B} ~ M_s~c^2 = \frac{\hbar c}{2 \pi k_B} ~\frac{1}{L_s}= \frac{1}{2 \pi k_B} ~\frac{c^3}{H \alpha'}
\label{eq:Ts}$$
$T_s$ is the critical string temperature in de Sitter space, as it is shown in Sec.\[sec:partition\] and Sec.\[sec:eBHS\] below.
The density $\rho_s(m, H)$ of mass levels and the level degeneracy $d_n(n)$ satisfy $$\rho_s(m, H) ~d \Bigg(\frac{m}{m_s}\Bigg)= d_n(n) ~dn
\label{eq:rhod}$$ ie. $$\rho_s(m, H) \simeq \frac{m}{m_s} \Bigg[ ~\frac{ d_n(n)}{g^{'}(n)} ~\Bigg]_{n=n(m)}
\label{eq:rhof}$$ where $\big( \frac{m}{m_s}\big)^2 \simeq g(n)$, $g(n)$ being read from the r.h.s of Eqs. (\[eq:mmsc\]) and (\[eq:mmso\]).
From Eqs. (\[eq:mmsc\]) - (\[eq:rhof\]), we derive the string mass density of levels in de Sitter background (closed strings):
$$\rho_s(m, H) \simeq \frac{(m/m_s)}{ \sqrt{1-\Big(\frac{m}{M_s} \Big)^2}}
\Bigg[ \Bigg(\frac{M_s}{m_s} \Bigg)^2 ~\frac{1}{2} \Big(1- \sqrt{1- \Big(\frac{m}{M_s}\Big)^2}~ \Big) \Bigg]^{-a'} \times
\label{eq:rhoMsc}$$
$$\exp\Bigg\{ b \Bigg( \frac{M_s}{m_s} \Bigg) \Bigg[ \frac{1}{2} \Big(1- \sqrt{1- \Big(\frac{m}{M_s}\Big)^2} \Big)
\Bigg]^{1/2} \Bigg\}$$
A similar formula holds for open strings. $m_s$ and $M_s$ are given by Eq. (\[eq:mMs\]). $\rho_s$ depends on $H$ through $M_s$ Eq. (\[eq:mMs\]). $M_{s}$, $(M_{s}/2)$, is the upper mass bound for closed, (open), strings.
For $H \rightarrow 0$ we recover the flat space-time solution
$$\rho_s(m) \simeq \Big( \frac{m}{m_s} \Big)^{-a} ~e^{\frac{b}{2} \big( \frac{m}{m_s} \big)} ~~~~, ~~~~a \equiv 2a'-~1
\label{eq:rhofc}$$
$\rho_s(m,H)$ Eq. (\[eq:rhoMsc\]) can be expressed in a more compact way as:
$$\rho_s (m, H)\simeq \Bigg (\frac{m}{\Delta_s M_s}\sqrt{\frac {2}{1 - \Delta_s}}\Bigg)~
\Bigg(\frac{M_s}{m_s}\sqrt{\frac{1 - \Delta_s}{2}}\Bigg)^{-a}~ e^{\Big( \frac{bM_s}{m_s}\sqrt{\frac{1 - \Delta_s}{2}}\Big)}
\label{eq:rhodm}$$
where $$\Delta_s \equiv\sqrt{1 -\Big(\frac{m}{M_s}\Big)^2}
\label{eq:deltasm}$$
Several expressions for the exact $\rho_s(m,H)$ are useful depending on the different behaviours we would like to highligth: the flat H=0 limit, the low mass, or the high mass behavior. Let us introduce the (zero order) string entropy in flat space time : $$S_s^{(0)}(m) = \frac{1}{2}~b k_B~ \Big(\frac{m}{m_s}\Big) =\frac{1}{2}~\frac{m~c^2}{t_s}
\label{eq:Ss0c}$$
where $t_s$ is the fundamental string temperature in flat space-time $$t_s =\frac{1}{bk_B}~ m_s~ c^2 = \frac{1}{bk_B}~\frac{\hbar~c}{\ell_{s}}
\label{eq:ts}$$
Therefore, from Eqs. (\[eq:rhoMsc\]) and (\[eq:Ss0c\]), the mass density of levels $\rho_s(m, H)$ for both open and closed strings can be expressed as :
$$\rho_s (m, H) =\Big( \frac{S_s^{(0)}}{k_B} \sqrt{f(x)} \Big)^{-a}~ e^{\Big(\frac{S_s^{(0)}}{k_B}\sqrt{f(x)}\Big)} ~~
\frac{1}{\sqrt{(1 - 4x^2)f(x)}}
\label{eq:rhoF}$$
$$a=\frac{(D-1)}{2} ~~\text{(open)},~~ a=D ~~\text{(closed)}, ~~~~~~f(x) = \frac{1 - \sqrt{1 - 4 x^2}}{2x^2}
\label{eq: FX}$$
$x$ being the dimensionless variable $$x(m, H)\equiv = \frac{1}{2}\Big(\frac{m}{M_s}\Big)=
\frac{m_s}{b M_s}\frac{S_s^{(0)}}{k_B}
\label{eq:X}$$
$S_s^{(0)}$ is given by Eq.(\[eq:Ss0c\]). In terms of $\Delta_s$ Eq.(\[eq:deltasm\]), we have:
$$\rho_s (m, H)\simeq \frac{1}{\Delta_s}\sqrt{\frac{1 + \Delta_s}{2}}~
\Bigg(\frac{S_s^{(0)}}{k_B}\sqrt{\frac{2}{1 + \Delta_s}}\Bigg)^{-a}~ e^{\Big( \frac{S_s^{(0)}}{k_B}\sqrt{\frac{2}{1 + \Delta_s}}\Big)}
\label{eq:rhodso}$$
$$\Delta_s \equiv\sqrt{1-4x^2}~~ ~~,~~~~~ f(x)= \frac{2}{1+\Delta_s}
\label{eq:deltas}$$
For small x, (small $H m\alpha '/c$), $f(x)$ can be naturally expressed as a power expansion in $x$. In particular, for $H=0$, we have $x=0$ and $f(x)=1$, and we recover the flat space time string solution: $$\rho_s(m) \simeq \Big( \frac{S_s^{(0)}}{k_B} \Big)^{-a} ~e^{\big( \frac{S_s^{(0)}}{k_B} \big)}
\label{eq:rhoS0f}$$ For $ x \ll 1 $, i.e $ m \ll M_{s} $, the corrections to the flat $(H=0)$ solution are given by: $$\rho _s(m, H))_{m\ll M_s} \sim \left( \frac{m}
{m_s}\right) ^{-a} ~e^{\frac{b}{2}\Big(\frac{m}{m_s}\Big)\Big[~1-~\frac{1}{8}( \frac{\alpha'H m}{c})^2~ + ~O\left(\frac{m}{M_s}\right)^3~\Big]}
\label{eq:rlH}$$
From Eqs. (\[eq:rhos\]), (\[eq:rhoF\]) we can read the full string entropy in de Sitter space :
$$S_s(m,H) = \hat {S_s}^{(0)}(m,H)
-a~k_B~\ln ~\big(\frac{\hat {S_s}^{(0)}(m, H)}{k_B}\big) - k_B ~\ln F(m,H)
\label{eq:SsHF}$$
$$\hat {S_s}^{(0)}(m,H)\equiv S_s^{(0)}\sqrt{f(x)}~~~~~,~~~~ F\equiv \sqrt{(1 - 4x^2)f(x)}
\label{eq:SsdsHF}$$
i.e : $$S_s(m,H) = \sqrt{f(x)} ~ S_s^{(0)}
-a~k_B~\ln \Big(\frac{\sqrt{f(x)}~S_s^{(0)}}{k_B}\Big) - k_B \ln \sqrt{f(x)}- k_B \ln~\sqrt{~1 - 4x^2~}
\label{eq:SsH}$$
The mass domain is $ 0 \leq m \leq M_{s}$, ie. $ 0 \leq x \leq 1/2 $, (which implies $ 0\leq \Delta_s \leq 1$, ie. $1 \leq f(x)\leq 2 $).\
All terms in the entropy except the first one have negative sign, (Eq.(\[eq:SsHF\]) or Eq.(\[eq:SsH\])). For $\Delta_s \neq 0$, (ie. $ m \neq M_{s}$), the entropy $S_s(m,H)$ of string states in de Sitter space is smaller than the string entropy for $H=0$. The effect of the Hubble constant is to reduce the entropy. For low masses $m \ll M_{s}$, the entropy is a series expansion in $(H m\alpha' /c \ll 1)$, like a low H expansion around the flat $H=0$ solution, $S_s^{(0)}$ being its leading term. But for high masses $m \rightarrow M_s$, that is $(H m \alpha^{'}/c) \rightarrow 1$, (i.e. $\Delta_s \rightarrow 0$), the situation is [*very different*]{} as we see it below.\
Moreover, Eq. (\[eq:SsHF\]) for $S_s(m,H)$ allows us to write in Section (\[sec:dual\]) the whole expression for the semiclassical de Sitter entropy $S_{sem}(H)$, as a function of the (Bekenstein-Hawking) de Sitter entropy $S_{sem}^{(0)}(H)$.
The String de Sitter Phase Transition\[sec:sdsps\]
==================================================
For $m \sim M_s$, the string mass density of levels Eq. (\[eq:rhoMsc\]) is $$\rho_s(m, H)_{m \sim M_s} \simeq \Big(\frac{M_s}{m_s}\Big)^{-a}\sqrt{\frac{M_s}{2(M_s-m)}} ~
\Bigg[ \frac{1}{2} \Big( 1 - \sqrt{\frac{2~(M_s-m)}{M_s}} \Big)
\Bigg]^{-\frac{(1+a)}{2}}
e^{\frac{b}{\sqrt{2}} \Big(\frac{M_s}{m_s}\Big) \Big( 1 - \sqrt{\frac{M_s-m}{2~M_s}} \Big)^{\frac{1}{2}}}
\label{eq:rhoMs}$$
Since $(M_s - m) \ll M_s$, a power expansion of the above equation in terms of the difference $(M_s-m)/M_{s}$ yields the leading order behaviour:
$$\rho_s(m, H)_{m \sim M_s} \sim \sqrt{\frac{M_s}{2~(M_s-m)}} ~~\Big(\frac{M_s}{m_s}\Big)^{-a}~
e^{\frac{b}{\sqrt 2}(\frac{M_s}{m_s})}
\label{eq:rl}$$
Thus, for $m \sim M_s$, the entropy behaves as: $$S_s(m,H)_{m \sim M_s} = k_B \ln \sqrt{\frac{M_s}{~(M_s-m)}}~-k_B\ln~2 ~+~k_B \frac{ b}{\sqrt{2}}~(\frac{M_s}{m_s})~-~ a k_B \ln~(\frac{M_s}{m_s})
\label{eq:SsMs}$$ Or, in terms of temperature : $$S_s(T,H)_{T \sim T_s} = k_B \ln \sqrt{\frac{T_s}{~(T_s-T)}}~-k_B\ln~2 ~+~k_B \frac{ b}{\sqrt{2}}~(\frac{T_s}{t_s})~-~ a k_B \ln~(\frac{T_s}{t_s})
\label{eq:SsTs}$$ $$T = \frac{1}{ 2 \pi k_B} m c^2.$$
We see that a phase transition takes place at $m = M_s$, ie $T = T_s$. This is a [**gravitational**]{} like phase transition: the square root [*branch point*]{} behaviour near the transition is analogous to the thermal self-gravitating gas phase transition of point particles [@13]. This is also the same behaviour of the microscopic density of states and entropy of strings with the spin modes included [@5]. As pointed out in [@4], this string behaviour is [*universal*]{}: this logarithmic singularity in the entropy (or pole singularity in the specific heat) holds in any number of dimensions, its origin is gravitational interaction in the presence of temperature, as Jeans’s instability at finite temperature but with a more complex structure.\
The transition occurs at the temperature $T_{s}$ Eq. (\[eq:Tsts\]) [*higher*]{} than the (flat space) string temperature $t_{s}$: $$T_s= \frac{b}{2\pi}\Bigg(\frac{M_s}{m_s}\Bigg) t_s = \frac{b}{2\pi}\Bigg(\frac{ L_{c\ell}}{\ell_s}\Bigg) t_s
\label{eq:Tsts}$$
This is so since in de Sitter background, the flat space-time string mass $m_s$, (Hagedorn temperature $t_s$) is the scale mass, (temperature), in the [*low*]{} $Hm$ regime. For high masses, the critical string mass, (temperature), in de Sitter background is $T_s$, instead of $m_s$, $(t_s)$. In de Sitter space, $H$ “pushes” the string temperature beyond the flat space (Hagedorn) value $t_s$ .\
By analogy with $t_s$, $T_s$ can be expressed as $$T_s = \frac{1} {bk_B} \sqrt{\frac{\hbar}{\alpha'_H c^2}}~~~~,~~~~ \alpha^{'}_H = \frac{\hbar}{c}\Big(\frac{2\pi}{b}~\frac{H \alpha'}{c}\Big)^2
\label{eq:TsEff}$$ That is, $H$, which acts in the sense of the string tension, does appear in the transition temperature as an “effective string tension” $(\alpha^{'}_H) ^{-1} $ : a smaller $\alpha^{'}_H$, (and thus a [*higher tension*]{}).
The effect of $H$ in the transition is similar to the effect of angular momentum. In 5 we have found a similar transition for strings in flat space-time but with the spin modes $(j)$ included: in that case, the transition occurs at a temperature $T_{j} = \sqrt{(j/\hbar)}~t_s$, [*higher*]{} than the Hagedorn temperature $t_{s}$, that is like an effective string constant $\alpha^{'}_j~ \equiv ~ \sqrt{\hbar/j}~ \alpha^{'}$, and thus, as a higher tension.
$\rho _s(m,H)$ Eqs. (\[eq:rlH\])-(\[eq:rl\]) expresses in terms of the typical mass scales in each domain: $m_s$ (as in flat space) for low masses, $M_s$ for high masses, to which corresponds the typical number of oscillating states: $N_s \sim \mbox{Int}[ \frac{L_{cl}}{L_s}] \sim \mbox{Int}[\frac{c^3}{\hbar \alpha ' H^2}]$, and there is the new factor $\Delta_s ^{-1}$ which is crucial for high masses. When $m > M_s$, the string does not oscillate (it inflates with the background, the proper string size is larger than the horizon [@8], [@15]. The meaning of the string de Sitter phase transition Eq.(\[eq:SsMs\]) or (\[eq:SsTs\]) is the following: when the string mass becomes $M_s$, it saturates de Sitter universe, the string size $L_s$ (Compton length for $M_s$) becomes $L_{c\ell}$, the string becomes “[*classical*]{} ” reflecting the classical properties of the background. $M_s$ is the mass of the background $M_{cl}$ Eq. (\[eq:Mcl\]), (with $\alpha '$ instead of $G/c^2$): for $m \rightarrow M_ s$ the string becomes the “[*background*]{} ” [@4]. Conversely, and interestingly enough, string back reaction supports this fact: $M_s$ is the mass of de Sitter background in its string regime, a de Sitter phase with mass $M_s$ Eq. (\[eq:mMs\]) and temperature $T_s$ Eq. (\[eq:Ts\]) is sustained by strings [@1]. ($L_s$, $M_s$, $T_s$) Eqs. (\[eq:mMs\])-(\[eq:Ts\]) are the [*intrinsic*]{} size, mass and temperature [*of*]{} de Sitter background in its string (high H) regime.
Partition Function of Strings in de Sitter Background and String Bound on the Semiclassical de Sitter Temperature \[sec:partition\]
===================================================================================================================================
The canonical partition function for a gaz of strings in de Sitter background is given by
$$\ln Z = \frac{V_{D-1}}{(2\pi)^{D-1}} \int^{M_s}_{m_0} d\Big( \frac{m}{m_s}\Big) \rho_s(m, H) ~
\int d^{D-1}k ~\ln \Bigg\{ \frac{1 + \exp \Big\{- \beta_{sem} \Big[(m^2 c^4 + \hbar^2 k^2 c^2)^{1/2}\Big] \Big\}}
{1 - \exp \Big\{- \beta_{sem} \Big[(m^2 c^4 + \hbar^2 k^2 c^2)^{1/2}\Big] \Big\}} \Bigg\}
\label{eq:Zk}$$
where supersymmetry has been considered for the sake of generality, $D-1$ is the number of space dimensions, $\rho_s(m, H)$ is the string density of mass levels in de Sitter space time Eq. (\[eq:rhoMsc\]), $\beta_{sem}= (k_B~ T_{sem})^{-1}$ with $T_{sem}$ being the semiclassical de Sitter temperature Eq. (\[eq:Tsem\]), ie. the Gibbons-Hawking temperature; $m_0$ is the lowest mass for which the asymptotic expression of $\rho_s(m, M_s)$ is valid, and $M_s$ Eq. (\[eq:mMs\]) is the upper bound for $m$ in de Sitter background. From Eq. (\[eq:Zk\]), we have $$\ln Z =\frac{4 V_{D-1}}{(2\pi)^{D/2}} \frac{c}{\beta_{sem}^{\frac{D-2}{2}} \hbar^{D-1}}
\sum_{n=1}^{\infty} \frac{1}{(2n-1)^{D/2}} ~
\int^{M_s}_{m_0} d\Big( \frac{m}{m_s}\Big)~ m^{D/2}~ \rho_s(m, H)~K_{D/2}
\Big( (2n-1)\beta_{sem} mc^2\Big)
\label{eq:Z}$$
Considering the asymptotic behaviour of the Bessel function $K_{\nu}(z)\sim \Big( \frac{\pi}{2z}\Big)^2 ~e^{-z}$, and the leading order $n=1$, $(\beta_{sem} ~m~c^2 \gg 1)$, yields\
$$\ln Z \simeq \frac{2 V_{D-1}}{(2\pi)^{ \frac{D-1}{2} }} ~\frac{1}{(\beta_{sem}\hbar^{2})^{\frac{D-1}{2}}}
\int^{M_s}_{m_0} d\Big( \frac{m}{m_s}\Big) \rho_s(m, H) ~m^{\frac{D-1}{2}} ~e^{-(\beta_{sem} m c^2)}
\label{eq:Zl}$$\
Let us see the behaviors of $\ln Z$ for low and high masses, i.e. $m \ll M_s$ and $m \sim M_s$ respectively. For $m\ll M_s$, the $\rho_s(m, H)$ leading behavior is equal to the flat space-time solution; from Eq. (\[eq:Zl\]),(for supersymmetric strings just multiply by a factor 2), we have for open strings:
$$(\ln Z)_{m\ll M_s} \sim \frac{2 V_{D-1}}{(2\pi)^{ \frac{D-1}{2}}}~
\frac{(m_s)^{\frac{D-3}{2}}}{(\beta_{sem}~\hbar^2)^{\frac{D-1}{2}}}~~
\frac{1}{(\beta_{sem}-\beta_{s})c^2}~e^{-(\beta_{sem}-\beta_{s})m_0 c^2}
\label{eq:ZmMs}$$
\
where $\beta_{s}=(k_B~t_s)^{-1}$, $t_s$ being the fundamental (flat space-time) string temperature Eq. (\[eq:ts\]). We see that for the low mass string spectrum, the canonical partition function shows a pole behaviour at $T_{sem}\rightarrow t_{s}$ , Eqs.(\[eq:Tsem\]) and (\[eq:ts\]). This is so, since for low masses, (low $Hm$ regime), the string mass (temperature) scale is the flat space-time mass $m_s$, ($t_s$). This single pole (Carlitz type [@12]) behavior near $t_s$ is universal for any space time dimension $D$. This is the same $T_{sem}\rightarrow t_{s}$ behavior as for strings in flat space time [@12] and as for strings in the Schwarzchild and Kerr black holes [@2], [@5].
For low temperatures $T_{sem} \ll t_{s}$, we recover the semiclassical (Q.F.T) non singular thermal behavior at the semiclassical (QFT) temperature $T_{sem}$: $$\ln Z \simeq V_{D-1} \Big( \frac{m_s}{2 \pi \beta_{sem} \hbar^2} \Big)^{\frac{D-1}{2}}~~ e^{-\beta_{sem}m_0 c^2}
\label{eq:ZTh}$$
From Eqs. (\[eq:rhoMs\]) and (\[eq:Zl\]) the leading behavior of $\ln Z$ for high masses ($m \sim M_s$) is given by\
$$(\ln Z)_{m \sim M_s} \sim \frac{V_{D-1}}{\left(\beta_{sem}\hbar c\right)^{D-1}}~~
\sqrt{\frac{\beta_{sem}-\beta_{sdS}} {\beta_{sem}}}
\label{eq:Zhl}$$
$$(\ln Z)_{T \sim T_s}\sim V_{D-1}\left(\frac{k_B T_{sem}}{\hbar c}\right)^{D-1}~~
\sqrt{1 - \frac{T_{sem}}{T_s}}
\label{eq:ZhT}$$
\
where $ \beta_{sdS} = (k_B T_s)^{-1} $, $T_s$ being the string de Sitter temperature Eq. (\[eq:Ts\]).
Eq.(\[eq:Zhl\]) shows a singular behavior for $\beta_{sem} \rightarrow \beta_{sdS}$ which is general for any space-time dimensions $D$; this is a square root branch point at $T_{sem}= T_s$. That is, a phase transition takes place for $T_{sem} \rightarrow T_s$, which from Eqs. (\[eq:Tsem\]), (\[eq:TM\]) and (\[eq:Ts\]) implies $M_{c\ell}\rightarrow m_{s}$, $L_{c\ell}\rightarrow \ell_{s}$.
Furthermore, we see from Eq. (\[eq:Zhl\]) that $T_{sem}$ has to be bounded by $T_s$, $(T_{sem} < T_s)$ . In fact, the low mass spectrum temperature condition $T_{sem}< t_{s}$ Eq. (\[eq:ZmMs\]), and the high mass spectrum condition $T_{sem}< T_{s}$ Eq. (\[eq:Zhl\]) both imply the following upper bound for the Hubble constant $H$ (Eqs. (\[eq:Lcl\]), (\[eq:Tsem\]), (\[eq:Ls\]), and (\[eq:Ts\])): $$L_{c\ell} > \ell_s, ~~\text{i.e.}, ~~H < \frac{c}{\ell_s}
\label{eq:Hb}$$
In the string phase transition $T_{sem} \rightarrow T_{s}$, $H$ reachs a maximum value sustained by the string tension $\alpha'^{-1}$ (and the fundamental constants $\hbar$, $c$ as well): $$H_s = c ~\sqrt{\frac{c}{\alpha'\hbar}}, ~~~~(\text{i.e.},~~\Lambda_s = \frac{1}{2 {\ell_s}^2}(D-1)(D-2))
\label{eq:Hmax}$$
The highly excited $m \rightarrow M_s$ string gaz in de Sitter space undergoes a phase transition at high temperature $T_{sem} \rightarrow T_{s}$, into a condensate stringy state. Eqs. (\[eq:Hmax\]) mean that the background itself becames a string state. In Section (\[sec:sdsps\]), we showed, from the microscopic dynamical density of states $\rho_s (m, H)$, that precisely at $T = T_s$, $(m = M_s)$, the string of mass $m$ in de Sitter space undergoes a phase transition at $m = M_s$ and becomes the background itself.\
QFT and string back reaction computations support this fact: de Sitter background is an exact solution of the semiclassical Einstein equations with the QFT back reaction of matter fields included, as well as a solution of the semiclassical Einstein equations with the string back reaction included [@1]: for $T_{sem} \ll T_s$, the curvature $R = R (T_{sem}, T_s)$, yields the QFT semiclassical curvature $R_{sem}$ (low H or semiclassical regime), and for $T_{sem}\rightarrow T_s$ it becomes a string state selfsustained by a string cosmological constant Eq.(\[eq:Hmax\]). The leading term of the de Sitter curvature in the quantum regime is given by $R_s = D \: (D-1)\: c / \ell_s^2$ plus negative corrections in an expansion in powers of ($R_{sem}/R_s$) [@1]. The two phases: semiclassical and stringy are dual of each other in the precise sense of the classical-quantum duality [@1], [@3],[@4].
The results of these Sections allow also to consider the string regimes of a black hole in a de Sitter (or asymptotically) de Sitter background. This allow to study the effects of the cosmological constant on the quantum string emission by black holes, and the string bounds on the semiclassical (Gibbons-Hawking) black hole-de Sitter (bhdS) temperature $T_{sem~bhdS}$ , this is done in Sections (VII)-(IX) below.
The Semiclassical Black Hole - de Sitter Background\[sec:BHS\]
==============================================================
The D-dimensional Schwarzschild - de Sitter space-time is described by the metric $$ds^{2}=-a(r) ~c^2~dt^{2} + a^{-1}(r) ~dr^2 + r^2~ d\Omega_{D-2}
\label{eq:mBHS}$$ where: $$a(r) = 1 - \frac{r_g}{r} - \Big( \frac{r}{L_{c\ell}}\Big)^2, ~~~~
r_g = \Bigg( \frac{16 \pi ~G~ M}{c^2(D-2) ~A_{D-2}}\Bigg)^{\frac{1}{D-3}},~~~~
A_{D-2} = \frac{2\pi ^{\frac{(D-1)}{2}}}{\Gamma \Big(\frac{(D-1)}{2}\Big)}
\label{eq:am}$$
$r_g$ being the Schwarzschild gravitational radius, and $L_{c\ell}$ is given by Eq. (\[eq:Lcl\]). For $D=4$, in terms of the cosmological constant $\Lambda$ Eq. (\[eq:H\]), one has $$a(r) = 1 - \Bigg(\frac{2 G M}{c^2}\Bigg)\frac{1}{r} - r^2 \frac{\Lambda}{3}, ~~~~~~r_g = \frac{2 G M}{c^2},~~~~~~\Lambda = 3 \Bigg(\frac{H}{c}\Bigg)^2
\label{eq:rg4}$$
The equation $a(r)=0$ has three real solutions: $r_h$ (black hole horizon), $r_c$ (cosmological horizon), and $r_{-}~=~-(r_h+r_c)$, which has to satisfy $$\frac{3}{\Lambda} = r_h^2 +r_c^2 + r_h~ r_c ~~~~\text{and}~~~~
\Bigg(\frac{2 G M}{c^2}\Bigg) \frac{3}{\Lambda} = r_h ~r_c \left( r_h +r_c\right)
\label{eq:rc}$$
The black hole surface gravity and the cosmological surface gravity are respectively: $$\mathcal{K}_{bhdS} =\frac{c^2}{2}~\Bigg \vert \frac{d a(r)}{dr} \Bigg \vert_{r=r_h} ~~, ~~~~~~
\mathcal{K}_c =\frac{c^2}{2}~\Bigg \vert \frac{d a(r)}{dr} \Bigg \vert_{r=r_c}
\label{eq:K}$$
From the above equations we have $$\mathcal{K}_{bhdS} =\frac{c^2}{2~r_h L_{c\ell}^2} ~ (r_c - r_h)~(r_h - r_-)
\label{eq:HHS}$$ and $$\mathcal{K}_c =\frac{c^2}{2~r_c L_{c\ell}^2}~(r_c - r_h)~(r_c - r_-)
\label{eq:KcS}$$\
being $r_h<r_c$, and $L_{c\ell}^2 = (3/\Lambda)$. Eq. (\[eq:HHS\]) can be written as well $$\mathcal{K}_{bhdS} = c^2 \Bigg( \frac{r_g}{2~r_h^2}-\frac{r_h}{L_{c\ell}^2} \Bigg)
\label{eq:Ka}$$
From Eqs. (\[eq:rc\]) there is a trivial black hole horizon solution $$r_h= \frac{1}{\sqrt\Lambda} = r_c~,~~~~\text{with} ~~~~~~\frac{GM}{c^2}= \frac{1}{3 \sqrt\Lambda},
\label{eq:rr}$$
and a more interesting one $$r_h \simeq r_g = \frac{2GM}{c^2}~.
\label{eq:rar}$$
From Eqs. (\[eq:Ka\]) and (\[eq:rar\]), the black hole surface gravity in the presence of $\Lambda$ is given by: $$\mathcal{K}_{bhdS} =\frac{c^2}{2~ r_g} ~ \Bigg( 1 - 2 ~\frac{r_g^2}{L_{c\ell}^2} \Bigg)~~~,
\label{eq:KH}$$ which for $\Lambda=0$ yields the Schwarzschild surface gravity $$\mathcal{K}_{bh} =\frac{c^2}{2 ~r_g}~ .
\label{eq:Krg}$$
The Hawking black hole temperature in de Sitter space is $$T_{sem~bhdS} = \frac{\hbar}{2 \pi k_B c}~~ \mathcal{K}_{bhdS}
\label{eq:TH}$$ which can be written as $$T_{sem~bhdS} = \frac{\hbar c}{2 \pi k_B}~~\frac{1}{L_{bhdS}}
\label{eq:TBH}$$ with $$L_{bhdS} =2~ r_g \Bigg( 1 - 2 ~\frac{r_g^2}{L_{c\ell}^2} \Bigg)^{-1}
\label{eq:LH}$$ Or, in terms of $H$ Eq. (\[eq:rg4\]): $$T_{sem~bhdS} = \frac{\hbar c}{2 \pi k_B}~~\frac{1}{2 r_g}~\Bigg( 1 - 2 ~ \Big(\frac{r_g~H}{c}\Big)^2\Bigg)
\label{eq:TBHS}$$ For $H=0$, one recovers the black hole Hawking temperature $$T_{sem~bh} = \frac{\hbar c}{4 \pi k_B~r_g} ~.
\label{eq:T}$$
With these expressions for the semiclassical bhdS background we are prepared to compute the quantum emission of strings by a Schwarzschild black hole in the de Sitter background.
Quantum String Emission by a Black Hole in de Sitter Background\[sec:eBHS\]
===========================================================================
The quantum field emission cross section $\sigma_{QFT} (k) $ of a given emitted species of particles in a mode $k$ by a black hole in de Sitter background is given by $$\sigma_{QFT}(k)=\frac{\Gamma_A}{e^{(\beta_{sem~bhdS} E(k))}-1}
\label{eq:sig}$$
where $\Gamma_A$ is the greybody factor (absorption cross section), and for the sake of simplicity, only bosonic states have been considered ; $\beta_{sem~ bhdS}=( k_B T_{sem~bhdS} )^{-1}$, and $T_{sem~bhdS}$ is given by Eq. (\[eq:TBH\]). The quantum field emission cross section of particles of mass $m$ is defined as
$$\sigma_{QFT}(m) = \int_{0}^{\infty} \sigma_{QFT} (k)~ d\mu (k)
\label{eq:sm}$$
where $d\mu (k)$ is the number of states between $k$ and $k+dk$:
$$d\mu(k) = \frac{2 V_{D-1}}{\Big(4\pi\Big)^{\frac{D-1}{2}}\Gamma \Big( \frac{D-1}{2} \Big) }~k^{D-2}~dk
\label{eq:mu}$$
From Eq. (\[eq:sm\]) we have $$\sigma_{QFT}(m)=\frac{V_{D-1}~\Gamma_A}{(2 \pi)^{\frac{D-1}{2}}}
\frac{\Big( mc^2\Big)^{\frac{D-2}{2}}~}{(\beta_{sem~bhdS})^{D/2}~ (\hbar c)^{D-1}} ~~
\times$$ $$\sqrt{\frac{2}{\pi}}~\sum_{n=1}^{\infty} \frac{1}{n^{D/2}}~
\Big\{ n\beta_{sem~bhdS} mc^2 K_{_{D/2}} (n\beta_{sem~bhdS}mc^2) + K_{_{D/2 - 1}} (n\beta_{sem~bhdS} mc^2) \Big\}
\label{eq:smD}$$
For large $m$ and the leading order $n=1$, $(\beta_{sem~bhdS}~ mc^2\gg1)$, we obtain with the asymptotic behavior of the Bessel function $K_{\nu}$: $$\sigma(m)_{QFT} \simeq \frac{V_{D-1}~\Gamma_A} {(2\pi )^\frac{D-1}{2}}~ ~
\frac{m^{\frac{D-1}{2}}} {\left(\beta_{sem~bhdS} ~\hbar^{2}\right)^ \frac{D-1}{2}}~ e^{-\beta_{sem~bhdS}~ mc^2}
\label{eq:smDl}$$
In the string analogue model, the string quantum emission cross section, $\sigma_{string}$, is given by $$\sigma_{string} \simeq \int_{m_0}^{M_s} \rho_{s}(m, H)~\sigma_{QFT}(m)~ d\Big(\frac{m}{m_s}\Big)
\label{eq:sD}$$
where $\rho_{s}(m, H)$ is given by Eq. (\[eq:rhoMsc\]).
For $m \ll M_s$, (away from the upper mass bound and temperature $T_s$ Eq.(\[eq:Ts\])), the $\rho_s(m, H)$ leading behaviour is given by the flat space solution $(H=0)$ Eq. (\[eq:rhofc\]). From Eqs. (\[eq:smDl\]), (\[eq:sD\]) and (\[eq:rhofc\]), (open strings), the leading contribution to the quantum string emission $\sigma_{string}$ for any D space-time dimensions is :
$$\sigma_{string}~(m\ll M_s) \sim \frac{V_{D-1}~\Gamma_A} {(2\pi )^\frac{D-1}{2}}~
\frac{ ~m_s^{\frac{D-3}{2}}} {\left(\beta_{sem~ bhdS}~\hbar^{2}\right)^{\frac{D-1}{2}}}~
\frac{ e^{-(\beta_{sem~bhdS}-\beta_{s})~m_0 c^2}}{\Big(\beta_{sem~bhdS} -\beta_{s}\Big)c^2}
\label{eq:smia}$$
For $m \ll M_s$, (which is a low $Hm$ regime), the string emission cross section shows the same singular behavior near $t_s$ as the low $Hm$ behavior of the canonical de Sitter partition function Eq. (\[eq:ZmMs\]), and as the quantum string emission by a (asymptotically flat) black hole [@2], [@5], here at the temperature $T_{sem~bhdS}$. This is so, since in the bhdS background, the string mass scale for low string masses (temperatures) is the Hagedorn (flat space) string temperature $t_s$. $T_{sem~bhdS} \rightarrow t_s$ is a high temperature behaviour for low $Hm \ll c/\alpha'$, $t_s$ is smaller than the string de Sitter temperature $T_s$.
For low temperatures $\beta_{sem~bhdS} \gg \beta_{s}$ we recover the semiclassical (QFT) Hawking emission at the temperature $T_{sem~bhdS}$:
$$\sigma_{string}\simeq \frac{V_{D-1}~\Gamma_A}{(2 \pi)^{\frac{D-1}{2}}}~
\frac{m_s^{\frac{D-3}{2}}}{\beta_{sem}~^{\frac{D+1}{2}}~ (\hbar c)^{D-1}}~e^{-\beta_{sem}m_0c^2}$$
For high masses ($m \sim M_s$) we have for the $\sigma_{string}$ leading behavior :\
$$\sigma_{string} ~~(m \sim M_s) \sim \frac{V_{D-1}~\Gamma_A}{\left(\beta_{sem~bhdS}~\hbar c\right)^{D-1}}~
\sqrt{\frac{\beta_{sem~bhdS}-\beta_{sdS}}{\beta_{sem~bhdS}}}
\label{eq:sim3}$$ $$\sigma_{string} ~~(T \sim T_s) \sim V_{D-1}~\Gamma_A~\left(\frac{k_B T_{sem~bhdS}}{\hbar c}\right)^{D-1}
\sqrt{1~-~\frac{T_{sem~bhdS}}{T_s}}
\label{eq:siT3}$$\
The black hole-de Sitter emission cross section shows a phase transition at $T_{sem~bhdS} = T_{s}$: the string emission by the black hole condensates into a de Sitter string state of string de Sitter temperature $T_s$. This is not like the flat (or asymptotically flat) space string phase transition (of Carlitz type [@12], [@2]), but this is a de Sitter type transition. Instead of featuring a single pole singularity in $(~T~ -~T_s~)$, the transition is a square root branch point. The branch point singular behavior at $T_{s}$ is valid for any D-dimensions and is like the one we found for the de Sitter canonical partition function Eq. (\[eq:Zhl\]) and for the de Sitter microscopic string density of states $\rho_s(m, H) $ Eq. (\[eq:rl\]) in the high $m$ (high $Hm \rightarrow c/\alpha'$) regime.\
\
The evaporation of a black hole in de Sitter space time from a semiclassical or quantum field theory regime (Hawking radiation) into a quantum string de Sitter regime (late stages), can be seen as well in the black hole decay rate. In the early evaporation stages, the semiclassical black hole in de Sitter background decays thermally as a grey body at the Hawking temperature $T_{sem~bhdS}$ Eq. (\[eq:TBHS\]), with the decay rate $$\Gamma_{sem} = \left| \frac{ d\ln M_{sem~bhdS} }{ d t}\right| \sim G~\left(T_{sem ~bhdS}\right)^{3} , ~~~~~~~~M_{sem~bhdS}= 2~\pi~T_{sem~bhdS}
\label{eq:decay}$$
($\hbar=c=k_{B}=1$). As evaporation proceeds,$T_{sem~bhdS}$ increases until it reaches the string de Sitter temperature $T_{s}$, the black hole itself becomes an excited string de Sitter state, decaying with a string width [@4],[@5] $\Gamma_{s}\sim \alpha'~T_{s}^{3}$ , $(G\sim \alpha')$ into all kind of particles, with pure (non mixed) quantum radiation. The implications of the limit $T_{sem~bhdS} = T_s$ are analyzed in the Section below.
String bounds for a black hole in de Sitter background\[sec:bBHS\]
==================================================================
The black hole-de Sitter (bh-dS) background tends asymptotically to de Sitter space-time. Black hole evaporation will be measured by an observer which is at this asymptotic region. Asymptotically, in the Schwarzschild black hole-de Sitter space time (bh-dS), $\rho_s(m,H)$ is equal to the string mass density of states in de Sitter space time Eq. (\[eq:rhoMsc\]). Then, for the partition function of a gaz of strings far from the black hole in bhdS space-time, we only need to substitute $\beta_{sem}$ in Section (\[sec:partition\]) by $\beta_{sem~bhdS}$, i.e., substitute the de Sitter temperature $T_{sem}$ Eq. (\[eq:Tsem\]) by the black hole temperature in de Sitter space $T_{sem~bhdS}$ Eq. (\[eq:TBHS\]). With this substitution, all results in Section (\[sec:partition\]) hold for bhdS as well. The condition $T_{sem~bhdS} < T_s$, (Eqs. (\[eq:Lcl\]), (\[eq:mMs\]), (\[eq:LH\])), yields now : $$\ell_s^2 < L_{c\ell}~L_{bhdS}
\label{eq:lLbhdS}$$ which implies the following condition $$H~\Big[1 - 2r_g^2\left(\frac{H}{c}\right)^2\Big] < \frac{2r_{g}c}{\ell_s^2}
\label{eq:HbS}$$
The bound is saturated ($T_{sem} = T_s$) for a gravitational radius which satisfies $$\left(\frac{r_g}{L_{c\ell}}\right)^{2}~+~r_g \frac{L_{c\ell}}{\ell_s^2}~-~\frac{1}{2} = 0
\label{eq:rgb}$$
yielding the physical solution
$$\label{eq:rgs}
r_g = \frac{1}{2}~\frac{L_{c\ell}^3}{\ell_s^2}~\Big[ - 1~+~\sqrt{ 1 + 2\left(\frac{\ell_s}{L_{c\ell}}\right)^4~}~\Big]$$
For $L_{c\ell} \gg\ell_s $ : $$r_g \simeq \frac{1}{2}~\frac {\ell_s^2} {L_{c\ell}}~\Big[1 + O(\frac {\ell_s}{L_{c\ell}})^2 ~\Big]~~~, ie.~~~ 2 r_g \simeq \frac {H}{c}\ell_s^2 = L_s
\label{eq:rga}$$
For $L_{c\ell}= \ell_s $ : $$2r_g = 0.73~\ell_s
\label{eq:rgb}$$
Eq. (\[eq:rgs\]) shows the relation between the Schwarzschild radius and the cosmological constant Eq. (\[eq:Lcl\]) when $T_{sem~bhdS} = T_s$ (string regime). We see that a black hole in de Sitter space allows an intermediate string regime, not present in the Schwarzschild black hole alone $(H=0)$, since in the bhdS background there are two characteristic string scales: $ L_s$ and $\ell_s$. In an asymtotically flat space-time, the black hole radius becomes $\ell_s$ in the string regime. In an asymptotically de Sitter space, when $T_{sem~bhdS}$ reaches $T_s$ , the black hole radius $r_g$ becomes the de Sitter string size $L_s$. If the de Sitter radius $L_{c\ell}$ reaches $L_s$, (which implies $L_{c\ell} = \ell_s$), then $r_g$ becomes determined by the scale $\ell_s$, as given by Eq.(\[eq:rgb\]).
Semiclassical (Q.F.T) and quantum (string) de Sitter regimes\[sec:dual\]
========================================================================
From the microscopic string density of mass states $\rho_s(m, H)$ Secs. (\[sec:qs\]) and (\[sec:sdsps\]), we have shown that for $m\rightarrow M_s$, i.e. $T\rightarrow T_s$, the string undergoes a phase transition into a semiclassical phase with mass $M_{cl}$ and temperature $T_{sem}$. Conversely, from the string canonical partition function Sec. (\[sec:partition\]) in de Sitter space and from the quantum string emission Sec. (\[sec:eBHS\]) by a black hole in de Sitter space, we have shown that for $T_{sem}\rightarrow T_s$, the semiclassical (Q.F.T) regime with Hawking-Gibbons temperature $T_{sem}$ undergoes a phase transition into a string phase at the string de Sitter temperature $T_{s}$ Eq. (\[eq:Ts\]). This means that in the quantum string regime, the semiclassical mass density of states $\rho_{sem}$ becomes the string mass density of states $\rho_s$ and the semiclassical entropy $S_{sem}$ becomes the string entropy $S_{s}$. Namely, a semiclassical de Sitter state, $(dS)_{sem}= (L_{c\ell}, M_{c\ell}, T_{sem}, \rho_{sem}, S_{sem})$, undergoes a phase transition into a quantum string state $(dS)_{s}$ = $(L_{s}, m , T_{s}, \rho_{s}, S_{s})$.
The sets $(dS)_{s}$ and $(dS)_{sem}$ are the same quantities but in different (quantum and semiclassical/classical) regimes. This is the usual classical/quantum duality but in the gravity domain, which is [*universal*]{}, not linked to any symmetry or isommetry nor to the number or the kind of dimensions. From the semiclassical and quantum de Sitter regimes $(dS)_{sem}$ and $(dS)_{s}$, we can write the full de Sitter entropy $S_{sem}(H)$, with quantum corrections included, such that it becomes the string entropy $S_s(m,H)$ Eq. (\[eq:S0\]) in the string regime: the full de Sitter entropy $S_{sem}(H)$ is given by
$$S_{sem}~(H) = \hat{S}_{sem}^{(0)}~(H)
-a~k_B~\ln ~(\frac{\hat{S}_{sem}^{(0)}~(H)}{k_B}) - k_B \ln~F(H)
\label{eq:SsemHF}$$
where $$\hat{S}_{sem}^{(0)}~(H)\equiv S_{sem}^{(0)}~(H) \sqrt{f(X)}~~~~,~~~~F(H)\equiv \sqrt{(1 - 4X^2)f(X)}
\label{eq:Fsem}$$
$$a=D~~ ,~~~~~f(X)= \frac{2}{1+\Delta},~~~~ \Delta \equiv\sqrt{1-4X^2}~=~ \sqrt{1 -\Big(\frac{\pi k_B}{S_{sem}^{(0)}(H)}\Big)^2}
\label{eq:Delta}$$
$$2 X(H)\equiv \frac{\pi k_B}{S_{sem}^{(0)}(H)}= \frac {M_{sem}}{M_{cl}}= \Big(\frac{m_{Pl}}{M_{cl}}\Big)^2
\label{eq:X}$$
\
$S_{sem}^{(0)}(H)$ is the usual Bekenstein-Hawking entropy of de Sitter space Eq. (\[eq:S0\]). $M_{cl}$ is the de Sitter mass scale Eq.(\[eq:Mcl\]), $M_{sem}$ is the semiclassical mass Eq.(\[eq:Msem\]). In terms of $S_{sem}^{(0)}(H)$, $S_{sem}(m,H)$ Eq.(\[eq:SsemHF\]) reads:
$$S_{sem}(H) = \sqrt{f(X)} S_{sem}^{(0)}(H)
-ak_B~\ln \Big( \sqrt{f(X)} \frac{S_{sem}^{(0)}(H)}{k_B} \Big) - k_B~\ln\sqrt{ f(X)}- k_B~\ln\sqrt{~1 - 4X^2~}
\label{eq:SsemH}$$
$\Delta$ Eq.(\[eq:Delta\]) with $X(H)$ Eq.(\[eq:X\]) describes $S_{sem}(H)$ in the mass domain $m_{pl} \leq M_{c\ell} \leq \infty$, that is, $0\leq X \leq 1/2$. (ie. $0\leq \Delta \leq 1$). The same formula but with $\hat{X}(H) = S_{sem}^{(0)}(H)/2\pi k_B$, instead of $X(H)$, describes $S_{sem}(H)$ in the mass domain $0 \leq M_{cl} \leq m_{pl}$.
Eq. (\[eq:SsemHF\]) provides the whole de Sitter entropy $S_{sem}(H)$ as a function of the Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$. $X \rightarrow 0$ means $M_{cl}\gg m_{Pl}$, that is $L_{c\ell}\gg \ell_{Pl}$ , (low $H \ll c/\ell_{Pl}$ or low curvature regime), in this case $\Delta \rightarrow 1$, $f(X)\rightarrow 1$ and $S_{sem}^{(0)}(H)$ is the leading term of $S_{sem}(H)$, with its logarithmic correction: $$S_{sem}(H) = S_{sem}^{(0)}(H) -ak_B~\ln \Big(\frac{S_{sem}^{(0)}(H)}{k_B}\Big)
\label{eq:SsemoH}$$
But for [**high**]{} Hubble constant, $H \sim c/\ell_{Pl}$, (ie. $M_{cl}\sim m_{Pl}$), $S_{sem}^{(0)}(H)$ is sub-dominant, a gravitational [**phase transition**]{} operates and the whole entropy $S_{sem}(H)$ is drastically [**different**]{} from the Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$, as we precisely see in the Section below
The de Sitter Gravitational Phase Transition \[subsec:EXT\]
===========================================================
For $\Delta \rightarrow 0$, that is for $ M_{c\ell}\rightarrow M_{sem}$, the entropy $S_{sem}(H)$ Eq. (\[eq:SsemHF\]) behaves as: $$S_{sem}(H)_{\Delta \sim 0} = k_B~ \ln \Delta ~+~O(1)
\label{eq:SsemMcl}$$
The Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$ is sub-leading, O(1), in this case, ($S_{sem}^{(0)}(H)_{\Delta = 0} = \pi k_B$).
In terms of the mass, or temperature: $$\Delta~ =~ \sqrt{1 - \Big(\frac{m_{Pl}}{M_{cl}} \Big)^4}~=~ \sqrt{1 - \Big(\frac{T_{sem}}{T} \Big)^2} ~~,
\label{eq:Deltasem}$$ where $$T = \frac{1}{2\pi k_B}M_{cl} c^2
\label{eq:T}$$
and $T_{sem}$ is the semiclassical (Gibbons-Hawking) de Sitter temperature Eq.(\[eq:Tsem\]).
In the limit $ M_{c\ell}\rightarrow M_{sem}$, which implies $ M_{cl} \rightarrow m_{Pl} $, $S_{sem}(H)$ is dominated by
$$S_{sem}(H)_{\Delta \rightarrow 0} = -~k_B~ \ln ~\Big(~\sqrt{2} \sqrt{1 - \frac{T_{sem}}{T}}~\Big)~~+~O(1)
\label{eq:SsemPl}$$
This shows that a [**phase transition**]{} takes place at $T \rightarrow T_{sem}$. This implies that the transition occurs for $M_{cl} \rightarrow m_{Pl}$, ie $T \rightarrow t_{Pl}$, (that is for high $H \rightarrow c/\ell_{Pl}$). This is a [**gravitational**]{} like transition, similar to the de Sitter string transition we analysed in Section (\[sec:sdsps\]): the signature of this transition is the square root [*branch point*]{} behavior at the critical mass (temperature) analogous to the thermal self-gravitating gas phase transition of point particles [@13], and to the string gaz in de Sitter space. This is also the same behaviour as that found for the entropy of the Kerr black hole in the high angular momentum $ J\rightarrow M^2G/c$ regime, (extremal transition) [@5]. This is [*universal*]{}, in any number of space-time dimensions.
A. B. acknowledges the Observatoire de Paris, LERMA, for the kind hospitality extended to him. M. R. M. acknowledges the Spanish Ministry of Education and Science (FPA04-2602 project) for financial support, and the Observatoire de Paris, LERMA, for the kind hospitality extended to her.
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|
---
author:
- 'L. De Maio'
- 'F. Dunlop$^{\dagger}$'
bibliography:
- 'ldfd.bib'
title: Sessile drop on oscillating incline
---
[**u**]{}
Introduction
============
Equilibrium of a drop pinned on an incline was studied by many authors, see [@DDH17] and references therein. Shape and motion of drops sliding down an inclined plane have also been studied, see [@LDL05] and references therein. Drops on vibrating horizontal surfaces have been the subject of much interest recently from experimental, theoretical or numerical points of view. The vibrations or oscillations of the substrate can be horizontal [@DCG05; @LLS04; @DCC06; @CK06], or vertical [@LLS06]. The effect of vibrations on hysteresis, pinning and depinning, was studied in particular by [@NBB04; @VSG07]. The effect of vibrations on the Cassie-Wenzel transition was studied in particular by [@BC09; @BPWE07]. A review of drop oscillations is given by [@MDCA14]. More recent experimental results and references are found in [@RE17].
Here we consider the case of an oscillating incline where the angle $\al(t)$ of the slope follows \[alphat\] (t)=[4]{}( t) while the circular basis of the drop remains fixed on the incline as a disc of radius $r$. We keep the frame of reference attached to the incline, so that the gravity vector oscillates: =(
g((t))0-g((t))
) We assume that the inertial pseudo-forces per unit volume, like the centrifugal force, are negligible with respect to gravity, which will be the case if $\om^2\,r\ll g$.
The Bond number is the ratio between gravity and capillarity, and we define it precisely as Bo=[g r\^2]{} where $\Delta\rho$ is the density difference between the two fluids and $\sigma$ is the interface tension. We are interested in moderate but significant drop deformations, with Bond number of order one, as shown on Fig. \[fig:iso\]. All the simulations presented here will be with $Bo=0.22$.
![ Water drop at equilibrium pinned on incline of angle $\al=\pi/4$. Bond number $Bo=0.22$. The drop is surrounded by oil.[]{data-label="fig:iso"}](diapo5.png "fig:"){width="\columnwidth"}\
For bond number of order one and pulsation $\om$ not much larger than the natural pulsation of the drop, the fluid velocity will vary from 0 to about $\om r$ over a distance $r$. This motivates a Reynolds number defined as Re\_=[ r\^2]{} with $\rho=\rho_{\rm water}$ and $\eta=\eta_{\rm water}$. In the same regime the quadratic term in the Navier-Stokes equation (\[NS\]) will be of order $\rho\om^2\,r$, the same as the centrifugal force per unit volume. Therefore it will be consistent, and will save some computing time, to neglect it (Stokes flow). The equation remains non-linear due to the interfacial tension force.
For pulsations $\om$ larger than the natural pulsation of the drop, the response of the drop and the actual velocity will be much smaller. A Reynolds number using the maximum measured velocity will always be less than 1 in our simulations. Viscosity plays an essential role in the present study, which does not allow short-cuts such as interface motion by curvature based on the Laplace-Young equation.
Diffuse interface and level set method
======================================
The sharp interface between immiscible fluids is replaced by a diffuse interface spreading over a few mesh elements across the physical interface. A level set function $\phi$, inspired by van der Waals, goes smoothly from zero to one when crossing the interface from fluid 1 into fluid 2. The mixture obeys the Navier-Stokes equation for an incompressible fluid, +(u)u&=&&&1.5cm++[**f**]{}\_[st]{}\[NS\]\
u&=&0 where ${\bf I}$ is the identity matrix, and we neglect the quadratic term in (\[NS\]). The density and dynamic viscosity are functions defined by =(1-)\_1+\_2=(1-)\_1+\_2 The surface tension force per unit volume ${\bf f}_{st}$ is \[st\] [**f**]{}\_[st]{}=(([**I**]{}-(\^T))) where $\sigma$ is the interfacial tension, $\nphi=\na\phi/|\na\phi|$ is a normal vector also defined in the bulk, and $\delta=6|\na\phi|\,\phi(1-\phi)$ is a smooth Dirac delta function concentrated near the interface, which is the level set $\{\phi=0.5\}$. Formula (\[st\]), being the divergence of a flux, can be integrated by parts in the weak form of the partial differential equation, and then requires just one derivative of $\phi$. It was shown by [@LNSZ94] to be a smooth approximation to the usual Laplace force $\sigma H\nphi\delta({\rm interface})$ where $H$ is the mean curvature of the interface and $\delta({\rm interface})$ is a true Dirac delta function supported by the interface.
The level set function $\phi$ obeys \[LS\] [t]{}+u=(-(1-)[||]{}) where $\ep$ in the diffusion term controls the interface thickness. It will be taken as $h/2$, half the mesh size. The parameter $\gamma$ is a constant with the dimension of a velocity, which we fix as $r\,\om/(2\pi)$ where $r$ is the initial radius of the drop. The level set method for two phase flow was developed in particular by [@OK05].
![Setup.[]{data-label="fig:diapo3"}](diapo3.png "fig:"){width="\columnwidth"}\
Setup
=====
The incline is designed with a circular hydrophilic patch of radius $r=2.5\,$mm and the remaining surface hydrophobic. The corresponding Young contact angles are set to 0 degree (perfectly hydrophilic) and 180 degrees (perfectly hydrophobic) respectively.
A water drop of volume $2\pi r^3/3$ is deposited on the hydrophilic patch. The vessel is filled with oil, and closed with no air inside. In the absence of gravity, the drop is a hemisphere, with contact angle $\pi/2$. This will also be the initial configuration in our simulations.
The vessel is intended to be large with respect to the water drop, so that friction occurs only near the drop. The Archimedes force, encapsulated in the pressure and gravity terms of the Navier-Stokes equation, does not depend upon the volume of the vessel. For simulation purposes, we have to use a simulation box of modest size. The effect of the box will be minimized if it has the symmetry of the problem at lowest order, hence a hemisphere with same center as the initial drop, and we choose its radius as four times the initial drop radius. On it we choose “slip” boundary conditions: impenetrable and frictionless, again to minimize the effect of having a relatively small simulation box.
The center of the hydrophilic patch is chosen as origin of coordinates and the $z$-axis perpendicular to the incline. The incline then starts oscillating around the $y$-axis according to (\[alphat\]). The plane $\{y=0\}$ is a plane of symmetry, allowing to make the study in a quarter of a sphere, see Fig. \[fig:diapo3\].
In the stationary regime, the contact angles at the front ($x=r$) and at the back ($x=-r$) will oscillate between a minimum angle $\theta^{\rm min}$ and a maximum angle $\theta^{\rm max}$. So long as the maximum contact angle remains strictly less than 180 degrees, the contact line cannot move into the hydrophobic region. So long as the minimum contact angle remains strictly larger than 0 degree, the contact line cannot move into the hydrophilic region. The role of the substrate is to ensure pinning.
For real substrates, the advancing angle $\theta^A$ of the hydrophobic material and the receding angle $\theta^R$ of the hydrophilic material will replace 180 degrees and zero degree respectively. The scope of our study is bounded by the conditions 0\^R<\^[min]{}<\^[max]{}<\^A. It implies bounds on Bond number and slope angle $\al$, which are satisfied in the present study. It would be interesting to go beyond and also study depinning. This is left to future work.
[|c|c|c|c|]{}\
& $\rho\,$\[kg/m$^3$\] & $\eta\,$\[Pas\] & $\sigma\,$\[N/m\]\
Engine oil & 888 & 0.079 &\
Water & 1000 & 0.001 &\
Interface & & & 0.031\
[*Comsol*]{}
============
We used the finite elements software [*Comsol*]{} (see https://www.comsol.com/) in mode [*Laminar Two-Phase Flow, Level Set*]{}, with fluid 1 as engine oil and fluid 2 as water, at 20$^\circ$C, see Table 1.
![Mesh.[]{data-label="fig:mesh"}](mesh.png "fig:"){width="\columnwidth"}\
The simulation box is a quarter of a sphere of radius $4r$. The outer sphere is not a physical boundary, and on it we choose [*slip*]{} boundary conditions:
u=0&=(),&=(+u\^T)0.5cm where $K$ is the viscous stress vector upon an infinitesimal surface of normal $\n$. The symmetry plane $\{y=0\}$ obeys the same boundary conditions, with also =0 The hydrophilic patch is a [*wetted wall*]{} with contact angle $\theta_w=0$, meaning a boundary condition (-\_W)=uwhere $\beta$ is a slip length equal to the mesh size $h$. The remaining part of the incline is a [*wetted wall*]{} with contact angle $\theta_w=\pi$. The mesh is built as shown on Fig. \[fig:mesh\], with maximal mesh size $h=0.4\,$mm and $h\sim 0.1\,$mm in the region of the interface, leading to 24560 degrees of freedom.
We impose at least one time step in every 1/40 of a period so as to be able to distinguish a sinusoidal response. With the chosen mesh, it turns out that [*Comsol*]{} does not need smaller time steps to satisfy its default tolerance. Each run for one value of $\om$ took about 20 hours with an Intel i7-3770CPU@3.40GHz x8.
![Trace of the drop on the symmetry plane $\{y=0\}$ at five times. $Bo=0.22$, $\om=0.1\,$s$^{-1}$.[]{data-label="fig:film"}](filmcontour.png "fig:"){width="\columnwidth"}\
Results
=======
The trace of the drop on the symmetry plane $\{y=0\}$ at different times is shown on Fig. \[fig:film\].
![Contact angles $\theta^r(t)$ at the front (red) and $\theta^{-r}(t)$ at the back (green), measured in degrees as (\[nnphi\]), for $Bo=0.22$, $\om=0.1\,$s$^{-1}$.[]{data-label="fig:thetarl"}](thetarl.png "fig:"){width="\columnwidth"}\
![Normalized abscissa of water center of mass $\bar x(t)$, as Eq. (\[xt\]) (red +), and sinusoidal fit of permanent regime, $A\,\sin(\om\,t-\varphi)$ as (\[sin\]) (green, continuous), for $Bo=0.22$, $\om=5\,$s$^{-1}$.[]{data-label="fig:om5x10T"}](om5x10T.png "fig:"){width="\columnwidth"}\
The contact angles $\theta^r(t)$ and $\theta^{-r}(t)$ at the front and the back are shown on Fig. \[fig:thetarl\]. These contact angles are measured as \[nnphi\] =() at $(x,y,z)=(r,0,0)$ and $(x,y,z)=(-r,0,0)$ respectively. When the incline is set in motion, at $t=0$, the liquid drop does not follow instantaneously, whence a start below 90 degrees. In the stationary regime a noticeable feature is that the contact line spends more time near the minimum than near the maximum.
Eq. (\[nnphi\]) is a measurement of the interface normal, pointing from oil into water, at a single point, which is a mesh vertex. Small numerical errors are clearly visible in Fig. \[fig:thetarl\]. A systematic error is also present: when the contact angle approaches $\theta^{\rm max}$, the contact line goes slightly into the hydrophobic region. Similarly when the contact angle approaches $\theta^{\rm min}$, the contact line goes slightly into the hydrophilic region. Measuring the contact angles at $x=\pm r$ underestimates the amplitude of oscillations.
Measuring contact angles, experimentally or numerically, is subject to debate, especially in dynamics. Fitting individual images of a film is tedious and systematic deviations may also be present if the fit is over a length where gravity produces bending. In dynamics the bending effect of gravity cannot be computed exactly. We have therefore chosen to analyse the data in terms of the motion of the centre of mass of the drop, whose definition is obvious and whose statistics is optimal.
The abscissa of the center of mass of water is recorded, normalized arbitrarily using the drop basis radius $r$ and the volume $\pi r^3/3$ of a quarter of a sphere of radius $r$: \[xt\] |x(t)=[dxdydz (x,y,z,t)xr\^4/3]{} The integral is over the simulation box, namely a quarter of a sphere of radius $4r$. After a transient, which lasts longer for larger $\om$, the system approaches a stable permanent regime, as shown on Figs. \[fig:om5x10T\], \[fig:om20x\]. The finite elements method does not conserve exactly the total mass of each fluid, and a small parasitic drift is often present in simulations, but it is not the case here.
We then use the [*gnuplot*]{} fit, a nonlinear least-squares Marquardt-Levenberg algorithm, and search for an amplitude $A$ and a phase lag $\varphi$ such that \[sin\] |x(t)-A(t-)0 tResults are like the example shown on Fig. \[fig:om5x10T\], where the error measured by the $rms$ of residuals over one period falls below 1% after a few periods (after 7 periods in the example shown). The resulting incertainties over $A$ and $\varphi$ are also below 1%.
A sinusoidal response with the same $\om$ as the incline angle was to be expected for a linear system. We used the Stokes equation with a non-linear surface tension force, and the transport equation (\[NS\]) is also non-linear.
Results are listed in Table 2, where the case $\om=0$ is in fact the limit as $\om\to0$, namely the stationary case $\al(t)=\pi/4\ \forall t>0$. Plots of $A$ and $\varphi$ versus $\om$ are given in Fig. \[fig:xphi\] and Fig. \[fig:xphiphi\]. They look much like a driven over-damped linear oscillator, with a notable exception: the amplitude of the permanent oscillations behaves like $\om^{-1}$ at large $\om$ instead of $\om^{-2}$ for the driven damped linear oscillator. The phase lag $\varphi(\om)$ is proportional to $\om$ at small $\om$ like a driven damped linear oscillator.
![Normalized abscissa of water center of mass $\bar x(t)$, Eq. (\[xt\]), for $\om=20\,$s$^{-1}$.[]{data-label="fig:om20x"}](om20.png "fig:"){width="\columnwidth"}\
[|c|c|c|]{}\
$\om\,$\[s$^{-1}$\] & $A$ & $\varphi\,$\[rad\]\
0 & 1.92 & 0\
0.05 & 1.89 & 0.275\
0.1 & 1.68 & 0.503\
0.2 & 1.25 & 0.79\
0.5 & 0.66 & 1.06\
1 & 0.387 & 1.17\
2 & 0.228 & 1.24\
5 & 0.105 & 1.41\
10 & 0.0564 & 1.55\
20 & 0.0286 & 1.66\
![Amplitude $A(\om)$ from (\[sin\]) with asymptote $0.58/\om$.[]{data-label="fig:xphi"}](xphi.pdf "fig:"){width="\columnwidth"}\
![Phase lag $\varphi(\om)$ from (\[sin\]) with line at $\pi/2$ and tangent at the origin $\varphi=5.5\om$.[]{data-label="fig:xphiphi"}](xphiphi.pdf "fig:"){width="\columnwidth"}\
Heuristics for $\om\to\infty$.
==============================
Let us first review the case of a driven solid oscillator, subject to a fluid friction force, obeying a differential equation of the form \[osc2\] x + [friction force]{} + [restoring force]{}=t As $\om\to\infty$ we don’t expect resonance. Therefore each term on the left-hand-side will be of order at most the order of the right-hand-side, namely $\OO(1)$. We expect a periodic permanent regime of period $T=2\pi/\om$. If the amplitude is $A$ then $\ddot x$ will be of order $A\om^2$, implying $A$ of order at most $\om^{-2}$. The restoring force will be $o(1)$, the velocity of order $A\om\sim\om^{-1}$ and the fluid friction force $o(1)$. Therefore, as $\om\to\infty$, the system tends to $\ddot x=\sin\om\,t$, leading to an amplitude $\om^{-2}$ and phase lag $\pi$, in agreement with the exact solution of the linear case.
Another driven system may obey a first order differential equation of the form \[osc1\] x + [restoring force]{}=t Again we expect a periodic permanent regime of period $T=2\pi/\om$, and no resonance, so that each term on the left-hand-side will be of order at most $\OO(1)$. If the amplitude is $A$ then $\dot x$ will be of order $A\om$, implying $A$ of order at most $\om^{-1}$. The restoring force will be $o(1)$. Therefore, as $\om\to\infty$, the system tends to $\dot x=\sin\om\,t$, leading to an amplitude $\om^{-1}$ and phase lag $\pi/2$, in agreement with the exact solution of the linear case.
We have studied a drop on an oscillating incline in a regime where the inertial forces, such as the centrifugal force, are negligible with respect to gravity and capillarity, both of same order for Bond number of order one. We thus have $\om^2r\ll g$. The acceleration term $\p u/\p t$ in the Navier-Stokes equation is of same order and therefore negligible. And the Reynolds number was always less than one so that the quadratic term in the Navier-Stokes equation could be neglected. Therefore a behaviour corresponding to (\[osc1\]) rather than (\[osc2\]) should be observed, leading to an amplitude $A\sim\om^{-1}$ rather than $A\sim\om^{-2}$ as $\om\to\infty$.
In simple words: a liquid drop can deform in many different ways and will do so as far as the shear stress $\na \u$ remains bounded. If $A$ is the amplitude of the motion of the center of mass in the frame of reference of the incline, then $\na \u$ is of order $A\om/r$, giving $A\sim\om^{-1}$ as $\om\to\infty$.
When $\om\to0$, acceleration is negligible in all cases, and both solid and liquid oscillators have an amplitude $\OO(1)$ and a phase lag $\OO(\om)$.
Conclusion
==========
A sessile millimetric droplet on an incline responds similarly to a driven damped linear oscillator to a sinusoidal oscillation of the angle of the incline. However, the amplitude of the drop deformation is proportional to $\om^{-1}$ at large $\om$, whereas a simple pendulum on an oscillating incline responds with an amplitude proportional to $\om^{-2}$ at large $\om$.
The diffuse interface modelisation imply diffusion times larger than the true physical times, but the discrepancy should go to zero with finer and finer meshes. Also, because there is more space for water (originally in the small sphere) to diffuse into oil (originally in the large sphere), than conversely, the level set 0.5, considered as the interface, shrinks a little during the first seconds. This effect should also go to zero with finer and finer meshes.
Beyond $\om\sim20\,$s$^{-1}$, in the oscillating frame of reference, one cannot neglect the inertial pseudo-forces. One can expect that including the centrifugal force in the Navier-Stokes equation would increase the drop deformation at large $\om$.
|
---
abstract: 'We study mesoscopic disorder fluctuations in an anisotropic gap superconductor, which lead to the spatial variations of the local pairing temperature and formation of superconducting islands above the mean-field transition. We derive the probability distribution function of the pairing temperatures and superconducting gaps. It is shown that above the mean-field transition, a disordered BCS superconductor with an unusual pairing symmetry is described by a network of superconducting islands and metallic regions with a strongly suppressed density of states due to superconducting fluctuations. We argue that the phenomena associated with mesoscopic disorder fluctuations may also be relevant to the high-temperature superconductors, in particular, to recent STM experiments, where gap inhomogeneities have been explicitly observed. It is suggested that the gap fluctuations in the pseudogap phase should be directly related to the corresponding fluctuations of the pairing temperature.'
author:
- Victor Galitski
bibliography:
- 'disorder.bib'
title: Mesoscopic gap fluctuations in an unconventional superconductor
---
Understanding the phase diagram and the properties of the high-temperature and other unconventional superconductors has been among the most complex problems of modern condensed matter physics. Most current theoretical approaches to the problem concentrate on strong correlation physics and usually assume that the effects of disorder are unimportant. However, there exist a number of recent experimental works, in particular STM studies of the high-$T_c$ cuprates,[@RMP.STM; @Yazdani; @E; @E2; @E3; @E4; @E5] which provide a tentative indication that at least in some materials disorder plays an important role in the local formation of the superconducting gap. In particular, Gomes et al. [@Yazdani] have studied the local development of the gap as a function of temperature in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ above the superconducting transition and up to a pseudogap temperature, where the gap inhomogeneities cease to exist. An important result of this experiment is that the real-space gap map observed was static and reproducible. This strongly suggests that the inhomogeneous gap formation is unlikely to be a phase-separation or superconducting fluctuation effect, but is due to some kind of disorder in the system.
Motivated by these experiments, we theoretically consider a disordered superconductor with an unusual pairing symmetry (e.g., a $d$-wave superconductor) and study mesoscopic variations of the local pairing temperature. We point out that the existence of the Griffiths-type [@Griffiths; @Grif_exp; @myGrif] phase in the superconducting phase diagram is specific to an anisotropic gap superconductor and should not occur in the conventional $s$-wave systems (due to Anderson theorem), unless they are extremely dirty or time-reversal symmetry is broken.[@SZ; @LS; @GaL] An important observation is that if the pairing gap is anisotropic, the Anderson theorem breaks down and the superconducting pairing temperature, $T_p$, is suppressed by disorder even if time-reversal symmetry is preserved (here and below we make a distinction between the pairing temperature, $T_p$, and the superconducting transition temperature, $T_c$, although in the framework of the weak-coupling BCS theory they are essentially the same). The impurities are positioned randomly in space and their density is a random variable. Thus, there always exist regions where the distribution of impurities is such that the local pairing temperature, $T_p({\bf r})$, is larger than the system-wide average value, $\langle{T_p}\rangle$, and the experimental temperature, $T$.[@IL] These regions form islands with a well-defined gap, which exist on the background of a metal, if $T_p({\bf r}) > T
> \langle{T_p}\rangle$. The width of the “mesoscopic fluctuation” region certainly depends on the strength of disorder and the only parameter, which may enter this dependence, is the dimensionless conductance. This defines a narrow window where the disorder-induced Griffiths phase co-exists with strong superconducting fluctuations. Therefore, the picture of impurity induced inhomogeneities in an anisotropic gap BCS superconductor is that of superconducting islands and metallic regions with strongly suppressed density of states.
We start with the following Hamiltonian with a built-in $l$-wave pairing ($l > 0$): $$\begin{aligned}
\label{H}
\hat{\cal H} =\!\!\!\! \int d^2{\bf r}\left\{
\hat\psi^{\dagger}({\bf r}) \left[ - \frac{{\bm \nabla}^2}{2m} -
\mu + U({\bf r}) \right] \hat\psi({\bf r}) -\lambda_l
\hat{b}^\dagger({\bf r}) \hat{b}({\bf r})\right\},\end{aligned}$$ where $U({\bf r})$ is a disorder potential, $\lambda_l$ is the $l$-wave interaction constant, and $b({\bf r})$ corresponds to an “$l$-wave Cooper pair” $\hat{b}({\bf r}) = \sum\limits_{{\bf
k},{\bf q}} \chi_l(\phi) \hat{\psi}({\bf k}+{\bf q}/2)
\hat{\psi}({\bf p}-{\bf q}/2) e^{i {\bf q}\cdot {\bf r}}$, with $\phi$ being the angle between the direction of the vector ${\bf
k}$ and the $x$-axis and $\chi_l(\phi)$ is the function, which enforces the $l$-wave symmetry of the gap in the mean-field.
The first step is to integrate out the fermions and express the action in terms of the order parameter $\Delta_{\bf k} =
\sum\limits_{{\bf k}, {\bf k}'} V({\bf k},{\bf k}') F({\bf k}' -
{{\bf q} / 2},{\bf k} + {{\bf q} / 2}) e^{i {\bf q} \cdot {\bf
r}}$, where according to Eq. (\[H\]) the interaction $V({\bf
k},{\bf k}') = -\lambda_l \chi_l(\phi) \chi_l(\phi')$ and $F$ is the standard Gor’kov’s Green’s function. In what follows we will concentrate on the spatial dependence of the gap and assume that its symmetry in the ${\bf k}$-space is preserved: $\Delta_{\bf
k}({\bf r}) = \Delta({\bf r}) \chi_l({\bf k}) \equiv \Delta_0
f({\bf r}) \chi_l({\bf k})$. Using these notations, we arrive at the following free energy for the system expressed in terms of the inhomogeneous order parameter $\Delta({\bf r})$ $$\begin{aligned}
\label{F} {\cal F}[\Delta,U] = && \!\!\!\!\!\!
\frac{1}{2}\int_{1,2} \Delta^*({\bf r}_1) A({\bf r}_1,{\bf r}_2)
\Delta ({\bf r}_2)\\
&&\!\!\!\!\!\!\!\!\!\!\!\! + \frac{1}{4}\int_{1,2,3,4}
\Delta^*({\bf r}_1) \Delta^*({\bf r}_2) B({\bf r}_1,{\bf r}_2,{\bf
r}_3,{\bf r}_4) \Delta({\bf r}_3)\Delta({\bf r}_4), \nonumber\end{aligned}$$ where $A({\bf r}_1,{\bf r}_2) = \lambda_l^{-1} \delta({\bf r}_1
- {\bf r}_2) - C({\bf r}_1,{\bf r}_2)$ and the Cooperon, ${\hat
C}$, is a random matrix expressed through the Green’s functions (before averaging over disorder) as follows $C({\bf r}_1,{\bf
r}_2) = T \sum\limits_{\varepsilon_n} G(\varepsilon_n;{\bf
r}_1,{\bf r}_2)G(-\varepsilon_n;{\bf r}_1,{\bf r}_2)$. As long as we are interested in the location of the classical (finite-temperature) phase transition, the dynamics of the order parameter and the Cooperon are not important. We present the Cooperon as a superposition of a local “mean-field” part and a disorder dependent correction $\hat{C} = \langle{\hat{C}}\rangle +
\delta \hat{C}$. The “mean-field” part is diagrammatically described by a simple Cooper bubble, without the disorder ladder (see Fig. 1a). Any disorder vertex correction vanishes due to the unusual symmetry of the gap. The line where the average Ginzburg-Landau coefficient vanishes $\langle{A}\rangle = 0$ determines the mean-field transition and leads to the well-known Abrikosov-Gor’kov’s equation [@AG] $$\label{AG} \ln{T_{p0} \over T_p} = \psi \left( {1 \over 2} + {1
\over 4 \pi T_p \tau} \right) - \psi \left( {1 \over 2} \right),$$ where $T_{p0}$ is the pairing temperature without disorder and $\tau$ is the scattering time. This equation implies that the pair-breaking effect of the conventional disorder potential in an anisotropic gap superconductor \[i.e., $\int d \phi \chi(\phi) = 0$\] is identical to that of a time-reversal perturbation in an $s$-wave superconductor.[@AIL] We reiterate that Eq. (\[AG\]) is a result of the averaging over disorder in the sample. Below we study mesoscopic corrections to this result, which qualitatively can be interpreted as local changes in the scattering time $\tau({\bf r})$ in Eq. (\[AG\]).
![\[FIG:UCF\] (a) The particle-particle bubble for an anisotropic gap superconductor. This Cooperon diagram contributes to the coefficient in the quadratic term of the Ginzburg-Landau expansion (\[F\]). All disorder vertex corrections to the Cooperon vanish.; (b) Pictorial representation for the coefficient in quartic term of the Ginzburg-Landau expansion (\[F\]). (c) One of the UCF-type diagrams, which contribute to the mesoscopic fluctuations of the transition point and mesoscopic gap fluctuations.](Fig1.eps){width="3.3in"}
We note that strictly speaking the nonlinear operator in the quartic term of Eq. (\[F\]) is also random, however its fluctuations can be neglected near the transition and its mean field value can be used $\langle{B}\rangle$. This coefficient is pictorially described by the Hikami-box diagram in Fig. 1b. A straightforward calculation of this diagram gives the following general result: $$\begin{aligned}
\label{B} \langle{B}\rangle = -{\nu \over 16 \pi^2 T^2}
\left[{ \alpha \over 12} \psi'''\left( {1 \over 2} + \alpha
\right) + \overline{|\chi_l|^4} \psi''\left( {1 \over 2} +
\alpha \right) \right],\end{aligned}$$ where $\nu$ is the density of states, $\alpha = (2 \pi T_p
\tau)^{-1}$, and the overline implies averaging over the Fermi surface. In the clean limit, Eq. (\[B\]) reproduces the result of Feder and Kallin [@Kallin], $\langle{B}\rangle = {7
\zeta(3) \nu / (8 \pi^2 T^2)}$. We note that the dirty limit is not reasonable in the context of an anisotropic gap superconductor, since it implies that the pairing temperature is suppressed to zero and there is no superconductivity. The maximum impurity concentration which allows for superconductivity (the quantum critical point) is $T_{p0} \tau_{\rm QPT} = \gamma /\pi
\sim 1$, where $\gamma \approx 1.781$ is the exponential of the Euler’s constant.
To find the local variations of the transition temperature, we consider the following eigenvalue problem for the random matrix $\delta \hat{C}$ $$\label{eigen} {1 \over \nu} \int d^2 {\bf r}' \delta C ({\bf
r},{\bf r}') \Delta ({\bf r}') = \epsilon \Delta({\bf r})$$ and define the probability distribution function (PDF) of its eigenvalues $\rho(\varepsilon) = \left\langle \delta \left(
\epsilon - \epsilon \left[ \delta \hat{C} \right] \right)
\right\rangle$. The averaging is performed over the PDF of the random Cooperon matrix, which we assume Gaussian $P \left[ \delta
{\hat C} \right] \propto \exp \left[ - {1 \over 2} \delta {\hat C}
* \hat{\hat{K}}^{-1} * \delta {\hat C} \right]$, where the asterisk implies a convolution over the two spatial variables and the operator $\hat{\hat{K}}$ corresponds to the correlator of two Cooperon operators, which in position representation has the form $K({\bf r}_1,{\bf r}_2;{\bf r}_3,{\bf r}_4) = \left\langle \delta
C ({\bf r}_1,{\bf r}_2)\delta C ({\bf r}_3,{\bf
r}_4)\right\rangle$. This disorder-averaged correlator can be calculated using the standard diagrammatic technique (see Fig. 1c). These diagrams are topologically equivalent to the universal conduction fluctuation (UCF) diagrams. However there are important differences: (i) First, here we are interested in the Cooper channel and (ii) Second, we are interested in the [*local*]{} physics, not in a long-wavelength behavior of the correlator.
Using the standard technique,[@NAL] we find the following expression for the correlator $$\begin{aligned}
\label{K1} K_1[ \{ {\bf r}_i \} ] = &&\!\!\!\! \delta\left({\bf
r}_1 - {\bf r}_4 \right) \delta\left({\bf r}_2 - {\bf r}_3 \right)
\left[ {2 \tau \overline{|\chi_l|^2} \over 4 \pi^2 D \nu }
\right]^2
\\ \nonumber && \!\!\!\!\!\!
\times {\int\limits_{\tau \to 0}^\infty}\int\limits_{\tau \to
0}^\infty {dt_1 dt_2 \over t_1 t_2 (t_1 + t_2)^2} \exp{\left[ -
{t_1 + t_2 \over 4 D t_1 t_2} \left| {\bf r}_1 - {\bf r}_3
\right|^2 \right]}.\end{aligned}$$ Here the index “1” implies that we consider only one among all possible UCF-type diagrams. However, they all contribute equally to the correlator of interest and lead to a combinatorial factor of $c$, which is equal to $c = 12$ in the orthogonal ensemble and $c = 6$ in the unitary ensemble (e.g., in the presence of a magnetic field). The PDF of the local transition temperatures and the corresponding gap amplitudes can be obtained using the optimal fluctuation method[@HL] $$\label{PDF.Tc} \left\langle \rho(\epsilon) \right\rangle \propto
\exp\left[ - {1 \over 2} {\epsilon^2 \over \left\langle f \otimes
f\, \left| \hat{\hat{K}} \right| \, f \otimes f \right\rangle}
\right],$$ where the eigenvalue $\epsilon$ has the physical meaning of a local pairing temperature fluctuation and $f({\bf
r})$ is a normalized function, which describes the spatial profile and the shape of a single disorder-induced superconducting \[if $\epsilon > (T - \langle T_p \rangle)/T$\] or metallic \[if $\epsilon < (T - \langle T_p \rangle)/T$\] puddle. Strictly speaking the latter function must be found from a non-linear integral equation $\epsilon f({\bf r}) = \Lambda \int_{1,2,3}
f^*({\bf r}_1) f^*({\bf r}_2) K({\bf r}_1,{\bf r}_2,{\bf r}_3,{\bf
r}) f({\bf r}_3) $ (where $\Lambda$ is a Lagrange multiplier which appears in the optimal fluctuation method; see Ref. \[\] for technical details). However, one can get a quantitatively reliable description of the PDF by considering the puddle function to be a Gaussian of a characteristic size $\xi$, i.e., $f(r) = \left( \pi \xi^2
\right)^{-1} \exp\left(-{r^2 \over 2 \xi^2}\right)$. In principle, one can study the distribution of puddle shapes by decomposing the function $f({\bf r})$ into spherical harmonics. We do not attempt a study of the puddle shapes here, but just point out that “higher-orbital momentum puddles” are less probable than spherically symmetric ones; the probability of finding a droplet with “momentum” $m$ scales as $p_m\propto p_0^{m}$, where $p_0$ is the probability of a spherical puddle. To find the latter we explicitly calculate the correlator in Eq. (\[PDF.Tc\]) and find $\left\langle f \otimes f\, \left| \hat{\hat{K}} \right| \, f
\otimes f \right\rangle = {3 c \over 4 \pi} {l^2 \over \xi^2} {1
\over g^2}$, where $\xi$ is the size of a puddle, $l$ is the mean free path, and $g = E_{\rm F} \tau / \pi$ is the dimensionless conductance. This leads to the following PDF of $T_p$’s (\[PDF.Tc\]): $$\label{PDF.Tc.res} \left\langle \rho(\xi,T_p) \right\rangle
\propto \exp \left[ - {4 \pi \over 3 c} \left( {\xi \over l}
\right)^2 g^2 \left( {T_p - \langle T_p \rangle \over \langle T_p
\rangle} \right)^2 \right].$$
![\[FIG:PDF\] (Color online) Plotted are distribution functions of the superconducting gap $\Delta$ for various temperatures, $T$. We assumed the following parameters: $g = 10$, $E_F/T_p \sim 40$, $T_p = 93~K$. The latter choice is motivated by the experimental work on Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, where inhomogeneities have been observed. The temperature behavior of the PDF following from the mesoscopic fluctuation theory is qualitatively similar to that observed in experiment. However, we note here that since our approach is based on the BCS theory, the theoretical results may not provide a quantitatively accurate description of the cuprates.](Fig2.eps){width="3in"}
We note that by applying a magnetic field, one can cross over from the orthogonal to the unitary ensemble and change the combinatorial factor in (\[PDF.Tc.res\]) from $c = 12$ to $c=6$, which may be experimentally testable and should manifest itself as a diminishing of the random $T_p(H)$ or gap distribution width exactly by the factor of two. To find the PDF of the superconducting gaps, we can just use the Ginzburg-Landau equation (\[F\]) and set $\Delta_0 = \sqrt{\nu \left( T_p - T \right) / (
\langle T_p \rangle \langle B \rangle )}$, where $\langle B
\rangle$ is given by Eq. (\[B\]). We note that it makes sense to consider a finite-size droplet with a well-defined local transition temperature or a gap, only if the size of the droplet is much larger than the coherence length $\xi \gg
\xi_{\Delta_{0}}$. The opposite limit corresponds to the case of a mesoscopic superconducting nanograin in which the notion of the gap is not well-defined (see Ref.\[\] for a review). Therefore, the smallest possible size of the puddle with a well defined gap $\Delta_0$ is $\xi_{\rm min} \sim v_{\rm
F}/{\Delta_0}$, which implies that the dimensionless ratio $\xi_{\rm min} /l \sim (T_p \tau)^{-1}$. The latter parameter is of order one and thus the PDF of the gaps takes the form (see also Fig. \[FIG:PDF\]) $$\label{PDF.gaps} P[\Delta] \sim {2 g \sqrt{\beta} b \Delta \over
\sqrt{\pi} \langle T_p \rangle^2} \exp \left[ - \beta g^2 \left(
{b \Delta^2 \over \langle T_p \rangle^2} + { T - \langle T_p
\rangle
\over \langle T_p \rangle } \right)^2 \right],$$ where in the $d$-wave case, $b \sim -\left[ {\alpha \over 3}
\psi'''\left({1 \over 2} + \alpha\right) + 3 \psi''\left({1 \over
2} + \alpha\right) \right]/(32 \pi^2)$, $\alpha = (E_F/\langle T_p
\rangle) (2 \pi^2 g)^{-1}$, and $\beta \sim 1$. Note that in Eq. (\[PDF.gaps\]) we have omitted a term proportional to $\propto \delta(\Delta)$, which describes normal regions. The physical picture, which emerges for such a disordered superconductor is that right above the mean-field transition temperature there exist rare superconducting islands separated by “normal regions” (which are still very close to the local transition temperature). We note that since the parameter $g
T_{p0}/E_{\rm F}$ is at best of order one, the “mesoscopic Griffiths phase” overlaps with the Ginzburg region of strong superconducting fluctuations.[@LV] This leads to the conclusion that the Griffiths phase is a mixture of superconducting islands and metallic regions with strongly suppressed density of states.
![\[FIG:dome\] (Color online) This figure is to illustrate the qualitative discussion in the text about the possibility of a dome-shaped doping dependence of the pairing temperature within the Abrokosov-Gor’kov theory. The figure shows a typical dependence of $T_p$ on “doping,” x, in the toy model of the Abrikosov-Gor’kov theory with both $T_{p0}(x)$ and $\tau(x)$ being dependent on the same doping parameter. The graphs are solutions of the following equation for $t_p$: $\ln{\left(x^{1-\alpha}/t_p\right)} = \psi\left(1/2 + x
\delta/t_p\right) - \psi(1/2)$, where $t_p$ is a dimensionless pairing temperature and $\delta$ characterizes the strength of disorder. The graphs correspond to $\alpha=1/2$ and $\delta = 0.1$ (red line), $0.15$ (blue line), and $0.2$ (green line).](Fig3.eps){width="3in"}
Even though our quantitative description directly applies only to a weakly coupled BCS superconductor, we believe that some aspects of the theory are relevant to the cuprates as well ( the importance of disorder effects for the cuprates have been discussed previously, see, e.g. \[\]). But first, we make the following curious observation: In a high-$T_c$ superconductor the major source of disorder is presumably the dopant atoms. But this access oxygen is also the source of the carriers, which lead to superconductivity in the first place. Thus, the “clean” pairing temperature in the Abrikosov-Gor’kov’s formula (\[AG\]) should explicitly depend on the doping level, $T_{p0}(x)$ (e.g. through a BCS-like high-energy cut-off or possibly via a locally modulated electron pairing interaction.[@H4]) The scattering time depends on $x$ too and contributes to superconductivity suppression. Thus there are two competing effects of the dopants: They both enhance and suppress superconducting properties. An interesting result, which follows from the simple Abrikosov-Gor’kov’s equation (\[AG\]), is that even if $T_{p0}(x)$ is monotonically increasing with $x$, but $T_{p0}(x)
\tau(x) \propto x^{-\alpha}$ with $\alpha > 0$, then the actual pairing temperature doping dependence has a dome-shaped form. An example of such a dependence for $\alpha =1/2$ is plotted in Fig. \[FIG:dome\].
The mesoscopic disorder fluctuations in the Abrikosov-Gor’kov theory may not correspond to the modulations of the transition temperature in a high-$T_c$ superconductor, but should be related to the modulations of the pseudogap temperature, $T_*$, which is believed to be the onset of Cooper pairing (i.e., to $T_p$, but not $T_c$). The “pseudogap” region $T_c<T<T_p$ presumably represents the regime of strong phase fluctuations [@Kivelson] and the superconducting transition is expected to be that of $XY$-type. In the latter scenario, the transition temperature is proportional to the superfluid density, which in turn is directly related to the local value of the gap. The mean field gap is determined by the deviation $(T_p - T)$ from the local pairing temperature (e.g., via a non-linear BCS-like self-consistency equation $\delta S/\delta \Delta = 0$). If $T_p({\bf r})$ fluctuates due to disorder, so does $\Delta({\bf r})$ and, quite generally, the local gap should “follow” local $T_p$. In fact, such a correlation has been observed in experiment.[@Yazdani] In the model of uncorrelated short range disorder, the only possible result for the corresponding PDF is Eq. (\[PDF.gaps\]), with $\Delta$ centered in the vicinity of the mean-field gap at the given temperature $P[\Delta] \propto \exp\left\{ - \beta g^2
\left[ 1 -\Delta/ \langle \Delta (T) \rangle \right]^2 \right\}$. In a clean material, the corresponding regime of mesoscopic fluctuations is very narrow unless there are other types of disorder effects, such as “structural disorder” (e.g., extended defects, warping of the 2D planes, etc.), which can be modelled as a random diffusion coefficient [@GaL] $\left\langle D({\bf
r})D({\bf 0}) \right\rangle = \langle D \rangle^2 a^2 \delta({\bf
r})$. These phenomena lead to qualitatively the same effect of random $T_p$ and $\Delta$, but, may occur at the length-scales much larger than the mean-free path and become important in a much wider range of parameters \[the width of the distribution is determined by $ (a/l) g^{-1}$ instead of $g^{-1}$\]. We also note that if indeed the local gap fluctuations observed in experiment [@Yazdani] are related to mesoscopic disorder effects, they should be correlated with the pinning properties,[@LO] i.e., the width of the gap distribution should be proportional to the critical current $j_c$ in the collective pinning regime [@GaL], $\left\langle \left(1 -
\Delta /\langle\Delta\rangle \right)^2 \right\rangle \propto
(j_c/j_{c0}) (H/H_{c2})$ (where $j_{c0}$ is the critical current in zero field).
The author acknowledges the Aspen Center for Physics for hospitality and the JQI for financial support.
|
---
author:
- Jeffrey Lin Thunder
title: Decomposable form inequalities
---
amstex =cmr10
‘=11 =msbm10 scaled 1100 =msbm10 =msbm10 scaled 800 == =@[@]{} @\#1[[@@[\#1]{}]{}]{} @@\#1[\#1]{} ‘=12
‘=11 =eufm10 scaled 1100 =eufm10 =eufm7 scaled 1100== =@[@]{} @\#1[[@@[\#1]{}]{}]{} @@\#1[\#1]{} ‘=12
**Abstract**
We consider Diophantine inequalities of the kind $|F(\ux )|\le m$, where $F(\uX )\in \bz [\uX]$ is a homogeneous polynomial which can be expressed as a product of $d$ homogeneous linear forms in $n$ variables with complex coefficients and $m\ge 1$. We say such a form is of finite type if the total volume of all real solutions to this inequality is finite and if, for every $n'$-dimensional subspace $S\subseteq \br ^n$ defined over $\bq$, the corresponding $n'$-dimensional volume for $F$ restricted to $S$ is also finite.
We show that the number of integral solutions $\ux\in\bz ^n$ to our inequality above is finite for all $m$ if and only if the form $F$ is of finite type. When $F$ is of finite type, we show that the number of integral solutions is estimated asymptotically as $m\rightarrow \infty$ by the total volume of all real solutions. This generalizes a previous result due to Mahler for the case $n=2$. Further, we prove a conjecture of W. M. Schmidt, showing that for $F$ of finite type the number of integral solutions is bounded above by $c(n,d)m^{n/d},$ where $c(n,d)$ is an effectively computable constant depending only on $n$ and $d$.
In this paper we consider forms in $n>1$ variables of the type $F(\uX)=\prod _{i=1}^dL_i(\uX)\in\bz [\uX ],$ where each $L_i(\uX)\in\bc [\uX ]$ is a linear form. For a positive integer $m$ we are interested in the integer solutions $\ux\in\bz ^n$ to the inequality $$|F(\ux )|\le m.\tag 1$$
Consider the case when $n=2$, $d>n$ and $F(\uX)$ is irreducible over $\bq$. Thue’s famous result in \[T\] is that the number of integer solutions to (1) in this case is finite. Later, Mahler in \[M\] estimated the number $N_F(m)$ ofsuch solutions as follows. Let $A(F)$ denote the area of the planar region$\{\ux\in\br ^2\: |F(\ux )|\le 1\},$ so that $m^{2/d}A(F)$ is the measure of the set of $\ux\in\br ^2$ that satisfy (1). (The hypothesis that $F$ is irreducible forces the discriminant to be nonzero, which implies that this area is finite.) Then $$\left |N_F(m)-m^{2/d}A(F)\right |=O\big (m^{1/(d-1)}\big )$$ as $m\rightarrow \infty$, where the implicit constants depend on $d$ and $F$. We also have a result due to Schmidt \[S3, Chap. III, Theorem 1C\] which states that for irreducible $F$, $N_F(m)\ll dm^{2/d}(1+\log m^{1/d})$ with an absolute implicit constant.
Other than results for the case $n=2$, little has been published on this question. Ramachandra in \[R\] proved that for norm forms of the type $N_{K/\bq}(X_1+\alpha X_2+\cdots +\alpha ^{n-1}
X_n)$, where $K=\bq (\alpha )$ is a number field of degree $d\ge 8n^6$ and $N_{K/\bq}$ denotes the norm from $K$ to $\bq$, one has $$|N_F(m)-m^{n/d}V(F)|=O(m^{\varepsilon +(n-1)/(d-n+2)})$$ for any $\varepsilon >0$ as $m\rightarrow\infty,$ where the implicit constant depends on both $F$ and $\varepsilon$, and $V(F)$ denotes the volume analogous to the area $A(F)$ above. Note that by the homogeneity of $F$, $m^{n/d}V(F)$ is the volume of the set of all real solutions to (1). Of course, one needs the subspace theorem to approach the general case. For norm forms, Schmidt showed in \[S1\] that the number of solutions to (1) is finite for all $m$ if and only if $F$ is a nondegenerate. Evertse has shown in \[E3\] that for nondegenerate norm forms $F$ of degree $d$ in $n$ variables, one has $$N_F(m)\le (16d)^{{1\over 3}(n+7)^3}m^{(n+\sum _{i=2}^{n-1}i^{-1})/d}\times
(1+\log m)^{{1\over 2}n(n+1)}.$$
The results above are of two different flavors. On the one hand the natural heuristic is that, in the absence of a compelling reason to the contrary, one expects that the volume of the region in $\br ^n$ defined by (1) should approximate the number of integral solutions to (1). The results of Mahler and Ramachandra above verify this in special cases. On the other hand, when $N_F(m)$ is finite one expects that it should be bounded above by a function independent of the specific coefficients of $F$. This was proven by Evertse in \[E1\] for the case $n=2$, and another result of Schmidt in \[S2\] confirms this in the general case of products of nondegenerate norm forms. Schmidt’s absolute upper bound above in the case $n=2$ appears to be the right order of magnitude in terms of $m$ (up to the logarithmic term). In fact, in \[S2\] Schmidt makes the conjecture that $N_F(m)\ll m^{n/d}$ for all nondegenerate norm forms of degree $d$ in $n$ variables, where the implicit constant depends only on $n$ and $d$. Evertse’s result above comes close to this.
When one tries to reconcile the heuristic with Schmidt’s conjecture, one is led to the conjecture that $V(F)\ll 1$ for nondegenerate norm forms. This was shown to be true in \[B\] for the case $n=2$, and was shown to be true for general forms in $d>n$ variables with nonzero discriminant in \[BT\].
Returning to our heuristic, what would be a “compelling reason" for $N_F(m)$ to [*not*]{} be approximated by the volume? One such reason comes immediately to mind. It is typically the case that, though the volume $V(F)$ may be finite, the lower dimensional volume of the region defined by (1) cut by a hyperplane is infinite. If such a hyperplane were defined over $\bq,$ then that rational hyperplane might contain infinitely many integral points. With this in mind, we say $F$ is of [*finite type*]{} if $V(F)$ is finite, and the same is true for $F$ restricted to any nontrivial rational subspace. Note in particular that if $F$ is of finite type, it does not vanish at any nonzero rational point. When $F$ is of finite type, then, we rule out this “compelling reason." Since $N_F(m)$ can be infinite if $F$ is a degenerate norm form, this could be a “compelling reason" as well. But degeneracy of a norm form is a rather algebraic concept, and it is not immediately clear what the connection is between this and the more geometric concept of the volume $V(F)$.
The purpose of this paper is to answer the following questions: When is $V(F)$ finite? More correctly, can one determine rather simply from a given factorization of $F$ whether $V(F)$ is finite or not? If $V(F)$ is finite, is $V(F)\ll 1$? When is $N_F(m)$ finite for all $m$? If $N_F(m)$ is finite, is it approximated by $m^{n/d}V(F)$? If $N_F(m)$ is finite, is $N_F(m)\ll m^{n/d}$? We will prove Schmidt’s conjecture and more. Here and from now on, all implicit constants in the $\ll$ notation depend only (and explicitly) on $n$ and $d$.
Let $F$ be a decomposable form of degree $d$ in $n$ variables with integral coefficients[.]{} If $V(F)$ is finite and $F$ does not vanish at a nonzero integral point[,]{} then $V(F)\ll 1$[. ]{}
Let $F$ be a decomposable form of degree $d$ in $n$ variables with integral coefficients[.]{} Then $N_F(m)$ is finite for all $m$ if and only if $F$ is of finite type[.]{} If $F$ is of finite type[,]{} then $N_F(m)\ll m^{n/d}.$
Apparently nondegenerate norm forms are of finite type. This could be shown more directly, though it is not a simple consequence of the definition of nondegenerate. The answer to our question regarding the finiteness of $V(F)$ requires further notation, so we leave it for the next section (see the proposition below). We only remark here that it is necessary that $d>n$ in order for $V(F)$ to be finite except for the case of a positive definite quadratic form in two variables.
Let $F$ be decomposable form of degree $d$ in $n$ variables with integral coefficients[.]{} If $F$ is of finite type[,]{} then there are $a(F),\ c(F)\in\bq$ satisfying $$1\le a(F)\le{d\over n}-{1\over n(n-1)}$$ and $${(d-n)\over d}\le c(F)< {d\choose n}(d-n+1)$$ such that $$|N_F(m)-m^{n/d}V(F)|\ll m^{(n-1)/(d-a(F))}(1+\log m)^{n-2}
\hofF ^{c(F)}.$$ If the discriminant is not zero[,]{} then we may take $a(F)=1$ and $c(F)=\break {d-1\choose n-1}-1.$
The quantities $a(F),\ c(F)$ and $\hofF$ appearing in Theorem 3 are explicitly defined in the next section. Note that $(n-1)/(d-a(F))<n/d$ in Theorem 3, so that the estimate for $N_F(m)$ given is not trivial. Theorem 3 is a broad generalization of Mahler’s result above.
This paper is organized as follows. Section 1 introduces some notation and defines some quantities connected to $F$ which will be used throughout. In the next section we derive some general results concerning the height $\hofF$. Sections 3 and 4 are the technical heart of the paper where we see that solutions to (1) lie in subsets of certain convex regions (these regions are parallelopipeds if $F$ factors over $\br$) and we garner pertinent information about these convex regions. Section 5 deals with the case when $V(F)$ is infinite. The next two sections are devoted to estimating volumes connected with (1) and analyzing the set of integral solutions to (1). The proofs of our theorems follow in the last section, using an inductive argument on the number of variables $n$.
Definitions and a linear programming result
===========================================
Throughout the rest of this paper, $F(\uX )=\prod _{i=1}^dL_i(\uX )\in\bz [\uX ]$ will denote a decomposable form of degree $d$ in $n$ variables with integral coefficients and $m\ge 1$ will be a fixed real number. The case where $F$ is a power of a positive definite quadratic form in two variables is exceptional and our questions posed in the introduction are trivially answered in this case, so from now on we will assume that $F$ is not such a form.
We will use the notion of “equivalent forms." If $F$ is a decomposable form in $n$ variables and $T\in\gln (\bz)$, then we can compose $F$ with $T$ to get a new form $G(\uX )=F\circ T(\uX )$. Since $\det (T)=\pm 1,$ we have $V(F)=V(G)$. Further, the integral solutions to (1) are in one-to-one correspondence (via $T^{-1}$) with the integral solutions to $|G(\ux)|\le m$. Because of this, we say two forms $F$ and $G$ are equivalent if there is a $T\in\gln (\bz )$ with $G=F\circ T$. The freedom to choose a representative from each equivalence class will be used to our advantage.
We now proceed with some definitions and notation. We will denote the usual $L_2$ norm of $\ux\in\bc ^n$ by $\| \ux \|$. We will denote the coefficient vector of a linear form $L_i(\uX )$ by $\uL _i\in\bc ^n$. Complex conjugation will be denoted by an overline: $\overline{\alpha}$. This notation will be extended to vectors as well, e.g., $\overline{\uL}.$ Elements of $\bc ^n$ will be viewed as $1\times n$ matrices (i.e., row vectors) and a superscript $^{tr}$ will denote the transpose of a matrix, so that $\uL ^{tr}$ is a column vector for a coefficient vector $\uL$.
We define the [*height*]{} of $F$ to be
${\displaystyle \hofF :=\prod _{i=1}^d\|\uL _i\|.}$
Note that $\hofF$ is actually independent of the particular factorization of $F$ used, though it is not preserved under equivalence.
Given a factorization of $F$, let $I(F)$ denote the set of all ordered $n$-tuples $(\uL _{i_1},\ldots ,\uL _{i_n})$ of linearly independent coefficient vectors. We let $b(\uL _i)$ denote the number of $n$-tuples in $I(F)$ where $\uL _i$ occurs and let $b(F)$ denote the maximum of these $b(\uL _i)$. Note that $b(F)$ is preserved under equivalence and is independent of the factorization used. Let $I'(F)\subset I(F)$ denote those $n$-tuples with $i_1<i_2<\cdots <i_n.$ Letting $|\cdot |$ denote the cardinality, we have $$|I(F)|=n!|I'(F)|\le n!{d\choose n},\tag 2$$ with equality if and only if the discriminant of $F$ is not zero.
Let $J(F)$ be the subset of $I(F)$ consisting of $n$-tuples that satisfy the following restriction: if $j<n,$ then either $\uL _{i_{j+1}}$ is proportional to $\overline{
\uL _{i_j}}$ or $\overline{\uL _{i_j}}$ is in the span of $\uL _{i_1},\ldots ,\uL _{i_j}.$ If $J(F)$ is not empty, we let $$a(F)=\max \left \{ \text{the number of $\uL _i$ in the span of
$\uL _{i_1},\ldots ,\uL _{i_j}$}\over j\right \},$$ where the maximum is over all $n$-tuples in $J(F)$ and $j=1,\ldots ,n-1.$ If $J(F)$ is empty, we leave $a(F)$ undefined. Note that the number of factors in the span of $\uL _{i_1},\ldots ,\uL _{i_n}$ is $d$ for all $n$-tuples in $I(F)$. We will see later (see Lemma 5 below) that $J(F)$ is in fact empty only when $I(F)$ is. Note that $a(F)\ge 1$ if it is defined, with equality if and only if the discriminant of $F$ is not zero.
We can now state our characterization of finite volume in terms of the factorization of $F$.
For a decomposable form $F$ as above[,]{} $V(F)$ is finite if and only if $a(F)$ is defined and less than $d/n$[.]{}
The proposition will be proven in Section 7 below.
We now continue with some definitions. Let $$c(F)=\cases {d-1\choose n-1}-1&\matrix \text{if the discriminant}\hfill\\ \noalign{\vskip-6pt}
\text{of $F$ is not zero,}\hfill\endmatrix
\\
{b(F)\over n!a(F)}\big (d-(n-1)a(F)\big ) -{1\over a(F)}&
\text{otherwise,}\endcases$$ whenever $a(F)$ is defined. This quantity occurs as an exponent on $\hofF$ in our arguments; we give it a name for notational convenience.
The [*semi-discriminant*]{} of $F$, which we denote by $S(F)$, is given by $$S(F):= \prod
\det (\uL ^{tr}_{i_1},\ldots ,\uL _{i_n}^{tr}),$$ where the product is over all $n$-tuples in $I(F)$ when $I(F)$ is not empty, and $S(F)=0$ otherwise. Unlike $\hofF$, the semi-discriminant can be dependent on the factorization. If $$F(\uX )=\prod _{i=1}^dL _i(\uX )=\prod _{i=1}^d\alpha _iL_i(\uX )$$ are two different factorizations of $F$, then the semi-discriminant for the first will equal that for the second if and only if $$\prod _{i=1}^d\alpha _i^{b(\uL _i)}=1.$$ Hence, the semi-discriminant is independent of the factorization if and only if $b(\uL _i)=b(F)$ for all $i$. This is not always the case, as the example $F(\uX )=X_1^2X_2X_3\cdots X_n$ shows. To deal with this nonuniqueness, we introduce a quantity which we call the [*normalized semi-discriminant*]{}, denoted by $NS(F)$ and defined by $$NS(F):=\prod {
\det (\uL ^{tr}_{i_1},\ldots ,\uL _{i_n}^{tr})\over
\|\uL _{i_1}\|\cdots\|\uL _{i_n}\|}= {S(F)\over \|\uL _1\|^{b(\uL _1)}\cdots
\|\uL _d\|^{b(\uL _i)}},$$ where the product is over all $n$-tuples in $I(F)$. Then $|NS(F)|$ is entirely determined by the form $F$. It is not preserved under equivalence.
We end this section with a simple linear programming result which will be needed later.
Let $k$ be a positive integer[.]{} Let $b_1\le \cdots \le b_k$ be a nondecreasing sequence of real numbers and let $A>0$[.]{} Then the minimum value of$x_1b_1+\cdots +x_kb_k$ subject to the restrictions $$\aligned x_i&\ge 0\hskip8pt \qquad\text{all $i$},\\
x_1+\cdots +x_j&\le jA \qquad\text{all $j$},\\
x_1+\cdots +x_k&=kA,\endaligned$$ is achieved when $x_i=A$ for all $i$.
We prove this by induction on $k$. The case $k=1$ is trivial, so assume $k>1$.
Suppose $x_1,\ldots ,x_k$ satisfy the restrictions given. Let $i$ be minimal such that $x_i>0$. If $i>1,$ then $$x_j'=\cases x_j&\text{if $j\neq i, i-1$},\\ x_i/2&\text{otherwise}\endcases$$ also satisfy the restrictions, and $x_1b_1+\cdots +x_nb_n\ge x_1'b_1+\cdots
+x_n'b_n$ since $b_{i-1}\le b_i$. This shows that there is a solution to our problem where $x_1>0$. On the other hand, it is well known that any solution to such a problem occurs at a vertex of the convex region determined by the restrictions. Such a vertex has $x_1=0$ or $A$, so the minimum can be achieved when $x_1=A.$
We now invoke the induction hypothesis, which says that the minimum value of $x_2b_2+\cdots +x_kb_k$ subject to the restrictions $$\aligned x_i&\ge 0 \phantom{j-1)A} \qquad\text{all $i>1$},\\
x_2+\cdots +x_j&\le (j-1)A\, \qquad\text{all $j>1$},\\
x_2+\cdots +x_k&=(k-1)A,\endaligned$$ is achieved when $x_i=A$ for all $i>1$.
Inequalities involving the height
=================================
For any factor $L_i(\uX )$ of $F(\uX )$[,]{} $\uL _i$ is proportional to a vector $\uL _i'$ with algebraic coefficients in a number field of degree no greater than $d$[,]{} and the field height $H(\uL _i')$ satisfies $H(\uL _i')\le \hofF.$ In particular[,]{} $\hofF\ge 1$[.]{}
See \[S3\] for a definition of $H(\uL )$. This is the usual field height (not absolute height) using $L_2$ norms at the infinite places.
Suppose first that $F$ is irreducible over $\bq$. It is known that $F(\uX )=aN_{K/\bq}\big (L(\uX )\big )$, where $a$ is a nonzero rational number, $K$ is a number field of degree equal to the degree of $F$ and $N_{K/\bq}$ denotes the norm from $K$ to $\bq$. Thus, any factor of $F$ is proportional to some conjugate of $L(\uX )$. The coefficient vectors of these conjugates all have the same field height (see the remark on p. 23 of \[S3\]). Further, by \[S3 Chap. III, Lemma 2A\], $\hofF={\rm cont}(F)H(\uL )$, where ${\rm cont}(F)$ denotes the content of $F$. Since the content of $F$ is a positive integer, we get $H(\uL )\le \hofF$. Since the field height function $H\ge 1$, the lemma is true when $F$ is irreducible over $\bq$.
In general, $$F(\uX )=\prod _{l=1}^kF_l(\uX ),$$ where each $F_l$ is a form with integral coefficients which is irreducible over $\bq$. Any linear factor $L_i(\uX )$ of $F$ is a factor of some $F_{l_i}$. By what we have shown, $\uL _i$ is proportional to an $\uL _i'$ with algebraic coefficients in a number field of degree no greater than the degree of $F_{l_i}$ and satisfying $H(\uL _i')\le {\Cal H}
(F_{l_i})$. The degree of $F_{l_i}$ is certainly no larger than the degree of $F$, and $$\hofF =\prod _{l=1}^k{\Cal H}(F_l).$$ We have shown that ${\Cal H}(F_l)\ge 1$ for all $l$, so $\hofF\ge{\Cal H}(F_{l_i})$ and the lemma is proven.
If $I(F)$ is not empty[,]{} then $$|NS(F)|\ge\hofF ^{-b(F)}.$$ For any $n$-tuple in $I(F)$ we have $${|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|\over\prod _{j=1}^n
\|\uL _{i_j}\|}\ge\hofF ^{-b(F)/n!}.\tag 3$$
Since $|NS(F)|$ is independent of the factorization used, we may choose any one we wish. First factor $F$ into a product of forms with integral coefficients which are irreducible over $\bq$, $$F(\uX )=\prod _{l=1}^kF_l(\uX ),$$ as in the proof of Lemma 2 above. Write each $F_l(\uX )$ as a rational multiple of a norm form as above in the proof of Lemma 2.
Since $F$ has rational coefficients, it is invariant under any element $\sigma$ of the Galois group of $\overline{\bq}$ over $\bq$, where $\overline{\bq}\subset\bc$ is the algebraic closure of $\bq$ in $\bc$. Thus, any element of the Galois group must take our factorization of $F$ to another, say $$\sigma (\uL _i)=\beta _i\uL _{\sigma '(i)},$$ where $\beta _i\in\bc ^{\times}$ and $\sigma '$ is an element of the permutation group of $\{1,\ldots ,d\}$. Also, $\sigma (S(F))$ is equal to the semi-discriminant with this factorization given by $\sigma$. For any $n$-tuple $(\uL _{i_1},\ldots ,\uL _{i_n})\in I(F)$, $$0\neq \sigma \big (\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})\big )=
\det (\uL _{\sigma '(i_1)}^{tr},\ldots ,\uL _{\sigma '(i_n)}^{tr})\times
\prod _{j=1}^n\beta _{i_j}.$$ Since $\sigma '$ is a permutation, in this manner we see that $b(\uL _i)=b(\uL _{\sigma '(i)}
)$ for any $i$. But the Galois group acts transitively on the factors of norm forms, so we conclude that $b(\uL _i)=b(\uL _j)$ whenever $L_i(\uX )$ and $L_j(\uX )$ are factors of the same irreducible $F_l(\uX )$, i.e., all the linear factors of a given $F_l$ have the same $b$ value. Let $b_l$ denote the $b$ value of the linear factors of $F_l$ for each $l=1,\ldots ,k$.
Suppose $\uL _{i_1},\ldots ,\uL _{i_{d'}}$ are the coefficient vectors of the linear factors of some $F_l$. Just like $F(\uX )$, $F_l(\uX )$ has integral coefficients and is invariant under $\sigma$. Hence, $$\prod_{j=1}^{d'} \beta _{i_j}=1
=\prod _{j=1}^{d'}\beta _{i_j}^{b_l}.$$ Taking into account the different factors $F_l$ of $F$, we are led to $$\prod _{i=1}^d\beta _i^{b(\uL _{\sigma '(i)})}=1.$$ As remarked in Section 1, this shows that the semi-discriminant $S(F)$ is the same for our initial factorization of $F$ and the factorization induced by $\sigma$. So our $S(F)$ is invariant under the Galois group of $\overline{\bq}$, and hence a rational number. It is nonzero since $I(F)$ is not empty.
Let $v$ be any place of $\overline{\bq}$. Then Hadamard’s inequality gives $${|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|_v\over\|\uL _{i_1}\|_v
\cdots \|\uL _{i_n}\|_v}\le 1,$$ where $\|\cdot\| _v$ denotes the usual $L_2$ norm if $v|\infty$ and the sup norm otherwise. In particular, $$|S(F)|_v\le \prod _{i=1}^d\|\uL _i\| ^{b(\uL _i)} _v.\tag 4$$ By the definition of $b_l$ we have $$\prod _{i=1}^dL _i(\uX )^{b(\uL _i)}=F_1(\uX )^{b_1}\cdots F_k(\uX )^{b_k}.
\tag 5$$ We let ${\bold F_l}$ denote the coefficient vector of $F_l$ for each $l=1,\ldots ,k$. If $v$ is non-archimedean, then Gauss’ lemma together with (4) and (5) gives $$|S(F)|_v\le\prod _{i=1}^d\|\uL _i\| _v^{b(\uL _i)}=\|{\bold F}_1\| _v^{b_1}
\cdots \|{\bold F}_k\|_v ^{b_k}\le 1.$$ This holds for any non-archimedean place, so $|S(F)|$ is a positive integer. In particular, $|S(F)|\ge 1$. By Lemma 3, ${\Cal H}(F_l)\ge 1$ for all $l$, so that by (5) $$\prod _{i=1}^d\|\uL _i\|^{b(\uL _i)}={\Cal H}(F_1)^{b_1}\cdots {\Cal H}(F_k
)^{b_k}\le
{\Cal H}(F_1)^{b(F)}\cdots {\Cal H}(F_l)^{b(F)}=\hofF ^{b(F)}.$$ Hence $|NS(F)|\ge\hofF ^{-b(F)}$ with this factorization of $F$.
As for (3), we note that $$|NS(F)|^{1/n!}= \prod
{|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|\over\prod _{j=1}^n
\|\uL _{i_j}\|}\ge\hofF ^{-b(F)/n!},$$ where the product is over all $n$-tuples of $I'(F)$. We saw above that each factor in this middle product is no greater than 1, thus each factor is bounded below by our lower bound for $|NS(F)|^{1/n!}.$
Bounds for linear factors
=========================
In this section our goal is to show that for any solution $\ux\in\br ^n$ of (1), there is an $n$-tuple in $I(F)$ with the product $|L_{i_1}(\ux )|\cdots |L_{i_n}(\ux )|$ relatively small. We start with a general result which says that $n$ linearly independent linear forms cannot simultaneously be small at $\ux$.
Let $\ux\in\br ^n\setminus \{\uo \}$ and let $
L_1(\uX ),\ldots ,L_n(\uX )$ be $n$ linearly independent linear forms[.]{} Suppose that $${|L_j(\ux )|\over \|\uL _j\|}\ge {|L_i(\ux )|\over \|\uL _i\|}$$ for $i=1,\ldots ,n$[.]{} Then $${|L _j(\ux )|\over\|\uL _j\|}
\ge {\|\ux\| |\det (\uL ^{tr}_1,\ldots ,\uL ^{tr}_n)|
\over n^{n/2}\prod _{i=1}^n\|\uL _i\|}.$$
Without loss of generality we may assume $\| \uL _i\| =1$ for all $i$ and $\|\ux\| =1$. Let $T$ denote the $n\times n$ matrix with rows $\uL _i$ and write $$\aligned {\frak m}&=\min _{\| \uy \| =1}\left \{\| T \uy ^{tr} \| \right \}
\qquad\text{and}\\
{\frak M}&=\max _{\|\uy\| =1}\left \{\|T \uy ^{tr}\|\right \}.\endaligned$$
Suppose $\| T\ux _1^{tr}\|={\frak m}$ and $\|\ux _1\| =1.$ Choose $\ux _2,\ldots ,
\ux _n\in \br ^n$, all of length 1, that also satisfy $|\det (\ux _1^{tr}, \ldots
,\ux _n^{tr})|=1.$ We then have $$\aligned |\det (T)|=|\det (T)||\det (\ux _1^{tr},\ldots ,\ux _n^{tr})|&=
|\det (T\ux _1^{tr},\ldots ,T\ux _n^{tr})|\\
&\le\prod _{l=1}^n\|T\ux _l^{tr}\|\\
&\le {\frak m}{\frak M}^{n-1}.\endaligned$$
Since $\| \uL _i\| =1$ for all $i$ we have ${\frak M}\le \sqrt{n}$, so that $${\frak m}\ge n^{(1-n)/2}|\det (T)|.$$ By the hypothesis, $|L_j(\ux )|\ge |L_i(\ux )|$ for all $i$, so that $$\sqrt{n}|L_j(\ux )|\ge\|T\ux ^{tr}\|\ge {\frak m}.$$ Combining these last two inequalities yields the lemma.
Suppose $I(F)$ is not empty[.]{} Then $a(F)$ is defined[.]{} If $a(F)<d/n$[,]{} then for every $\ux\in\br^n$ there is an $n$[-]{}tuple in $J(F)$ such that $${\prod _{j=1}^n|L_{i_j}(\ux )|\over|\det (\uL _{i_1}^{tr},\ldots ,
\uL _{i_n}^{tr})|}\ll \left ({|F(\ux )|\over
\|\ux \|^{d-na(F)}}\right )^{1/a(F)}\hofF^{c(F)}.\tag 6$$
Suppose $I(F)$ is not empty and let $\ux\in\br ^n.$ We define minima $\lambda _1\le\lambda _2\le\cdots\le\lambda
_n$ and choose indices $i_1,\ldots ,i_n$ as follows. Let $$\lambda _1=\min\{|L_i(\ux )|/\|\uL _i\|\},$$ where the minimum is over all factors $L_i(\uX )$ of $F(\uX )$. Choose $i_1$ such that $$|L_{i_1}(\ux )|/\|\uL _{i_1}\|=\lambda _1.$$ We then continue recursively, letting $$\lambda _{j+1}=\min\{|L_i(\ux )|/\|\uL _i\|\}\ge \lambda _j,$$ where the minimum is over all factors $L_i(\uX )$ where $\uL _i$ is not in the span of $\uL_{i_1},\ldots ,
\uL_{i_j}$, for $j=1,\ldots ,n-1$. We choose $i_{j+1}$ such that $\uL _{i_{j+1}}$ is not in the span of $\uL _{i_1},\ldots ,\uL _{i_j}$ and $$|L_{i_{j+1}}(\ux )|/\|\uL _{i_{j+1}}\|=\lambda _{j+1},$$ with the stipulation that $\uL_{i_{j+1}}$ is proportional to $\overline{\uL _{i_j}}$ if $\overline{\uL _{i_j}}$ is not in the span of $\uL _{i_1},\ldots ,\uL _{i_j}$. (Note that if this were the case, then $\lambda _{j+1}=\lambda _j$, so that such a choice for $i_{j+1}$ is possible.) These minima are well defined since $I(F)$ is not empty, implying that the set of all $\uL _i$ has rank $n$. By construction, $(\uL _{i_1},\ldots ,\uL _{i_n})\in J(F)$, so $a(F)$ is defined.
Now suppose $a(F)<d/n$ and $\ux\in\br ^n$. If $F(\ux )=0,$ then (6) trivially holds since $\lambda _1=0$. So we may as well assume $F(\ux )\neq 0$, which implies that $\lambda _1>0$. Let $a_1$ be the number of $\uL _i$ which are linearly dependent on $\uL _{i_1}$. For $j>1$ let $a_j$ be the number of $\uL _i$ which are in the span of $\uL _{i_1},\ldots ,\uL _{i_j}$ but not in the span of $\uL _{i_1},\ldots ,
\uL _{i_{j-1}}$. If $\uL _i$ is in the span of $\uL _{i_1},\ldots ,\uL _{i_j}$ but not in the span of $\uL _{i_1},\ldots ,
\uL _{i_{j-1}}$, then $|L_i(\ux )|/\|\uL _i\|\ge\lambda _j$ by the definition of $\lambda _j$. Thus, $${|F(\ux )|\over \hofF}=\prod _{i=1}^d{|L_i(\ux )|\over \|\uL _i\|}\ge
\prod _{j=1}^n\lambda _j^{a_j}.\tag 7$$
By definition, $a_1+\cdots +a_j$ is the number of $\uL _i$ in the span of $\uL _{i_1},\ldots ,\uL _{i_j}$ for $1\le j\le n$. This implies that $$a_1+\cdots +a_j\le ja(F)\qquad 1\le j<n\tag 8$$ and $a_1+\cdots +a_n=d$. Let $s=a_1+\cdots +a_{n-1}$, so $s\le (n-1)a(F)$ by (8). Since $\lambda _n\ge \lambda _{n-1},$ we see that $$\lambda _n^{a_n}\lambda _{n-1}^{a_{n-1}}\ge\lambda _n^{a_n-((n-1)a(F)-s)}
\lambda _{n-1}^{a_{n-1}+((n-1)a(F)-s)}.$$ Define $a_j'$ by $$a_j'=\cases a_n-((n-1)a(F)-s)=d-(n-1)a(F)&\text{for $j=n$,}\\
a_{n-1}+((n-1)a(F)-s)&\text{for $j=n-1$,}\\
a_j&\text{otherwise.}\endcases$$ Then (7) and (8) hold with $a_j'$ in place of $a_j$, and also $$a_1'+\cdots +a_{n-1}'=(n-1)a(F).\tag 9$$
Because of (8) and (9), we can use Lemma 1 with $k=n-1$, $b_j=\log\lambda _j$ and $A=a(F)$. We get $$\prod _{j=1}^{n-1}\lambda _j^{a_j'}\ge\prod _{j=1}^{n-1}\lambda _j^{a(F)}.$$ This and (7) imply that $$\aligned {|F(\ux )|\over \hofF}=\prod _{i=1}^d{|L_i(\ux )|\over \|\uL _i\|}&
\ge
\prod _{j=1}^n\lambda _j^{a_j'}\\
&=
\lambda _n^{d-na(F)+a(F)}\prod _{j=1}^{n-1}\lambda _j^{a_j'}\\
&\ge
\lambda _n^{d-na(F)}\prod _{j=1}^n\lambda _j^{a(F)}\\
&=\lambda _n^{d-na(F)}
\left (\prod _{j=1}^n{|L_{i_j}(\ux )|\over \|\uL _{i_j}\|}\right )^{a(F)}.
\endaligned$$ By Lemma 4, $$\lambda _n={|L_{i_n}(\ux )|\over \|\uL _{i_n}\|}\gg \|\ux\|
{|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|\over
\prod _{j=1}^n\|\uL _{i_j}\|}.$$ So by (3) $$\aligned {|F(\ux )|\over \hofF}&\ge
\lambda _n^{d-na(F)}
\left (\prod _{j=1}^n{|L_{i_j}(\ux )|\over \|\uL _{i_j}\|}\right )^{a(F)}\\
&\gg \|\ux\|^{d-na(F)}
\left ({|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|\over
\prod _{j=1}^n\|\uL _{i_j}\|}\right )^{d-na(F)}
\left (\prod _{j=1}^n{|L_{i_j}(\ux )|\over \|\uL _{i_j}\|}\right )^{a(F)}\\
&= \|\ux\|^{d-na(F)}
\left ({|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|\over
\prod _{j=1}^n\|\uL _{i_j}\|}\right )^{d-(n-1)a(F)} \\
&\qquad \times\
\left ({\prod _{j=1}^n|L_{i_j}(\ux )|\over
|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|}
\right )^{a(F)}\\
&\ge \|\ux \|^{d-na(F)}\hofF ^{-b(F)\big (d-(n-1)a(F)\big )/n!}
\left ({\prod _{j=1}^n|L_{i_j}(\ux )|\over
|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|}
\right )^{a(F)}.\endaligned$$ This proves (6) in the case where the discriminant is zero.
When the discriminant is not zero we can do somewhat better. First of all, we have $a(F)=1$. Letting $L_{i_1},\ldots ,L _{i_n}$ be as above, we see that $${|F(\ux )|\over \hofF}\ge \prod _{l\neq i_1,\ldots ,i_n}
{|L_l(\ux )|\over \|\uL _l\|}\times
\prod _{j=1}^n{|L_{i_j}(\ux )|\over \|\uL _{i_j}\|}.$$ By Lemma 4, $$\aligned {|F(\ux )|\over \hofF}&\gg
\prod _{l\neq i_1,\ldots ,i_n}
{\|\ux\|\cdot |\det (\uL _l^{tr},\uL _{i_1}^{tr},\ldots ,\uL _{i_{n-1}}^{tr})|
\over\|\uL _l\|\cdot \|\uL _{i_1}\|\cdots \|\uL _{i_{n-1}}\|}
\prod _{j=1}^n{|L_{i_j}(\ux )|\over \|\uL _{i_j}\|}\\
&=\|\ux\| ^{d-n}
\prod _{l\neq i_1,\ldots ,i_{n-1}}
{|\det (\uL _l^{tr},\uL _{i_1}^{tr},\ldots ,\uL _{i_{n-1}}^{tr})|
\over\|\uL _l\|\cdot \|\uL _{i_1}\|\cdots \|\uL _{i_{n-1}}\|}\times
{\prod _{j=1}^n|L_{i_j}(\ux )|\over |\det(\uL _{i_1}^{tr},\ldots ,\uL _{i_n}
^{tr})|}.\endaligned$$ As with (3), Hadamard’s inequality and our bound for $|NS(F)|$ in Lemma 3 give $$\prod _{l\neq i_1,\ldots ,i_{n-1}}
{|\det (\uL _l^{tr},\uL _{i_1}^{tr},\ldots ,\uL _{i_{n-1}}^{tr})|
\over\|\uL _l\|\cdot \|\uL _{i_1}\|\cdots \|\uL _{i_{n-1}}\|}\ge |NS(F)|^{1/n!}
\ge
\hofF ^{-b(F)/n!}.$$ Since the discriminant is not zero, each $\uL _i$ occurs in the same number of$n$-tuples in $I(F)$, i.e., $b(\uL _i)=b(F)$ for each $i$. Hence $$db(F)=\sum _{i=1}^db(\uL _i)=n|I(F)|.$$ By (2) then, $${b(F)\over n!}={n|I(F)|\over dn!}={n\over d}{d\choose n}={d-1\choose n-1}.$$ This proves the case when the discriminant is not zero.
The estimate in Lemma 5 is not so good when $\hofF$ is large in comparison to $m$ or $\|\ux\|$. In such a situation we will use the following, which generalizes \[S3 Chap. IV, Lemma 6A\].
Suppose $I(F)$ is not empty and $\hofF$ is minimal among forms equivalent to $F$[.]{} Suppose further that $F$ does not vanish at any nonzero integral point[.]{} Then for every $\ux\in\br ^n$ there is an $n$[-]{}tuple in $I'(F)$ with $${|F(\ux )|^{n/d}\over \hofF ^{1/d}}
\gg {\prod _{j=1}^n
|L_{i_j} (\ux )|\over |\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|}.$$
If $F(\ux )=0$ the statement is trivial, so assume otherwise. By homogeneity of the quantities $${\prod _{j=1}^n|L_{i_j} (\ux )|\over
|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|},$$ we may use any factorization of $F$. Let $F(\uX )=\prod _{l=1}^kF_l(\uX )=\prod _{i=1}^d
L_i(\uX )$ be the factorization of $F$ in the proof of Lemma 3, and introduce a new factorization $F(\uX )=\prod _{i=1}^dL_i'(\uX )$ given by $$L_i'(\uX )={|F(\ux )|^{1/d}\over |L_i(\ux )|}L_i(\uX )$$ for each $i$. By hypothesis, $${\Cal H}(F\circ T)\ge\hofF =\prod _{i=1}^d\|\uL '_i\|\tag 10$$ for any $T\in\gln (\bz )$.
There are $r_1$ real linear factors and $r_2$ pairs of complex conjugate linear factors of $F$, say. Arrange the indices so that $\uL '_i\in\br ^n$ for $i\le r_1,$ $\uL '_i\in\bc ^n$ for $r_1<i\le d=r_1+2r_2$ and $\uL '_{i+r_2}=\overline{
\uL '_i}$ for $r_1<i\le r_1+r_2$. Let $\be ^d\subset\br ^{r_1}\oplus
\bc ^{2r_2}$ be the set of $\ux =(x_1,\ldots ,x_d)$ where $x_{i+r_2}=\overline{x_i}$ for $r_1<i\le r_1+
r_2.$ Then $\be ^d$ is $d$-dimensional Euclidean space via the usual hermitian inner product on $\bc ^d$.
Let $M$ be the $d\times n$ matrix given by $$M:=\pmatrix \uL '_1\\ \vdots \\ \uL '_d\endpmatrix =
(\um _1^{tr},\ldots ,\um _n^{tr}).$$ Then $\um _j\in\be ^d$ for all $1\le j\le
n.$ Moreover, $$\|\wedge _{j=1}^n\um _j\| ^2=
\sum _{I'(F)}
|\det\big ((\uL '_{i_1})^{tr},\ldots ,(\uL '_{i_n})^{tr}\big )|^2,\tag 11$$ where the sum is over all $n$-tuples in $I'(F)$. The interplay between (10) and (11) which deal with lengths of the rows and columns of $M$, respectively, will be used to get our result.
Let $\lambda _1\le\cdots \le\lambda _n$ be the successive minima of the $n$-dimensional lattice $\Lambda =\oplus _{j=1}^n\bz\um _j\subset\be ^d$ with respect to the unit ball. Then by Minkowski’s theorem, $$\lambda _1^2\cdots\lambda _n^2 \ll\det (\Lambda )^2=
\|\wedge _{j=1}^n\um _j\|^2.\tag 12$$ We need a lower bound for $\lambda _1^2\cdots \lambda _n^2$. We first get a lower bound on $\lambda _1^2\cdots\lambda _{n-1}^2$. We then get a lower bound on $\lambda _n$ and finish the proof.
Let $\uz _1,\ldots ,\uz _n$ be a basis for $\Lambda$ satisfying $\|\uz _j\|\le
j\lambda _j$ for each $j$. Write $$MT=(\uz _1^{tr},\ldots ,\uz _n^{tr}),$$ where $$T=(\ua _1^{tr},\ldots ,\ua _n^{tr})\in\gln (\bz ),$$ and write $\uz _j=(z_{j,1},\ldots ,z_{j,d})$ for $1\le j\le n$. Note that $z_{j,i}=L_i'(\ua _j)$ for $j=1,\ldots ,n$ and $i=1,\ldots ,d$.
Since $F(\ua _1)\neq 0$ by construction, we have $|F(\ua _1)|\ge 1.$ The arithmetic-geometric inequality thus gives $$\aligned (\lambda _1)^2\ge\|\uz _1\|^{2}&\ge
d\left (\prod _{j=1}^{d}|z_{1,i_j}|^2\right )^{1/d}\\
&=
d\left (\prod _{j=1}^{d}|L'_{i_j}(\ua _1)|\right )^{2/d}\\
&=
d\left (|F(\ua _1)|\right )^{2/d}\\
&\ge d.\endaligned$$ In particular, $$\prod _{j=1}^{n-1}\lambda _j^2\ge\lambda _1^{2(n-1)}\ge 1.\tag 13$$
We need a better bound for $\lambda _n$. For this, we use another application of the arithmetic-geometric inequality together with (10), getting $$\aligned
n^3\lambda _n^2\ge\sum _{j=1}^n(j\lambda _j)^2\ge
\sum _{j=1}^n\|\uz _j\|^2
&=\sum _{j=1}^n\sum _{i=1}^d|z_{j,i}|^2\\
&=\sum _{i=1}^d\sum _{j=1}^n|z_{j,i}|^2\\
&=\sum _{i=1}^d\|(z_{1,i},\ldots ,z_{n,i})\|^2\\
&\ge d\left (\prod _{i=1}^d\|(z_{1,i},\ldots ,z_{n,i})\|^2\right )^{1/d}\\
&= d\left ({\Cal H}(F\circ T)\right )^{2/d}\\
&\ge d(\hofF )^{2/d}.
\endaligned$$
Our bound for $\lambda _n$ together with the bound (13) yields $$\prod _{j=1}^n\lambda _j^2\ge n^{-3}d\hofF^{2/d}.$$ By (11) and (12), we get $$\sum _{I'(F)}
|\det\big ((\uL '_{i_1})^{tr},\ldots ,(\uL '_{i_n})^{tr}\big )|^2
\gg \hofF ^{2/d}.$$ There are no more than ${d\choose n}$ summands here by (2). The largest summand thus satisfies $$|\det \big ((\uL _{i_1}')^{tr},\ldots ,(\uL _{i_n}')^{tr}\big )|
\gg \hofF ^{1/d}.$$ Finally, we have $$|\det \big ((\uL _{i_1}')^{tr},\ldots ,(\uL _{i_n}')^{tr}\big )|
={|\det (\uL _{i_1}^{tr},\ldots ,\uL _{i_n}^{tr})|\cdot |F(\ux )|^{n/d}\over
\prod _{j=1}^n
|L_{i_j}(\ux )|}.$$
-6pt
Auxiliary results
=================
-6pt
By Lemmas 5 and 6, any solution to (1) satisfies an inequality of the form $${\prod _{j=1}^n|L_{i_j}(\ux )|\over|\det (\uL _{i_1}^{tr},\ldots ,
\uL _{i_n}^{tr})|}\ll A,$$ where $A$ is some given bound. Our goal here is to get information on the solutions to such inequalities. Specifically, we show that such solutions lie in convex sets. Further, given bounds for the lengths of such solutions considered, there are upper bounds for the number of such convex sets. Lastly, we determine upper bounds for both the volume and the number of integral points in such convex sets.
Let $K _1(\uX ),\ldots ,K _n(\uX )\in\bc [\uX ]$ be $n$ linearly independentlinear forms in $n$ variables[.]{} Denote the corresponding coefficient vectors by$\uK _1,\ldots ,\uK _n$. Let $A,B,C>0$ with $C>B$ and let $D>1$[.]{} Consider the set of $\ux\in\br ^n$ satisfying $${\prod _{i=1}^n|K_i(\ux )|\over |\det(\uK _1^{tr},\ldots ,\uK _n^{tr}
)|}\le A\tag 14$$ and also $B\le\|\ux\|\le C.$ If $BC^{n-1}\ge D^{n-1}n!n^{n/2}A$[,]{} then this set lies in the union of less than $$n^3\left (\log _D\big (BC^{n-1}/n!n^{n/2}A\big )\right )^{n-2}$$ convex sets of the form $$\gather
\{\uy\in\br ^n\: |K_i'(\uy )|\le a_i\ \text{for $i=1,\ldots ,n$}\}, \\
|\det \big ((\uK _1')^{tr},\ldots ,(\uK _n')^{tr}\big )|=1,\tag 15 \\
\|\uK '_i\| =1\qquad i=1,\ldots ,n,\endgather$$ with $$\prod _{i=1}^na_i<D^nn!n^{n/2}{CA\over B}.$$ If $BC^{n-1}<D^{n-1}n!n^{n/2}A,$ then this set lies in the union of no more than $n!$ convex sets of this form[.]{}
The proof of \[S3 Chap. IV, Lemma 7A\] shows that the solutions to (14) can be partitioned into $n!$ subsets, and for each such subset there exist pairwise orthogonal linear forms $K '_1(\uX),\ldots ,K ' _n(\uX)$ (these depend on the subset) such that all solutions $\ux$ in that subset satisfy $$[\_[i=1]{}\^n|K\_i’()||((’\_1)\^[tr]{},…,(\_n’) \^[tr]{})|]{}n!A.$14'$ $$ After possibly rescaling, we may assume that $\|\uK '_i\|=1$ for each $i$. This implies the modulus of the determinant is 1 as well.
Let $\ux$ be a solution to ($14'$) of length at least $B$. By Lemma 4, for some $i_0$ (depending on $\ux$, of course) we have $|K_{i_0}(\ux )|\ge n^{-n/2}B$. This leaves us with $$\prod _{i\neq i_0}|K_i'(\ux )|\le {n!n^{n/2}A\over B}.$$
Write $|K_i'(\ux )|=D^{-n_i}C$ for each $i\neq i_0$. If $\|\ux\|\le C,$ then $n_i\ge 0$, and by the above estimate $\sum _{i\neq i_0}
n_i\ge\log _D\big (BC^{n-1}/n^{n/2}n!A\big )$. Let $[\cdot]$ denote the greatest integer function. Then $$\sum_{i\neq i_0}[n_i]>\sum _{i\neq i_0}(n_i-1)\ge
\log _D\left ({BC^{n-1}\over D^{n-1}n!n^{n/2}A}\right ).$$ For the time being, denote the quantity $\log _D\left ({BC^{n-1}\over D^{n-1}n!n^{n/2}A}\right )$ by $Q$. If $Q\ge 0$, then we can find nonnegative integers $z_i\le [n_i]$ for each $i\neq i_0$ that satisfy $\sum _{i\neq i_0}z_i=[Q]$. Further, our solution $\ux$ satisfies $|K _i'(\ux )|\le D^{-z_i}C$ for $i\neq i_0$ since $-z_i\ge -n_i$. To make the notation uniform, we set $z_{i_0}=0$, so that $|K _i'(\ux )|\le D^{-z_i}C$ for all $i$. If $Q<0$, then we simply set all $z_i=0$.
Summarizing what we have accomplished so far, we see that the solutions $\ux$ to (14) with $B\le\|\ux\|\le C$ lie in the union of $n!$ subsets, and for each such subset there are pairwise orthogonal linear forms $K_1'(\uX),
\ldots ,K_n'(\uX )$ with $\|\uK _i'\|=1$ such that all solutions in that subset lie in convex sets of the form $$\{\uy\in\br ^n\: |K _i'(\uy )|\le D^{-z_i}C\ \text{for all $i$}\},$$ where the $z_i$’s are nonnegative integers, at least one of which is 0, satisfying $$\sum _{i=1}^nz_i=\max \{[Q],0\}.$$ Letting $a_i =D^{-z_i}C$, we have $$\prod _{i=1}^na_i\le C^nD^{-[Q]}<C^nD^{1-Q}=D^nn!n^{n/2}{CA\over B}.$$
It remains to estimate the number $n$-tuples $(z_1,\ldots ,z_n)$ which satisfy the above conditions. Towards that end, for a nonnegative integer $a$ denote the number of $n$-tuples $(z_1,\ldots ,z_n)\in\bz ^n$ satisfying $z_i\ge 0$ and $\sum z_i=a$ by $f(n,a)$. Clearly $f(1,a)=1$. We claim that $$f(n,a)\le {(a+n-1)^{n-1}\over (n-1)!}.$$ We see this by induction on $n$. Assuming $n\ge 2$ and our claim is true for $n-1$, $$\aligned \noalign{\vskip5pt} f(n,a)=\sum _{i=0}^af(n-1,a-i)
&\le\sum _{i=0}^a{(a-i+n-2)^{n-2}\over (n-2)!}\\ \noalign{\vskip5pt}
&=\sum _{j=0}^a{(j+n-2)^{n-2}\over (n-2)!}\\ \noalign{\vskip5pt}
&\le {1\over (n-2)!}\int_0^{a+1}(x+n-2)^{n-2}\, dx\\ \noalign{\vskip5pt}
& \le {(a+n-1)^{n-1}\over (n-1)!}.\endaligned$$
Suppose $Q\ge 0$. Then by this claim the number of $n$-tuples $(z_1,\ldots ,z_n)$ of nonnegative integers with $z_{i_0}=0$, say, satisfying $\sum _{i=1}^nz_i=[Q]$ is no greater than $([Q]+n-2)^{n-2}/(n-2)!$. (When $z_{i_0}=0$, we use the case $n-1$ of our claim.) Taking into account the $n$ different possibilities for $i_0$, we see that the total number of possible $n$-tuples we must consider is no greater than $n([Q]+n-2)^{n-2}/(n-2)!<n(Q+n-1)^{n-2}/(n-2)!.$ Of course, if $Q<0$ we have the one $n$-tuple where $z_i=0$ for all $i$.
Now if $Q\ge 0$, we have $${n(Q+n-1)^{n-2}\over (n-2)!}={n\big (\log _D(BC^{n-1}/n!n^{n/2}A)\big )^{n-2}
\over (n-2)!}.$$ Also, $Q\ge 0$ if and only if $BC^{n-1}\ge D^{n-1}n!n^{n/2}A$. Taking into account the $n!$ different subsets and using $n\cdot n!/(n-2)!<n^3$ completes the proof.
We will also use the following variation of Lemma 7, which does away with the lower bound condition $\|\ux\|\ge B$ at the expense of a higher power of the logarithmic term in the number of convex sets.
Let $K _1(\uX ),\ldots ,K _n(\uX )\in\bc [\uX ]$ and $\uK _1,\ldots ,\uK _n$ be as in Lemma [7.]{} Let $A,C>0$ and $D>1$[.]{} If $C^{n}\ge D^nn!A$[,]{} then the solutions $\ux$ to [(14)]{} with $\|\ux\|\le C$ lie in the union of less than $$n\left (\log _D\big (C^n/n!A\big )\right )^{n-1}$$ convex sets of the form [(15)]{} with $$\prod _{i=1}^na_i<D^nn!A.$$ If $C^n<D^nn!A,$ then such solutions lie in the union of no more than $n!$ convex sets of this form[.]{}
The proof goes essentially the same way as for Lemma 7. The difference is that we do not invoke Lemma 4. Again we have $n!$ subsets where all solutions in the subset satisfy (14$'$). Let $\ux$ be such a solution with$\|\ux\|\le C$ and write $|K_i'(\ux )|=D^{-n_i}C$ with $n_i\ge 0$ for each $i$. This time we have $\sum _{i=1}^n
n_i\ge\log _D\big (C^n/n!A\big )$, so that $$\sum_{i=1}^n[n_i]>\log _D\left ({C^n\over D^nn!A}\right ).$$ This time denote the quantity $\log _D\left ({C^n\over D^nn!A}\right )$ by $Q$. As before, if $Q\ge 0$, then we can find nonnegative integers $z_i\le [n_i]$ for each $i$ that satisfy $\sum _{i=1}^nz_i=[Q]$ and our solution $\ux$ satisfies $|K _i'(\ux )|\le D^{-z_i}C$ for all $i$. If $Q<0$ we set all $z_i=0$ again, so that $$\sum _{i=1}^nz_i=\max \{[Q],0\}.$$ Set $a_i=D^{-z_i}C$ again.
We are now in the same position as with Lemma 7. The difference is that here we do not say one of the exponents $z_{i_0}$ is zero, and $$\prod _{i=1}^na_i\le C^nD^{-[Q]}<C^nD^{1-Q}=D^nn!A.$$ Using the claim in the proof of Lemma 7, the number of $n$-tuples $(z_1,\ldots ,z_n)$ of nonnegative integers satisfying $\sum _{i=1}^nz_i=[Q]$ is no greater than $$([Q]+n-1)^{n-1}/(n-1)!<(Q+n)^{n-1}/(n-1)!$$ when $Q\ge 0.$ Using $${(Q+n)^{n-1}\over (n-1)!}={\big (\log _D(C^n/n!A)\big )^{n-1}
\over (n-1)!}$$ and $Q\ge 0$ if and only if $C^n\ge D^nn!A$ completes the proof.
We need estimates for the number of integer points and also the volume of the set of all points in convex sets of the form (15). (When the $K_i(\uX )$s are real linear forms these convex sets are simply parallelopipeds.) The following lemmas will provide the needed estimates.
Let ${\Cal C}\subset\br ^n$ be a convex body [(]{}convex[,]{} closed[,]{} bounded and symmetric about the origin[)]{} and let $\Lambda\subset\br ^n$ be a lattice[.]{} Suppose there are $n$ linearly independent lattice points in ${\Cal C}$[.]{} Then there are $\uy _1,\ldots ,\uy _n\in {\Cal C}$ such that the number of lattice points in ${\Cal C}$ is no greater than $$3^n2^{n(n-1)/2}{|\det (\uy _1^{tr},\ldots ,\uy _n^{tr})|
\over \det (\Lambda )}.$$
By the homogeneity of the upper bound here we may assume$\Lambda =\bz ^n$.
The proof is by induction on $n$. If $n=1$, then ${\Cal C}$ is an interval centered at the origin, say $[-y_1,y_1]$. Since ${\Cal C}$ contains a nonzero integer point by hypothesis, we have $y_1\ge 1$. Thus, the number of integer points in ${\Cal C}$ is no greater than $2y_1 +1\le 3y_1.$
Now assume $n>1$ and let $\uz _1,\ldots \uz _n$ be $n$ linearly independent integer points in ${\Cal C}$. Let $V$ be the span of the first $n-1$ of them and let $\Lambda ^-=\bz ^n\cap V.$ Then $\Lambda ^-$ is a primitive sublattice and there is a $\uz _n'\in\bz ^n$ with $\bz ^n=\Lambda ^-\oplus \bz\uz _n'$.
Any integer point $\uz$ in ${\Cal C}$ may be written as a sum $\uz =\uz ^-+a\uz _n'$ where $\uz ^-\in\Lambda ^-$ and $a\in\bz$. Further, since $\uz _n$ is an integer point in ${\Cal C}$ but not in $V$, we see that $a\neq 0$ is possible here. By Cramer’s rule $$|a|={|\det \big ((\uz _1')^{tr},\ldots, (\uz _{n-1}')^{tr},\uz ^{tr}\big )|
\over |\det\big ((\uz _1')^{tr},\ldots, (\uz _n')^{tr}\big )
|}=|\det \big ((\uz _1')^{tr},\ldots ,(\uz _{n-1}')^{tr},\uz ^{tr}\big )|,$$ where $\uz _1',\ldots ,\uz _{n-1}'$ form a basis for $\Lambda ^-$ (so that $\uz _1'\ldots ,\uz _n'$ is a basis for $\bz ^n$).
For any $a$ we estimate the number of $\uz ^-\in\Lambda ^-$ with $\uz ^-+a\uz _n'\in {\Cal C}$ as follows. Let $\{\uz _1^-,\ldots ,\uz _N^-\}$ be the set of all such $\uz ^-$. Then the set of differences $(\uz _i^- +a\uz _n')-(\uz _1
^- +a\uz _n')$ is a set of $N$ distinct integer points in $\Lambda ^-\cap
2{\Cal C}$ by convexity. Note that $\Lambda ^-$ contains $n-1$ linearly independent lattice points in ${\Cal C}\cap V$, namely $\uz _1,\ldots ,
\uz _{n-1}$. Thus, by the induction hypothesis there are $\uy _1^-,
\ldots ,\uy _{n-1}^-\in 2{\Cal C}\cap V$ such that the number of $\uz ^-\in
\Lambda ^-$ with $\uz ^-+a\uz_n'\in {\Cal C}$ is no greater than $$3^{n-1}2^{(n-1)(n-2)/2}
{\|\uy _1^-\wedge\cdots\wedge\uy _{n-1}^-\|\over \det (\Lambda ^-)}
.$$ The important thing to note here is the uniformity of this bound; it does not depend on $a$.
Now let $|a_0|$ be maximal such that there is a $\uz ^-\in\Lambda ^-$ with $\uz ^-+a_0\uz _n'\in {\Cal C}$ and let $\uy _n$ be this lattice point in $
{\Cal C}$. Let $\uy _i={1\over 2}\uy _i ^-\in {\Cal C}$ for $i=1,\ldots ,n-1$. We then have $2|a_0|+1$ possible values of $a$ to consider above, and we now see that the number of integer points in ${\Cal C}$ is no greater than $$\aligned & 3^{n-1}2^{(n-1)(n-2)/2}
{\|\uy _1^-\wedge\cdots\wedge\uy _{n-1}^-\|\over \det (\Lambda ^-)}
\big (2|a_0|+1\big )\\
&\qquad \le\ 3^{n-1}2^{(n-1)(n-2)/2}
{\|\uy _1^-\wedge\cdots\wedge\uy _{n-1}^-\|\over \det (\Lambda ^-)}
3|a_0|\\
&\qquad =\ 3^n2^{n(n-1)/2}{\|\uy _1\wedge\cdots\wedge\uy _{n-1}\|\over \det (\Lambda ^-)}
|\det \big ((\uz _1')^{tr},\ldots ,(\uz _{n-1}')^{tr},(\uy _n)^{tr}\big )|\\
&\qquad =\ 3^n2^{n(n-1)/2}
|\det (\uy _1^{tr},\ldots ,\uy _n^{tr})|.\endaligned$$
Though we do not need it, the proof of Lemma 8 actually shows that the $\uy _i$’s in ${\Cal C}$ satisfy $2^{n-i}\uy _i\in\Lambda$ as well.
Let ${\Cal C}$ be a convex body of the form [(15).]{} Then either all integral points in ${\Cal C}$ lie in a proper subspace[,]{} or the number of such points is no greater than
${\displaystyle 3^n2^{n(n-1)/2}n!\prod _{i=1}^na_i.}$
The volume of ${\Cal C}$ is no greater than
${\displaystyle 2^nn!\prod _{i=1}^na_i.}$
Choose $\uy _1,\ldots ,\uy _n\in {\Cal C}$ with $|\det
(\uy _1^{tr},
\ldots ,\uy _n^{tr})|$ maximal (this is clearly possible since ${\Cal C}$ is bounded). Let ${\Cal P}$ be the region $${\Cal P}=\{\uy =a_1\uy _1+\cdots +a_n\uy _n\: |a_i|\le 1\
\text{for all $i$}\}.$$We claim that ${\Cal P}\supseteq {\Cal C}$. Indeed, if there were a $\uy _0\in {\Cal C}\setminus
{\Cal P}$, then without loss of generality $\uy _0=\sum c_i\uy _i$ with $c_1>1$. But then $$|\det (\uy _0^{tr},\uy _2^{tr},\ldots ,\uy _n^{tr})|=|\det \big ((c_1\uy _1)
^{tr},\uy _2^{tr},\ldots ,\uy _n^{tr}\big )|
>|\det (\uy _1^{tr},\ldots ,\uy _n^{tr})|,$$ which contradicts the assumption on the $\uy _i$’s. Since the volume of ${\Cal P}$ is $2^n|\det (\uy _1^{tr},\ldots ,\uy _n^{tr})|,$ we see that there exist $\uy _1,\ldots ,\uy _n\in{\Cal C}$ with $$2^n|\det (\uy _1^{tr},\ldots ,\uy _n^{tr})|\ge \text{Vol}({\Cal C}).$$
Finally, if we denote the $n\times n$ matrix with rows $\uK _1',\ldots ,
\uK _n'$ by $T$, then for any $\uy _1,\ldots ,\uy _n\in{\Cal C}$ we have $$\aligned |\det (\uy _1^{tr},\ldots ,\uy _n^{tr})|=
|\det (\uy _1^{tr},\ldots ,\uy _n^{tr})|\times
|\det (T)|&=|\det\big (T\uy _1^{tr},\ldots ,T\uy _n^{tr}\big )|\\
&\le n!\prod _{i=1}^n\max _{1\le j\le n}\{|K_i'(\uy _j )|\}\\
&\le n!\prod _{i=1}^na_i.\endaligned$$ Lemma 9 follows from this estimate, the estimate given above, and Lemma 8.
-6pt
The infinite volume case
========================
-6pt
This section is devoted entirely to showing that the volume $V(F)$ is infinite if $a(F)$ is undefined or at least $d/n$. This is one half of the proposition. We will also show that if $a(F)$ is undefined or at least $d/n$, then (1) has infinitely many integral solutions for $m$ sufficiently large. Since none of this depends on the particular factorization of $F$ used, we’ll assume that $\overline{L _i(\uX )}$ is a factor for all $i$, i.e., the complex linear factors occur in conjugate pairs. We break up our argument into a series of three lemmas.
If $a(F)$ is undefined or at least $d/n$[,]{} then there is a $k<n$ and $k$ coefficient vectors $\uL _{i_1},\ldots ,\uL _{i_k}$ which satisfy the following conditions[:]{}
[1)]{} they are linearly independent[;]{}
[2)]{} there are at least $kd/n$ coefficient vectors $\uL _i$ in their span[;]{}
[3)]{} for all indices $j$[,]{} if $\overline{\uL _{i_j}}$ is not in the span of $\uL _{i_1},\ldots ,\uL _{i_j}$[,]{} then $j<k$ and $\uL _{i_{j+1}}=
\overline {\uL _{i_j}}.$
Suppose first that $a(F)$ is undefined. Then by Lemma 5 $I(F)$ is empty, i.e., the rank of $(\uL _1^{tr},\ldots ,\uL _d^{tr})$ is less than $n$. Let $k$ be this rank. Choose an $\uL _{i_1}$. If $k=1$, then all $\uL _i$, in particular $\overline{\uL _{i_1}}$, are in the span of $\uL _{i_1}$. If $k>1$, then choose an $\uL _{i_2}$ which is linearly independent of $\uL _{i_1}$, with the stipulation that $\uL _{i_2}=\overline{\uL _{i_1}}$ if this is a possible choice. Continue on in this fashion, getting $\uL _{i_1},\ldots ,\uL _{i_k}$. They satisfy conditions 1 and 3 by construction. There are $d>kd/n$ factors in their span, so condition 2 is satisfied as well.
Now suppose that $a(F)$ is defined and at least $d/n$. Then there is an $n$-tuple $(\uL _{i_1},\ldots ,\uL _{i_n})\in J(F)$ and a $j<n$ where $\uL _{i_1},\ldots ,\uL _{i_j}$ have at least $jd/n$ coefficient vectors in their span (by the definition of $a(F)$). Let $j_0$ be the least such index where this is true. By the definition of $J(F)$, if $\overline{\uL _{i_{j_0}}}$ is in the span of $\uL _{i_1},\ldots ,\uL _{i_{j_0}}$, then these $j_0$ coefficient vectors satisfy all three conditions above with $k=j_0$.
Suppose $\overline{\uL _{i_{j_0}}}$ is not in the span of $\uL _{i_1},\ldots ,\uL _{i_{j_0}}$. If $j_0=1$, then there are at least $d/n$ coefficient vectors $\uL _i\not\in\br ^n$ proportional to $\uL _{i_1}$ and at least $d/n$ additional coefficient vectors $\overline{\uL _i}$ proportional to $\overline{\uL _{i_1}}$. In this case we let $k=2$ and use $\uL _{i_1}$ and $\overline{\uL _{i_1}}$. (Note that $n>2$ since $F$ is assumed not to be a power of a positive definite quadratic form in two variables.) Now suppose $j_0>1$. Note that condition 3 is still satisfied for all $j<j_0$ by the definition of $J(F).$ Also, by the minimality of $j_0$, there are fewer than $(j_0-1)d/n$ coefficient vectors in the span of $\uL _{i_1},\ldots ,\uL _{i_{j_0-1}}$. Consider for a moment the collection of $\uL _i$ which are not in the span of these $j_0-1$ coefficient vectors, but are in the span of $\uL _{i_1},\ldots ,\uL _{i_{j_0}}.$ We could replace $\uL _{i_{j_0}}$ with any of these and the span would remain the same. If $\overline{\uL _i}$ is in the span of $\uL _{i_1},\ldots ,\uL _{i_{j_0}}$ for one of these $\uL _i$, then we replace $\uL _{i_{j_0}}$ with $\uL _i$ and let $k=j_0$ as above. If not, then there are more than $(j_0d/n) - (j_0-1)d/n =d/n$ of these $\uL _i$, so there are more than $d/n$ coefficient vectors $\overline{\uL _i}$ which are not in the span of $\uL _{i_1},\ldots ,\uL _{i_{j_0}}$. This shows that $j_0d/n$ must be less than $d-(d/n)=(n-1)d/n$, i.e., $j_0<n-1$. In this case we let $k=j_0 +1<n$ and let $\uL _{i_{k}}=\overline{\uL _{i_{j_0}}}$. Then conditions 1 and 3 are satisfied. Further, in addition to the at least $j_0d/n$ coefficient vectors in the span of $\uL _{i_1},\ldots ,\uL _{i_{j_0}}$, we have more than $d/n$ additional coefficient vectors $\overline{\uL _i}$ in the span of $\uL _{i_1},\ldots ,\uL _{i_{j_0}},\uL _{i_k}$. This shows that condition 2 holds as well.
Suppose $a(F)$ is either undefined or at least $d/n$[.]{} Let $k$ and $\uL _{i_1},\ldots ,\uL _{i_k}$ be as in Lemma [10.]{} Then there are linearly independent $\uK _1,\ldots ,\uK _k\in\br ^n$ which share the same span as $\uL _{i_1},\ldots ,\uL _{i_k}$[.]{}
Suppose $0\le l<k$ and $\uK _1,\ldots ,\uK _l\in\br ^n$ have been chosen so that their span is equal to the span of $\uL _{i_1},
\ldots ,\uL _{i_l}.$
If $\overline{\uL _{i_{l+1}}}$ is in the span of $\uL _{i_1},\ldots ,\uL _{i_{l+1}}$, then it is in the span of $\uK _1,\ldots , \uK _l,\break \uL _{i_{l+1}}$. In this case write $$\overline{\uL _{i_{l+1}}}=(a+ib)\uL _{i_{l+1}}+\uz,$$ where $a,b\in\br$ and $\uz\in\bc ^n$ is in the span of $\uK _1,\ldots ,\uK _l.$ Note that both the real and imaginary parts of $\uz$ are in the span of $\uK _1,\ldots ,\uK _l$ since the $\uK _i$s are real. A short computation shows that $$\aligned \Re (\uz )+(a-1)\Re (\uL _{i_{l+1}})&=b\Im (\uL _{i_{l+1}})\\
\Im (\uz ) +b\Re (\uL _{i_{l+1}})&=-(a+1)\Im (\uL _{i_{l+1}}).\endaligned$$ If both $b=0$ and $a=-1,$ then we let $\uK _{l+1}=\Im (\uL _{i_{l+1}}).$ Otherwise we let $\uK _{l+1}=\Re (\uL _{i_{l+1}}).$ In either case the span of $\uK _1,\ldots ,\uK_l,\uL _{i_{l+1}}$ is equal to the span of $\uK _1,\ldots ,\uK _l,\uK _{l+1}$. If $\overline{\uL _{i_{l+1}}}$ is not in the span of $\uL _{i_1},\ldots ,\uL _{i_{l+1}}$, then $\uL _{i_{l+2}}=
\overline{\uL _{i_{l+1}}}.$ We let $\uK _{l+1}=\Re (\uL _{i_{l+1}})$ and $\uK _{l+2}=\Im (\uL _{i_{l+1}})$ in this case. Then the span of $\uL _{i_1},\ldots ,\uL _{i_{l+2}}$ is equal to the span of $\uK _1,\ldots ,\uK_{l+2}$.
Proceeding in this fashion until $l=k$ yields the lemma.
Suppose $a(F)$ is either undefined or at least $d/n$[.]{} Let $k$ be as in Lemma [10.]{} Then there is an orthonormal basis $\uK '_1,\ldots ,\uK '_n\in\br ^n$ of $\br ^n$ such that[,]{} for all $\ux\in\br ^n$ and $0<a\le b$ satisfying $$|K'_i(\ux )|\le a\qquad i=1,\ldots ,k$$ and $$|K_i'(\ux )|\le b\qquad
i=k+1,\ldots ,n,$$ we have $$|F(\ux )|^{n/d}\le n^n\hofF ^{n/d}a^kb^{n-k}.$$ Further[,]{} $V(F)$ is infinite and $N_F(m)$ is infinite for all $m$ sufficiently large[.]{}
Get $\uK _1,\ldots ,\uK _k$ as in Lemma 11. Let $\uK _1',\ldots ,\uK _k'$ be an orthonormal basis for their span, and enlarge this collection to an orthonormal basis $\uK '_1,\ldots ,\uK _n'$ of $\br ^n$. Let $\ux$, $a$ and $b$ be as in the statement of the lemma. Now at least $kd/n$ of the coefficient vectors $\uL _i$ are in the span of $\uK '_1,\ldots ,\uK _k'$, and the corresponding factors of $F(\uX )$ satisfy $$|L_i(\ux )|\le n\|\uL _i\|\max _{1\le j\le k}\{|K_j'(\ux )|\}=n\|\uL _i\|a.$$ There are no more than $d-kd/n =(n-k)d/n$ factors $L_i(\uX )$ which remain, and they satisfy $$|L_i(\ux )|\le n\|\uL _i\|\max _{1\le j\le n}\{|K_j'(\ux )|\}=n\|\uL _i\|b.$$ Thus, $$|F(\ux )|=\prod _{i=1}^d|L_i(\ux )|\le n^d\hofF a^{kd/n}b^{(n-k)d/n},$$ and the first part of the lemma is proven.
For $\ux\in\br ^n$ write $\ux =\sum _{i=1}^nx_i\uK '_i$. For any $a\le 1,$ the set of $\ux$ satisfying $$|x_i|\le\cases a^{-k/(n-k)}&\text{if $i>k$,}\\
a&\text{if $i\le k$}\endcases$$ is contained in the set of $\ux$ satisfying $|F(\ux )|\le n^d\hofF $ by the first part of the lemma. Letting ${\frak m}$ denote $\displaystyle{\max _{1\le i\le k}\{|x_i|\}}$ in what follows, we see that $$\aligned
&\hskip-1in \idotsint\limits_{|x_i|\le 1}\left [\quad\idotsint
\limits_{|x_j|\le {\frak m}^{-k/(n-k)}}
\prod _{j=k+1}^ndx_j\right ]\prod _{i=1}^kdx_i\\ \noalign{\vskip5pt}
&\qquad = 2^{n-k}
\idotsint\limits_{|x_i|\le 1}
{\frak m}^{-k}
\prod _{i=1}^kdx_i\\ \noalign{\vskip5pt}
&\qquad \ge\idotsint\limits_{\|(x_1,\ldots ,x_k)\|\le 1}
\|(x_1,\ldots ,x_k)\|^{-k}
\prod _{i=1}^kdx_i\\ \noalign{\vskip5pt}
&\qquad =kV(k)\int _0^1r^{-1}dr\\ \noalign{\vskip5pt}
&\qquad =\infty,\endaligned$$ where $V(k)$ denotes the volume of the unit ball in $\br ^k$. Thus the volume of the set of $\ux\in\br ^n$ with $|F(\ux )|\le n^d\hofF$ is infinite. By homogeneity, this shows that $V(F)$ is infinite.
Finally, let $0<a\le b$ satisfy $a^kb^{n-k}=1$. Then the parallelopiped defined by $|K_j'(\ux )|\le a$ for $1\le j\le k$ and $|K_j'(\ux )|\le b$ for $j>k$ has volume $2^n$. By Minkowski’s theorem there is a nontrivial integral point in such a parallelopiped. Letting $a\rightarrow 0,$ we get infinitely many nonzero integral points contained in such parallelopipeds. Thus, there are infinitely many integral $\ux$ with $|F(\ux )|\le n^d \hofF.$
Small solutions
===============
Let $B_0\ge 1$. Any solution $\ux \in \br ^n$ to (1) with $\|\ux\|\le B_0$ will be called a [*small solution*]{}. We will use $B_0=m^{1/(d-a(F))}$ in our proofs of the theorems, but since most of our estimates up until that point will not require “small" to be dependent on $m$, we will leave $B_0$ variable when possible. In this section we will bound both the volume of all small real solutions to (1) and the number of small integral solutions, and we will also also compare the volume of all small solutions with the number of small integral solutions. As a notational convenience, let $S_0$ denote the cardinality of the set of small integral solutions and let $V_0$ denote the volume of all small solutions.
Suppose $I(F)$ is not empty[,]{} $\hofF$ is minimal among forms equivalent to $F$[,]{} $\hofF >1$ and $F$ has no nontrivial integral zeros[.]{} Then $$V_0\ll m^{n/d}\left (1+{\log B_0\over\log\hofF }\right )^{n-1}$$ and $$S_0\ll
m^{n/d}\left (1+{\log B_0\over\log\hofF }\right )^{n-1} +B_0^{n-1}
\left (1+{\log B_0\over\log\hofF }\right )^{n-1}.$$
According to Lemma 6, for any solution $\ux\in\br ^n$ to (1) there is an $n$-tuple in $I'(F)$ with $$\prod _{j=1}^n{|L_{i_j}(\ux )|\over |\det (\uL _{i_1}^{tr},\ldots ,\uL _{
i_n}^{tr})|}\ll {m^{n/d}\over \hofF ^{1/d}}.$$ Set $A=m^{n/d}/\hofF ^{1/d}$, $C=B_0$ and $D=\hofF^{1/nd}$ in Lemma 7$'$. We see that the solutions $\ux$ to the above inequality with $\|\ux\|\le C$ lie in $$\ll 1+
(\log _DC)^{n-1} \ll
\left (1+{\log B_0\over\log\hofF }\right )^{n-1}$$ convex sets of the form (15) with $$\prod _{j=1}^na_i\ll D^nA= m^{n/d}.$$ By Lemma 9, such a convex set has volume $\ll m^{n/d}.$ There are no more than ${d\choose n}$ $n$-tuples to consider here by (2), so we get our bound for $V_0.$
As for $S_0$, we estimate exactly as above. The difference is that our convex sets may not contain $n$ linearly independent integral points; they may lie in a proper rational subspace. So it remains to estimate the number of integral points in these proper subspaces. By (2) again, there are $$\ll \left (1+{\log B_0\over\log\hofF }\right )^{n-1}$$ such subspaces to deal with. We claim that for any proper rational subspace of $\bq ^n$ of dimension $n'$, the number of integral points in the subspace with length at most $B_0$ is $\ll B_0^{n'}.$ Our proof will be complete once we show this claim.
We prove our claim by induction on $n'$. If $n'=1$ the result is obvious. Now suppose $W$ is a proper rational subspace of dimension $n'>1.$ Let $\Lambda$ be the lattice of integral points in $W$. If $\Lambda$ doesn’t contain $n'$ linearly independent points of length no more than $B_0,$ then we apply the induction hypothesis to the proper subspace of $W$ these small lattice points span (and use $B_0\ge 1$) to show that $\Lambda$ contains $\ll B_0^{n'}$ lattice points of length at most $B_0$.
Suppose $\Lambda$ contains $n'$ linearly independent lattice points of length at most $B_0$. Let $T\in\gln (\br )$ be an orthonormal transformation taking $W$ to the span of the first $n'$ canonical basis vectors of $\br ^n$. Let ${\Cal C}\subset \br ^{n'}$ be the set of points of length at most $B_0$. Since $T(\Lambda )$ is a lattice containing $n'$ linearly independent lattice points in ${\Cal C}$ and $T$ is orthonormal, Lemma 8 gives $$|{\Cal C}\cap T(\Lambda )|\ll {B_0^{n'}\over \det (T(\Lambda ))}=
{B_0^{n'}\over \det (\Lambda )}.$$ It is well known that $\det (\Lambda )\ge 1$, so we see that the number of integral points in $W$ with length at most $B_0$ is $\ll B_0^{n'}$. Our claim follows by induction, whence our proof of Lemma 13 is complete.
For the purposes of Theorem 3, we need to compare the number of integral small solutions with the total volume of all small solutions. It proves convenient here to use the sup norm rather than the Euclidean norm. So let $V_0'$ denote the volume of all solutions to (1) with sup norm at most $B_0$, and similarly for $S_0'$.
With the notation above[,]{} we have $$|S_0'-V_0'|\le dn(2B_0+1)^{n-1}.$$
Let $\mu$ denote the usual Lebesgue measure on $\br$ and let $\nu$ denote the $\sigma$-finite measure gotten from the characteristic function of $\bz$, that is, $\nu (E)$ is the number of integer points in the set $E$ for any Borel set $E\subseteq \br$. Let $\chi$ be the characteristic function of the set $$\{\uy\in\br ^n\: |F(\uy )|\le m\ \text{and $|y_i|\le B_0$ for all $i$}\}.$$ What we want to do here is estimate the difference between the integrals of $\chi$ with respect to the product measures $\mu ^n$ and $\nu ^n$. The lemma follows from the case $I=\{1,\ldots ,n\}$ of the following claim:
For any nonempty subset $I\subseteq \{1,\ldots ,n\}$ and fixed values $y_i\in\br$ for $i\not\in I,$ we have $$\multline
\left |\int\cdots\int\chi (y_1,\ldots ,y_n)\prod _{i\in I}d\mu (y_i)-
\int\cdots\int\chi (y_1,\ldots ,y_n)\prod _{i\in I}d\nu (y_i)\right |\\
\le
d|I|(2B_0+1)^{|I|-1},\endmultline$$ where $|I|$ denotes the cardinality of $I$. The major point of this estimate is that it is independent of the particular choices of $y_i\in\br$ for $i\not\in I$. We prove this claim (and whence Lemma 14) by induction on the cardinality of $I$.
Suppose that $I=\{i_0\}$ and $y_i\in\br$ are fixed for $i\neq i_0$. Then $$F(y_1,\ldots ,Y_{i_0}, \ldots ,y_n)\in\br [Y_{i_0}]$$ is a polynomial in one variable of degree no greater than $d$. This implies that the set $$E= \{y_{i_0}\in\br \: |F(y_1,\ldots ,y_n)|\le m\ \text{and $|y_i|\le B_0$
for all $i$}\}$$ is a (possibly empty) union of no more than $d$ nonintersecting closed intervals. Now $$\int\chi (y_1,\ldots ,y_n)d\mu (y_{i_0})=\int _Ed\mu (y_{i_0})$$ and similarly for the $\nu$ measure. Further, the difference between the length of a closed interval and the number of integer values therein is between $-1$ and $1$. This shows the case $|I|=1$ of the claim.
Now suppose $|I|>1$. We will use the induction hypothesis twice and the Fubini-Tonelli theorem to show the claim holds for $I$. Choose $i_0\in I$. Then by the Fubini-Tonelli theorem and the triangle inequality
[$$\multline
\left |\idotsint\chi (y_1,\ldots ,y_n)\prod _{i\in I}d\mu (y_i)-
\idotsint\chi (y_1,\ldots ,y_n) \prod_{i\in I}d\nu (y_i)\right |\\
\le
\left |\int
\left [\int\cdots \int \chi (y_1,\ldots ,y_n)
\!\prod _{i\in I, i\neq i_0}\! d\mu (y_i)-
\int\cdots\int\chi (y_1,\ldots ,y_n)
\!\prod _{i\in I, i\neq i_0}\!d\nu (y_i)\right] \, d\mu (y_{i_0})\right |\\
+
\left |\int\cdots\int
\left [\int\chi (y_1,\ldots ,y_n)\, d\mu (y_{i_0})
-\int\chi (y_1,\ldots ,y_n)\, d\nu (y_{i_0})\right] \prod _{i\in I, i\neq i_0}d\nu (y_i)\right |.
\endmultline$$]{} Using the induction hypothesis on $I\setminus\{i_0\}$ and the fact that $\chi$ is the characteristic function of a set contained in the cube $\{\uy\in\br ^n\: |y_i|\le B_0\}$ gives
[$$\multline
\left |\int\left [\int\cdots\int \chi (y_1,\ldots ,y_n)
\!\prod _{i\in I, i\neq i_0}\! d\mu (y_i)-
\int\cdots\int\chi (y_1,\ldots ,y_n)
\!\prod _{i\in I, i\neq i_0}\! d\nu (y_i)\right ]\, d\mu (y_{i_0})\right |\\
\le
\int \left |\int\cdots\int\chi (y_1,\ldots ,y_n)
\prod _{i\in I, i\neq i_0}d\mu (y_i)-
\int\cdots\int\chi (y_1,\ldots ,y_n)
\prod _{i\in I, i\neq i_0}d\nu (y_i)\right |\, d\mu (y_{i_0})\\
\le\int _{[-B_0,B_0]}d(|I|-1)(2B_0+1)^{|I|-2}\, d\mu (y_{i_0})\\
=2B_0d(|I|-1)(2B_0+1)^{|I|-2}< d(|I|-1)(2B_0+1)^{|I|-1}.\endmultline$$]{}
Similarly,
[$$\multline
\left |\int\cdots\int
\left [\int\chi (y_1,\ldots ,y_n)\, d\mu (y_{i_0})
-\int\chi (y_1,\ldots ,y_n)\, d\nu (y_{i_0})\right] \prod _{i\in I, i\neq i_0}d\nu (y_i)\right |\\
\le
\int\cdots\int\left |\int\chi (y_1,\ldots ,y_n)\, d\mu (y_{i_0})-
\int\chi (y_1,\ldots ,y_n)\, d\nu (y_{i_0})\right |
\prod _{i\in I, i\neq i_0}d\nu (y_i)\\
\le\int _{[-B_0,B_0]}\cdots\int _{[-B_0,B_0]}d
\prod _{i\in I, i\neq i_0}d\nu (y_i)\\
=d(2[B_0]+1)^{|I|-1}\le d(2B_0+1)^{|I|-1}.\endmultline$$]{}
Adding these two estimates together finishes our proof of the claim.
Estimating large solutions
==========================
Throughout this section we will assume that $a(F)$ is defined and less than $d/n$ (this forces $d>n$). It is appropriate at this time to note some inequalities involving $a(F)$ and $c(F)$ under this assumption. By definition, $ka(F)\in\bz$ for some $k<n$, so that $kna(F)\le kd -1$ and $$1\le a(F)\le {d\over n}-{1\over n(n-1)}.\tag 16$$ Using this, we get $$n-d\le {na(F)-d\over a(F)}\le {1\over 1-n}.\tag 17$$ If the discriminant of $F$ is not zero, then $$1\le {d-1\choose n-1}-1=c(F).$$ If the discriminant of $F$ is zero, then by (2), (16) and (17) $$\aligned c(F)&= {b(F)\over n!}\times {d-(n-1)a(F)\over a(F)}-{1\over a(F)
}\\
&< |I'(F)|(d-n+1)\\
&\le {d\choose n}(d-n+1).\endaligned$$ Here we also used $b(F)/n!\le |I'(F)|,$ which is clear from the definitions. Using $b(F)/n!\ge 1$ (which is also clear from the definitions), and $a(F)<d/n$ gives $$\aligned c(F)&= {b(F)\over n!}\times {d-(n-1)a(F)\over a(F)}-{1\over a(F)
}\\
&\ge{d-na(F)\over a(F)}+{a(F)-1\over a(F)}\\
&>{(d-n)\over d}.\endaligned$$ Thus, $${(d-n)\over d}\le c(F)\le {d\choose n}(d-n+1).\tag 18$$
For indices $l\ge 0$ let $B_l=e^lB_0$ and $C_l=e^{l+1}B_0$. Let $$A_0=m^{1/a(F)}B_0^{(na(F)-d)/a(F)}\hofF ^{c(F)}$$ and for $l\ge 0$ let $A_l=e^{(na(F)-d)l/a(F)}A_0$. Recall that $m,B_0\ge 1$ by hypothesis and $\hofF\ge 1$ by Lemma 3. By (16), (17), and (18) $$A_l=e^{(na(F)-d)l/a(F)}A_0
\ge B_0^{n-d}e^{l(n-d)}.\tag 19$$ Let $V_{l+1}$ denote the total volume of the set of solutions $\ux\in\br ^n$ to (1) with $B_l\le\|\ux\|\le C_l$.
If $I(F)$ is not empty and $a(F)<d/n$[,]{} then $$\sum _{l=1}^{\infty}V_l\ll
\hofF ^{c(F)}m^{1/a(F)}B_0^{(na(F)-d)/a(F)}(1+\log B_0)^{n-2}.$$
Set $m=B_0=1$. Clearly $V_0\ll 1$. By Lemma 15 $\sum _{l=1}^{\infty}V_l<\infty$ whenever $a(F)$ is defined and less than $d/n$. This together with Lemma 12 proves the proposition.
By Lemma 5, for any solution $\ux\in\br ^n$ to (1) with $B_l\le\|\ux\|$ there is an $n$-tuple in $I'(F)$ with $$\align {\prod _{j=1}^n|L_{i_j}(\ux )|\over|\det (\uL _{i_1}^{tr},\ldots ,
\uL _{i_n}^{tr})|}&\ll \left ({|F(\ux )|\over
\|\ux \|^{d-na(F)}}\right )^{1/a(F)}\hofF^{c(F)}\tag 20\\
&\le\left ({m\over B_{l}^{d-na(F)}}\right )^{1/a(F)}\hofF ^{c(F)}\\
&=m^{1/a(F)}B_0^{(na(F)-d)/a(F)}e^{(na(F)-d)l/a(F)}\hofF ^{c(F)}\\
&=A_l.\endalign$$
We will estimate using Lemma 7. We have $$\multline \max \left \{n!,n^3\big (\log (B_lC_l^{n-1}/n!n^{n/2}A_l)\big )^
{n-2}\right \}\\
\le\max \left \{n!,n^3\big (\log (B_0^de^{n(l+1)}/e^{l(n-d)}
)\big )^{n-2}\right \}\\
\ll (1+l+\log B_0)^{n-2}
\endmultline \tag 21$$ by (19). Setting $A=A_l,\ B=B_l,\ C=C_l$ and $D=e$ in Lemma 7, we see by (21) that the solutions $\ux$ to (20) with $B_l\le\|\ux\|\le C_l$ are contained in $\ll (1+l+\log B_0)^{n-2}$ convex sets of the form (15) with $$\prod _{i=1}^na_i\ll {C_lA_l\over B_l}\ll A_l\le e^{l/(1-n)}
A_0$$ by (17). According to Lemma 9, the volume of such a convex set is $\ll e^{l/(1-n)}A_0.$ Taking into account the total number of possible $n$-tuples in $I'(F)$ using (2), we find that $$V_{l+1}\ll e^{l/(1-n)}A_0(1+l+\log B_0)^{n-2}\le {(1+l)^{n-2}\over
(e^{1/(n-1)})^l}A_0
(1+\log B_0)^{n-2}.$$
We thus have $$\aligned\noalign{\vskip5pt} \sum _{l=0}^{\infty}V_{l+1}
&\ll A_0(1+\log B_0)^{n-2}
\sum _{l=0}^{\infty}{(l+1)^{n-2}\over (e^{1/(n-1)})^l}\\ \noalign{\vskip5pt}
&\ll \hofF ^{c(F)}m^{1/a(F)}B_0^{(na(F)-d)/a(F)}(1+\log B_0)^{n-2}.\\
\noalign{\vskip5pt}
\endaligned$$
When estimating the number of integer solutions to (1) of length greater than $B_0$, we proceed very much as in the proof of Lemma 15. However, since we are counting integer solutions as opposed to estimating volumes, we must also account for the possibility that all solutions in a given convex set of the form (15) lie in a proper subspace, so that Lemma 9 cannot be used in a manner similar to our use of it in the proof above. Our goal is to reach the point where we may estimate the remaining (extremely large) integer solutions using a quantitative version of the subspace theorem.
Suppose $I(F)$ is not empty and $a(F)<d/n.$ Then the integral solutions $\ux$ to [(1)]{} with $B_0\le \|\ux\|$ lie in the union of a set of cardinality $S$ satisfying $$S
\ll m^{1/a(F)}B_0^{(na(F)-d)/a(F)}(1+\log B_0)^{n-2}\hofF ^{c(F)}$$ and $$\ll \big (1+\log m+\log\hofF \big )\big (1+\log m+\log\hofF+\log B_0 \big )
^{n-2}$$ proper rational subspaces[.]{}
Exactly as in the proof of Lemma 15, any integral solution $\ux$ to (1) with $B_l\le \|\ux\|\le C_l$ satisfies (20) for some $n$-tuple in $I'(F)$. We apply Lemma 7 again, getting the same convex sets of the form (15) as in the proof of Lemma 15. When those sets contain $n$ linearly independent lattice points, we estimate the number of such points using Lemma 9 exactly as we estimated the $V_{l+1}$ above. These points make up the set of cardinality $S$.
By (21), our solutions $\ux$ to (20) with $B_l\le\|\ux\|\le C_l$ lie in the union of $\ll (l+1+\log B_0)^{n-2}$ convex sets of the form (15). Taking into account the different possible $n$-tuples, we see that those solutions $\ux$ with $B_0\le\|\ux\|\le C_{l}$ not already accounted for in $S$ lie in the union of $\ll (l+1)(l+1+\log B_0)^{n-2}$ proper rational subspaces. We need to determine how large $l$ should be so that solutions of length at least $C_l$ can be dealt with using the subspace theorem.
If $${l+1\over 2(n-1)}\ge \log m +{d\choose n}(d-n+1)\log (\hofF),$$ then by Lemma 2, (16), (17) and (18) we have $$\aligned C_l^{(d-na(F))/2a(F)}\ge C_l^{1/2(n-1)}&\ge e^{(l+1)/2(n-1)}\\
&\ge m\hofF ^{{d\choose n}(d-n+1)}\\
&\ge m^{1/a(F)}\hofF ^{c(F)}.\endaligned$$ By Lemma 5 and (17), for any solution $\ux$ to (1) with $\|\ux\|\ge C_l$ there is an $n$-tuple in $I'(F)$ satisfying $$\aligned{\prod _{j=1}^n|L_{i_j}(\ux )|\over|\det (\uL _{i_1}^{tr},\ldots ,
\uL _{i_n}^{tr})|}&\ll
\left ({|F(\ux )|\over \|\ux\| ^{d-na(F)}}\right )
^{1/a(F)}\hofF ^{c(F)}\\
&\le {1\over \|\ux\| ^{(d-na(F))/2a(F)}}
\left ({m\over C_l^{(d-na(F))/2}}\right )^{1/a(F)}\hofF ^{c(F)}\\
&\le \|\ux\| ^{-1/2(n-1)}
\left ({m\over C_l^{(d-na(F))/2}}\right )^{1/a(F)}\hofF ^{c(F)}.\endaligned$$ Let $l_0$ be least such that $\|\ux\|\ge C_{l_0}$ implies that $${\prod _{j=1}^n|L_{i_j}(\ux )|\over|\det (\uL _{i_1}^{tr},\ldots ,
\uL _{i_n}^{tr})|}<
\|\ux\| ^{-1/2(n-1)}.$$ By what we showed above, $l_0\ll 1+\log m+\log\hofF .$ Let $l_1$ be the least such that $C_{l_1}\ge m^{1/d}C_{l_0}$ and $C_{l_1}\ge m^{1/d}\hofF$. Then $l_1\ll 1+\log m+\log\hofF$, too.
The integral solutions $\ux$ to (1) with $B_0\le\|\ux\|\le C_{l_1}$ either lie in our set of cardinality $S$ or $$\multline \ll l_1(l_1+1+\log B_0)^{n-2}\\
\ll\big (1+\log m+\log\hofF \big )
\big (1+\log m+\log \hofF +\log B_0)^{n-2}\endmultline$$ proper rational subspaces. Since the solutions to $F(\ux )=0$ lie in no more than $d$ proper subspaces, we restrict ourselves for what remains to integral solutions $\ux$ to (1) with $|F(\ux )|\ge 1$ and $\|\ux\|\ge C_{l_1}$. Let $\ux$ be such a solution and write $\ux =g\ux '$ for some primitive integer point $\ux '$ and some integer $g\ge 1$. By the homogeneity of $F$, $$m\ge |F(\ux )|=|F(g\ux ')|=g^d|F(\ux ')|\ge g^d,$$ so that $g\le m^{1/d}$. Thus, $$\|\ux '\|\ge m^{-1/d}\|\ux\|\ge m^{-1/d}C_{l_1}\ge \max \{C_{l_0},\hofF\}$$ and $\ux '$ is a primitive solution to (1).
By the definition of $l_0$, we have $${\prod _{j=1}^n|L_{i_j}(\ux ')|\over|\det (\uL _{i_1}^{tr},\ldots ,
\uL _{i_n}^{tr})|}<\|\ux '\| ^{-1/2(n-1)}$$ for some $n$-tuple in $I'(F)$. By Lemma 2 we may assume each $\uL _{i_j}$ here is defined over a number field of degree at most $d$ and has field height at most $\hofF\le\|\ux '\|.$ By a version of the quantitative subspace theorem due to Evertse \[E2, Corollary\], the set of such primitive integral $\ux '$ lies in the union of $\ll 1$ proper subspaces. Taking into account the number of possible $n$-tuples using (2), we see that the integral solutions $\ux$ to (1) with $\|\ux\|\ge C_{l_1}$ lie in $\ll 1$ proper rational subspaces. This completes the proof.
Proof of the theorems
=====================
As remarked above, to prove our theorems we set $B_0=m^{1/(d-a(F))}$, giving $$\align m^{1/a(F)}B_0^{(na(F)-d)/a(F)}&=m^{1/a(F)}m^{(na(F)-d)/a(F)(d-a(F))}
\tag 22\\
&=m^{(d-a(F)+na(F)-d)/a(F)(d-a(F))}\\
&=m^{(n-1)/(d-a(F))}.\endalign$$
By the proposition[,]{} it suffices to prove that $V(F)\ll 1$ when $I(F)$ is not empty and $a(F)<d/n$[.]{} Moreover[,]{} by homogeneity we need only show that $m^{n/d}V(F)\ll m^{n/d}$ for some positive $m$[.]{} We may assume $\hofF$ is minimal among forms equivalent to $F$ since $V(F)$ is invariant under equivalence.
Suppose first that $\hofF =1$. In this case we set $m=1$, too. Clearly $V_0$ is no larger than the volume of the unit ball in $\br ^n$. By (22) and Lemma 15 we have $$\sum _{l=1}^{\infty}V_l\ll 1;$$ thus, $$V(F)=\sum _{l=0}^{\infty}V_l\ll 1.$$
Now suppose $\hofF >1.$ By (16) we have $${n-1\over d-a(F)}\le {n-1\over d-{d\over n}+{1\over n(n-1)}}={n\over
d+{1\over (n-1)^2}}.\tag 23$$ Choose $m$ so that $$\hofF ^{c(F)}m^{(n-1)/(d-a(F))}= m^{{n\over d+1/2(n-1)^2}}.$$ Then $\log m\gg\ll\log\hofF$ and $$\hofF ^{c(F)}m^{(n-1)/(d-a(F))}(1+\log m )^{n-2}\ll m^{n/d}$$ by (18) and (23). By Lemma 13, $$\aligned V_0&\ll m^{n/d}\left (1+{\log B_0\over \log\hofF}\right )^{n-1}\\
&\ll m^{n/d}\left (1+{\log m\over\log\hofF }\right )^{n-1}\\
&\ll m^{n/d}.\endaligned$$ By Lemma 15 and (22), $$\multline \sum _{l=1}^{\infty} V_l\ll \hofF ^{c(F)}
m^{1/a(F)}B_0^{(na(F)-d)/a(F)}\\ =
\hofF ^{c(F)}m^{(n-1)/(d-a(F))}(1+\log m)^{n-2}\ll m^{n/d}.\endmultline$$ Thus, $$m^{n/d}V(F)=\sum _{l=0}^{\infty}V_l\ll m^{n/d}.$$
Suppose $W$ is a proper rational subspace of $\br ^n$ of dimension $n'$. Then there is a $T\in\gln (\bz )$ with $$T:W\cap\bz ^n\rightarrow \{(z_1,\ldots ,z_n)\in\bz ^n\:
z_i=0\ \text{for $i>n'$}\}.$$ Then $G:=F\circ T^{-1}$ is an equivalent form, and $F$ restricted to $W$ is equivalent to $G$ restricted to $\br ^{n'}$. In this manner, we see that considering integral solutions to (1) for $F$ restricted to a proper rational subspace is equivalent to considering integral solutions to (1) for a form in fewer variables. With this in mind, we will prove that $N_F(m)\ll m^{n/d}$ when $F$ is of finite type by induction on $n$. But we first deal with the simpler case when $F$ is not of finite type.
Suppose that $F$ is not of finite type. Then there is some nontrivial subspace $W$ defined over $\bq$ where the volume of solutions to (1) in $W$ is infinite. Let $n'\ge 1$ be the dimension of $W$. If $n'=1,$ then $F$ vanishes on $W$. Trivially $N_F(m)$ is infinite for all $m$ in this case. Suppose $n'>1$ and get a form $F'(\uX )\in\bz [\uX ]$ in $n'$ variables where the $\ux$ in $W$ are in one-to-one correspondence with $\ux '\in\br ^{n'}$ via a $T\in\gln (\bz )$ with $F(\ux ) =F'(\ux ')$ as above. Since $V(F')$ is infinite by hypothesis, the proposition shows that $a(F')$ is either undefined or at least $d/n'$. Lemma 12 shows that $N_{F'}(m)$ is infinite for all sufficiently large $m$. Thus, there are infinitely many solutions $\ux\in W\cap
\bz ^n$ to (1) for all sufficiently large $m$. This shows that $N_F(m)$ is infinite for all sufficiently large $m$ when $F$ is not of finite type.
Now suppose $F$ is of finite type. Interestingly, our argument for the first step in the induction where $n=2$ is the same as our argument for $n>2$ using the induction hypothesis. Rather than present the same argument twice, then, we will simply assume that $n\ge 2$ and that the number of integral solutions to (1) restricted to a proper subspace of dimension $n'<n$ is $\ll m^{n'/d}$. The number of solutions to (1) restricted to any proper 1-dimensional rational subspace is $\ll m^{1/d}$, since $F$ is not identically $0$ on such a subspace, so our assumption in the case $n=2$ is correct. Finally, without loss of generality we may assume $\hofF$ is minimal among forms equivalent to $F$.
By the proposition, $I(F)$ is not empty and $a(F)<d/n$. Suppose first that $$\hofF ^{c(F)}m^{(n-1)/(d-a(F))}\le m^{{n\over d+1/2(n-1)^2}}.$$ Then (18) and (23) show that $\log\hofF\ll\log m$, and we also have $$\hofF ^{c(F)}m^{(n-1)/(d-a(F))}(1+\log m)^{n-2}\ll m^{n/d}.$$ By Theorem 1, $V_0'\le m^{n/d}V(F)\ll m^{n/d}$, so (22), (23) and Lemma 14 give $S_0'\ll m^{n/d}.$ Further, (22), (23), and Lemma 16 show that the integral solutions of length at least $B_0$ lie in the union of a set of cardinality $\ll m^{n/d}$ and $\ll (1+\log m)^{n-1}$ proper subspaces. By the induction hypothesis (or the trivial 1-dimensional case when $n=2$), these proper subspaces contribute $$\ll m^{(n-1)/d}(1+\log m)^{n-1}\ll m^{n/d}$$ integral solutions. So $N_F(m)\ll m^{n/d}.$
Now suppose $$\hofF ^{c(F)}m^{(n-1)/(d-a(F))}\ge m^{{n\over d+1/2(n-1)^2}} ,$$ so that $\log\hofF\gg 1+\log m$ by (18) and (23). Let $C_{l_1}$ be as in the proof of Lemma 16. As shown in the proof of Lemma 16, $l_1\ll (1+\log m+\log \hofF)$.
By Lemma 6, if $\ux$ is a solution to (1), then there is a $n$-tuple in $I'(F)$ such that $${\prod _{j=1}^n|L_{i_j}(\ux )|\over|\det (\uL _{i_1}^{tr},
\ldots ,\uL _{i_n}^{tr})|}\ll m^{n/d}\hofF ^{-1/d}.$$ We use Lemma $7'$ with $A=m^{n/d}\hofF ^{-1/d}$, $C=C_{l_1}$ and $D=\hofF ^{1/nd}$. We have $$\log _D\left ({C^n\over n!A}\right )\ll {\log C_{l_1}\over\log\hofF }\ll
{\log m+l_1\over\log\hofF }\ll 1,$$ so the set of all such $\ux$ with $\|\ux\|\le C_{l_1}$ lie in $\ll 1$ convex sets of the form (15) with $$\prod _{i=1}^na_i\ll D^nA=m^{n/d}.$$ By Lemma 9, if such a set contains $n$ linearly independent integral points, it contains $\ll m^{n/d}$ of them. Taking into account the number of possible $n$-tuples via (2), we see that the integral solutions $\ux$ to (1) with $\|\ux \|\le C_{l_1}$ lie in the union of $\ll 1$ proper rational subspaces and a set of cardinality $\ll m^{n/d}$. As shown in the proof of Lemma 16, all integral solutions $\ux$ to (1) with $\|\ux\|\ge C_{l_1}$ lie in $\ll 1$ proper subspaces. By the induction hypothesis (or the trivial 1-dimensional case if $n=2$), all our proper subspaces contain $\ll m^{(n-1)/d}$ integral solutions total. So $N_F(m)\ll m^{n/d}$.
By Lemma 15 and (21) we have $$m^{n/d}V(F)-V_0'\le \sum _{l=1}^{\infty}V_l\ll \hofF ^{c(F)}m^{(n-1)/
(d-a(F))}(1+\log m)^{n-2}.$$ By Lemma 14, we get $$|S_0'-m^{n/d}V(F)|
\ll \hofF ^{c(F)}m^{(n-1)/
(d-a(F))}(1+\log m)^{n-2}.$$ As we saw in the proof of Theorem 2, the number of integral solutions to (1) restricted to any proper subspace is $\ll
m^{(n-1)/d}.$ By Lemma 16 and (21) then, the number of integral solutions to (1) with length at least $B_0$ is $$\ll \hofF ^{c(F)}m^{(n-1)/(d-a(F))}(1+\log m)^{n-2}.$$ From this, we get $$N_F(m)-S_0'\ll
\hofF ^{c(F)}m^{(n-1)/(d-a(F))}(1+\log m)^{n-2}.$$ Theorem 3 follows.
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|
---
abstract: 'The out-of-equilibrium production of dark matter (DM) from standard model (SM) species in the early universe (freeze-in mechanism) is expected in many scenarios in which very heavy beyond the SM fields act as mediators. In this conference, I have talked about the freeze-in of scalar, fermionic and vector DM though the exchange of moduli fields [@chowdhury_moduli_2019], which are in the low-energy spectrum of many extra-dimensions and string theory frameworks. We have shown that the high temperature dependencies of the production rate densities in this model, as well as the possibility of having moduli masses at the post-inflationary reheating scale, make it crucial to consider the contribution of the freeze-in prior the start of the standard radiation era for a correct prediction of the DM relic density.'
author:
- 'Maíra Dutra$^{[0000-0003-4851-242X]}$'
title: The moduli portal to dark matter particles
---
Introduction
============
The close relationship between the couplings dark matter (DM) particles might have to particles belonging to the Standard Model of Particle Physics (SM) and their evolution through the early universe makes the DM puzzle an open problem in the interface between particle physics and cosmology.
Direct detection searches seek to detect nuclear or electronic recoils from DM scatterings, and the current status of no positive signals but more and more sensitive detectors mean that the SM-DM coupling need to be weaker and weaker. However, very weak couplings might imply that DM and SM particles were never in thermal equilibrium in the early universe. The out-of-equilibrium production of DM from SM species in the early universe, which is the so-called *freeze-in mechanism* [@chung_production_1998; @hall_freeze-production_2010; @bernal_dawn_2017], is expected in many scenarios in which very heavy beyond the SM (BSM) fields act as mediators. If the masses of these heavy mediators are close to the scale of the post-inflationary reheating, it is important to take into account the freeze-in during a reheating period in which entropy is being injected into the thermal bath. Here we shed light on this matter, presenting handy formulae that can be useful for any scenario.
Moduli fields are scalars which would be present in the effective limit of many string theory frameworks. Since they would need to be very feebly coupled to SM fields, it is interesting to investigate whether their feeble interactions with dark and visible matter would be enough to produce dark matter via freeze-in. In this conference, I have presented a recent study on the moduli portal to dark matter [@chowdhury_moduli_2019].
The model
=========
We have considered a complex modulus field, $\T = t + i a$, whose presence at temperatures below some cut-off scale $\Lambda$ would appear as corrections to the free Lagrangians[^1]. We consider BSM scalar, fermion and vector fields as feebly interacting massive particle (FIMP) candidates, which are DM candidates produced via freeze-in. Our effective Lagrangian connecting the modulus field to dark and standard scalars, fermions and vectors, here generically denoted by $\Phi, \Psi$ and $X_\mu$, reads $$\Lag_{eff} \supset
\begin{cases}
\left(1 + \frac{\a_i}{\Lambda}t\right) (|D_\mu \Phi|^2 - \mu_\Phi^2) & \text{(scalars)} \\
\frac{1}{2}\left(1 + \frac{\a_{L,R}^i}{\Lambda}t + i \frac{\beta_{L,R}^i}{\Lambda}a\right) \bar{\Psi}_{L,R} i \slashed{D} \Psi_{L,R}
& \text{(fermions)} \\
-\frac{1}{4} \left(1+ \frac{\a_i}{\Lambda}t\right) X_{\mu \nu} X^{\mu \nu} - \frac{\beta_i}{\Lambda}a \, X_{\mu \nu} \tilde X^{\mu \nu}
&\text{(vectors)} \\
\end{cases}$$
Couplings to the real and imaginary components of the modulus are denoted respectively by $\a$ and $\b$, and the tensors $X_{\m \n}$ and $\tilde X_{\m \n}$ are respectively the field strength and dual field strength of $X_\m$.
In order to avoid imaginary contributions to the mass and kinetic terms of the scalars, we assume that only the real component of the modulus interact with the SM Higgs and the scalar FIMP candidate. In the case of the scalar FIMP, though, we do not assume that the modulus change the mass term. The interactions with fermions are in principle chiral and a chirality flip would give us explicit dependence on the fermion mass in the amplitudes. For this reason, SM fermions cannot produce the FIMPs above EWSB, since they are massless. For the interactions of moduli with vectors, we have a Higgs-like operator for the real component and a Peccei-Queen operator for the axial component.
![Schematic Feynman diagrams leading to the freeze-in production of our FIMP candidates, scalar ($\phi$), fermionic ($\psi$) and vector ($V_\mu$) fields, out of annihilations of all the SM bosons ($H, G_\mu^a,B_\mu, W_\mu^i$) via exchange of the real ($t$) and axial ($a$) components of a modulus field.[]{data-label="fig:diagrams"}](figures/modulimodel.pdf)
As we are going to emphasize in the next section, the squared amplitudes of the freeze-in processes give us valuable information about the freeze-in temperature. In our case, the FIMP candidates are produced from $s$-channel annihilations of Higgs bosons and SM gauge bosons, having $t$ and $a$ as mediators, as depicted in Fig. \[fig:diagrams\]. The squared amplitudes are given by: $$\label{M0}
|\M|^2_0 = \frac{\a_\dm^2 \lambda (s,\a_i)}{\Lambda^4} \frac{ s^4 \Big( 1 - \frac{2m_\dm^2}{s} \Big)^2}{(s-\mre^2)^2+\mre^2 \Gre^2}$$
$$\begin{split}\label{M12}
|\M|^2_{1/2} = & \frac{\a_\dm^2 \lambda (s,\a_i)}{\Lambda^4} \frac{ m^{2}_\dm s^3 \Big( 1 - \frac{4m_\dm^2}{s} \Big)}{(s-\mre^2)^2+\mre^2 \Gre^2}
+ \frac{\beta_\dm^2 \lambda (s,\b_i)}{\Lambda^4} \frac{ m_\dm^2 s^3}{(s-\mim^2)^2+\mim^2 \Gim^2}
\end{split}$$
$$\begin{split}\label{M1}
|\M|^2_{1} = & \frac{\a_\dm^2 \lambda (s,\a_i)}{\Lambda^4}\frac{ s^4 \Big( 1 - \frac{4 m_\dm^2}{s} + \frac{6m_\dm^4}{s^2} \Big)}{(s-\mre^2)^2+\mre^2 \Gre^2}
+ \frac{\beta_\dm^2 \lambda (s,\b_i)}{\Lambda^4} \frac{ s^4 \Big( 1 - \frac{4 m_\dm^2}{s} \Big)}{(s-\mim^2)^2+\mim^2 \Gim^2} \,.
\end{split}$$
Above, $\lambda (s,\a_i), \lambda (s,\b_i)$ are the sums over Higgs and SM gauge bosons contributions, which can be a function of the Mandelstam variable $s$. For further details, see [@dutra_origins_2019].
Results
=======
Production rate densities and evolution of FIMP relic density
-------------------------------------------------------------
The freeze-in temperature is determined by the interactions between FIMPs and the SM species involved. In particular, the temperature-dependence of the production rate densities tell us if the freeze-in happens at the lowest or highest scale of a given cosmological period. This is what I want to emphasize in this section.
The rate at which the number of DM particles change in a comoving volume $a^3$, with $a$ the scale factor, is given by the Boltzmann fluid equation $\dot N_\dm = R_\dm (t) a^3$, with $N_\dm = n_\dm a^3$ the total number of DM particles and $R_\dm (t)$ the time/temperature-dependent interaction rate density, which in the case of the freeze-in only account for production and not for loss of FIMPs. On the other hand, the Hubble rate $H(t)$ determines how the scale factor varies with time, $H(t) = \dot a/a$, and since this quantity is proportional to the total energy density of the universe, different species dominating the expansion lead to different final total number of DM particles, as well as different time-temperature relations.
For a $12 \to 34$ process, the production rate density of species $3$, in the limiting case where species $1$ and $2$ have Maxwell-Boltzmann distributions, is given by [@dutra_origins_2019] $$\begin{split}
R_3^{12\to 34} \equiv n_1^{eq} n_2^{eq} \la \sigma v \ra = \frac{\S_{12}\S_{34}}{32(2\pi)^6}\int & ds \frac{\sqrt{\lambda(s,m_3^2,m_4^2)}}{s} \int d\Omega_{13} |\M|^2 \\
& \times
\frac{2T}{\sqrt{s}} \sqrt{\lambda(s,m_1^2,m_2^2)} K_1\left(\frac{\sqrt{s}}{T}\right)\,,
\end{split}$$
![Production rate densities of our FIMP candidates as functions of the inverse of the SM bath temperature.[]{data-label="fig:rates"}](figures/rates.pdf)
In Fig.\[fig:rates\], I show the production rate densities of a scalar, fermionic and vector FIMPs (blue, green and red curves respectively), as functions of the inverse of temperature. Generic features of the production rates are that, the higher the temperature, the more DM is produced. Also, notice that the rates start to vanish when the temperature of the thermal bath becomes smaller than the DM mass, which is the famous Boltzmann suppression. We can recognize the presence of the poles of both components of the modulus, when the temperature of the thermal bath equals their masses. We can also notice the weaker temperature dependence of the production rate of a fermionic FIMP, due to the chirality flip. For completeness, I point out in Fig. \[fig:rates\] the analytic approximations of the production rates in the limiting cases where the mediators are much lighter, of the same order and much heavier than the temperature of the SM thermal bath.
As previously stated, the relic density of DM depends on which species dominates the cosmic expansion. While it is usual to assume that DM production happened during a radiation-dominated era, in which $H(T) \propto T^2$, this might not be the case in general. In inflationary theories, the universe had undergone a period of entropy production called *reheating* in which it cools down slower and $H(T) \propto T^4$. Such a period would happen from a moment when the temperature of the SM bath reaches a maximal value $T_\max$ up to the moment in which there is no more entropy production, the so-defined reheat temperature $T_\rh$. We do not know the scale of $T_\rh$, which could be as low as $4 \times 10^{-3}$ GeV [@hannestad_what_2004] and as high as $7 \times 10^{15}$ GeV [@rehagen_low_2015]. A general study of freeze-in through heavy portals should therefore take into account the possibility that the masses of mediators are at the reheating scale. So, as long as we have a thermal bath of SM radiation, it starts producing DM. In this context, the relic density of dark matter today receives a contribution from the reheating period and from the radiation era [@dutra_origins_2019]: $$\begin{split}\label{relicsplit}
\Omega_\dm^0 h^2
= \frac{m_\dm}{2.16 \times 10^{-28}} \Big( & \int_{T_0}^{T_\rh} dT \frac{g_s^*}{g_s \sqrt{g_e}} \frac{R_\dm (T)}{T^6} + \\
&+ 1.6\,c\,B_\rad \,g_\rh^{-3/2}\, T_\rh^7 \int_{T_\rh}^{T_\max} dT g_e^* \frac{R_\dm (T)}{T^{13}} \Big) \, .
\end{split}$$
It is easy to see from the equation above that if $R_\dm \propto T^n$ for $n<5$, the production during radiation era is infrared (happens at the lightest scale available) and if $n>5$, it is ultraviolet (happens at the highest scale available, $T > T_\rh$). The same is true for the production during reheating, where the power of temperature is $n = 12$.
From the temperature dependence of the production rates pointed out in Fig.\[fig:rates\], we can understand that the freeze-in of all the FIMP candidates happen at the highest scale of radiation era, the reheating temperature ($n>5$), except for the case where the moduli are produced on-shell, which can make the relic density raise again after levelling-off. Neglecting the production during reheating in such a model lead to an underestimation in the relic density of many orders of magnitude, as it was explicitly shown in the case of an on-shell exchange of spin-2 fields [@bernal_spin-2_2018].
![Evolution of the relic density of our FIMP candidates.[]{data-label="fig:relic"}](figures/relics.pdf)
In Fig. \[fig:relic\], we see the resulting evolution of the relic density, for the same set of free parameters of Fig. \[fig:rates\]. These are the solutions of the coupled set of Boltzmann fluid equations for the evolution of DM, SM radiation and inflaton (driving the reheat period). We have fixed $T_\rh = 10^{11}$ GeV and $T_\max = 10^{13}$ GeV. In the presence of an on-shell production of a mediator, the production of DM is enhanced and that is why we see the relic density of vector and fermionic DM getting enhanced close to the pole of the axial modulus. Of course, the final relic density needs to agree to the Planck results, as I am going to show in the next section.
Agreement with Planck results
-----------------------------
![Contours of good relic density in our free parameter space.[]{data-label="fig:contours"}](figures/approx_fig2full.pdf "fig:") ![Contours of good relic density in our free parameter space.[]{data-label="fig:contours"}](figures/approx_fig3full.pdf "fig:")
We can now see the values of the new physics scale $\Lambda$ and FIMP mass providing a good relic density of DM today, as inferred by the Planck satellite [@aghanim_planck_nodate]. In Fig. \[fig:contours\], the reheating and maximal temperatures are set to $10^{10}$ and $10^{12}$ GeV. Since the relic density increases with DM mass, we need smaller overall couplings in order to not overproduce the FIMPs and therefore $\Lambda$ needs to be raised for a given relic density value. Also, due to the exponential Boltzmann suppression, the thermal bath cannot produce dark matter much before the time of maximal temperature and $\Lambda$ is sharply lowered as to compensate the suppressed rates. In the upper panel, the pole of the axial modulus is reached inside the radiation era. Since it enhances the relic densities of vector and fermionic DM candidates, we need higher values of $\Lambda$. Notice that the curves for the fermionic DM depend strongly on the DM mass, due to the chirality flip. In the lower panel, the exchange of a heavier real modulus suppresses more the relic densities, so that we can have lower values of $\Lambda$, but still at intermediate scales.
Concluding remarks
==================
In this conference, I have discussed the case in which heavy moduli fields exchange between visible and dark matter are the underlying physics of the feeble couplings necessary for the freeze-in to happen. Such fields appear in many structural extensions of the SM, and our results are expected to be embedded in more realistic realizations. We have seen that if the temperature dependencies of the production rates of FIMPs is strong enough, which can be achieved in effective models with derivative couplings, FIMPs would have already been produced at the start of the radiation era. As an interesting outcome, in a wide range of our parameter space a good relic density is “naturally” achieved for scalar, fermionic and vector FIMP candidates with the moduli masses at intermediate scales, and for reasonable scales of new physics.
### Acknowledgments.
I want to thank the co-authors of the work presented in this conference, Debtosh Chowdhury, Emilian Dudas and Yann Mambrini. I also acknowledge the support from the Brazilian PhD program “Ciências sem Fronteiras”-CNPQ Process No. 202055/2015-9 during the development of this work and the current support of the Arthur B. McDonald Canadian Astroparticle Physics Research Institute.
[6]{} D. Chowdhury, E. Dudas, M. Dutra and Y. Mambrini, “Moduli Portal Dark Matter,” Phys. Rev. D [**99**]{}, no. 9, 095028 (2019) doi:10.1103/PhysRevD.99.095028 \[arXiv:1811.01947 \[hep-ph\]\]. D. J. H. Chung, E. W. Kolb and A. Riotto, “Production of massive particles during reheating,” Phys. Rev. D [**60**]{}, 063504 (1999) doi:10.1103/PhysRevD.60.063504 \[hep-ph/9809453\]. L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, “Freeze-In Production of FIMP Dark Matter,” JHEP [**1003**]{}, 080 (2010) doi:10.1007/JHEP03(2010)080 \[arXiv:0911.1120 \[hep-ph\]\]. N. Bernal, M. Heikinheimo, T. Tenkanen, K. Tuominen and V. Vaskonen, “The Dawn of FIMP Dark Matter: A Review of Models and Constraints,” Int. J. Mod. Phys. A [**32**]{}, no. 27, 1730023 (2017) doi:10.1142/S0217751X1730023X \[arXiv:1706.07442 \[hep-ph\]\]. M. Dutra, “Origins for dark matter particles : from the ‘WIMP miracle’ to the ‘FIMP wonder’,” tel-02100637, 2019SACLS059. S. Hannestad, “What is the lowest possible reheating temperature?,” Phys. Rev. D [**70**]{}, 043506 (2004) doi:10.1103/PhysRevD.70.043506 \[astro-ph/0403291\]. T. Rehagen and G. B. Gelmini, “Low reheating temperatures in monomial and binomial inflationary potentials,” JCAP [**1506**]{}, 039 (2015) doi:10.1088/1475-7516/2015/06/039 \[arXiv:1504.03768 \[hep-ph\]\]. N. Bernal, M. Dutra, Y. Mambrini, K. Olive, M. Peloso and M. Pierre, “Spin-2 Portal Dark Matter,” Phys. Rev. D [**97**]{}, no. 11, 115020 (2018) doi:10.1103/PhysRevD.97.115020 \[arXiv:1803.01866 \[hep-ph\]\]. N. Aghanim [*et al.*]{} \[Planck Collaboration\], “Planck 2018 results. VI. Cosmological parameters,” arXiv:1807.06209 \[astro-ph.CO\].
[^1]: We here consider corrections up to the first order in the cut-off scale $\Lambda$.
|
---
abstract: 'We develop a nonstandard concept of atomic clocks where the blackbody radiation shift (BBRS) and its temperature fluctuations can be dramatically suppressed (by one to three orders of magnitude) independent of the environmental temperature. The suppression is based on the fact that in a system with two accessible clock transitions (with frequencies $\nu^{}_1$ and $\nu^{}_2$) which are exposed to the same thermal environment, there exists a “synthetic” frequency $\nu^{}_{\mathrm}{syn}$ $\propto$ ($\nu^{}_1-\varepsilon^{}_{12}\nu^{}_2$) largely immune to the BBRS. As an example, it is shown that in the case of $^{171}$Yb$^+$ it is possible to create a clock in which the BBRS can be suppressed to the fractional level of 10$^{-18}$ in a broad interval near room temperature (300$\pm 15$ K). We also propose a realization of our method with the use of an optical frequency comb generator stabilized to both frequencies $\nu^{}_1$ and $\nu^{}_2$. Here the frequency $\nu^{}_{\mathrm}{syn}$ is generated as one of the components of the comb spectrum and can be used as an atomic standard.'
author:
- 'V. I. Yudin[^1], A. V. Taichenachev, M. V. Okhapkin[^2], and S. N. Bagayev'
- 'Chr. Tamm, E. Peik, N. Huntemann, T. E. Mehlstäubler, and F. Riehle'
title: Atomic clocks with suppressed blackbody radiation shift
---
The main progress in modern fundamental metrology is connected with the development of atomic clocks. The most promising frequency standards today are based on single trapped ions [@Rosenband08] and on ensembles of neutral atoms confined to an optical lattice at the magic wavelength [@aka08; @ludlow08]. It is believed that these clocks can provide frequency references with unprecedented small systematic uncertainties in the 10$^{-17}$-10$^{-18}$ range. This progress will probably lead to a redefinition of the unit of Time and to new fundamental tests of physical theories in particular in the fields of General Relativity, cosmology, and unification of the fundamental interactions [@Turyshev09; @Peik08].
The largest effect that contributes to the systematic uncertainty of many atomic clocks is the interaction of the thermal blackbody radiation with the atomic eigenstates. This effect was first considered in 1982 for cesium atomic clocks [@Itano], but remains up to now a major problem for many modern atomic time and frequency standards. At present there exist three approaches to tackle the blackbody radiation shift (BBRS) problem. The first one is the use of cryogenic techniques to suppress this shift to a negligible level. This approach is pursued for the mercury ion clock [@Oskay], for the Cs fountain clock [@Levi2010], and for the Sr optical lattice clock [@Middelmann]. The second approach is the precise temperature stabilization of the experimental setup in combination with theoretical and/or semiempirical numerical calculations of the shift at given temperature [@Angstmann06; @Safronova10]. The third approach is based on the choice of an atom or ion where both levels of the reference transition have approximately the same BBRS. Here the most promising candidate is $^{27}$Al$^+$ with a fractional BBRS of the reference transition frequency of $\sim$10$^{-17}$ [@Rosenband08; @Chou], followed by $^{115}$In$^+$ [@Becker; @Safronova]. However, the latter approach limits the choice of candidates for tests of fundamental theories.
In the present paper we propose an alternative method allowing us to suppress the BBRS and its fluctuations in atomic frequency standards by one to three orders of magnitude without using cryogenic techniques and precise temperature stabilization. Our approach is based on the use of two reference transitions in an identical thermal environment. We show that in such a system there exists a combined frequency for which the BBRS is significantly suppressed over a wide temperature range. For instance, a trapped $^{171}$Yb$^+$ ion meets this condition in a straightforward way, because $^{171}$Yb$^+$ has at least three suitable reference transitions: an electric-quadrupole and an electric-octupole optical transition [@Tamm09; @Hosaka09; @Sherstov10], and a magnetic-dipole radiofrequency (rf) transition between the ground-state hyperfine sublevels. Apart from laboratory standards, the proposed method can be particularly useful in cases where it is impossible to control the environmental temperature with sufficient accuracy or to use cryogenic techniques, for instance in transportable frequency standards or in space-based clocks that approach the Sun in order to test the local position invariance underlying General Relativity [@Turyshev09].
Our approach is based on the fact that for the large majority of transitions in atoms or ions that are of interest as frequency standard reference transitions, the temperature dependence $\Delta(T)$ of the BBRS is very well approximated by the law $\propto T^4$. Consider now two clock transitions with frequencies $\nu^{(0)}_1$ and $\nu^{(0)}_2$ exposed to the same thermal environment, i.e., located in the same probe volume. We assume that $\nu^{(0)}_1 < \nu^{(0)}_2$. The effect of the BBRS on each transition frequency can be represented as: $$\label{omegaT}
\nu^{}_{j}(T) \approx \nu_{j}^{(0)}+a^{}_j
\left(\frac{T}{T_0}\right)^4\quad (j=1,2)\,,$$ where $a^{}_j$ is an individual characteristic of the transition $j$ determined by the atomic structure and $T_0$ is the mean temperature of the clock operation. Let us introduce the coefficient $\varepsilon^{}_{12}=a^{}_1/a^{}_2$. As is easily seen, the following superposition does not experience the BBRS: $\nu^{}_1(T)-\varepsilon^{}_{12}\nu^{}_2(T)=\nu_1^{(0)}-\varepsilon^{}_{12}\nu_2^{(0)}$. In compliance with this we define a new “synthetic” frequency $\nu^{}_{\mathrm}{syn}$ as $$\label{omegaG}
\nu^{}_{\mathrm}{syn}=R[\nu^{}_1(T)-\varepsilon^{}_{12}\nu^{}_2(T)]
=R[\nu_1^{(0)}-\varepsilon^{}_{12}\nu_2^{(0)}]\,,$$ where $R$ is some numerical multiplier whose value can be chosen freely. Thus, one can use the frequency $\nu^{}_{\mathrm}{syn}$ as a new clock output frequency which is immune to the BBRS and to fluctuations in the operating temperature, while the thermal shifts $a^{}_jT^4$ of the working frequencies $\nu_j$ can be large.
One possibility is to independently measure both frequencies $\nu^{}_{1,2}(T)$ and to use for the clock operation the synthetic frequency according to Eq. (\[omegaG\]) (assuming, for example, $R=\pm 1$). In this case, obviously the synthetic frequency does not directly correspond to any frequency of a physical signal. Another approach is to synthesize this frequency as a real physical signal by means of an optical frequency comb generator. Let us consider the situation where two modes of the frequency comb generator are stabilized to the two optical frequencies $\nu^{}_{1,2}(T)=f^{}_0+n^{}_{1,2}f_{r}$ at a given temperature $T$ (see Fig.\[comb\_fig\]). As a result, the parameters of the comb spectrum, i.e., the pulse repetition rate $f_{r}$ and the offset frequency $f_0$ are unambiguously determined and the frequency of the $m$-th mode equals: $$\begin{aligned}
\label{m_mode}
\nu_m(T)&=&f_0+mf_{r}=\nonumber\\
&&\frac{m(\nu^{(0)}_2-\nu^{(0)}_1)+n^{}_2\nu^{(0)}_1-n^{}_1\nu^{(0)}_2}{n^{}_2-n^{}_1}+
\\
&&\frac{m(a^{}_2-a^{}_1)+n^{}_2a^{}_1-n^{}_1a^{}_2}{n^{}_2-n^{}_1}\left(\frac{T}{T_0}\right)^4\,.\nonumber\end{aligned}$$ From this expression one can define a number $m=m^{}_0$ for which the coefficient of the temperature-dependent term is zero: $$\label{m0}
m^{}_0=\frac{n^{}_1a^{}_2-n^{}_2a^{}_1}{a^{}_2-a^{}_1}
=\frac{n^{}_1-\varepsilon^{}_{12}n^{}_2}{1-\varepsilon^{}_{12}}\,.$$ This shows that the BBRS is suppressed for the frequency $\nu_{m_0}$. After a simple transformation we see that the frequency $\nu_{m_0}$ is the synthetic frequency defined in Eq. (\[omegaG\]), $$\label{omega_m0}
\nu^{}_{m_0}=\nu^{\mathrm}{(comb)}_{\mathrm}{syn}=\frac{\nu_1^{(0)}-\varepsilon^{}_{12}\nu_2^{(0)}}{1-\varepsilon^{}_{12}}\,,$$ as it should be. Here, $m_0$ is the natural number closest to the value of the right-hand side of Eq. (\[m0\]). For this it is necessary to satisfy the condition $m^{}_0>0$ that is equivalent to $\nu^{\mathrm}{(comb)}_{\mathrm}{syn}>0$ in Eq. (\[omega\_m0\]).
Apart from the frequency $\nu_{m_0}$ which is a component of the optical spectrum of the frequency comb generator, in our system one can also define the much smaller frequency $$\label{omega_low}
\frac{\nu_{m_0}}{m^{}_0}=f_{r}+\frac{f^{}_0}{m^{}_0}\,$$ which corresponds to a rf standard at $\nu_{m_0}$/$m^{}_0$. Since the frequencies $f_{r}$ and $f^{}_0$ can be extracted from a stabilized comb generator with negligible error, one can use them to synthesize $\nu_{m_0}$/$m^{}_0$. This synthesized radiofrequency has the same immunity to BBRS as $\nu_{m_0}$. It is interesting to note that the radiofrequency given in Eq. (\[omega\_low\]) is well-defined in our system even if $m^{}_0<0$ in Eq. (\[m0\]), i.e., if the basic frequency component $\nu_{m_0}$ exists only virtually.
As was shown above, in the case of a frequency comb stabilized to two BBR-shifted clock transitions with frequencies $\nu^{}_1(T)$ and $\nu^{}_2(T)$, there exists a frequency component $\nu_{m_0}$ (for $m^{}_0>0$) for which the thermal shift and the sensitivity to temperature fluctuations vanish. This frequency component can serve as an atomic frequency standard. In practice, the BBRS is strongly suppressed for a range of comb frequencies around $\nu_{m_0}$. The residual shift of frequency components $\nu_{m_0\pm l}$ near $\nu_{m_0}$ is given by: $$\begin{aligned}
\label{omega_pm}\nonumber
\nu^{}_{m_0\pm l}&=&\nu_{m_0}\pm lf_{r} \\
&=&\nu_{m_0}\pm l\frac{\nu^{(0)}_2-\nu^{(0)}_1}{n^{}_2-n^{}_1}\pm
l\frac{a^{}_2-a^{}_1}{n^{}_2-n^{}_1}\left(\frac{T}{T_0}\right)^4.\end{aligned}$$ This indicates that the suppression is effective as long as $(n^{}_2-n^{}_1)\gg l$. For example, for frequencies $\nu^{(0)}_1$ and $\nu^{(0)}_2$ in the optical range, the comb mode index difference $(n^{}_2-n^{}_1)$ will typically be in the range of 10$^5$ or higher (see the discussion for the case of $^{171}$Yb$^+$ below). On the whole, the choice of the exact value of the synthetic frequency $\nu^{\mathrm}{}_{\mathrm}{syn}$ is to some extent arbitrary if one takes into account that the coefficient $\varepsilon^{}_{12}$ is only known with limited accuracy and that Eq. (1) is an approximation that neglects higher-order terms in the temperature dependence of the BBRS [@Farleywing]. Including higher-order terms the BBRS can be expresssed as: $$\label{DT}
\Delta^{(j)}(T)=a^{}_j\left(\frac{T}{T_0}\right)^4+b^{}_j
\left(\frac{T}{T_0}\right)^6+...\quad (j=1,2)\,.$$ From this we can estimate a basic limitation of the possibility to suppress the BBRS and its temperature dependence. Usually, near room temperature $T_0=300$ K the contribution of the higher terms \[$b^{}_j (T/T_0)^6+...$\] is a factor of 10$^{}$ to 10$^{3}$ smaller than that of the main $T^4$-term [@Itano; @Pal'chikov]. This indicates that here it would not be useful to suppress the $T^4$-dependence of $\nu^{}_{\mathrm}{syn}$ to better than one to three orders of magnitude because higher-order contributions to the BBRS remain uncompensated. For example, in order to achieve a suppression factor of 10$^2$ for the $T^4$-dependence, it would be sufficient to know the coefficient $\varepsilon^{}_{12}$ with relative uncertainty of 10$^{-2}$.
It may be noted that apart from theoretical calculations the coefficient $\varepsilon^{}_{12}$ can be determined by purely experimental means. To do this we can apply a quasistatic electric field (or the field of an infrared laser) to determine the shifts of the reference transition frequencies $\nu^{}_1$ and $\nu^{}_2$ due to the differences in the static polarizabilities of the involved atomic energy levels. From a practical point of view it is very advantageous that we do not need to know the magnitude of the electric field at the place of the atoms because we only have to determine the ratio $a_1/a_2$. If a frequency comb generator is stabilized to $\nu^{}_1$ and $\nu^{}_2$ as shown in Fig.1, the frequency $\nu_{m_0}$ can be identified in a direct way as the frequency component which does not experience any scalar Stark shift in the applied quasistatic field.
As an example that permits the practical realization of the ideas presented above, we consider the ion $^{171}$Yb$^+$. As shown in Fig.\[Yb\_fig\], the level system of $^{171}$Yb$^+$ provides two narrow-linewidth transitions from the ground state in the visible spectral range which can be used as reference transitions of an optical frequency standard: the quadrupole transition $^2$S$_{1/2}(F=0) \to ^2$D$_{3/2} (F=2), \lambda \approx 436$ nm and the octupole transition $^2$S$_{1/2}(F=0) \to
^2$F$_{7/2}(F=3), \lambda\approx 467$ nm. More detailed information on the spectroscopy of these transitions can be found in [@Schneider; @Tamm09; @Hosaka09; @Sherstov10]. It may be noted that the case of $^{171}$Yb$^+$ is especially attractive because here both clock transitions lie in a technically convenient frequency range and experience exactly the same thermal environment if probed in one ion.
The BBRS of the quadrupole and octupole transitions of Yb$^+$ were calculated in Ref. [@Lea06]. This calculation is based on calculated oscillator strengths and experimental lifetime and polarizability data. The room-temperature BBRS of the quadrupole transition is calculated as $a_{\mathrm}{quad} = -0.35(7)$ Hz (fractional shift $5.1(1.1) \times 10^{-16}$) and that of the octupole transition as $a_{\mathrm}{oct} = -0.15(7))$ Hz (fractional shift $2.4(1.1) \times 10^{-16}$). The relatively small value of $a_{\mathrm}{oct}$ and the large relative uncertainty is due to nearly equal shifts of the $^2$S$_{1/2}$ and $^2$F$_{7/2}$ levels.
Using the results of Ref. [@Lea06], for $^{171}$Yb$^+$ we find that $\varepsilon^{}_{12} = a_{\mathrm}{oct}$/$a_{\mathrm}{quad} = 0.43(22)$. We expect that the large uncertainty of this value can be reduced to less than 1% by improved atomic structure calculations or by a direct measurement of $\varepsilon^{}_{12}$ as discussed above. In the following, we will not take into account the present uncertainty because our conclusions remain qualitatively unchanged for all values of $\varepsilon^{}_{12}$ in this uncertainty range. In particular, we find the synthetic frequency $\nu^{\mathrm}{(comb)}_{\mathrm}{syn}\approx 607$ THz, corresponding to a wavelength $\lambda_{\mathrm}{syn}\approx 494$ nm. This frequency lies sufficiently close to the initial reference transitions at 436 nm and 467 nm that it can be generated as a spectral component of a femtosecond comb generator that is locked to the reference transitions as shown in Fig. 1.
The higher-order contributions in Eq.(\[DT\]) to the BBRS of the octupole reference transition are negligible compared to that of the quadrupole transition. For the latter, we find $b/a\approx
0.1$ at $T_0=300$ K. As a result, we estimate that the BBRS can be suppressed to the fractional level of $2.7\times 10^{-17}$ at 300 K with variations at the level of $\pm 5 \times 10^{-18}$ if the ambient temperature varies in a broad interval of $\pm 15$ K.
It is also possible to estimate the frequency interval around $\nu^{\mathrm}{(comb)}_{\mathrm}{syn}$ where the components of the comb spectrum have a similar level of suppression of the thermal shift and of its fluctuations. For a suppression factor of 10$^2$, this interval has a width of the order of 1000 GHz. Thus, for $d\sim
100$ MHz, the indicated interval contains 10$^4$ comb modes, each of which could be used as a stable frequency reference.
Other variants of BBRS-free optical frequency standards at a synthethic frequency can be conceived based on transitions $^1$S$_0 \to ^3$P$_0$ in alkaline-earth-like neutral atoms confined in an optical lattice. Consider, for instance, the combination of the reference transitions of strontium ($\nu^{}_1
\approx 429$ THz, $\lambda \approx 698$ nm) and ytterbium ($\nu^{}_2 \approx 518$ THz, $\lambda \approx 578$ nm). Using the calculations in Ref. [@Porsev], in this case we obtain $\varepsilon^{}_{12}\approx 1.69$ and an estimated synthetic frequency $\nu^{\mathrm}{(comb)}_{\mathrm}{syn}\approx 648$ THz ($\lambda_{\mathrm}{syn} \approx 463$ nm). The technical realization of this variant requires the operation of two lattice-based clocks with different atoms (Sr and Yb) in the same vacuum chamber.
So far we have considered examples where both reference transition frequencies lie in the optical region. However, it is also possible to realize schemes where the high frequency $\nu^{}_{2}$ is optical, but the lower frequency $\nu^{}_{1}$ corresponds to a fine- or hyperfine-structure splitting so that it lies in the terahertz or microwave range. In contrast to the above example of ion Yb$^+$, which seems unique because it provides two optical reference transitions, in this case one can find many appropriate schemes that use a single atomic species. Also the ion $^{171}$Yb$^+$ offers the possibility of using the ground-state hyperfine transition $F=0\to F=1$ at $\nu^{}_1$=12.6 GHz (see Fig.\[Yb\_fig\]) as a low-frequency reference transition. For the combination with the octupole transition $^2$S$_{1/2}(F=0) \to
^2$F$_{7/2}(F=3)$ at $\nu^{}_2$=642 THz, a numerical estimate yields $\varepsilon^{}_{12}$$\approx$6.6$\times$10$^{-5}$ and a synthetic frequency $\nu^{\mathrm}{}_{\mathrm}{syn}=-(\nu^{}_1-\varepsilon^{}_{12}\nu^{}_2)\approx$30 GHz. (We expect that our estimate on $\varepsilon^{}_{12}$ is accurate to $\pm 20$ %, but it is not principal for further results.) Here the $T^6$-contribution to the BBRS (see Eq.(\[DT\])) limits the BBRS suppression at the fractional level of 7.5$\times$10$^{-19}$ at $T$=300 K with variations of $\pm$2$\times$10$^{-19}$ in the temperature interval of 300$\pm 15$ K. However, we should also take into account the shift of the ground-state hyperfine levels ($\propto T^2$) due to the magnetic blackbody radiation field [@Itano]. The corresponding BBRS of the hyperfine frequency $F=0\to F=1$ for $^{171}$Yb$^+$ is: $$\label{MD}
\Delta^{(1)}_{\mathrm}{magn}(T)=-1.616\times
10^{-7}\times\left(\frac{T({\mathrm}{K})}{300}\right)^2\; {\mathrm}{Hz}.$$ For $\nu^{\mathrm}{}_{\mathrm}{syn}$=30 GHz this shift results in a fractional level of 5.4$\times$10$^{-18}$ at $T$=300 K with a variation of $\pm$5$\times$10$^{-19}$ for 300$\pm 15$ K. Since the magnetic BBRS contribution can be readily calculated with an accuracy of less than 1%, it is possible to apply a corresponding correction to $\nu^{\mathrm}{}_{\mathrm}{syn}$ with an uncertainty contribution of less than 10$^{-19}$.
For $^{171}$Yb$^+$ we have thus shown the possibility to create a synthetic-frequency-based atomic clock with a fractional uncertainty contribution due to BBRS of $<$1.5$\times$10$^{-18}$ in a broad interval of 300$\pm 15$ K. To achieve this, we need to know the coefficient $\varepsilon^{}_{12}$ with a relative accuracy in the range of 0.1-0.2%. In order to reduce the BBRS uncertainty contribution to less than 10$^{-17}$, the value of $\varepsilon^{}_{12}$ needs only be known with a relative accuracy of 3%. We also have pointed out that for $^{171}$Yb$^+$ the combination of the octupole optical clock transition with the ground-state hyperfine transition can yield a much better BBRS suppression than the combination of the octupole and quadrupole optical clock transitions. The use of the quadrupole transition yields a lower BBRS suppression because the upper level $^2$D$_{3/2}$ is connected to the $^2$P$_{1/2}$ level by a strong infrared transition at 2.44 $\mu$m, which produces a relatively large $T^6$-contribution to the BBRS. The final comparison of the two options for BBRS suppression should also include detailed estimates on the magnitudes of other systematic uncertainty contributions in the considered experimental setup.
The concept of a synthetic atomic frequency standard based on two reference transitions can also be extended to the case that both reference frequencies $\nu{}_{1,2}$ lie in the microwave range. Atomic fountain clocks are based on reference transitions in the microwave range between the ground-state hyperfine sublevels of alkali atoms. For a synthetic atomic fountain frequency standard, for instance the combination $^{87}$Rb ($\nu^{}_1 \approx
6.8$ GHz) and $^{133}$Cs ($\nu^{}_2 \approx 9.2$ GHz) can be considered. Here, at the synthetic frequency $(\nu^{}_1-\varepsilon^{}_{12}\nu^{}_2)\approx 1.9$ GHz it is possible to suppress the fractional BBRS of the individual standards by two orders of magnitude. It is interesting to note that nearly optimal conditions for the efficient suppression of the BBRS are realized in the dual Rb$/$Cs fountain clock described in Ref. [@Guena] because here both reference transitions are exposed to the same thermal environment.
We finally note that the $^{171}$Yb$^+$ optical frequency standard is a very sensitive system for a search for temporal variations of the fine structure constant $\alpha$ [@Dzuba; @Lea]. The frequencies of the quadrupole and octupole reference transitions of Yb$^+$ have significant contributions from relativistic effects and would undergo changes with different sign in consequence of a change of $\alpha$. The synthetic frequency that eliminates the BBRS retains this sensitivity. The $\alpha$-dependence of an atomic transition frequency may be expressed generally as $\nu=\nu_0 + qx$, where $x\equiv (\alpha/\alpha_0)^2-1$, $\nu_0$ defines the frequency at the present value of the fine structure constant, $\alpha_0$, and $q$ quantifies the sensitivity to changes of $\alpha$ [@Dzuba]. The $q$ parameter for the synthetic frequency is simply given by $q_{\mathrm}{syn}=(q_1-\varepsilon^{}_{12}q_2)/(1-\varepsilon^{}_{12})$. For Yb$^+$, with $q$ parameters as given in Ref. [@Dzuba], $q_{\mathrm}{syn}$ amounts to about -3220 THz. Comparison with the Yb$^+$ synthetic frequency $\nu^{\mathrm}{(comb)}_{\mathrm}{syn}\approx 607$ THz indicates the strong sensitivity. In a test for variations of $\alpha$, the synthetic frequency would have to be compared to an “anchor” reference transition with small $q$ value, like the $^1$S$_0\rightarrow ^3$P$_0$ transition in Al$^+$.
In conclusion, we have proposed and developed the concept of an atomic frequency standard where the frequency shift due to the ambient blackbody radiation and related fluctuations of the output frequency can be suppressed by one to three orders of magnitude without using cryogenic techniques. We also expect that our results will stimulate refined atomic structure calculations on Yb$^+$ and other atomic systems that are of interest in this context. Such calculations can yield precise values for the frequency synthesis parameter $\varepsilon^{}_{12}$ and determine limitations of the achievable BBRS suppression.
We thank U. Sterr for useful discussions. This work was supported by QUEST, DFG/RFBR (grant 10-02-91335), RFBR (grant Nos. 10-02-00406, 11-02-00775, and 11-02-01240), RAS, Presidium SB RAS, and by the federal programs “Development of scientific potential of higher school 2009-2010” and “Scientific and pedagogic personnel of innovative Russia 2009-2013”.
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[^1]: e-mail address: viyudin@mail.ru
[^2]: also PTB
|
---
abstract: 'Previous results indicated that high $p_T$ particle suppression in Au+Au interactions is a final state effect, since R$_{dA}$ ratios were compatible with unity, albeit within large experimental errors. It is important to test this conclusion to higher precision since the modification of structure functions may be involved. Recent d+Au data taken in 2008 improve the integrated luminosity by about a factor of thirty compared to the 2003 data. A more precise measurement of both $\pi^0$ and $\gamma$ at higher $p_T$ will shed new light on whether the initial state in the heavy nuclei is modified.'
address: '$^{a}$ University of California – Riverside, 900 University Ave, Riverside CA 92521, USA'
author:
- 'Ondřej Chvála$^{a}$ for the PHENIX collaboration'
title: 'Measurement of $\pi^{0}$ and $\gamma$ in d+Au collisions at $\sqrt{s_{NN}}=$200GeV by PHENIX experiment'
---
Introduction {#intro}
============
The PHENIX experiment at RHIC is designed for high rate measurement of electromagnetic probes, specifically direct photons $\gamma$ and neutral pions $\pi^{0}$ at mid-rapidity ($|\eta|<$0.35). The high event rate allows unique access to rare probes, such as particles produced at very high transverse momentum ($p_T$).
Early results for the nuclear modification factors R$_{AA}$, the spectrum in Au+Au collisions over the spectrum in p+p interactions scaled by the respective nucleon overlap integrals, showed suppression of $\pi^{0}$ and charged hadron production at high-$p_{T}$ ($p_{T} \gtrsim$6GeV/c) [@Adcox:2001jp; @Adcox:2002pe; @Adler:2003qi]. This result, combined with the apparent lack of suppression for high-$p_T$ direct $\gamma$ production [@Adler:2005ig] has been interpreted as an indication of the formation of a dense strongly interacting medium, the sQGP, in heavy ion collisions.
Furthermore, the preliminary analysis of the high statistics 2004 Au+Au run showed a possible decrease of direct photon R$_{AA}$ above 14GeV/c of transverse momentum [@Reygers:2008pq]. Possible explanations for this observation include the isospin effect (the difference of partonic content between protons and neutrons), the EMC effect (modifications of the distribution of partons in the heavy nuclei), and the suppression of the direct photons originating from fragmenting partons which are quenched by the medium.
RHIC measurements of particle production in d+Au collisions from 2003 show no suppression of produced particles within large experimental error bars [@Adler:2003ii; @Peressounko:2006qs], indicating little or no modification of the initial state in gold nuclei. This result confirmed the attribution of the large suppression observed in central Au+Au interactions to final state effects in the sQGP medium.
The final analysis of the 2003 data set revealed some suppression of $\pi^{0}$ production at the highest $p_T$ in the most central d+Au interactions [@Adler:2006wg], however the experimental uncertainties are too large to estimate cold nuclear matter effects quantitatively [@Barnafoldi:2008ec; @Zhang:2008ek].
RHIC year 2008 data set {#2008}
=======================
The RHIC run in year 2008 improved the total integrated luminosity of the d+Au sample by a factor of $\approx$30 compared to the 2003 run: 1.65$\times$10$^9$ minimum bias (MB) events, and 3.68$\times$10$^9$ events from a high-$p_{T}$ photon (ERT) trigger were recorded. There were also 0.53$\times$10$^9$ MB triggered and 1.17$\times$10$^9$ ERT triggered p+p collisions taken in 2008. The uncorrected $\pi^{0}$ yields are shown in Fig. \[fig:pi0\].
The improvement in total d+Au integrated luminosity can be appreciated from the relative statistical errors in the $\pi^{0}$ measurement using PHENIX PbSc EMCal. The error at $p_{T}=$15GeV/c is 32% using the 2003 sample, and is reduced to 3.5% in the recent high statistics run.
![Raw mid-rapidity yields per event of $\pi^{0}$ production from the 2008 data sets: p+p minbias, d+Au minbias, and three centrality bins in d+Au collisions using PHENIX PbSc EMCal. The spectra from ERT triggered runs (red) are scaled to match MB yields (blue) in 6–13 GeV/c region of $p_{T}$. Only statistical errors are shown.[]{data-label="fig:pi0"}](run8_pi_scaled_pid0.eps){width="105.00000%"}
Conclusions
===========
Previous results from data taken in year 2003 indicate that the high $p_{T}$ suppression observed in Au+Au interactions is a final state effect since the R$_{dA}$ ratios are consistent with unity, albeit within large experimental errors. It is important to test this conclusion to higher precision. Recent d+Au data taken in 2008 improve the integrated luminosity by about a factor of thirty compared to the 2003 run. The 2008 PHENIX measurement will shed more light on the origin of cold nuclear matter effects.
[00]{}
K. Adcox [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [**88**]{}, 022301 (2002) \[arXiv:nucl-ex/0109003\]. K. Adcox [*et al.*]{} \[PHENIX Collaboration\], Phys. Lett. B [**561**]{}, 82 (2003) \[arXiv:nucl-ex/0207009\]. S. S. Adler [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [**91**]{}, 072301 (2003) \[arXiv:nucl-ex/0304022\]. S. S. Adler [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [**94**]{}, 232301 (2005) \[arXiv:nucl-ex/0503003\]. K. Reygers \[PHENIX Collaboration\], J. Phys. G [**35**]{}, 104045 (2008) \[arXiv:0804.4562 \[nucl-ex\]\]. S. S. Adler [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [**91**]{}, 072303 (2003) \[arXiv:nucl-ex/0306021\]. D. Peressounko \[PHENIX Collaboration\], Nucl. Phys. A [**783**]{}, 577 (2007) \[arXiv:hep-ex/0609037\].
S. S. Adler [*et al.*]{} \[PHENIX Collaboration\], Phys. Rev. Lett. [**98**]{}, 172302 (2007) \[arXiv:nucl-ex/0610036\].
G. G. Barnafoldi, G. Fai, P. Levai, B. A. Cole and G. Papp, \[arXiv:0805.3360 \[hep-ph\]\]. B. W. Zhang and I. Vitev, arXiv:0810.3194 \[nucl-th\].
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---
address:
- '(1) Institut für Astronomie, Universität Wien, Türkenschanzstra[ß]{}e 17, A-1180 Wien, Austria'
- '(2) Department of Mathematics, University of Évora, R. Romão Ramalho 59, 7000 Évora, Portugal'
- '(3) MPI für extraterrestrische Physik, Giessenbachstra[ß]{}e 1, D-85748 Garching, Germany'
author:
- 'D. BREITSCHWERDT (1), M.A. DE AVILLEZ (1,2), M.J. FREYBERG (3)'
title: 'Keynote Lecture: Galactic and Extragalactic Bubbles'
---
[: ISM: general - ISM: evolution - ISM: bubbles - Galaxies: ISM - X-rays: ISM]{}
Introduction
============
The term “bubble” in science is not well defined. A convenient operational definition could be that of a closed two-dimensional surface in three-space, separating two media of different (physical) properties, with lower density inside than outside. Consequently there exists a vast range of topics in the literature from electron bubbles in superfluid helium to bubbles in avalanches. In astrophysics, interstellar bubbles are an already classical subject in ISM research, and seem to have experienced a renaissance each time a new process for significant energy injection has been found. After the discovery of the Stroemgren (1939) sphere, generated by stellar Lyman continuum photons, it became clear that the rise in temperature in the H[ii]{} region, $T_{II}$, with respect to the neutral ambient medium, $T_{I}$, would imply a strong pressure imbalance of the order $\sim T_{II}/T_{I}$, and thus a shock wave would be driven outwards (Oort 1954). The major effect is the growth of the H[ii]{} region in size due to the decrease in electron density $n_e$ inside and hence in recombination rate ($\propto
n_e^2$), allowing stellar photons to propagate further out. The net result is a bubble filled with low density ionized gas. Another, even more powerful energy injection mechanism emerged after the discovery of P Cygni profiles in stellar spectra (Morton 1967), revealing the existence of hypersonic stellar winds smashing into the ISM at speeds of $\sim 2000$ km/s and a canonical mass loss rate of $10^{-6} \, \msol/{\rm yr}$, or even higher for Wolf-Rayet stars. Like in the case of the solar wind this leads to a two-shock structure, separated by a contact discontinuity that isolates the wind from the ISM material, but is in reality unstable to perturbations and allows mixing and mass loading of the stellar wind flow. Observationally, these bubbles can be detected in X-rays, since the temperature behind the inner termination shock rises according to mass and momentum conservation (energy conservation for a monatomic gas does not give any new information) to $T \approx
\frac{3}{32} \bar m \frac{V_w^2}{k_B} = 5.4 \times 10^7 \, {\rm K}$, neglecting ambient thermal pressure with respect to ram pressure. Under these conditions radiative cooling of the bubble is negligible, and the bubbles are in their so-called energy driven phase. Adiabatic cooling due to $p-dV$ work on the surrounding medium, however, is significant. Sweeping up and compressing ISM gas slows down the outer shock, which can eventually suffer severe radiative losses, as the downstream density is considerably higher than behind the stellar wind termination shock. Moreover the dense outer shell (or part of it) is photoionized and thus well observable in the optical. Diffuse soft X-ray emission from stellar wind bubbles has been observed from NGC 6888 with ASCA SIS (Wrigge et al. 1998) and from S 308 with XMM-[*Newton*]{} EPIC pn (Chu et al. 2003), revealing temperatures of $1.5 \times 10^6$ K and $8 \times
10^6$ K for NGC 6888, and $1.1 \times 10^6$ K, for S 308. As we shall see, mass loading must play an important rôle for decreasing the temperature thereby enhancing the cooling and thus the X-ray emissivity.
Already 70 years ago, Baade & Zwicky (1934) suggested that stellar core collapse resulting in a neutron star could release a sufficient amount of gravitational energy to power a supernova (SN) explosion. The effect of the kinetic energy of $10^{51}$ erg, which is only $\sim 1$% of the released neutrino energy, is dramatic. After a free expansion phase, in which a shell with mass similar to the ejecta mass is swept up, the pressure difference between shocked ISM and ejecta drives a reverse shock into the supernova remnant (SNR), reheating the ejecta as it propagates inwards. The transfer of energy to the ISM is now determined by the adiabatic Sedov-Taylor phase, lasting several ten thousands of years until the cooling time of the outer shock becomes less than the dynamical time scale, and the SNR enters the radiative phase before eventually merging with the ISM. The widespread O[vi]{} line in the ISM is thought to be the signature of old SNRs.
Young star clusters, like the four million years old NGC 2244 exciting the Rosette Nebula, can blow holes into the emission nebulae by the *combined* action of several stellar winds. Since the discoveries of huge H[i]{} shells, so-called supershells, either in the Milky Way (e.g. Heiles 1979, 1980) or in M 31 (Brinks & Shane 1984), or by direct observation of huge X-ray emitting cavities (e.g. Cash et al. 1980) in the Orion, Eridanus or Cygnus regions, it is understood that O- and B-stars in concert can create *superbubbles* (SBs). Although stellar winds are the initial contributors over the first few million years, it is obvious that SN explosions dwarf their energy input over the SB life time of a few tens of million years. The SBs range in sizes from a few tens to a few hundreds of parsecs. The most prominent ones are undoubtedly the Local Bubble (LB), in which our solar system is immersed, and which is still not well understood, and the adjacent Loop I SB (LI); both will be discussed in some detail in this review. Extreme examples of bubbles with respect to their energy input on galactic scales are those driven by (nuclear) starbursts - often called superwinds -, which in case of NGC 3079 (Cecil et al. 2002) show clear signatures of an outflowing bubble both with Chandra and HST. Although driven by star formation processes, but for lack of space, we will discuss neither planetary nebulae, which exhibit a prominent white dwarf blown wind bubble, nor pulsar wind bubbles driven by energetic particles, nor bipolar outflows and jets, which show some additional features such as Mach disks. Why is it important to study bubbles? Apart from being interesting astrophysical objects, they are part of the interstellar matter cycle, enriching the ISM with metals, and, most importantly, they are the major energy sources of the ISM, controlling its structure and evolution, with SNRs and SBs being the major contributors.
In Section \[obs\] we present some recent observations of the LB and the LI SB, taken in X-rays and in H[i]{}. Then, in Section \[anal\] some analytical work, mainly similarity solutions, and their limitations are discussed, and in Section \[num\] numerical high resolution simulations of the LB and LI are shown, finishing off with our conclusions in Section \[conc\].
Observation of nearby superbubbles {#obs}
==================================
Since we want to describe bubbles in the *young local universe*, the closest examples are undoubtedly the LB, in which our solar system is immersed, and the neighbouring LI SB, whose outer shell is most likely in contact with the LB shell (cf.Egger & Aschenbach 1995). The centre of the LI bubble is approximately 250 pc away, and the bubble radius is about 170 pc. Its proximity is most impressively seen in a ROSAT All Sky Survey (RASS) multispectral image (Freyberg & Egger 1999), which shows it as the largest coherent X-ray structure in the sky (see Fig. \[rass\_rgb\]), centred roughly on the Galactic Centre direction and covering a solid angle of $7/6 \, \pi$ (Breitschwerdt et al. 1996).
Although the existence of local X-ray emission was realized already soon after the observation of the diffuse soft X-ray background (SXRB) (Bowyer et al. 1968), the idea received considerable support by H[i]{} observations, revealing a local *cavity* (e.g. Frisch & York, 1983).
The simplest explanation is given by the so-called “displacement” or Local Hot Bubble model (Sanders et al., 1977; Tanaka & Bleeker, 1977), in which it is assumed that the solar system is immersed in a bubble of diffuse hot plasma in collisional ionization equilibrium, that displaces H[i]{}, and has an average radius of $100$ pc. More recent observations of the Local Cavity (see Fig. \[lb-na1\]), using Na[i]{} as a sensitive tracer of H[i]{} (Lallement et al. 2003), show a more complex 3D structure of the hydrogen deficient hole in the Galactic disk, as well as a clear indication of opening up into the halo like a chimney. As a natural result, also the X-ray brightness of the bubble will vary with direction, especially if entrained clouds are shocked and evaporated. Indeed, observations of the SXRB exhibit a distinct patchiness in emission, as has been reported from ROSAT PSPC observations (Snowden et al. 2000), and more recently by a mosaic of observations of the Ophiuchus cloud (Mendes et al. 2005), which was used to shadow the SXRB and thus allow to disentangle cloud fore- and background emission.
The absorbing column density in direction to the Ophiuchus cloud, which is a well-known star forming region at a distance of 150 pc in direction of the Galactic Centre, contains up to $10^{22} {\rm
cm}^{-2}$ H atoms, efficiently blocking out background radiation up to 1 keV. This is demonstrated nicely by a deep shadow in diffuse X-rays, which correlates very well with the IRAS 100 $\mu$ contours (s. Fig. \[oph-im\]). Our analysis of the spectral composition of the fore- and background radiation (s. Fig. \[oph-sp\]) shows convincingly that the major fraction of the emission below 0.3 keV (most likely unresolved carbon lines) is generated in the foreground. In addition we observe a significant local fraction of oxygen lines (O[vii]{} and O[viii]{}) between 0.5 and 0.7 keV, as well as lines between 0.7-0.9 keV (probably iron). These results give strong observational evidence that the gas inside the LB is *not* in collisional ionization equilibrium, in disagreement with the classical Local Hot Bubble model. A small fraction of the foreground emission stems also from LI. However, Fig. \[oph-sp\] shows that the off-cloud spectrum contains iron lines, which are absent in the on-cloud spectrum. Therefore the excitation temperature in the LI SB must be significantly higher than in the LB, allowing to disentangle spectrally the respective contributions. This is not surprising as LI is still an active SB in contrast to the LB, as we shall see in section \[num\].
The spectral interpretation of the SXRB is far from trivial. First of all it is at present unclear to what extent different sources contribute. These are: (i) diffuse local emission from the LB (and possibly the LI SB, although its major component has a distinctly higher temperature), (ii) diffuse Galactic emission from the hot ISM (and other SBs) and unresolved point sources (e.g. X-ray binaries), (iii) a diffuse Galactic halo component (presumably from the Galactic fountain/wind), (iv) a diffuse extragalactic component (thought to consist of the WHIM = Warm Hot Intergalactic Medium and unresolved point sources). Shadowing the darkest regions of the Milky Way, i.e. nearby Bok globules with extinctions of $A_V \sim
30 - 50$ mag, give unmistakably in case of Barnard 68 *two temperature components* of the local emission (and thus *inconsistent* with the standard Local Hot Bubble model!): $k
T_1 \approx 0.14\pm 0.04$ keV, and $T_2 \approx 0.20\pm 0.06$ keV (Freyberg et al. 2004). How is this possible? Several (not mutually exclusive) explanations have to be further investigated: (i) the LB is an old SB emitting X-rays from a gas not in ionization equilibrium (cf. Breitschwerdt & Schmutzler 1994) thus mimicking a multi-temperature plasma, (ii) there is a significant contribution from heliospheric plasma which undergoes charge exchange reactions with highly ionized solar wind atoms (Lallement 2004). At present it is unclear what the quantitative contribution of the latter process is (values $\leq 75$% in the disk and $\leq 50$% in the halo have been advocated). Such a very local emission should in principle exhibit seasonal variations. We have obtained two exposures of the Ophiuchus cloud, which partially overlap and are 6 months apart. The differences in emission measure and the spectrum are within the noise level. Although this is no counterargument it does not support the hypothesis of a substantial time-dependent variation of the heliospheric contribution. Further studies are needed to pin down this crucial component.
Analytical treatment of superbubble (SB) evolution {#anal}
==================================================
The dynamics of SBs has been worked out analytically by McCray & Kafatos (1987), based on earlier work by Pikel’ner (1968), Dyson & deVries (1972), Weaver et al. (1977) on stellar winds.
A basic principle, which is used in aerodynamics for constructing models in the wind channel, is the scaling of hydrodynamic flows if there are no specific length or time scales entering the problem. Strictly speaking this is never fulfilled, because there are always boundary layers or time-dependent changes in the flow, but for studying the large-scale asymptotic behaviour of the flow this ansatz works remarkably well. If we e.g. neglect the initial switch-on phase of a SB, if we assume that the stellar source region is much smaller than the bubble, and if the discontinuous energy supply during the SN explosion phase can be approximated by a continuous injection of mass, momentum and energy, then *similarity solutions* are reasonably well applicable.
Mathematically speaking, the transformation to a similarity variable $\xi = (r/A) t^{-\alpha}$, projects the family of solutions of a PDE system to a one-dimensional family, with all hydrodynamic variables depending only on the dimensionless similarity variable $\xi$. A flow is said to be self-similar if its properties at any point $x_1$ and instance of time $t_1$ can be recovered by a similarity transformation at some other point in spacetime $(x_0, t_0)$. The exponent $\alpha$ can already be derived from dimensional analysis. The physical quantities determining the SB dynamics are the energy injection rate, $L_{\rm SB}$, (with mass and momentum injection being negligible with respect to the shell mass and momentum during the energy driven phase) and the ambient density, $\rho_0$. Note that it is implicitly assumed that the pressure of the ambient medium can be neglected with respect to the interior pressure of the bubble. This is certainly valid until the shock becomes weak, in which case counterpressure has to be included. Then, $\xi =
\left(L_{\rm SB}/\rho_0\right)^{-1/5} r t^{-3/5}$, is the only possibility to form a dimensionless quantity. Using this similarity variable, it is now possible to construct the complete flow solutions in terms of variables $u^\prime(\xi), \rho^\prime(\xi)$ and $P^\prime(\xi)$, obeying matching conditions for boundaries in the flow at which these variables change discontinuously, like at the termination shock (where the “wind” ejecta are decelerated), the contact discontinuity (separating the wind from the ISM flow), and the outer shock (propagating into the ISM). The integration of the resulting ODE system is a straightforward but tedious exercise that can be carried out with the help of an integral, representing the conservation of the total energy of the system. We can simplify the procedure considerably by making a few additional, but well justified, assumptions about the flow in the different regions. Firstly, the ejecta gas, having a high kinetic energy, is compressed and heated by the strong termination shock, converting 3/4 of its initial bulk motion into heat. Therefore the temperature and the speed of sound in this bubble region are so high, that radiative cooling can be neglected and the pressure remains uniform for long time. On the other hand the pressure in the swept-up shell is also uniform due to its thinness, or in more physical terms, because the sound crossing time is much less than the dynamical time scale. This is because the outer shock can cool efficiently, as the ISM density is orders of magnitude higher than the ejecta gas density, if the latter one is assumed to be smoothly distributed. In essence, we are allowed to assume spatially constant density and pressure in the wind bubble and the shell, respectively.
As it turns out, the assumption of constant energy injection rate $\lsb$ can be relaxed without violating the similarity argument. In reality we are dealing with an OB association, in which the stars are distributed according to some initial mass function (IMF) given by $\Gamma = {d\log\zeta(\log m) \over d\log m}$; $\zeta$ denotes the number of stars per unit logarithmic mass interval per unit area with $\Gamma = -1.1 \pm 0.1$ for stars in Galactic OB associations with masses in excess of 7 $\msol$ (Massey et al. 1995). This translates into a number $N(m) \, dm$ of stars in the mass interval $(m, m+dm)$ (calibrated for some mass interval $N_0 = N(m_0)$), i.e.$
N(m) dm = N_0 \left(m\over m_0\right)^{\Gamma-1} dm \,.
$ It can be transformed into a time sequence, if we express the stellar mass by its main sequence life time, $\tau_{\rm ms}$. For stars within the mass range $7\msol \leq m \leq 30 \, \msol$ this can be empirically approximated by $\tau_{\rm ms} = 3\times 10^7 \,
(m/[10 \msol])^{-\eta}$ yr (Stothers 1972), with $\eta = 1.6$. Since this defines $m$ as a function of time $\tau$, implicitly assuming that the energy input can be described as a continuous process, we obtain ${m(\tau)} ={\rm M}_\odot \, \left(\tau \over C\right)^{-1/\eta}
$, with $C = 3.762 \times 10^{16}$ s.
Let then $L_{\rm SB}(t)$ be the energy input per unit time due to a number of successive SN explosions with a constant energy input of $E_{\rm SN} = 10^{51}$ erg each, so that the cumulative number of SNe between stellar masses $m$ and $m_{\rm max}$ reads $$\begin{aligned}
\tilde N_{\rm SN}(m) = \int_m^{\rm mmax} N(m^\prime ) dm^\prime =
\frac{N_0 \, m_0}{\Gamma}\left[\left(m^\prime \over
m_0\right)^\Gamma\right]_{m}^{\rm mmax} \,.\end{aligned}$$ Then we have $$\begin{aligned}
L_{\rm SB} &=& {d \over dt} (\tilde N_{\rm SN} E_{\rm SN})= E_{\rm SN}
{d \tilde N_{\rm SN} \over dt}
= E_{\rm SN} {d \tilde N_{\rm SN} \over dm} {dm \over d\tau} {d\tau \over dt}\\
&=&\frac{N_0 E_{\rm SN} \msol \, K^{1-\Gamma}}{\eta C} \,
\left(\frac{\tau_0+t}{C}\right)^{-(\Gamma/\eta + 1)} \,,
\label{eninp1}\end{aligned}$$ using the previous equations, and putting $m_0 = K \, \msol$. Since $\tau = t + \tau_0$, where $t$ is the time elapsed since the first explosion, i.e. $\tau_0 = \tau_{\rm MS}(m_{\rm max})$, we have $d\tau/dt=1$. With the above values for $\Gamma$ and $\eta$, we obtain the useful formula $L_{\rm SB} = L_0 \, t_7^{\delta}$, where $\delta = -(\Gamma/\eta + 1) = -0.3125$ and $t_7 = t/10^7$ yr. $L_0$ depends on the richness of the stellar cluster. Thus we see that, depending on the stellar IMF, the energy input rate by SN explosions is a mildly decreasing function of time. Although the number of core collapse SNe increases as the higher masses of the cluster become depopulated, the increasing time interval between explosions more than compensates this effect.
If the ambient medium is further assumed to have a constant ambient density, or one which varies with distance like $\rho \propto
r^{-\beta}$, in which case the similarity variable has to be transformed to $\alpha = 3/(5-\beta)$, the system can be cast into the following form: $$M_{\rm sh}(r) = \int_0^r \rho(r^\prime) d^3 r^\prime \,, \quad
E_{\rm th}(r) = 1/(\gamma -1) \int_0^r p(r^\prime) d^3 r^\prime \,.
\label{mascon1}$$ and the energy input is shared between kinetic and thermal energy. Using $\gamma = 5/3$ for the ratio of specific heats, observing that the bubble pressure $P_b$ remains uniform, and applying spherical symmetry, conservation of momentum and energy $${d\over dt} (M_{\rm sh} \dot R_b) = 4 \pi R_b^2 P_b \,, \quad
{d E_{\rm th} \over dt} = L_{\rm SB}(t) - 4 \pi R_b^2 \dot R_b P_b \,,
\label{mom}$$ yields the solution $$\begin{aligned}
R_b &=& A t^\alpha \,; \quad \alpha = {\delta + 3 \over 5 - \beta} \,,\\
A &=& \left\{{(5-\beta)^3 (3-\beta) \over (7 \delta - \beta - \delta
\beta + 11) (4 \delta + 7 - \delta \beta - 2
\beta)}\right\}^{1/(5-\beta)} \times \left\{{L_0 \over 2\pi (\delta
+ 3) \rho_0} \right\}^{1/(5-\beta)} \,.
\label{simsol1}\end{aligned}$$ Since the swept-up shell is usually thin, the bubble and shell radius can be treated as equal during the energy driven phase and are denoted by $R_b$. The similarity variable in the case considered here is given by $\alpha = (2-\Gamma/\eta)/(5-\beta)$ . For simplicity, the ambient density is assumed to be constant ($\beta=0$), although, as we shall see in our numerical simulations, this assumption becomes increasingly worse with time. On scales of ten parsec, the ISM cannot be assumed to be homogeneous any more. As the cold and warm neutral media are observed to be rather filamentary in structure, high pressure flows will be channelled through regions of low density and pressure. It should be mentioned here, that it is not only the pressure difference between the bubble and the ambient medium that determines the expansion, as it is sometimes argued, but also the *inertia* of the shell is a crucial factor (see Eq. \[mom\]). Therefore mass loading of the flow is an important factor. Unfortunately, some convenient assumptions, like e.g. the bubble behaves isobaric, do not hold any more. Pittard et al. (2001a,b) have shown that similarity flow can be maintained provided the mass loading rate scales as $\dot \rho
\propto r^{(5-7\beta)/3}$ in case of conductive evaporation or $\dot
\rho \propto r^{(-2\beta-5)/3}$ for hydrodynamic mixing according to the Bernoulli effect. It was assumed that in the former case clumps passed through the outer shock as it expanded into a clumpy medium and evaporated in the hot bubble, whereas in the latter case the clumps were thought to be ejected by the central source itself. Here it is possible for strong mass loading that the wind flow is slowed down considerably due to mass pick-up, and in the extreme case even a termination shock transition can be avoided.
Berghöfer & Breitschwerdt (2002) have studied the evolution of the LB under the assumption that 20 SNe from the Pleiades moving subgroup B1 exploded according to their main sequence life times with masses between 20 and 10 ${\rm M}_{\odot}$. Using the above similarity solutions, the radius and the expansion velocity of the bubble evolve as $$R_b = 251 \left(2 \times 10^{-24} {\rm g}/{\rm cm}^3 \over
\rho_0\right)^{1/5}
t_7^{0.5375} \, {\rm pc} \,,
\dot R_b = 13.22 \left(2 \times 10^{-24} {\rm g}/{\rm cm}^3
\over \rho_0\right)^{1/5} t_7^{-0.4625}\, {\rm km/s} \,.
\label{bubrad1}$$ As a result of a decreasing energy input rate the exponent in the expansion law of the radius, $\alpha= 43/80=0.5375$, in Eq. (\[bubrad1\]) is between a Sedov ($\mu=0.4$) and a stellar wind ($\mu=0.6$) type solution. Thus the present radius of the LB will be 289 pc and 158 pc and its velocity is 11.7 km/s and 6.4 km/s, if the ambient density is $\rho_0 = 2 \times 10^{-24} \, {\rm
g}/{\rm cm}^3$ and $\rho_0 = 4 \times 10^{-23} \, {\rm g}/{\rm
cm}^3$, respectively (for details see Berghöfer & Breitschwerdt 2002). In the latter case the value of the ambient density would correspond roughly to that of the cold neutral medium. There are several reasons why we may have overestimated the size of the LB in the similarity solutions above. Firstly, the mass inside the bubble is significantly higher than the pure ejecta mass, as can be inferred from the ROSAT X-ray emission measures; when assuming bubble parameters of $R_b = 100$ pc and $n_b = 5 \times 10^{-3} \,
{\rm cm}^{-3}$ (e.g. Snowden et al. 1990) a mass of at least 600 ${\rm M}_\odot$ is derived, and using non-equilibrium ionization plasma models (Breitschwerdt & Schmutzler 1994) it is even more than a factor of five higher. The contribution of ejecta is only of the order of 100 ${\rm
M}_\odot$, and the bulk of the bubble mass is therefore due to hydrodynamic mixing of shell material, heat conduction between shell and bubble and evaporation of entrained clouds; hence the flow must be mass-loaded. The net effect is to reduce the amount of specific energy per unit mass, because the material mixed in is essentially cold, thus increasing the rate of radiative cooling. Secondly, the stellar association has probably been surrounded by a molecular cloud with a density in excess of $n_0 = 100 \, {\rm cm}^{-3}$ with subsequent break-out of the bubble and dispersal of the parent cloud (Breitschwerdt et al. 1996). Thirdly, the number of SN explosions could be less; here we have assumed that all 20 SNe have occurred inside the LB. This need not be the case as the subgroup B1 does not move through the centre of the LB. ROSAT PSPC observations have revealed an annular shadow centered toward the direction ($l_{\rm
II} = 335^\circ$, $b_{\rm II} = 0^\circ$), which has been interpreted as an interaction between the LB and the neighbouring LI SB (Egger & Aschenbach 1995). The trajectory of the cluster B1 may have partly crossed the LI region. Alternatively and more likely, part of the thermal energy might have been liberated into the Galactic halo, since there is some evidence that the LB is open toward the North Galactic Pole (see Lallement et al. 2003). It should also be mentioned that due to small number statistics the true number of SNe can vary by a factor of 2. Finally, although there is no stringent evidence, it would be very unusual, if the LB would not be bounded by a magnetic field, whose tension and pressure forces would decrease the size of the LB.
Given these uncertainties and the fact that the simple analytic model discussed above can only be considered as an upper limit, the direct comparison with observations is not convincing. The bubble radius and shell velocity are rather insensitive to the energy input rate and the ambient density (due to the power of $1/5$) and therefore not well constrained, but depend more sensitively on the expansion time scale. Thus we can only assert with some confidence that the age of the LB should be between $1 - 2 \times 10^7$ yr.
The most serious drawback of analytical solutions in general and of similarity solutions in particular, is the assumption of homogeneity of the ambient medium on scales exceeding about 10 pc. To see this, consider the area coverage of the disk by hot gas, which is $\xi_{\rm SN} \sim \nu_0/2 \, \tau_{\rm SN}\, (R_{\rm SN}/R_{\rm
gal})^2 \approx 0.67$ due to SNe, and $\xi_{\rm SB} \sim \nu_0/2 \,
\tau_{\rm SB}\, (R_{\rm SB}/R_{\rm gal})^2 \approx 0.9$ due to SBs, respectively, assuming that half of the explosions go off randomly, and half in a clustered fashion within a star forming disk of 10 kpc radius and a disk SN rate of $\nu_0 = 2$ per century. Here we used the final SN radius according to McKee & Ostriker (1977), being $R_{\rm SN}\approx 55$ pc after $\tau_{\rm SN} \approx 2.2 \times
10^6$ yr and the SB radius from the paper of McCray & Kafatos (1987) of $R_{\rm SB} \approx 212$ pc after $\tau_{\rm SB} \approx
10^7$ yr, for a typical cluster with 50 OB stars. The overturning rate of a typical patch of ISM will then roughly be between $3.4
\times 10^6$ yr and $1.1 \times 10^7$ yr for SNe and SBs, respectively. Since we did not take into account overlapping of remnants and bubbles these values are lower limits. As will be shown in the next section, a roughly constant star formation rate and hence SN rate for a Galactic initial mass function (IMF) will lead to an ISM background medium that is highly irregular in density and temperature (and even pressure variations within an order of magnitude are observed) and it bears a high level of turbulence.
Numerical simulations of the Local Bubble (LB) and Loop I (LI) evolution {#num}
========================================================================
We have performed high resolution 3D simulations of the Galactic disk and halo (Avillez & Breitschwerdt 2004, 2005; see also this volume) on a grid of $1\, {\rm kpc} \times 1 \, {\rm kpc}$ in the plane and $z=\pm 10 \,
{\rm kpc}$ perpendicular to it. Using AMR technique, we obtained resolution of scales down to 1.25 pc for MHD, and 0.625 pc for pure hydrodynamical (HD) runs. These simulations, which revealed many new features of the ISM, e.g. low volume filling factor of hot gas in the disk, establishment of the fountain flow even in the presence of a disk parallel magnetic field, more than half of the mass in classical thermally unstable regions, serve as a *realistic background medium* for the expansion of the LB and the LI SB. We took data cubes of HD runs and picked up a site
![Temperature map (cut through Galactic plane) of a 3D Local Bubble simulation, 14.4 Myr after the first explosion; LB is centered at (175, 400) pc and Loop I at (375, 400) pc.[]{data-label="fig_temp"}](mavillez_fig1a.ps){width="0.8\hsize"}
with enough mass to form the 81 stars, with masses, $M_*$, between 7 and 31 ${\rm M}_{\odot}$, that represent the Sco-Cen cluster inside the LI SB; 39 massive stars with $14 \leq M_* \leq 31 \, {\rm
M}_{\odot}$ have already exploded, generating the LI cavity. At present the Sco-Cen cluster (arbitrarily located at $(375,400)$ pc has 42 stars to explode within the next 13 Myrs). We followed the trajectory of the moving subgroup B1 of Pleiades (see Berghöfer & Breitschwerdt 2002), whose SNe in the LB went off along a path crossing the solar neighbourhood. As a result, we observe that the locally enhanced SN rates produce coherent LB and LI structures (due to ongoing star formation) within a highly disturbed background medium (see Fig. \[fig\_temp\]). The successive explosions heat and pressurize the LB, which at first looks smooth, but develops internal temperature and density structure at later stages. After 14 Myr the 20 SNe that occurred inside the LB fill a volume roughly corresponding to the present day size (see Fig. \[fig\_temp\], bubbles are labelled by LB and L1). The cavity is still bounded by an outer shell, which exhibits holes due to Rayleigh-Taylor instabilities, as has been predicted analytically by Breitschwerdt et al. (2000), and it will start to fragment in $\sim 3$ Myr from now.
![ O[vi]{} column density averaged over angles (left panel) indicated in Fig. \[fig\_temp\] and maximum column density (right panel) as a function of LOS path length at $14.1 \leq t\leq 15$ Myr of Local and Loop I bubbles evolution.[]{data-label="fig3_ovi"}](OVI-Average.ps "fig:"){width="0.45\hsize"} ![ O[vi]{} column density averaged over angles (left panel) indicated in Fig. \[fig\_temp\] and maximum column density (right panel) as a function of LOS path length at $14.1 \leq t\leq 15$ Myr of Local and Loop I bubbles evolution.[]{data-label="fig3_ovi"}](OVI-Max.ps "fig:"){width="0.45\hsize"}\
It has been argued that a crucial test of any LB model is the column density of the interstellar ion O[vi]{} (Cox 2004), whose discovery back in the 70’s led to the establishment of the hot intercloud medium. So far all models have failed to reproduce the fairly low O[vi]{}-value, most recently measured with FUSE (Oegerle et al. 2004), to be $N_{\rm OVI} \simeq 7\times 10^{12}$ cm$^{-2}$. To compare this with our simulations we have calculated the average and maximum column densities of O[vi]{}, i.e., $\langle \mbox{N(O{\sc vi})} \rangle$ and $\mbox{N}_{\mbox{max}}\mbox{(O{\sc vi})} $ along 91 lines of sight (LOS) extending from the Sun and crossing LI from an angle of $-45\deg$ to $+45\deg$ (s. Fig. \[fig\_temp\]). Within the LB (i.e., for a LOS length $l_{LOS}\leq 100$ pc) $\langle \mbox{N(O{\sc
vi})} \rangle$ and $\mbox{N}_{\mbox{max}}\mbox{(O{\sc vi})} $ decrease steeply from $5\times10^{13}$ to $3\times 10^{11}$ cm$^{-2}$ and from $1.2\times 10^{14}$ to $1.5\times 10^{12}$ cm$^{-2}$, respectively, for $14.1 \leq t\leq 15$ Myr (Fig. \[fig3\_ovi\]), because no further SN explosions occur and recombination is taking place. For LOS sampling gas from outside the LB (i.e., $l_{LOS}>100$ pc) $\langle \mbox{N(O{\sc vi})} \rangle >
6\times 10^{12}$ and $\mbox{N}_{\mbox{max}}\mbox{(O{\sc vi})} >
5\times 10^{13}$ cm$^{-2}$. We have made histograms of column densities obtained in the 91 LOS for $t=14.5$ and 14.6 Myr, which show that for $t=14.6$ Myr all the LOS have column densities smaller than $10^{12.9}$ cm$^{-2}$, while for $t=14.5$ Myr 67% of the lines have column densities smaller than $10^{13}$ cm$^{-2}$ and in particular 49% of the lines have $\mbox{N(O{\sc vi})}\leq 7.9\times
10^{12}$ cm$^{-2}$. Noting that in the present model at 14.5 Myr the O[vi]{} column densities are smaller than $1.7\times 10^{13}$ cm$^{-2}$ and $\langle \mbox{N(O{\sc vi})} \rangle = 8.5\times
10^{12}$ cm$^{-2}$ (see the respective lines in both panels of Fig. \[fig3\_ovi\]), we are thus able to reproduce the measured $\langle \mbox{N(O{\sc vi})} \rangle$ values, provided that the age of the LB is $\sim 14.7^{+0.5}_{-0.2}$ Myrs.
Conclusions {#conc}
===========
Galactic and extragalactic interstellar bubbles are still an active area of research. Despite the widespread belief that H[ii]{} regions, SNRs, stellar wind bubbles and superbubbles are fully understood in theory, it has to be emphasized that *real bubbles*, observed in the Galaxy, such as the Local or LI superbubbles, or in external galaxies, such as in the LMC, are often poorly fitted by standard similarity solutions. The reason lies in the inapplicability of major assumptions, e.g., that the ISM is homogeneous, and that the bubbles are either in an energy or momentum conserving phase. High resolution 3D simulations in a highly structured and turbulent background medium offer a much better description and include physical processes such as mass loading and turbulent mixing on a fundamental level. Although this drains heavily on computer resources, the increased precision of observations in the near future will warrant such an effort.
Acknowledgments {#acknowledgments .unnumbered}
===============
DB would like to thank Thierry Montmerle and Almas Chalabaev for their excellent organization, for financial help and patience with the manuscript. It was a great pleasure to be in La Thuile. He also thanks Verena Baumgartner for proofreading of the text.
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|
---
abstract: 'The paradigm of layered networks is used to describe many real-world systems – from biological networks, to social organizations and transportation systems. While recently there has been much progress in understanding the general properties of multilayer networks, our understanding of how to control such systems remains limited. One fundamental aspect that makes this endeavor challenging is that each layer can operate at a different timescale, thus we cannot directly apply standard ideas from structural control theory of individual networks. Here we address the problem of controlling multilayer and multi-timescale networks [focusing on two-layer multiplex networks with one-to-one interlayer coupling. We investigate]{} the practically relevant case when the control signal is applied to the nodes of one layer. We develop a theory based on disjoint path covers to determine the minimum number of inputs ($N_{\textrm{i}}$) necessary for full control. We show that if both layers operate on the same timescale then the network structure of both layers equally affect controllability. In the presence of timescale separation, controllability is enhanced if the controller interacts with the faster layer: $N_{\textrm{i}}$ decreases as the timescale difference increases up to a critical timescale difference, above which $N_{\textrm{i}}$ remains constant and is completely determined [by]{} the faster layer. We show that the critical timescale difference is large if Layer I is easy and Layer II is hard to control in isolation. In contrast, control becomes increasingly difficult if the controller interacts with the layer operating on the slower timescale and increasing timescale separation leads to increased $N_{\textrm{i}}$, again up to a critical value, above which $N_{\textrm{i}}$ still depends on the structure of both layers. This critical value is largely determined by the longest path in the faster layer that does not involve cycles. [By identifying the underlying mechanisms that connect timescale difference and controllability for a simplified model, we provide crucial insight into disentangling how our ability to control real interacting complex systems is affected by a variety of sources of complexity.]{}'
author:
- Márton Pósfai
- Jianxi Gao
- 'Sean P. Cornelius'
- 'Albert-László Barabási'
- 'Raissa M. D’Souza'
title: 'Controllability of [multiplex]{}, multi-timescale networks'
---
Introduction {#sec:introduction}
============
Over the past two decades, the theory of networks proved to be a powerful tool for understanding individual complex systems [@ALB02; @NEW03]. However, it is now increasingly appreciated that complex systems do not exist in isolation, but interact with each other [@KIV14; @BOC14]. Indeed, an array of phenomena – from cascading failures [@BUL10; @BRU12] to diffusion [@GOM13] – can only be fully understood if these interactions are taken into account. Traditional network theory is not sufficient to describe the structure of such systems, so in response to this challenge, the paradigm of multilayer networks is being actively developed. Here we study a fundamental, yet overlooked aspect of multilayer networks: each individual layer can operate at a different timescale. Particularly, we address the problem of controlling multilayer, multi-timescale systems [focusing on two-layer multiplex networks]{}. Recently significant efforts have been made to uncover how the underlying network structure of a system affects our ability to influence its behavior [@WAN02; @SOR07; @LIU11; @WAN12; @YUA13; @COR13; @POS13; @GAO14; @IUD15]. However, despite the appearance of coupled systems from infrastructure to biology, the existing literature – with a few notable exceptions [@CHA14; @MEN16; @YUA14; @ZHA16] – has focused on control of networks in isolation, and the role of timescales remains unexplored.
Control of multilayer networks is important for many applications. For example, consider a CEO aiming to lead a company consisting of employees and management. Studying the network of managers or the network of employees in isolation does not take into account important interactions between the different levels of hierarchy of the company. On the other hand, treating the system as one large network ignores important differences between the dynamics of the different levels, e.g. management may meet weekly, while employees are in daily interaction. In general, the interaction of timescales plays an important role in organization theory [@ZAH99]. Or consider gene regulation in a living cell. External stimuli activate signaling pathways which through a web of protein-protein interactions affect transcription factors responsible for gene expression. The activation of a signaling pathway happens on the timescale of seconds, while gene expression typically takes hours [@ALB02b]. As a third example, consider an operator of an online social network who wants to enhance the spread of certain information by interacting with its users. However, a user may subscribe to multiple social networking services and may opt to share news encountered in one network through a different one – out of reach of the operator. The dynamics of user interaction on different websites can be very different depending on user habits and the services offered [@LER10; @KWA10; @BAK12]. For example the URL shortening service Bit.ly reports that the half-life of shared links depends on the social networking platform used: half the clicks on a link happened within 2.8 hours after posting on Twitter, within 3.2 hours on Facebook and within 7.4 hours on Youtube [@BIT11].
Common features of these examples are that (i) each interacting subsystem is described by a separate complex network; (ii) the dynamics of each subsystem operate on a different, but often comparable timescale and (iii) the external controller directly interacts with only one of the subsystems. Here we study the control properties of a model that incorporates these common features, yet remains tractable. [More specifically, we study discrete-time linear dynamics on two-layer multiplex networks, meaning that we assume one-to-one coupling between the nodes of the two layers. This choice ensures both analytical tractability and the isolation of the role of timescales from the effect of more complex multilayer network structure. Identifying the underlying mechanisms that govern the controllability of this simple model provides crucial insight into disentangling how our ability to control real interacting complex systems is affected by a variety of sources of complexity.]{}
So far only limited work investigated [controllability]{} of multilayer networks. Menichetti et al. investigated the controllability of two-layer multiplex networks governed by linear dynamics such that the dynamics of the two layers are not coupled, but the input signals in the two layers are applied to the same set of nodes [@MEN16]. Yuan et al. identified the minimum number of inputs necessary for full control of diffusion dynamics, allowing the controller to interact with any layer [@YUA14]. Zhang et al. investigated the controllable subspace of multilayer networks with linear dynamics without timescale separation if the controller is limited to interact with only one layer; showing that it is more efficient to directly control peripheral nodes than central ones [@ZHA16]. Here we also limit the controller to one layer, yet by exploring the minimum input problem, we offer a direct metric which allows us to compare our findings to previous results for single-layer networks [@LIU11]. More so, the key innovation of our work is that we take into account the timescale of the dynamics of each layer, a mostly overlooked aspect of multilayer networks.
It is worth mentioning the recent work investigating the related, but distinct problem of controllability of networks with time-delayed linear dynamics [@QIA16]. The key difference between time-delay and timescale difference is that for time-delayed dynamics the state of a node will depend on some previous state of its neighbors; however, the typical time to change the state of a node remains the same throughout the system. While in case of timescale difference, the typical time needed for changes to happen can be different in different parts of the system.
In the next section, we introduce a simple model that captures some common properties of multilayer networks and we describe the problem setup. In Sec. \[sec:min\_input\_prob\], we develop a theory to determine the minimum number of inputs required for controlling multiplex, multi-timescale networks with discrete-time linear dynamics relying on graph combinatorial methods. In Sec. \[sec:results\], we use networks with tunable degree distribution to systematically uncover the role of network structure and timescale separation. We study three scenarions: no timescale separation, Layer I operates faster and Layer II operates faster. Finally, in Sec. \[sec:conclusions\] we provide a discussion of our results and we outline open questions.
Model definition {#sec:model_definition}
================
We aim to study the controllability of coupled complex dynamical systems with the following properties: (i) each subsystem (layer) is described by a complex network; (ii) the operation of each layer is characterized by a different timescale and (iii) the controller only interacts directly with one of the layers. We propose a model that satisfies these requirements and yet is simple enough to remain tractable. We focus on two-layer [multiplex]{} systems, meaning that there is a one-to-one correspondence between the nodes of the two layers.
The model is defined by a weighted directed two-layer multiplex network $\mathcal M$ which consists of two networks $\mathcal L_{\textrm{I}}$ and $\mathcal L_{\textrm{II}}$ called layers and a set of links $E_{{\textrm{I}},{\textrm{II}}}$ connecting the nodes of the different layers. Each layer $\mathcal L_\alpha$ (where $\alpha\in\{{\textrm{I}},{\textrm{II}}\}$) consists of a set of nodes $V_\alpha=\{v_1^\alpha,v_2^\alpha,\ldots,v_N^\alpha\}$ and a set of links $E_\alpha$, where a directed link $(v_i^\alpha,v_j^\alpha,w_{ij}^\alpha)\in E_\alpha$ is an ordered node pair and a weight representing that node $v_i^\alpha$ influences node $v_j^\alpha$ with strength $w^\alpha_{ij}$. The two layers are connected by link set $E_{{\textrm{I}},{\textrm{II}}}=\{(v_i^{\textrm{I}},v_i^{\textrm{II}}, w_i^{\textrm{I,II}})\vert i=1,2\ldots,N\}$, in other words, there is directed one-to-one coupling from Layer I to Layer II (Fig. \[fig:3examples\]a). Although the links are weighted, the exact values of the weights do not have to be known for our purposes.
Our goal is to control the system by only interacting directly with Layer I. We study linear discrete-time dynamics $$\label{eq:dynamics}
\begin{aligned}
{{\bf x}}_{\textrm{I}}(t) &= {{\bf A}}_{\textrm{I}}{{\bf x}}_{\textrm{I}}(t-\tau_{\textrm{I}}) + {{\bf B}}{{\bf u}}(t-\tau_{\textrm{I}})\\
{{\bf x}}_{\textrm{II}}(t) &= {{\bf A}}_{\textrm{II}}{{\bf x}}_{\textrm{II}}(t-\tau_{\textrm{II}}) + \Delta_{\tau_{\textrm{I}}}(t){{\bf D}}{{\bf x}}_{\textrm{I}}(t - \tau_{\textrm{I}})
\end{aligned}
\begin{aligned}
\quad\text{ if }&(t\bmod\tau_{\textrm{I}})=0,\\
\quad\text{ if }&(t\bmod\tau_{\textrm{II}})=0,
\end{aligned}$$ where ${{\bf x}}_{\textrm{I}}(t)$ and ${{\bf x}}_{\textrm{II}}(t)\in \mathbb{R}^N$ represent the state of nodes in Layer I and II; the matrices ${{\bf A}}_{\textrm{I}}$ and ${{\bf A}}_{\textrm{II}}\in \mathbb{R}^{N\times N}$ are the transposed weighted adjacency matrices of Layer I and II, capturing their internal dynamics. The weighted diagonal matrix ${{\bf D}}\in \mathbb{R}^{N\times N}$ captures how Layer I affects Layer II.
Vector ${{\bf u}}(t)\in \mathbb R^M$ provides the set of independent inputs and the matrix ${{\bf B}}\in \mathbb{R}^{N\times M}$ defines how the inputs are coupled to the system. To differentiate between the function ${{\bf u}}(t)$ and an instance of the function at a given time step, we refer to a component $u_i(t)$ of vector ${{\bf u}}(t)$ as an independent input, and we call its value at time step $t^\prime$, $u_i(t=t^\prime)$, a signal.
Finally, $\tau_{\textrm{I}}, \tau_{\textrm{II}}\in \{1,2,\ldots\}$ are the timescale parameters of each subsystem, meaning that the state of Layer I is updated according to Eq. (\[eq:dynamics\]) every $\tau_{\textrm{I}}$th time step; and Layer II is updated every $\tau_{\textrm{II}}$th time step. And $$\Delta_{\tau_{\textrm{I}}}(k) = \left\{ \begin{aligned}
1 \quad\text{ if }&(k\bmod \tau_{\textrm{I}})=0,\\
0 \quad\text{ if }&(k\bmod \tau_{\textrm{I}})\neq 0,
\end{aligned}\right.$$ is the Kronecker comb, meaning that Layer I directly impacts the dynamics of Layer II if the two layers simultaneously update. We investigate three scenarios: (i) the subsystems operate on the same timescale, i.e. $\tau_{\textrm{I}}=\tau_{\textrm{II}}=1$; (ii) Layer I updates faster, i.e. $\tau_{\textrm{I}}=1$ and $\tau_{\textrm{II}}>1$; and (iii) Layer II updates faster $\tau_{\textrm{I}}>1$ and $\tau_{\textrm{II}}=1$.
We seek full control of the system [as defined by Kalman [@KAL60]]{}, meaning that with the proper choice of ${{\bf u}}(t)$, we can steer the system from any initial state to any final state in finite time. To characterize controllability, we aim to design a matrix ${{\bf B}}$ such that the system is controllable and the number of independent control inputs, $M$, is minimized. The minimum number of inputs, $N_{\textrm{i}}$, serves as our [measure of how difficult it is to control the system]{}.
To find a robust and efficient algorithm to determine $N_{\textrm{i}}$, we rely on the framework of structural controllability [@LIN74]. We say that a matrix ${{\bf A}}^*$ has the same structure as ${{\bf A}}$, if the zero/nonzero elements of ${{\bf A}}$ and ${{\bf A}}^*$ are in the same position, and only the value of the nonzero entries can be different, in other words, in the corresponding network the links connect the same nodes, only the link weights can differ. A linear system of Eq. (\[eq:dynamics\]) defined by matrices $({{\bf A}}_{\textrm{I}}, {{\bf A}}_{\textrm{II}}, {{\bf D}}, {{\bf B}})$ is structurally controllable if there exists matrices with the same structure $({{\bf A}}_{\textrm{I}}^*, {{\bf A}}_{\textrm{II}}^*, {{\bf D}}^*, {{\bf B}}^*)$ such that the dynamics defined by $({{\bf A}}_{\textrm{I}}^*, {{\bf A}}_{\textrm{II}}^*, {{\bf D}}^*, {{\bf B}}^*)$ are controllable [according to the definition of Kalman]{}. Note that ultimately we are interested in controllability and not structural controllability. Yet, structural controllability is a useful tool because (i) if a linear system is structurally controllable, it is controllable for almost all link weight combinations [@LIN74] and (ii) determining structural controllability can be mapped to a graph combinatorial problem allowing for efficient and numerically robust algorithms.
Minimum input problem {#sec:min_input_prob}
=====================
Before addressing the minimum input problem of multiplex networks, we revisit the case of single-layer networks by providing an alternative explanation of the Minimum Input Theorem of Liu et al. [@LIU11]. This new approach readily lends itself to be extended to multiplex, multi-timescale networks. Thus providing the basis for Sec. \[sec:multiplex\_net\], in which we develop an algorithm to determine $N_{\textrm{i}}$ for two-layer multiplex networks.
Single-layer networks {#sec:single-layer_net}
---------------------
The linear discrete-time dynamics associated to a single-layer weighted directed network $\mathcal L$ are formulated as $$\label{eq:single-layer_dynamics}
{{\bf x}}(t + 1) = {{\bf A}} {{\bf x}}(t) + {{\bf B}}{{\bf u}}(t),$$ where ${{\bf x}}(t)$, ${{\bf A}}$, ${{\bf B}}$ and ${{\bf u}}(t)$ are defined similarly as in Eq. (\[eq:dynamics\]) (Fig. \[fig:singlenet\]a). To obtain a graph combinatorial condition for structural controllability we rely on the dynamic graph $\mathcal D_{T}$, which represents the time evolution of a system from $t=0$ to $t=T$ [@MUR87; @PFI13; @POS14b]. Each node $v_i$ in $\mathcal L$ is split into $T+1$ copies $\{v_{i,0},v_{i,1},\ldots,v_{i,T}\}$, each copy $v_{i,t}$ represents the state of node $v_i$ at time step $t$. We add a directed link $(v_{i,t} \rightarrow v_{j,t+1})$ for $t=0,1,\ldots,T-1$ if they are connected by a directed link $(v_{i}\rightarrow v_{j})$ in the original network, representing that the state of node $v_{j}$ at time $t+1$ depends on the state of its in-neighbours at the previous time step. To account for the controller, for each independent input we create $T$ nodes $u_{i,t}$ ($i=1,2,\ldots, M$; $t=0,1,\ldots,T-1$) each representing a control signal (i.e. the value of the $i$th input at time step $t$). We draw a directed link $(u_{i,t}\rightarrow v_{j,t+1})$ for $t=0,1,\ldots,T-1$ if $b_{ji}\neq 0$, where $b_{ji}$ is an element of matrix ${{\bf B}}$.
According to Theorem 15.1 of Ref. [@MUR87], a linear system $({{\bf A}}, {{\bf B}})$ is structurally controllable if and only if in the associated dynamic graph $\mathcal D_N$ node sets $U=\{u_{i,t}\vert i=1,2,\ldots,M;t=0,1\ldots,N-1\}$ (green nodes in Fig. \[fig:singlenet\]b) and $V_N=\{v_{i,t=N}\vert i=1,2\ldots,N\}$ (blue nodes) are connected by $N$ disjoint paths (red links), i.e. there exists a set of disjoint paths $C=\{P_1,P_2,\ldots,P_N\}$ such that $U$ contains the set of starting points and $V_T$ is the set of endpoints. A path $P$ of length $l$ between node $v_{i_0}$ and $v_{i_l}$ is a sequence of $l$ consecutive links $[(v_{i_0}\rightarrow v_{i_1}),(v_{i_1}\rightarrow v_{i_2}),\ldots,(v_{i_{l-1}}\rightarrow v_{i_l})]$ such that each node is traversed only once. Node $v_{i_0}$ is the starting point and $v_{i_l}$ is the endpoint of $P$. Two paths $P_1$ and $P_2$ are disjoint if no node is traversed by both $P_1$ and $P_2$, a set of paths is disjoint if all paths in the set are pairwise disjoint.
A possible interpretation of this result is that if a $P_i$ path has starting point $u_{j,t_0}$ and endpoint $v_{k,t_1}$, we say that the signal $u_{j}(t_0)$ is assigned to set $x_k(t_1)$, the state of node $v_k$ at time $t_1$, through path $P_i$. Therefore we refer to path $P_i$ as a control path. The clear meaning of the dynamic graph and the control paths makes this condition useful to formulate proofs and to interpret results. However, it is rarely implemented to test controllability of large networks, because the size of the dynamical graph grows as $N^2$, rendering such algorithms too slow. In the following, we provide a condition that only requires the dynamic graph $\mathcal D_1$ as input; therefore it is more suitable for practical purposes.
It was shown in Refs. [@MUR87; @LIU11; @COM13] that a linear system $({{\bf A}}, {{\bf B}})$ is structurally controllable if and only if (i) in $\mathcal D_1$ we can connect nodes $U\cup V_0=\{u_{i,t=0}\vert i=1,2,\ldots,M\}\cup\{v_{i,t=0}\vert i=1,2\ldots,N\}$ (green nodes in Fig. \[fig:singlenet\]c) and nodes $V_1=\{v_{i,t=1}\vert i=1,2\ldots,N\}$ (blue nodes) via $N$ disjoint paths (red links) and (ii) all nodes are accessible from the inputs. This result can be understood as a self-consistent version of the previous condition involving $\mathcal D_N$: Instead of keeping track of the entire control paths as we previously did, we concentrate on a single time step. Consider the dynamic graph $\mathcal D_1$ representing the time evolution of the system from $t=0$ to $t=1$, and assume that the system is controllable. By definition we can set the state of each node independently at $t=0$; therefore we can treat them as control signals to control the system at a later time step. Now let us aim to control the system at $t=1$, according to our previous condition, it is necessary that $N$ disjoint paths exist between nodes $U\cup V_0=\{u_{i,t=0}\vert i=1,2,\ldots,M\}\cup\{v_{i,t=0}\vert i=1,2\ldots,N\}$ and nodes $V_1=\{v_{i,t=1}\vert i=1,2\ldots,N\}$. This is exactly requirement (i), together with the accessibility requirement (ii) it is a sufficient and necessary condition. Note that $D_1$ is a bipartite network (each link is connected to exactly one node in $U\cup V_0$ and one node in $V_1$) and each disjoint path in $\mathcal D_1$ is a single link.
The minimum input problem aims to identify the minimum number of inputs that guarantee controllability for a given network, in other words, the goal is to design a ${{\bf B}}\in\mathbb{R}^{N\times M}$ for a given ${{\bf A}}$ such that $M$ is minimized. For this we consider the dynamic graph $\mathcal D_1$ without nodes representing control signals. We then find a maximum cardinality matching, where a matching is a set of links that do not share an endpoint. The matching is a set of disjoint paths connecting node sets $V_0$ and $V_1$. Controllability requires $N$ disjoint paths between $U\cup V_0$ and $V_1$; therefore $N_{\textrm{i}} = N - N_{\textrm{match}}$, where $N_{\textrm{match}}$ is the size of the maximum matching (if $N_{\textrm{match}}=N$, $N_{\textrm{i}}=1$). Allowing the inputs to be connected to multiple nodes we can guarantee that all nodes are accessible from the inputs. Thus we recovered the Minimum Input Theorem of Liu et al. [@LIU11].
In summary, by relying on a self-consistent condition for structural controllability we re-derived the known result that identifying $N_{\textrm{i}}$ is equivalent to finding a maximum matching in $\mathcal D_1$. In the next section we show that this new self-consistent approach lends itself to be extended to the multiplex, multi-timescale model defined by Eq. (\[eq:dynamics\]), allowing us to derive analogous method to identify $N_{\textrm{i}}$.
Multiplex networks {#sec:multiplex_net}
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To find the minimum number of inputs $N_{\textrm{i}}$ for multiplex, multi-timescale networks, we first extend the definition of the dynamic graph. We define the dynamic graph $\mathcal D_{\tau_{\textrm{II}}}$ such that it captures the time evolution of a multiplex system defined by $({{\bf A}}_{\textrm{I}}, {{\bf A}}_{\textrm{II}}, {{\bf D}}, {{\bf B}})$ and Eq. (\[eq:dynamics\]) from $t=0$ to $t=\tau_{\textrm{II}}$. For sake of brevity, we assume that $\tau_{\textrm{I}} = 1$ and $\tau_{\textrm{II}} \geq 1$, the case of $\tau_{\textrm{I}} > 1$ and $\tau_{\textrm{II}} = 1$ is treated similarly (Fig. \[fig:3examples\]d). Each node $v_i^{\textrm{I}}$ in Layer I is split into $\tau_{\textrm{II}}+1$ copies $\{v^{\textrm{I}}_{i,0},v^{\textrm{I}}_{i,1},\ldots,v^{\textrm{I}}_{i,\tau_{\textrm{II}}}\}$; each node $v_i^{\textrm{II}}$ in Layer II is split into two copies $\{v^{\textrm{II}}_{i,0},v^{\textrm{II}}_{i,\tau_{\textrm{II}}}\}$, because Layer II does not update during the intermediate time steps. We draw a link from $v^{\textrm{I}}_{i,t}$ to $v^{\textrm{I}}_{j,t+1}$ ($t=0,1,\ldots,\tau_{\textrm{II}}-1$) if they are connected in Layer I by a directed link $(v^{\textrm{I}}_{i}\rightarrow v^{\textrm{I}}_{j})$, and similarly we connect $v^{\textrm{II}}_{i,0}$ to $v^{\textrm{II}}_{j,\tau_{\textrm{II}}}$ if they are connected in Layer II. In addition we draw a link between each pair $v^{\textrm{I}}_{i,0}$ and $v^{\textrm{II}}_{i,\tau_{\textrm{II}}}$ to account for the interconnectedness.
As a natural extension of self-consistent approach introduced in Sec. \[sec:single-layer\_net\], assume that the system is controllable. If the system is controllable, we can set the state of each node independently at $t=0$. To control the system at $t=\tau_{\textrm{II}}$, all nodes at $t=\tau_{\textrm{II}}$ in $\mathcal D_{\tau_{\textrm{II}}}$ (blue nodes in Fig. \[fig:3examples\]) have to be connected to a node at $t=0$ or to a control signal (green nodes) via a disjoint path (red links). In other words, a linear two-layer system $({{\bf A}}_{\textrm{I}}, {{\bf A}}_{\textrm{II}}, {{\bf D}}, {{\bf B}})$ is structurally controllable only if there exists $2N$ disjoint paths in the dynamic graph connecting node set $U\cup V_0 =\{u_{i,t}\vert i=1,2,\ldots,M;t=0,1,\ldots,\tau_{\textrm{II}}-1\}\cup \{v^{\textrm{I}}_{i,0}\vert i=1,2,\ldots,N\}\cup\{v^{\textrm{II}}_{i,0}\vert i=1,2,\ldots,N\}$ and node set $V_{\tau_{\textrm{II}}}=\{v^{\textrm{I}}_{i,\tau_{\textrm{II}}}\vert i=1,2,\ldots,N\}\cup\{v^{\textrm{II}}_{i,\tau_{\textrm{II}}}\vert i=1,2,\ldots,N\}$. In other words, a linear two-layer system $({{\bf A}}_{\textrm{I}}, {{\bf A}}_{\textrm{II}}, {{\bf D}}, {{\bf B}})$ is structurally controllable only if there exists $2N$ disjoint paths in the dynamic graph connecting node set $U\cup V_0=\{u_{i,t}\vert i=1,2,\ldots,M;t=0,1,\ldots,\tau_{\textrm{II}}-1\}\cup \{v^{\textrm{I}}_{i,0}\vert i=1,2,\ldots,N\}\cup\{v^{\textrm{II}}_{i,0}\vert i=1,2,\ldots,N\}$ and node set $V_{\tau_{\textrm{II}}}=\{v^{\textrm{I}}_{i,\tau_{\textrm{II}}}\vert i=1,2,\ldots,N\}\cup\{v^{\textrm{II}}_{i,\tau_{\textrm{II}}}\vert i=1,2,\ldots,N\}$.
To test whether the system is controllable by $M$ independent inputs, we need to find a ${{\bf B}}\in\mathbb R^{N\times M}$ such that the system is controllable. We do not have to check all possibilities, because if such ${{\bf B}}$ exists, then the system is also controllable for ${{\bf B}}^\prime\in\mathbb R^{N\times M}$ where ${{\bf B}}^\prime$ has no zero elements; therefore, we only check the case when each input is connected to each node in Layer I. Given matrices $({{\bf A}}_{\textrm{I}}, {{\bf A}}_{\textrm{II}}, {{\bf D}}, {{\bf B}}^\prime)$, we now have to count the number of disjoint paths connecting $U\cup V_0$ and $V_{\tau_{\textrm{II}}}$ in the corresponding dynamic graph $\mathcal D_{\textrm{II}}$. We find these paths using maximum flow: We set the capacity of each link and each node to 1, we then find the maximum flow connecting source node set $U\cup V_0$ to target node set $V_{\tau_{\textrm{II}}}$ using any maximum flow algorithm of choice. If the system is structurally controllable, the maximum flow equals to $2N$; if it is less than $2N$, additional inputs are needed.
We can now identify the minimum number of inputs $N_{\textrm{i}}$ by systematically scanning possible values of $M$. A simple approach is to first set $M=1$, and test if the system is controllable. If not, increase $M$ by one. Repeat this until the smallest $M$ yielding full control is found. Significant increase in speed is possible if we find the minimum value of $M$ using bisection. We initially know that $N_{\textrm{i}}^{\textrm{upper}}=N \geq N_{\textrm{i}} \geq N_{\textrm{i}}^{\textrm{lower}}=1$. We set $M=(N_{\textrm{i}}^{\textrm{upper}}+N_{\textrm{i}}^{\textrm{lower}})/2$, and test if the system is controllable. If yes, we set $N_{\textrm{i}}^{\textrm{upper}}=M$; if no, we set $N_{\textrm{i}}^{\textrm{lower}}=M$. We repeat this until $N_{\textrm{i}}^{\textrm{upper}}=N_{\textrm{i}}^{\textrm{lower}}$, which provides $N_{\textrm{i}}$. For implementation, we used Google OR-tools and igraph python packages [@CSA06; @ORTOOLS].
The one-to-one coupling between Layer I and Layer II guarantees that full control is possible with at most $N$ independent inputs; therefore we often normalize $N_{\textrm{i}}$ by $N$, i.e. $n_{\textrm{i}} = N_{\textrm{i}} / N$.
Note that in the above argument we rely on the test of structural controllability based on the dynamic graph, which was originally introduced for single-timescale networks [@MUR87]. The sufficiency of the condition relies on the fact that the zero is the only degenerate eigenvalue of a matrix ${{\bf A}}$ if the nonzero elements of ${{\bf A}}$ are uncorrelated. However, this might not remain true for the spectrum of ${{\bf A}}^\tau$, where $\tau>1$, due to correlations arising in the nonzero elements of ${{\bf A}}^\tau$. If a $\lambda\neq 0$ eigenvalue has larger geometric multiplicity than the multiplicity of $0$, $N_{\textrm{i}}$ would be larger than predicted by the dynamic graph; if a $\lambda\neq 0$ eigenvalue has larger geometric multiplicity than $1$ but smaller than the multiplicity of zero, it does not affect $N_{\textrm{i}}$, but may require connecting an input to multiple nodes [@YUA13]. In the $\tau_{\textrm{I}}>0$ and $\tau_{\textrm{II}}=1$ case, a control signal is only injected into Layer II every $\tau_{\textrm{I}}$ time step (Fig. \[fig:3examples\]d); therefore, the spectrum of ${{\bf A}}_{\textrm{II}}^{\tau_{\textrm{I}}}$ becomes relevant. However, we are interested in large and sparse complex networks whose spectra is dominated by the zero eigenvalue [@YUA13]. Therefore it is reasonable to expect that the spectrum of ${{\bf A}}^\tau$ will be dominated by zero eigenvalues as well. Meaning that the minimum number of inputs is correctly given by this graph combinatorial condition. Furthermore the one-to-one coupling between the layers guarantees that control is possible by only interacting with Layer I directly.
So far, we developed a method to characterize controllability of a [multiplex]{}, multi-timescale system based on the underlying network structure and the timescale of each of its layers. In the next section, we rely on these tools to systematic study how network characteristics and timescales affect $N_{\textrm{i}}$.
Results {#sec:results}
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In this section we investigate how different timescales and the degree distribution of each layer affect controllability. For timescales, we consider three scenarios: (i) the subsystems operate on the same timescale, i.e. $\tau_{\textrm{I}}=\tau_{\textrm{II}}=1$; (ii) Layer I updates faster, i.e. $\tau_{\textrm{I}}=1$ and $\tau_{\textrm{II}}>1$; and (iii) Layer II updates faster $\tau_{\textrm{I}}>1$ and $\tau_{\textrm{II}}=1$. To uncover the effect of degree distribution, we consider layers with Poisson (ER) or scale-free (SF) degree distribution, the latter meaning that the distribution has a power-law tail.
We generate scale-free layers using the static model [@GOH01]: We start with $N$ unconnected nodes. Each node $v_i$ is assigned two hidden parameters $w_\text{in}(i)=i^{-\zeta_\text{out}}$ and $w_\text{out}(i)=i^{-\zeta_\text{out}}$, where $i=1,2,\ldots,N$. The weights are then shuffled to eliminate any correlations of the in- and out-degree of individual nodes and between layers. We then randomly place $L$ directed links by choosing the start- and endpoint of the link with probability proportional to $w_\text{in}(i)$ and $w_\text{out}(i)$, respectively. For large $N$ this yields the degree distribution $$P^{\text{SF}}_\text{in/out}(k) = \frac{\left[c(1-\zeta_\text{in/out})^{1/\zeta_\text{in/out}}\right]}{\zeta_\text{in/out}}\frac{\Gamma(k-1/\zeta_\text{in/out},c[1-\zeta_\text{in/out}])}{\Gamma(k+1)},$$ where $c=L/N$ is equal to the average degree, and $\Gamma(n,x)$ is the upper incomplete gamma function. For large $k$, $P_\text{in/out}^{\text{SF}}(k)\sim k^{-(1+1/\zeta_\text{in/out})} = k^{-\gamma_\text{in/out}}$, where $\gamma_\text{in/out}=1+1/\zeta_\text{in/out}$ is the exponent characterizing the tail of the distribution.
To reduce the number of parameters we only study layers with symmetric degree distribution, e.g. $P(k)=P_{\textrm{in}}(k)=P_{\textrm{out}}(k)$; however, the in- and out-degree of a specific node can be different.
No timescale separation ([$\tau_{\textrm{I}}=\tau_{\textrm{II}}=1$]{.nodecor}) {#sec:noseparation}
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In the special case when both layers operate on the same timescale, i.e. $\tau_{\textrm{I}}=\tau_{\textrm{II}}=1$ (Fig. \[fig:3examples\]b), there is no qualitative difference between the dynamics of the layers. The reason why the system cannot be treated as a single large network is that we are only allowed to directly interact with Layer I. Recently Iudice et al. developed methodology to identify $N_{\textrm{i}}$ if the control signals can only be connected to a subset of nodes [@IUD15]. However, the one-to-one coupling between the layers enables us to find $N_{\textrm{i}}$ using a simpler approach.
Finding $N_{\textrm{i}}$ for a single-layer network is equivalent to finding a maximum matching of the network [@LIU11]. A matching is a set of directed links that do not share starting or end points, and a node is unmatched if there is no link in the matching pointing at it. Liu et al. showed that full control of a network is possible if each unmatched node is controlled directly by an independent input; therefore $N_{\textrm{i}}$ is provided by the minimum number of unmatched nodes. To determine $N_{\textrm{i}}$ for a two-layer network, we first find a maximum matching of the combined network of Layer I and Layer II. If there are no unmatched nodes in Layer II, we only have to interact with Layer I; therefore we are done. If a node $v^{\textrm{II}}_i$ is unmatched in Layer II, $v^{\textrm{I}}_i$ is necessarily matched by some node $v^{\textrm{I}}_j$, otherwise the size of the matching could be increased by adding $(v_i^{\textrm{I}}\rightarrow v_i^{\textrm{II}})$. By taking out the link $(v^{\textrm{I}}_j\rightarrow v^{\textrm{I}}_i)$ from the matching and including $(v^{\textrm{I}}_i\rightarrow v^{\textrm{II}}_i)$ the size of the maximum matching does not change, and we moved the unmatched node from Layer II to Layer I. We repeat this for all unmatched nodes in Layer II. (Note that it may be necessary to connect inputs to additional nodes so that all nodes are reached by the control signals. Due to the one-to-one coupling between the layers this too can be accomplished by interacting only with Layer I.) This simplified method allows faster identification of $N_{\textrm{i}}$ using the Hopcroft-Karp algorithm [@HOP73] and analytically solving $n_{\textrm{i}}=N_{\textrm{i}} /N$ for random networks based on calculating the fraction of always matched nodes as described in Appendix \[app:sec:analytical\] [@ZDE06; @JIA13; @JIA14; @POS14].
First, we measure $n_{\textrm{i}}$ while fixing the average degree of Layer II ($c_{\textrm{II}}$) and varying the average degree of Layer I ($c_{\textrm{I}}$). For both ER-ER and SF-SF networks, we find that $n_{\textrm{i}}$ decreases for increasing values of $c_{\textrm{I}}$ and converges to $n_{\textrm{i}}^{\textrm{II}}=N_{\textrm{i}}^{\textrm{II}}/N$, the normalized number of inputs needed to control Layer II in isolation (Fig. \[fig:secIIIA\]a). The latter observation is easily understood: $n_{\textrm{i}}$ is determined by the fraction of unmatched nodes in the combined network of the two layers; if $c_{\textrm{I}}$ is high enough, Layer I is perfectly matched; therefore all unmatched nodes are in Layer II. Based on the same argument, $n_{\textrm{i}}^{\textrm{I}}$ also serves as a lower bound for $n_{\textrm{i}}$.
Varying both $c_{\textrm{I}}$ and $c_{\textrm{II}}$ for ER-ER and both $\gamma_{\textrm{I}}$ and $\gamma_{\textrm{II}}$ for SF-SF with constant average degrees $c_{\textrm{I}}=c_{\textrm{II}}$, we find that dense networks require less inputs than sparse networks (Fig. \[fig:secIIIA\]b) and degree heterogeneity makes control increasingly difficult (Fig. \[fig:secIIIA\]c) – in line with results for single-layer networks [@LIU11]. We also observe that $n_{\textrm{i}}$ is invariant to exchanging Layer I and Layer II. This is explained by the fact that the size of the maximum matching is invariant to flipping the direction of all links, and on the ensemble level this is the same as swapping the two layers for networks with $P(k_{\textrm{in}})=P(k_{\textrm{out}})$.
In summary, for no timescale separation controllability is equally affected by the network structure of both layers, and $n_{\textrm{i}}$ is greater or equal to the number of inputs necessary to control any of its layers in isolation. Similarly to single-layer networks, networks with low average degree and high degree heterogeneity require more independent inputs than sparse homogeneous networks.
Layer I updates faster ([$\tau_{\textrm{I}}=1$, $\tau_{\textrm{II}}>1$]{.nodecor})
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In the previous section we found that the network structure of the two layers equally affect $n_{\textrm{i}}$ if $\tau_{\textrm{I}}=\tau_{\textrm{II}}=1$. This is not the case if the timescales are different, for example if Layer I updates faster than Layer II, we expect that we need fewer inputs than in the same timescale case by the virtue of having more opportunity to interact with the faster system (Fig. \[fig:3examples\]b). In this section we systematically study this effect using the algorithm described in Sec. \[sec:multiplex\_net\] and analytical arguments.
By measuring $n_{\textrm{i}}$ for ER-ER and SF-SF networks as a function of $\tau_{\textrm{II}}$, we find that $n_{\textrm{i}}$ monotonically decreases with increasing $\tau_{\textrm{II}}$ (Fig. \[fig:secIIIB\]a), confirming our expectations. For both ER-ER and SF-SF networks $n_{\textrm{i}}(\tau_{\textrm{II}})$ converges to $n_{\textrm{i}}^{\textrm{I}}=N_{\textrm{i}}^{\textrm{I}}/N$ which is the normalized number of inputs needed to control Layer I in isolation. This can be understood by the following argument: Suppose that $\tau_{\textrm{II}}=N$, the maximum number of time steps needed to impose control on any network with $N$ nodes [@KAL63]. We use the state of Layer I at $t=0$ to set the state of Layer II at $t=N$, and we have $N$ time steps to impose control on Layer I as if it was just by itself. For a given network we define the critical timescale parameter $\tau^{\textrm{c}}_{\textrm{II}}$ as the minimum value of $\tau_{\textrm{II}}$ for which $n_{\textrm{i}}(\tau_{\textrm{II}})=n_{\textrm{i}}^{\textrm{I}}$. Above the critical timescale separation, Layer I completely determines $n_{\textrm{i}}(\tau_{\textrm{II}})$ independent of the structure of Layer II, in other words, the [multiplex]{} nature of the system no longer plays a role in determining $n_{\textrm{i}}$.
Measuring $\tau_{\textrm{II}}^{\textrm{c}}$ we find that for both ER-ER and SF-SF networks $\tau_{\textrm{II}}^{\textrm{c}}$ monotonically increases with increasing $c_{\textrm{I}}$ for fixed $c_{\textrm{II}}$, and decreases with increasing $c_{\textrm{II}}$ for fixed $c_{\textrm{I}}$ (Fig \[fig:secIIIB\]b). That is $\tau_{\textrm{II}}^{\textrm{c}}$ is the highest if Layer I is dense and Layer II is sparse. SF-SF networks have significantly lower $\tau_{\textrm{II}}^{\textrm{c}}$ than ER-ER networks with the same average degree.
To understand the observed pattern we provide an approximation to calculate $\tau_{\textrm{II}}^{\textrm{c}}$. We call a node $v^{\textrm{I}}_i$ externally controlled if in the dynamic graph $v^{\textrm{I}}_{i,\tau_{\textrm{II}}}$ is connected to an external signal $u_{j,t}$ via a disjoint control path (e.g. nodes $v^{\textrm{I}}_A$ and $v^{\textrm{I}}_B$ in Fig. \[fig:3examples\]c), and the number of such nodes is denoted by $N_{\textrm{e}}(\tau_{\textrm{II}})$. We have previously shown that we require $N_{\textrm{i}}^{\textrm{I}}$ independent inputs at $\tau_{\textrm{II}}^{\textrm{c}}$. For each independent input and each time step, we have one control signal $u_{i,t}$; therefore we need timescale parameter $$\label{eq:tauIIc}
\tau_{\textrm{II}}^{\textrm{c}}=\lceil N_{\textrm{e}}(\tau_{\textrm{II}}^{\textrm{c}})/N_{\textrm{i}}^{\textrm{I}}\rceil$$ to insert enough signals required by the $N_{\textrm{e}}(\tau_{\textrm{II}}^{\textrm{c}})$ externally controlled nodes, where $\lceil\cdot\rceil$ is the ceiling function. Equation (\[eq:tauIIc\]) is not yet useful as it contains $\tau_{\textrm{II}}^{\textrm{c}}$ on both side. Observing that $N_{\textrm{e}}(\tau_{\textrm{II}})$ is a monotonically increasing function of $\tau_{\textrm{II}}$ and $N_{\textrm{e}}(\tau_{\textrm{II}}=1)= N_{\textrm{i}}(\tau_{\textrm{II}}=1)$, we can write $$N_{\textrm{i}}(\tau_{\textrm{II}}=1) \leq N_{\textrm{e}}(\tau_{\textrm{II}}^{\textrm{c}})\leq N.$$ In the special case when Layer II is fully connected, $\tau_{\textrm{II}}^{\textrm{c}}=1$ and $N_{\textrm{e}}(\tau_{\textrm{II}}^{\textrm{c}}=1)= N_{\textrm{i}}(\tau_{\textrm{II}}=1)$. In the case when Layer II is entirely disconnected, i.e. is composed of isolated nodes, $N_{\textrm{e}}(\tau_{\textrm{II}})=N=N_{\textrm{i}}(\tau_{\textrm{II}}=1)$. These two opposite limiting cases suggest that it is reasonable to approximate $N_{\textrm{e}}(\tau_{\textrm{II}}^{\textrm{c}})$ by its lower bound: $$\label{eq:tauIIc_approx}
\tau_{\textrm{II}}^{\textrm{c}}\approx\lceil N_{\textrm{i}}(\tau_{\textrm{II}}=1)/N_{\textrm{i}}^{\textrm{I}}\rceil,$$ which entirely depends on quantities that we can easily measure or analytically compute. We find that Eq. (\[eq:tauIIc\_approx\]) preforms remarkably well: Figure \[fig:secIIIB\]b compares direct measurements of $\tau_{\textrm{II}}^{\textrm{c}}$ to approximations obtained by using measurements and analytically computed values of $N_{\textrm{i}}(\tau_{\textrm{II}}=1)$ and $N_{\textrm{i}}^{\textrm{I}}$. The approximation based on measurements out performs the analytical calculations, because the analytical results provide the expectation value of the numerator and denominator for ER and SF network ensembles[;]{} and therefore the ceiling function is applied to the fraction of averages, instead of averaging after applying the ceiling function. [To further test the Eq. (\[eq:tauIIc\_approx\]), we fix $n_{\textrm{i}}(\tau_{\textrm{II}}=1)$ and $n^{\textrm{I}}_{\textrm{i}}$ and we analytically calculate $c_{\textrm{I}}$ and $c_{\textrm{II}}$ for SF-SF networks with varying degree exponent $\gamma=\gamma_{\textrm{I}}=\gamma_{\textrm{II}}$ using the framework developed in Appendix \[app:sec:analytical\]. Then we generate SF-SF networks and measure $\tau_{\textrm{II}}^{\textrm{c}}$ as a function of $\gamma$. The approximation predicts that $\tau_{\textrm{II}}^{\textrm{c}}$ remains constant, in line with our observations (Fig. \[fig:secIIIB\]c).]{}
The good performance of Eq. (\[eq:tauIIc\_approx\]) is partly due to the role of the ceiling function, as it is insensitive to changes in the numerator that are small compared to $N_{\textrm{i}}^{\textrm{I}}$. Indeed, errors are more pronounced if $N_{\textrm{i}}(\tau_{\textrm{II}}=1)/N_{\textrm{i}}^{\textrm{I}}$ is close to an integer (e.g. data point $c_{\textrm{I}}=4.5$ and $c_{\textrm{II}}=1$ in Fig. \[fig:secIIIB\]b for ER-ER), or $N_{\textrm{i}}(\tau_{\textrm{II}}=1)\gg N_{\textrm{i}}^{\textrm{I}}$ (e.g. data points $n_{\textrm{i}}^{\textrm{I}}=0.084$ in Fig. \[fig:secIIIB\]c).
What we learn from this approximation is that $\tau_{\textrm{II}}^{\textrm{c}}$ depends only indirectly on the degree distribution of Layer I and Layer II through the control properties of the system without timescale separation – $N_{\textrm{i}}(\tau_{\textrm{II}}=1)$ and $N^{\textrm{I}}_{\textrm{i}}$. In Sec. \[sec:noseparation\], we showed that $N_{\textrm{i}}(\tau_{\textrm{II}}=1)\geq N^{\textrm{II}}_{\textrm{i}}$, therefore $\tau_{\textrm{II}}^{\textrm{c}}$ is expected to be large if Layer I is easy to control (e.g. it is dense and has homogeneous degree distribution) and Layer II is hard to control (e.g. it is sparse and has heterogeneous degree distribution).
In summary, if Layer I updates faster, timescale separation enhances controllability up to a critical timescale parameter $\tau_{\textrm{II}}^{\textrm{c}}$, above which $n_{\textrm{i}}(\tau_{\textrm{II}})=n^{\textrm{I}}_{\textrm{i}}$ and is completely determined by Layer I. The critical timescale parameter $\tau_{\textrm{II}}^{\textrm{c}}$ largely depends on the controllability of the system without timescale separation, it is expected to be large if Layer I is easy and Layer II is hard to control.
Layer II updates faster ([$\tau_{\textrm{I}}>1$, $\tau_{\textrm{II}}=1$]{.nodecor})
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Finally we investigate the case when Layer II operates faster than Layer I, i.e. $\tau_{\textrm{I}}>1$ and $\tau_{\textrm{II}}=1$ (Fig. \[fig:3examples\]d). Measurements show that $n_{\textrm{i}}$ monotonically increases in function of $\tau_{\textrm{I}}$ for both ER-ER and SF-SF networks, and $n_{\textrm{i}}$ remains constant if $\tau_{\textrm{I}}\geq \tau^{\textrm{c}}_{\textrm{I}}$, where $\tau^{\textrm{c}}_{\textrm{I}}$ is defined for a single network (Fig. \[fig:secIIIC\]a). To understand these results consider the following argument: Some nodes of Layer II are internally controlled, meaning that the state of these nodes at $t=\tau_{\textrm{I}}$ is set by the state of nodes within Layer II at $t=0$ connected to them via disjoint control paths (node $v^{\textrm{II}}_C$ in Fig. \[fig:3examples\]d); while the rest of the nodes of Layer II have to be controlled by nodes of Layer I. The maximum number of internally controlled nodes is set by the number of disjoint paths of length $\tau_{\textrm{I}}$. A directed open path traversing $l$ links in Layer II yields a path in the dynamic graph of at most length $l$; therefore if $\tau_{\textrm{I}}>l$ the path can no longer be used for control. For example, in Fig. \[fig:3examples\]a path $(v_B^{\textrm{II}}\rightarrow v_A^{\textrm{II}})$ consists of a single link; therefore, we can use it for control if $\tau_{\textrm{I}}=1$ (Fig. \[fig:3examples\]b) and it is no longer useful if $\tau_{\textrm{I}}>1$ (Fig. \[fig:3examples\]d). However, a cycle can support a path in the dynamic graph of any length, e.g. the self-loop $(v_C^{\textrm{II}}\rightarrow v_C^{\textrm{II}})$ in Fig. \[fig:3examples\]. This predicts that $$\label{eq:tauI_ni_upperbound}
n_{\textrm{i}}(\tau_{\textrm{I}}=\infty)\geq 1-n_{\textrm{cycle}},$$ where $n_{\textrm{cycle}}=N_{\textrm{cycle}}/N$ is the maximum fraction of nodes that can be covered with cycles in Layer II. Furthermore, it also means that $$\label{eq:tauIc_upperbound}
\tau_{\textrm{I}}^{\textrm{c}}\leq l_{\textrm{max}}+1,$$ where $l_{\textrm{max}}$ is the maximum length of a control path that does not involve cycles, a quantity that only depends on the structure of Layer II. We provide the formal definition $l_{\textrm{max}}$ and algorithms to measure $n_{\textrm{cycle}}$ and $l_{\textrm{max}}$ in Appendix \[app:sec:algorithms\].
Both $l_{\textrm{max}}$ and $n_{\textrm{cycle}}$ only depend on Layer II, furthermore both strongly depend on whether Layer II contains a strongly connected component (SCC) or not. Uncorrelated random directed networks – both ER and SF – undergo a percolation transition at $c=1$ [@SCH02]. If $c< 1$, the network is composed of small tree components, meaning the $n_{\textrm{cycle}}=0$ and $l_{\textrm{max}}$ is equal to the diameter $D$ of the network. If the system is in the critical point $c=1$, the size of the largest component $S$ diverges as $N\rightarrow \infty$, but the relative size $S/N$ remains zero. The largest component contains a small number of cycles; therefore $D$ is only approximately equal to $l_{\textrm{max}}$. If $c>1$, a unique giant SCC emerges which contains cycles; therefore $n_{\textrm{cycle}}>0$ and $l_{\textrm{max}}$ is no longer directly connected to the diameter. Rigorous mathematical results show that the diameter of the ER model scales as $D\sim \log(N)$ for $c\neq 1$, and $D\sim N^{1/3}$ for $c=1$, the latter corresponding to percolation transition point [@NAC08], suggesting that the critical timescale parameter $\tau_{\textrm{I}}^{\textrm{c}}$ also depends on $N$. Indeed, Figure \[fig:secIIIC\_N\] shows that $\tau_{\textrm{I}}^{\textrm{c}}$ monotonically increases with $N$ for both ER-ER and SF-SF networks.
We now scan possible values of $c_{\textrm{I}}$ while keeping $c_{\textrm{II}}$ and $N$ fixed, we find that $n_{\textrm{i}}(c_{\textrm{I}})$ and $\tau_{\textrm{I}}^{\textrm{c}}(c_{\textrm{I}})$ quickly converges to its respective lower and upper bound provided by Eqs. (\[eq:tauI\_ni\_upperbound\]) and (\[eq:tauIc\_upperbound\]) (Fig. \[fig:secIIIC\]b-c). Varying $c_{\textrm{II}}$ and keeping $c_{\textrm{I}}$ fixed shows more intricate behavior: $\tau_{\textrm{I}}^{\textrm{c}}(c_{\textrm{II}})$ increases, peaks and decreases again (Fig. \[fig:secIIIC\]d). This is explained by changes in the structure of Layer II: For small $c_{\textrm{II}}$ the network is composed of small components with tree structure, increasing $c_{\textrm{II}}$ agglomerates these components, thus increasing $l_{\textrm{max}}$. For large $c_{\textrm{II}}$, a giant SCC exists supporting many cycles, as $c_{\textrm{II}}$ increases more and more nodes can be covered with cycles reducing $l_{\textrm{max}}$. At the critical point $c_{\textrm{II}}^*=1$ the giant SCC emerges, and the largest component consists of $N^\alpha$ nodes ($0<\alpha<1$) with only few cycles, providing the peak of $\tau_{\textrm{I}}^{\textrm{c}}(c_{\textrm{II}})$. Although $c_{\textrm{II}}^*=1$ for both ER and SF networks in the $N\rightarrow\infty$ limit, finite size effects delay the peak of $\tau_{\textrm{I}}^{\textrm{c}}$ for SF-SF networks. Below the transition point, $\tau_{\textrm{I}}^{\textrm{c}}$ is smaller for ER-ER networks than for SF-SF networks with the same average degree. In contrast, above the transition point SF-SF networks have larger $\tau_{\textrm{I}}^{\textrm{c}}$. A likely explanation is that the cycle cover of SF networks is smaller than the cycle cover of ER networks with the same average degree, thus more nodes can potentially participate in the longest control path that does not involve cycles.
The number of inputs above the critical timescale parameter $n_{\textrm{i}}(\tau_{\textrm{I}}=\infty)$ is also affected by the cycle cover of Layer II (Fig. \[fig:secIIIC\]e): For $c_{\textrm{II}}<1$, Layer II does not contain cycles yielding $n_{\textrm{i}}(\tau_{\textrm{I}}=\infty)=1$; for large $c_{\textrm{II}}$, Layer II can be completely covered with cycles, and $n_{\textrm{i}}(\tau_{\textrm{I}}=\infty)$ is determined by $n_i^{\textrm{I}}$, the number of inputs needed to control Layer I in isolation.
In summary, if Layer II updates faster, timescale separation reduces controllability up to a critical timescale parameter $\tau^{\textrm{c}}_{\textrm{I}}$. For the model networks, the value of $\tau^{\textrm{c}}_{\textrm{I}}$ depends on whether Layer II has a giant SCC; $\tau^{\textrm{c}}_{\textrm{I}}$ has the highest value at the percolation threshold of Layer II. If Layer II does not contain a giant SCC, degree heterogeneity decreases $\tau^{\textrm{c}}_{\textrm{I}}$; above the percolation threshold homogeneous networks have lower $\tau^{\textrm{c}}_{\textrm{I}}$. For all timescale parameters, it remains true that ER-ER networks require less independent inputs than SF-SF networks with the same average degree.
Conclusions {#sec:conclusions}
===========
Here we explored controllability of interconnected complex systems with a model that incorporates common properties of these systems: (i) it consists of two layers each described by a complex network; (ii) the operation of each layer is characterized by a different, but often comparable timescale and (iii) the external controller only interacts with one layer directly. [We focused on two-layer multiplex networks, meaning that we assume one-to-one coupling between the nodes of the two layers. Our motivation for this choice was to ensure analytical tractability and to isolate the specific role of timescales from the effect of more complex multilayer network structure. Results obtained for more general multilayer networks will ultimately be shaped by a variety of features such as complex interconnectivity structure, correlations in network structure and details of dynamics. However, even by studying multiplex networks, we uncovered nontrivial phenomena, attesting that without understanding each individual effect, it is impossible to fully understand a system as a whole.]{}
Using structural controllability we were able to solve the model, thereby directly linking controllability to a graph combinatorial problem. We investigated the effect of network structure and timescales by measuring the minimum number of independent inputs needed for control, $N_{\textrm{i}}$. Overall we found that dense networks with homogeneous degree distribution require less inputs than sparse heterogeneous networks, in line with previous results for single-layer networks [@LIU11]. We showed that if we control the faster layer directly, $N_{\textrm{i}}$ decreases with increasing timescale difference, but only up to a critical value. Above the critical timescale difference, $N_{\textrm{i}}$ is completely determined by the faster layer and we do not have to take into account the multiplex structure of the system. This critical timescale separation is expected to be large if the faster layer would be easy to control and the slower layer would be hard to control in isolation. If we interact with the slower layer, control is increasingly difficult for increasing timescale difference, again up to a critical value, above which $N_{\textrm{i}}$ still depends on the structure of both layers. In this case the critical timescale difference largely depends on the longest control path that does not involve cycles in the faster layer.
Although our model offers only a stylized description of real systems, it is a tractable first step towards understanding the role of timescales in control of interconnected networks. [By identifying the network characteristics that affect important measures of controllability, such as minimum number of inputs needed for control and critical timescale difference, our results serve as a starting point for future work that aims]{} to relax some of the model’s assumptions. Some of these extensions are relatively straightforward using the tool set developed here, for example, the effect of higher order network structures can be studied by adding correlations to the underlying networks. Other extensions are more challenging, e.g. if the interconnection between the layers is incomplete or the layers contain different number of nodes, the minimum input problem is computationally more difficult; therefore investigating such systems would require development of efficient approximation schemes. [Structural control theory does not take the link weights into account; therefore answering questions that depend on the specific strength of the connections require the development of different tools. For example, for continuous-time systems the timescales are encoded in the strength of the interactions; or the minimum control energy also depends on value of the link weights.]{}
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Yang-Yu Liu, Philipp Hövel and Zsófia Pénzváltó for useful discussions. We gratefully acknowledge support from the US Army Research Office Cooperative Agreement No. W911NF-09-2-0053 and MURI Award No. W911NF-13-1-0340, and the Defense Threat Reduction Agency Basic Research Awards HDTRA1-10-1-0088 and HDTRA1-10-1-00100.
Analytical solution for [$\tau_{\textrm{I}}=\tau_{\textrm{II}}=1$]{.nodecor} {#app:sec:analytical}
============================================================================
In this section we derive an analytical solution of $n_{\textrm{i}} = N_{\textrm{i}} /N$ in case of $\tau_{\textrm{I}}=\tau_{\textrm{II}}=1$ for two-layer random networks with predefined degree distribution as defined in Sec. \[sec:results\]. This network model is treelike in the $N\rightarrow\infty$ limit; therefore it lends itself to the generating function formalism. The approach described here is based on calculating the fraction of nodes that are matched in all possible maximum matchings [@JIA13]. This solution is substantially simpler than the one described in Ref. [@LIU11]; however, it only applies to bipartite networks (or to bipartite representations of directed networks), and cannot be generalized to unipartite networks.
We aim to calculate the expected size of the maximum matching of the following undirected bipartite network $\mathcal B$. Layer I $\mathcal{L}_{\textrm{I}}$ and Layer II $\mathcal{L}_{\textrm{II}}$ are generated independently either using the ER or the SF model; $V_{\textrm{I}}$ and $E_{\textrm{I}}$ are the node and link sets of $\mathcal{L}_{\textrm{I}}$ and $V_{\textrm{II}}$ and $E_{\textrm{II}}$ are the node and link sets of $\mathcal{L}_{\textrm{II}}$. Each node in $v^{\textrm{I}}_i\in V_{\textrm{I}}$ is split into two copies $v^{\textrm{I}}_{i,0}\in V^{\textrm{I}}_{0}$ and $v^{\textrm{I}}_{i,1}\in V^{\textrm{I}}_{1}$, we draw a link $(v^{\textrm{I}}_{i,0}-v^{\textrm{I}}_{j,1})$ if there exists a link $(v^{\textrm{I}}_{i}\rightarrow v^{\textrm{I}}_{j})$ in $\mathcal{L}_{\textrm{I}}$. We treat $\mathcal{L}_{\textrm{II}}$ similarly. We then add links $(v^{\textrm{I}}_{i,0}-v^{\textrm{II}}_{i,1})$ for all $i$. That is all links in $\mathcal B$ connect exactly one node in $V^{\textrm{I}}_{0}\cup V^{\textrm{II}}_{0}$ to one node in $V^{\textrm{I}}_{1}\cup V^{\textrm{II}}_{1}$. Nodes in $V^{\textrm{I}}_{0}\cup V^{\textrm{I}}_{1}$ belong to Layer I, and nodes in $V^{\textrm{II}}_{0}\cup V^{\textrm{II}}_{1}$ belong to Layer II. The network $\mathcal B$ is the undirected version of the dynamical graph $\mathcal D_1$ without control signals.
In general, multiple possible maximum matchings may exist in a network. We first calculate the fraction of nodes that are matched in all possible maximum matchings. It was shown in Ref. [@JIA13] that in any network $\mathcal G$ a node $v$ is always matched if and only if at least one of its neighbors is not always matched in $\mathcal G\setminus v$, where $\mathcal G\setminus v$ is the network obtained by removing node $v$ from $\mathcal G$. We translate this rule to a set of self-consistent equations to calculate the expected fraction of always matched nodes in our random network model in the $N\rightarrow\infty$ limit. We provide comments on the issues of applying the rule proven for finite networks to infinite ones at the end of this section.
To proceed we define a few probabilities. We randomly select a link $e$ connecting two nodes $v^{\textrm{I}}_{i,0}\in V^{\textrm{I}}_{0}$ and $v^{\textrm{I}}_{j,1}\in V^{\textrm{I}}_{1}$. Let $\theta^{\textrm{I}}_{0}$ be the probability that $v^{\textrm{I}}_{i,0}$ is always matched in $\mathcal B\setminus e$, and $\theta^{\textrm{I}}_{1}$ be the probability that $v^{\textrm{I}}_{j,1}$ is always matched in $\mathcal B\setminus e$. Similarly we randomly select a link $e$ connecting a node $v^{\textrm{I}}_{i,0}\in V^{\textrm{I}}_{0}$ with a node $v^{\textrm{II}}_{i,1}\in V^{\textrm{II}}_{1}$. Let $\theta^{\textrm{I,II}}_{0}$ be the probability that node $v^{\textrm{I}}_{i,0}$ is always matched in $\mathcal B\setminus e$, and $\theta^{\textrm{I,II}}_{1}$ be the probability that node $v^{\textrm{II}}_{i,1}$ is always matched in $\mathcal B\setminus e$. The probabilities $\theta^{\textrm{II}}_{0}$ and $\theta^{\textrm{II}}_{1}$ are defined similarly. According to the rule described above these quantities can be determined by the following set of equations: $$\label{eq:app:theta}
\begin{split}
\theta^{\textrm{I}}_{0} &= 1 - H^{\textrm{I}}(\theta^{\textrm{I}}_{1})\theta^{\textrm{I,II}}_{1}, \\
\theta^{\textrm{I}}_{1} &= 1 - H^{\textrm{I}}(\theta^{\textrm{I}}_{0}),\\
\theta^{\textrm{I,II}}_{0} &= 1 - G^{\textrm{I}}(\theta^{\textrm{I}}_{1}),\\
\theta^{\textrm{I,II}}_{1} &= 1 - G^{\textrm{II}}(\theta^{\textrm{II}}_{0}),\\
\theta^{\textrm{II}}_{0} &= 1 - H^{\textrm{II}}(\theta^{\textrm{II}}_{1}),\\
\theta^{\textrm{II}}_{1} &= 1 - H^{\textrm{II}}(\theta^{\textrm{II}}_{0})\theta^{\textrm{I,II}}_{0},
\end{split}$$ where $G^{\textrm{I/II}}(x)=\sum_{k=0}^\infty P^{\textrm{I/II}}(k)x^k$ are the generating functions of the degree distributions and $H^{\textrm{I/II}}(x)=\sum_{k=1}^\infty k/{\left\langle k\right\rangle} P^{\textrm{I/II}}(k)x^{k-1}$ are the generating functions of the excess degree distributions.
If we remove a node $v$ which is not always matched, the size of the maximum matching does not decrease. However, if $v$ is matched in all maximum matchings, the number of matched nodes will decrease by two. Therefore to count the size of the maximum matching, we first count the number of nodes that are always matched. By doing so, we have double counted the case when an always matched node is matched by another always matched one. This case occurs for each link $e$ that connects two nodes that are not always matched in $\mathcal G\setminus e$. Combining these two contributions, the expected number of links in the matching is $$\begin{split}
N_{\textrm{match}} =& N[1 - G^{\textrm{I}}(\theta^{\textrm{I}}_{1})\theta^{\textrm{I,II}}_{1}] + N[1 - G^{\textrm{I}}(\theta^{\textrm{II}}_{0})] + N[1 - G^{\textrm{II}}(\theta^{\textrm{I}}_{1})] + N[1 - G^{\textrm{II}}(\theta^{\textrm{II}}_{0})\theta^{\textrm{I,II}}_{0}] -\\
-& c_{\textrm{I}}N(1-\theta^{\textrm{I}}_{0})(1-\theta^{\textrm{I}}_{1}) - N(1-\theta^{\textrm{I,II}}_{0})(1-\theta^{\textrm{I,II}}_{1}) - c_{\textrm{II}}N(1-\theta^{\textrm{II}}_{0})(1-\theta^{\textrm{II}}_{1}),
\end{split}$$ where the first four terms count the number of nodes that are always matched in $V^{\textrm{I}}_{0}$, $V^{\textrm{I}}_{1}$,$V^{\textrm{II}}_{0}$ and $V^{\textrm{II}}_{1}$, respectively; and the last three terms correct the double counting. The expected number of independent inputs needed is determined by the number of unmatched nodes in $V^{\textrm{I}}_{1}$ and $V^{\textrm{II}}_{1}$: $$N_{\textrm{i}} = 2N - N_{\textrm{match}}.$$ Due to the links between Layer I and Layer II, the size of the maximum matching is at least $N$, meaning that $N_{\textrm{i}}\leq N$. Therefore we normalize $N_{\textrm{i}}$ by $N$, yielding $$\begin{split}
n_{\textrm{i}} = & G^{\textrm{I}}(\theta^{\textrm{I}}_{1})\theta^{\textrm{I,II}}_{1} + G^{\textrm{I}}(\theta^{\textrm{II}}_{0}) + G^{\textrm{II}}(\theta^{\textrm{I}}_{1}) + G^{\textrm{II}}(\theta^{\textrm{II}}_{0})\theta^{\textrm{I,II}}_{0} - 2 +\\
+& c_{\textrm{I}}(1-\theta^{\textrm{I}}_{0})(1-\theta^{\textrm{I}}_{1}) + (1-\theta^{\textrm{I,II}}_{0})(1-\theta^{\textrm{I,II}}_{1}) + c_{\textrm{II}}(1-\theta^{\textrm{II}}_{0})(1-\theta^{\textrm{II}}_{1}).
\end{split}$$
### Comments on matchings in the configuration model {#sec:app:infintematching .unnumbered}
The method we described to calculate the expected size of the maximum matching does not work for unipartite ER or SF networks generally. The reason for this is that above a critical average degree $c^*$ a densely connected subgraph forms, which is referred to as the core of the network (sometimes leaf removal core or computational core) [@BAU01; @COR06; @LIU12]. To derive Eq. (\[eq:app:theta\]), we assume that the neighbors of a randomly selected node $v$ are independent of each other in $\mathcal B\setminus v$ and removing a single node does not influence macroscopic properties, e.g. $\theta$. The effect of the core is that these assumptions no longer hold and removing just a few nodes may drastically change the number of always matched nodes. Possible way of circumventing this problem is to introduce a new category of nodes: in addition to keeping track of nodes that are sometimes matched and always matched, we separately account for nodes that are almost always matched [@ZDE06].
The reason why the calculation works for bipartite networks is that a core in the bipartite network will have two sides: all nodes on one side will be always matched and all nodes on other will be some times matched [@JIA13; @JIA14; @POS14]. If the expected size of the core on the two sides is different, finite removal of nodes will not change macroscopic properties. If the expected size of the two sides of the core is the same, removal of finite nodes may change which side is always matched and which side is sometimes matched [@JIA13]. However, this does not change expected fraction of matched nodes; therefore does not interfere with the calculations.
Algorithms {#app:sec:algorithms}
==========
Cycle cover ([$N_{\textrm{cycle}}$]{.nodecor}) {#app:sec:ncycle}
----------------------------------------------
To find the maximum cycle cover of a directed network $\mathcal L$, we assign weight $0$ to each link in $\mathcal L$; and we add a self-loop with weight $1$ to each node that does not already have a self-loop. Then we find the minimum weight maximum directed matching in $\mathcal L$ augmented with self-loops by converting the problem to a minimum cost maximum flow problem. The maximum matching is guaranteed to be perfect, because each node has a self-loop. The minimum weight perfect matching in the directed network corresponds to a perfect cycle cover where the number of self-loops with weight $1$ is minimized. Therefore the maximum cycle cover in $\mathcal L$ without extra self-loops is $$N_{\textrm{cycle}}= N-W,$$ where $W$ is the sum of the weights of the links in the minimum weight perfect matching.
Longest control path not involving cycles ([$l_{\textrm{max}}$]{.nodecor}) {#supp:sec:lmax}
--------------------------------------------------------------------------
In this section we provide the algorithm to measure the longest control path not involving cycles $l_{\textrm{max}}$ of Layer II of a two-layer network for the case $\tau_{\textrm{I}}\geq 1$ and $\tau_{\textrm{II}}=1$. The algorithm itself serves as the precise definition of $l_{\textrm{max}}$.
Given a two-layer directed network $\mathcal M$, let $N_{\textrm{cycle}}$ be the maximum number of nodes that can be covered by node disjoint cycles in Layer II. To measure $l_{\textrm{max}}$, first we construct the dynamical graph $\mathcal D_{l}^{\textrm{II}}$ representing the time evolution of the Layer II between time $t=0$ and $t=l$ as if it would be isolated as defined in Sec. \[sec:single-layer\_net\]. We search for disjoint control paths connecting nodes at time step $t=0$ with nodes at time step $t=l$, e.g. each control path connects a node $v^{\textrm{II}}_{i,0}$ with $v^{\textrm{II}}_{j,l}$. The maximum number of such paths $N_{\textrm{path}}(l)$ provides the maximum number of internally controlled nodes if $\tau_{\textrm{I}}=l$. To determine $N_{\textrm{path}}(l)$ we convert the problem to a maximum flow problem: We set the capacity of each link and each node in $\mathcal D_{l}^{\textrm{II}}$ to 1. We then find the maximum flow connecting source node set $V^{\textrm{II}}_0=\{v^{\textrm{II}}_{i,0}\vert i=1,2,\ldots,N\}$ to target node set $V^{\textrm{II}}_l=\{v^{\textrm{II}}_{i,l}\vert i=1,2,\ldots,N\}$ using a maximum flow algorithm of choice. The maximum flow provides $N_{\textrm{path}}(l)$. And $l_{\textrm{max}}$ is defined as one less than the smallest value of $l$ such that $$N_{\textrm{path}}(l)=N_{\textrm{cycle}}.$$
Figures \[app:fig:lmax-example-1\] and \[app:fig:lmax-example-2\] provide two examples to illustrate the calculation of $l_{\textrm{max}}$.
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![[**Structural controllability of two-layer [multiplex]{} networks.**]{} [**(a)**]{} A two-layer network. [**(b-c)**]{} To determine $N_{\textrm{i}}$, we construct the dynamic graph representing the time evolution of the system from $t_0=0$ to $t_1=\max(\tau_{\textrm{I}},\tau_{\textrm{II}})$. The system is controllable [only]{} if all nodes at $t_1$ (blue) are connected to nodes at $t_0$ or nodes representing control signals (green) via disjoint paths (red). [**(b)**]{} In case of no timescale separation ($\tau_{\textrm{I}}=\tau_{\textrm{II}}=1$), each disjoint control path consists of a single link, yielding $N_{\textrm{i}}=2$. [**(c)**]{} If Layer I updates twice as frequently as Layer II ($\tau_{\textrm{I}}=1$, $\tau_{\textrm{II}}=2$), we are allowed to inject control signals at time steps $t=0$ and $1$, reducing the number of inputs to $N_{\textrm{i}}=1$. [**(d)**]{} On the other hand, if Layer II is faster ($\tau_{\textrm{I}}=2$, $\tau_{\textrm{II}}=1$), Layer II needs to support longer control paths, yielding $N_{\textrm{i}} = 3$.[]{data-label="fig:3examples"}](figure_secI)
![[**Structural controllability of single-layer networks.**]{} [**(a)**]{} A single-layer network, we apply inputs to nodes $v_A$ and $v_B$. [**(b)**]{} The dynamic graph $\mathcal D_N$ representing the time evolution of the dynamics from $t=0$ to $t=N$. The system is controllable, because we can connect the set of nodes representing control signals (green) to the set of nodes at $t=N$ (blue) via disjoint paths (red). [**(c)**]{} The dynamic graph $\mathcal D_1$ representing the time evolution of the dynamics from $t=0$ to $t=1$. The system is controllable, because we can connect the control signals and nodes at $t=0$ (green) to the set of nodes at $t=1$ (blue) via disjoint paths (red), and all nodes are accessible from control signals.[]{data-label="fig:singlenet"}](figure_singlenet)
![[**No timescale separation.**]{} [**(a)**]{} Number of inputs $n_{\textrm{i}}$ in function of $c_{\textrm{I}}$ for ER-ER and SF-SF ($\gamma_{\textrm{I}}=\gamma_{\textrm{II}}=2.5$) networks. The circles represent simulations, the continuous line is the analytical solution, and the dashed line is the analytical solution of $n^{\textrm{II}}_{\textrm{i}}$, the number of independent inputs necessary to control Layer II in isolation [@LIU11]. [**(b)**]{} $n_{\textrm{i}}$ for ER-ER networks with varying average degrees $c_{\textrm{I}}$ and $c_{\textrm{II}}$. In both layers $P(k)=P(k_{\textrm{in}})=P(k_{\textrm{out}})$, therefore the heatmap is symmetric with respect to the diagonal. Increasing $c$ in either layer enhances controllability. [**(c)**]{} $n_{\textrm{i}}$ for SF-SF networks with $c_{\textrm{I}}=c_{\textrm{II}}=4.0$ and varying degree exponents $\gamma_{\textrm{I}}$ and $\gamma_{\textrm{II}}$. Increasing degree heterogeneity in either layer increases $n_{\textrm{i}}$. Each data point is the average over $10$ randomly generated networks with $N=10,000$. The standard deviation of the measurements remains below 0.01.[]{data-label="fig:secIIIA"}](figure_secIIIA)
![[**Layer I updates faster.**]{} [**(a)**]{} Number of inputs $n_{\textrm{i}}$ for single ER-ER and SF-SF ($\gamma_{\textrm{I}}=\gamma_{\textrm{II}}=2.5$) networks with $N=10,000$ and varying timescale parameter $\tau_{\textrm{II}}$. The number of inputs $n_{\textrm{i}}$ monotonically decreases with increasing $\tau_{\textrm{II}}$, and for $\tau_{\textrm{II}}\geq \tau_{\textrm{II}}^{\textrm{c}}$, $n_{\textrm{i}}=n_{\textrm{i}}^{\textrm{I}}$. [**(b)**]{} The critical timescale parameter $\tau_{\textrm{II}}^{\textrm{c}}$ for ER-ER and SF-SF ($\gamma_{\textrm{I}}=\gamma_{\textrm{II}}=2.5$) networks with varying average degree $c_{\textrm{I}}$ and $c_{\textrm{II}}$. The crosses represent direct measurements of $\tau_{\textrm{II}}^{\textrm{c}}$; the squares represent the approximation obtained by applying Eq. (\[eq:tauIIc\_approx\]) to measurements of $n_{\textrm{i}}(\tau_{\textrm{II}}=1)$ and $n^{\textrm{I}}_{\textrm{i}}$; and the dashed line is an approximation obtained using analytically calculated expectation values of $n_{\textrm{i}}(\tau_{\textrm{II}}=1)$ and $n^{\textrm{I}}_{\textrm{i}}$. [[**(c)**]{} We measure $\tau_{\textrm{II}}^{\textrm{c}}$ for SF-SF networks with the same $n_{\textrm{i}}(\tau_{\textrm{II}}=1)$ and $n^{\textrm{I}}_{\textrm{i}}$ as a function of $\gamma=\gamma_{\textrm{I}}=\gamma_{\textrm{II}}$. Equation (\[eq:tauIIc\_approx\]) predicts that $\tau_{\textrm{II}}^{\textrm{c}}$ remains constant (dashed line), in line with our observations.]{} For (b-c), each data point is the average over $10$ randomly generated networks with $N=10,000$ and error bars represent the standard deviation.[]{data-label="fig:secIIIB"}](figure_secIIIB)
![[**Layer [II]{} updates faster.**]{} [**(a)**]{} Number of inputs $n_{\textrm{i}}$ for single ER-ER and SF-SF ($\gamma_{\textrm{I}}=\gamma_{\textrm{II}}=2.5$) networks with $N=10,000$ and varying timescale parameter $\tau_{\textrm{I}}$. The number of inputs $n_{\textrm{i}}$ monotonically increases with increasing $\tau_{\textrm{I}}$; for $\tau_{\textrm{I}}\geq \tau_{\textrm{I}}^{\textrm{c}}$, $n_{\textrm{i}}=n_{\textrm{i}}(\tau_{\textrm{I}}=\infty)$. [**(b)**]{} $\tau_{\textrm{I}}^{\textrm{c}}$ as a function of $c_{\textrm{I}}$. For $c_{\textrm{I}}\leq 1$, $\tau_{\textrm{I}}^{\textrm{c}}$ quickly reaches its upper bound; for $c_{\textrm{I}}>1$, the convergence is somewhat delayed. [**(c)**]{} $n_{\textrm{i}}(\tau_{\textrm{I}}=\infty)$ as a function of $c_{\textrm{I}}$. Increasing $c_{\textrm{I}}$ facilitates control, until $n_{\textrm{i}}$ reaches its lower bound. [**(d)**]{} $\tau_{\textrm{I}}^{\textrm{c}}$ as a function of $c_{\textrm{II}}$ with fix $c_{\textrm{I}}=4.0$. The peak of $\tau_{\textrm{I}}^{\textrm{c}}$ corresponds to the critical point where the giant strongly connected component in Layer II emerges. [**(e)**]{} $n_{\textrm{i}}(\tau_{\textrm{I}}=\infty)$ as a function of $c_{\textrm{II}}$ with fix $c_{\textrm{I}}=4.0$. For $c_{\textrm{II}}<1$, Layer II does not contain cycles, therefore $n_{\textrm{i}}(\tau_{\textrm{I}}=\infty)=1$; for large $c_{\textrm{II}}$, Layer II can be completely covered with cycles, and $n_{\textrm{i}}(\tau_{\textrm{I}}=\infty)$ is determined by $n_{\textrm{i}}^{\textrm{I}}$. For (b-e), each data point is the average over $10$ randomly generated networks with $N=10,000$ and error bars represent the standard deviation.[]{data-label="fig:secIIIC"}](figure_secIIIC)
![[**Layer II updates faster – Network size effects.**]{} Critical timescale parameter for ER-ER networks and SF-SF networks with varying network size $N$. [**(a)**]{} Layer II has no giant strongly connected component ($c_{\textrm{II}}=0.5<1$), $l_{\textrm{max}}$ equals the diameter $D$ of Layer II which scales as $D\sim \log N$ for ER networks, and the diameter of SF networks is smaller than the diameter of ER networks with the same average degree. The fact that $l_{\textrm{max}}+1\geq\tau_{\textrm{I}}^{\textrm{c}}$ suggest that $\tau_{\textrm{I}}^{\textrm{c}}\sim\log(N)$. [**(b)**]{} At the critical point $c_{\textrm{II}}=1.0$ the diameter of ER networks scales as $D\sim N^{1/3}$, suggesting that $\tau_{\textrm{I}}^{\textrm{c}}$ scales as a powerlaw of $N$. [**(c)**]{} Above the critical point ($c_{\textrm{II}}=4.0>1$) there is no direct connection between $D$ and $\tau_{\textrm{I}}^{\textrm{c}}$, nonetheless observations suggest $\tau_{\textrm{I}}^{\textrm{c}}\sim \log N$. In contrast with the $c_{\textrm{II}}\leq 1$ case, $\tau_{\textrm{I}}^{\textrm{c}}$ increases more rapidly for SF-SF networks than for ER-ER networks. Each data point is the average over $100$ randomly generated networks with $c_{\textrm{I}}=4.0$ and error bars represent the standard deviation.[]{data-label="fig:secIIIC_N"}](figure_secIIIC_N)
![$l_{\textrm{max}}$[ **– Example 1.**]{} [**(a)**]{} A directed network with tree structure; therefore not containing cycles. The diameter $D=2$ is the length of the longest path. [**(b)**]{} We count the maximum number of disjoint control paths $N_{\textrm{path}}(l)$ which connect nodes at time step $0$ with nodes at time step $l$. We find that $l^\prime=3$ is the smallest value of $l$ such that $N_{\textrm{path}}(l)=N_{\textrm{cycle}}=0$; therefore $l_{\textrm{max}}=2$. There are no cycles; therefore $l_{\textrm{max}}=D$.[]{data-label="app:fig:lmax-example-1"}](app-lmax-example-1)
![$l_{\textrm{max}}$[ **– Example 2.**]{} [**(a)**]{} A directed network containing a cycle. The size of the maximum cycle cover is $N_{\textrm{cycle}}=1$. [**(b)**]{} We count the maximum number of disjoint control paths $N_{\textrm{path}}(l)$ which connect nodes at time step $0$ with nodes at time step $l$. We find that $l^\prime=2$ is the smallest value of $l$ such that $N_{\textrm{path}}(l)=N_{\textrm{cycle}}=1$; therefore $l_{\textrm{max}}=1$. $N_{\textrm{path}}(l)$ remains non-zero for $l>l_{\textrm{max}}$, showing that cycles can support control paths of any length.[]{data-label="app:fig:lmax-example-2"}](app-lmax-example-2)
|
---
abstract: 'We discuss the effect of small perturbation on nodeless solutions of the nonlinear [Schr[ö]{}dinger]{} equation in 1+1 dimensions in an external complex potential derivable from a parity-time symmetric superpotential that was considered earlier \[Phys. Rev. E 92, 042901 (2015)\]. In particular we consider the nonlinear partial differential equation $\{ \, {{\rm i}}\, \partial_t + \partial_x^2 + g |\psi(x,t)|^2 - V^{+}(x) \, \} \, \psi(x,t) = 0$, where $V^{+}(x) = \qty( -b^2 - m^2 + 1/4 ) \, \sech^2(x) - 2 i \, m \, b \, \sech(x) \, \tanh(x)$ represents the complex potential. Here we study the perturbations as a function of $b$ and $m$ using a variational approximation based on a dissipation functional formalism. We compare the result of this variational approach with direct numerical simulation of the equations. We find that the variational approximation works quite well at small and moderate values of the parameter $b m$ which controls the strength of the imaginary part of the potential. We also show that the dissipation functional formalism is equivalent to the generalized traveling wave method for this type of dissipation.'
author:
- Fred Cooper
- 'John F. Dawson'
- 'Franz G. Mertens'
- Edward Arévalo
- 'Niurka R. Quintero'
- Bogdan Mihaila
- Avinash Khare
- Avadh Saxena
bibliography:
- 'johns.bib'
date: ', EDT'
title: 'Response of exact solutions of the nonlinear [Schr[ö]{}dinger]{} equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry '
---
\[s:Intro\]Introduction
=======================
The topic of balanced loss and gain or parity-time (${\mathcal{PT}}$) symmetry and its relevance for physical applications on the one hand, as well as its mathematical structure on the other, have drawn considerable attention from both the physics and the mathematics community. The original proposal of Bender and his collaborators [@r:Bender:2007nr; @0305-4470-39-32-E01; @1751-8121-41-24-240301; @1751-8121-45-44-440301] towards the study of such systems was made as an alternative to the postulate of Hermiticity in quantum mechanics. Keeping in perspective the formal similarity of the [Schr[ö]{}dinger]{} equation with Maxwell’s equations in the paraxial approximation, it was realized that such ${\mathcal{PT}}$ invariant systems can in fact be experimentally realized in optics [@Makris2011; @0305-4470-38-9-L03; @PhysRevLett.100.103904; @PhysRevLett.101.080402; @PhysRevLett.103.123601; @PhysRevB.80.235102; @PhysRevA.81.022102; @r:Ruter:2010mz; @PhysRevLett.103.093902; @r:Regensburger:2012gf]. Subsequently, these efforts motivated experiments in several other areas including ${\mathcal{PT}}$ invariant electronic circuits [@PhysRevA.84.040101; @1751-8121-45-44-444029], mechanical circuits [@r:Bender:2013ly], and whispering-gallery microcavities [@r:Peng:2014ul].
Concurrently, the notion of supersymmetry (SUSY) originally espoused in high-energy physics has also been realized in optics [@PhysRevLett.110.233902; @r:Heinrich:2014qf]. The key idea is that from a given potential one can obtain a SUSY partner potential with both potentials possessing the same spectrum, except possibly for one eigenvalue [@0038-5670-28-8-R01; @r:CooperKhareSukhatmePR]. Therefore, an interplay of SUSY with ${\mathcal{PT}}$ symmetry is expected to be quite rich and is indeed very useful in achieving transparent as well as one-way reflectionless complex optical potentials [@0305-4470-33-1-101; @doi:10.1142/S0217751X01004153; @Bagchi2000285; @Ahmed2001343; @PhysRevA.89.032116].
A previous paper [@PhysRevE.92.042901] explored the interplay between ${\mathcal{PT}}$ symmetry, SUSY and nonlinearity. That paper derived exact solutions of the general nonlinear [Schr[ö]{}dinger]{} (NLS) equation in 1+1 dimensions when in a ${\mathcal{PT}}$-symmetric complex potential [@0038-5670-28-8-R01; @r:CooperKhareSukhatmeBOOK]. In particular, they considered the nonlinear partial differential equation $$\label{eqn1}
\qty{
i \,
\partial_t
+
\partial_x^2
-
V^\pm(x)
+
g | \psi(x,t) |^{2\kappa}
} \, \psi(x,t)
=
0 \>,$$ for arbitrary nonlinearity parameter $\kappa$, with $$\label{eqn1x}
V^{\pm}(x)
=
W_1^2(x)\mp W_1'(x) - (m-1/2)^2 \>,$$ and the partner potentials arise from the superpotential $$\label{eqn1y}
W_1(x)
=
\qty( m - 1/2 ) \, \tanh{x}
-
i b \, \sech{x} \>,$$ giving rise to
\[VpVm\] $$\begin{aligned}
V^{+}(x)
&=
\qty( -b^2 - m^2 + 1/4 ) \, \sech^2(x)
\label{eqn7a} \\
& \quad
-
2 i \, m \, b \, \sech(x) \, \tanh(x),
\notag \\
V^{-}(x)
&=
\qty( - b^2 - (m-1)^2 + 1/4 ) \, \sech^2(x)
\label{eqn8a} \\
& \quad
-
2 i \, \qty( m - 1 ) \, b \, \sech(x) \, \tanh(x) \>.
\notag\end{aligned}$$
For $m=1$, the *complex* potential $V^{+}(x)$ has the same spectrum, apart from the ground state, as the *real* potential $V^{-}(x)$ and this fact was used in the numerical study of the stability of the bound state solutions of the NLS equation in the presence of $V^{+}(x)$ (see Ref. [@PhysRevE.92.042901]). In a recent complementary study [@Cooper:2017aa] of this system of nonlinear [Schr[ö]{}dinger]{} equations in ${\mathcal{PT}}$ symmetric SUSY external potentials, the stability properties of the bound state solutions of NLS equation in the presence of the external real SUSY partner potential $V^{-}(x)$ were investigated. The stability regime of these solutions, which depended on the parameters $(b, \kappa)$, was compared to the stability regime of the related *solitary* wave solutions to the NLS equation in the absence of the external potential. Because the NLS equation in the presence of $V^{-}(x)$ is a Hamiltonian dynamical system, in Ref. [@Cooper:2017aa] they were able to use several variational methods to study the stability of the solutions when they undergo certain small deformations, and showed that these variational methods agreed with a linear stability analysis based on the Vakhitov-Kolokolov (V-K) stability criterion [@r:Comech:2012uk; @Vakhitov:1973aa] as well as numerical simulations that have recently been performed.
In Ref. [@PhysRevE.92.042901] we determined the exact solutions of the equation for $m=1$ for $V^{+}(x)$, which was complex. We studied numerically the stability properties of these solutions using linear stability analysis. We found some unusual results for the stability which depended on the value of $b$. What was found for $m=1$ (and $\kappa=1$) was that the eigenvalues of the linear stability matrix became complex for $0.56 < b < 1.37$.
At that time we had not yet formulated a variational approach for deriving the NLS equation in the presence of complex potentials. Recently we have developed such an approach and have applied it to the response of the solutions of the NLS equation to weak external complex periodic potentials. Using four variational parameters we were able to successfully predict the time evolution of these solitary waves when compared to direct numerical simulation of the NLS equation in the presence of these complex potentials [@PhysRevE.94.032213]. Given this new tool we would like to return to the original problem of the stability of the exact solutions found in Ref. [@PhysRevE.92.042901] and see how well this variational approach agrees with numerical simulations as a function of the strength of the dissipative part of the potential which is proportional to $bm$. In this paper we focus on the external potential $V^{+}(x)$ which is symmetric in $b \leftrightarrow m$.
Here we will compare the numerical simulations with the results of our collective coordinate (CC) approximation. We will also look at the linear stability analysis that arises from studying the linearization of the CC ordinary differential equations (ODEs). For the case of a real external potential, studying the eigenvalues of this reduced stability analysis predicted the correct stability regime [@PhysRevE.85.046607].
This paper is structured as follows. In Sec. \[s:SSModel\] we review the non-hermitian SUSY model that we studied in Ref. [@PhysRevE.92.042901] and add the self-interactions of the NLS equation to the linear model. In Sec. \[s:general\] we give some of the exact low order moment equations for this problem. In Sec. \[s:CCs\] we introduce our collective coordinate approach, whereas in Sec. \[s:FourPar\] we use a four parameter trial wave function that we considered in an earlier study of soliton behavior in complex periodic external potentials, and derive equations for the four CC’s. In Sec. \[s:6cc\] we expand the number of CC’s to six and derive equations for the six CC’s. In Sec. \[s:LinearRes\] we study the linear response theory of the six CC approximation. In Sec. \[s:Numerical\] we present our numerical strategy for solving the NLS equation starting from a perturbed exact solution. In Sec. \[s:results\] we compare the four and six CC approximations with direct numerical simulations. In Sec. \[s:conclusions\] we present our main conclusions. Finally in Appendix A we provide the definitions of various integrals and in Appendix B we show that for this problem our variational approach is equivalent to the generalized traveling wave method [@PhysRevE.82.016606].
\[s:SSModel\]NLS equation in the presence of a Non-Hermitian Supersymmetric external potential
==============================================================================================
We were interested in studying the NLS equation in the presence of a complex external potential and were intrigued by the fact that as a result of $\mathcal{PT}$ symmetry, there existed complex potentials whose SUSY partners were real and had explicitly known spectra of bound states. This led us to study the external potential defined by the $\mathcal{PT}$ symmetric SUSY superpotential $W_1(x)$ given by Eq. . This superpotential gives rise to supersymmetric partner potentials given by Eqs. . For the case $m=1$, $V^{-}(x)$ is the well known P[ö]{}schl-Teller potential [@r:Poschl:1933ek; @r:Landau:1989jt]. The relevant bound state eigenvalues assume an extremely simple form as $$\label{eqn9}
E_n^{(-)}
=
-\frac{1}{4} \, \qty[ \, 2 b - 2 n - 1 \, ]^2 \>.$$ Such bound state eigenvalues only exist when $n < b - 1/2$. We notice that for the ground state (n=0) to exist requires $b > 1/2$. The existence of a first excited state (n=1) requires $b > 3/2$. Here we consider the general $V^{+}(x)$ arising from the superpotential $W_1(x)$ depending on $m,b$ as an external potential modifying the nonlinear [Schr[ö]{}dinger]{} equation. Rewriting the external potential given in Eq. as, $$\label{e:Vp}
V^{+}(x)
=
V_1(x) + {{\rm i}}\, V_2(x) \>,$$ we have
\[e:VV\] $$\begin{aligned}
V_1(x)
&=
- ( \, b^2 + m^2 - 1/4 ) \, \sech^2(x) \>,
\label{e:VVa} \\
V_2(x)
&=
- 2 \, m b \tanh(x) \sech(x) \>.
\label{e:VVb}\end{aligned}$$
Note this potential is invariant under the exchange of $b$ and $m$. We are interested in the stability properties of the exact solutions of the NLS equation in this external potential: $$\label{e:NLS equation-1}
\{
i \,
\partial_t
+
\partial_x^2
+
g | \psi(x,t) |^{2 \kappa}
-
[\, V_1(x) + i V_2(x) \,] \,
\} \, \psi(x,t)
=
0 \>.$$ This equation can be obtained from a generalized Euler-Lagrange equation using a dissipation functional [@PhysRevE.94.032213], $$\label{e:euler-lagrange}
\frac{\delta \Gamma}{\delta \psi^{\ast}}
=
- \frac{\delta {\mathcal{F}}}{\delta \psi_t^{\ast}} \>,$$ where
\[e:defGF\] $$\begin{aligned}
\Gamma
&=
{\!\int\!}\dd{t}
\Bigl \{ \,
\frac{{{\rm i}}}{2} {\!\int\!}\dd{x}
[\, \psi^{\ast} \psi_t - \psi \psi_t^{\ast} \,]
-
H \,
\Bigr \} \>,
\label{e:defGF-a} \\
H
&=
{\!\int\!}\dd{x}
\Bigl \{ \,
| \psi_x |^2
-
\frac{g \, | \psi |^{2\kappa +2}}{\kappa + 1}
+
V_1(x) \, | \psi |^2 \,
\Bigr \} \>,
\label{e:defGF-b} \\
{\mathcal{F}}
&=
{\!\int\!}\dd{t} F
=
{{\rm i}}{\!\int\!}\dd{x} \dd{t} V_2(x) \,
[\, \psi_t \, \psi^{\ast} - \psi_t^{\ast} \, \psi \,] \>.
\label{e:defGF-c}\end{aligned}$$
Localized solutions to Eq. exist for arbitrary values of $\kappa, m, b$. Here we use $\psi_0(x,t)$ to denote the exact solution to the NLS equation in the external potential, $$\label{psi0wf}
\psi_0(x,t)
=
A_0 \, \sech^{1/\kappa}(x)\, {{\rm e}}^{{{\rm i}}[\, Et + \phi(x)\,] } \,$$ where $$\phi(x)
=
\frac{4 b m \kappa}{\kappa + 2} \,
\tan^{-1}[\tanh(x/2)\,] \>,$$ with $E = 1/\kappa^2$, and $$\label{gasq}
g A_0^{2 \kappa}
=
\frac{[4 b^2 \kappa^2-(\kappa + 2)^2]\, [4 m^2 \kappa^2 -(\kappa+2)^2]}
{4 \kappa^2 (\kappa+2)^2} \>.$$ We notice when $mb=0$ the potential is real and that solutions exist for $m^2+ b^2-1/4 < (\kappa+1)/\kappa^2$. There are two regimes where $A_0^2$ is positive and so a solution exists when $m \neq 0$. This form of the solution reflects the fact that the potential $V^+$ is invariant under the interchange $m \leftrightarrow b$.
In a previous paper [@r:Dawson:2017td] we studied the stability of these solutions for $m=0$ (real external potential) and for arbitrary $\kappa$. In that paper, we also considered two other cases where exact solutions exist. For the case of $g = -1$ and attractive potential, for $V_2=0$, all the solutions that were allowed were stable. Solutions also exist for $V_2 \neq 0$ and are given by Eqs. and with $g =-1$. For $g=1$ and a repulsive real potential we found the solutions for $V_2=0$ were translationally unstable. Solutions again exist when $V_2 \neq 0$ for this case. We will not discuss these solutions further here.
Here we will confine ourselves to $\kappa=1, g=1$ and an attractive external potential $V_1$ and study the domain of applicability of the variational methods we have developed previously to the case of increasing the dissipation by allowing $m$ to vary. In particular for the case we will concentrate on here ($\kappa=1$) we have that $$\label{gAsq}
g A_0^{2} = (4 b^2 -9)(4m^2 -9) / 36 \>,$$ so that when $m^2 < 9/4$ we need that $b^2 < 9/4$ for there to be a solution. Also if we confine ourselves to an attractive potential so that we avoid the known translational instability associated with repulsive potentials [@r:Dawson:2017td], then we also require $b^2 + m^2 > 1/4$. Note that $g A^2_0$ is independent of $g$. For $\kappa=1$ we have
\[e:NLS equation-4\] $$\begin{aligned}
\phi(x)
&=
( 4 \, m \, b/3 ) \, \tan^{-1}[\, \tanh(x/2) \,] \>,
\label{e:NLS equation-4a} \\
\partial_x \phi(x)
&=
(2/3) \, m \, b \, \sech(x) \>.
\label{e:NLS equation-4b}\end{aligned}$$
\[s:general\]Some General Properties of the NLS equation in complex potentials
==============================================================================
We are interested in solitary wave solutions that approach zero exponentially at $\pm \infty$. For these solutions we define the mass density $\rho(x,t) = \abs{\psi(x,t)}^2$, and the mass or norm $M(t)$ as $$\label{eqMass1}
M(t)
=
{\!\int\!}\dd{x} \rho(x,t)
=
{\!\int\!}\dd{x} \abs{\psi(x,t)}^2 \>.$$ In addition, we define the current as: $$j(x,t)
= i \,
[\,
\psi(x,t) \, \psi_x^\ast(x,t)
-
\psi^{\ast}(x,t) \, \psi_x(x,t) \,
] \>.$$ Multiplying the NLS equation by $\psi^{\ast}(x,t)$ and subtracting the complex conjugate of the resulting equation, we obtain $$\label{cont}
\pdv{\rho(x,t)}{t}
+
\pdv{j(x,t)}{x}
=
2 V_2(x) \, \rho(x,t).$$ Integrating over space, and assuming that $j(+\infty,t) - j(-\infty,t) = 0$, we find $$\label{mdot}
\dv{M(t)}{t}
=
2 {\!\int\!}\dd{x} V_2(x) \, \rho (x,t) \>.$$ Note that $M$ is conserved when $V_2(x)=0$. If we instead multiply the NLS equation by $\psi^\ast$ and add the complex conjugate of the resulting equation, we get $$\begin{aligned}
\label{add}
& {{\rm i}}\, (\, \psi^\ast \psi_t - \psi \psi^\ast_t \,)
\\
& \qquad
=
-
2 g \rho^2
-
\psi^\ast \psi_{xx}
-
\psi \psi^\ast_{xx}
+
2 V_1(x) \, \rho \>,
\notag\end{aligned}$$ which when we integrate over space, leads to the virial theorem: $$\begin{aligned}
& \frac{{{\rm i}}}{2}
{\!\int\!}\dd{x}
(\, \psi^{\ast} \psi_{t} - \psi_t^\ast \psi \,)
-
{\!\int\!}\dd{x}
\qty [\,
\abs{\psi_x}^2
-
g \, \abs{\psi}^{4} \,
]
\\
& \qquad\qquad
=
{\!\int\!}\dd{x} V_1(x) \, \abs{\psi}^2 \>.
\notag\end{aligned}$$ The average position $q(t)$ can be defined through the first moment of $x$ as follows: $$M_1(t)
=
{\!\int\!}\dd{x} x \, \rho(x,t) = q(t) M(t) \>.$$ Multiplying the continuity equation by $x$ and integrating over all space we find: $$\dv{M_1}{t}
=
2 \, P(t)
+
2 {\!\int\!}\dd{x} \,
x \, V_2(x) \, \rho(x,t) \>,$$ where the momentum $$\begin{aligned}
P(t)
&=
\frac{1}{2} {\!\int\!}\dd{x} j(x,t)
\\
&=
\frac{{{\rm i}}}{2}
{\!\int\!}\dd{x}
\qty[\,
\psi^\ast(x,t) \, \psi_x(x,t)
-
\psi^{\ast}_x(x,t) \psi(x,t) \,] \>.
\notag\end{aligned}$$ Here, we assumed that $$\label{e:assumedlimits}
\lim_{x\rightarrow \infty} x j(x,t)
-
\lim_{x\rightarrow -\infty} x j(x,t)
= 0 \>.$$ Assuming that the density is a function of $y = x - q(t)$ and $t$, we find $$\begin{aligned}
\dv{t} \, \qty[\, M(t) \,q(t) \,]
&=
2 \, P(t)
+
2 {\!\int\!}\dd{y} y \, V_2(y+q(t)) \, \rho(y,t)
\\
& \qquad
+
2 \, q(t) {\!\int\!}\dd{x} \, V_2(x) \, \rho(x-q(t),t) \>.\end{aligned}$$ We recognize the last term as $q(t) \dd{M(t)}/\dd{t}$, so that we finally have: $$\label{dotq1}
M(t) \,
\dv{q(t)}{t}
=
2 \, P(t)
+
2 {\!\int\!}\dd{y} y \, V_2(y+q(t)) \, \rho(y,t) \>.$$ Taking the time derivative of the momentum $P (t)$, using the equations of motion for $\psi$ and $\psi^{\ast}$, and integrating by parts, we find $$\label{pdot}
\dv{P(t)}{t}
=
-
{\!\int\!}\dd{x} \rho (x,t) \pdv{V_1(x)}{x}
+
{\!\int\!}\dd{x} j(x,t ) \, V_2(x) \>.$$ Here $$\pdv{V_1(x)}{x}
=
2 \, \qty( \, b^2 + m^2 - 1/4 \,) \,
\tanh(x) \, \sech^2(x) \>.$$ Note that in our case $V_1(x) $ is an even function of $x$ and $V_2(x)$ is an odd function. In our study we will assume $\rho(x,t) = \tilde \rho (y, t)$ where $y(t) = x-q(t)$. That is, the functional form of $\rho$ will be maintained if it is given a slight perturbation away from the origin. If it stays at the origin ($q(t) =0$) and only changes its width and amplitude under perturbation, then we see that since $\rho$ is an even function of $y$ and $V_2(x)$ is an odd function of $x$, the mass is conserved. One can in a systematic fashion obtain the equations for the higher moments of $ \langle x^n \, \hat{p}^m \rangle $, where $\hat{p} = -i \partial/\partial x$. It can be demonstrated that the four and six collective coordinate approximations we derive in this paper will satisfy a particular subset of four or six moment equations [@PhysRevE.82.016606].
\[s:CCs\]Collective coordinates
===============================
The time dependent variational approximation relies on introducing a finite set of time-dependent real parameters in a trial wave function that one hopes captures the time evolution of a perturbed solution. By doing this one obtains a simplified set of ordinary differential equations for the collective coordinates in place of solving the full partial differential equation for the NLS equation. By judiciously choosing the collective coordinates, they can be simply related to the moments of $x$ and ${\hat p}= -i \partial/\partial x $ averaged over the density $\rho(x,t)$.
That is, we set $$\begin{aligned}
\label{e:VT-1}
\psi(x,t)
&\mapsto
\tilde{\psi}[\,x,Q(t)\,]
\\
Q(t)
&=
\{\, Q^1(t),Q^2(t),\dotsc,Q^{2n}(t) \,\} \in \mathbb{R}^{2n} \>.
\notag\end{aligned}$$ The success of the method depends greatly on the choice of the the trial wave function $\tilde{\psi}[\,x,Q(t)\,]$. The generalized Euler-Lagrange equations lead to Hamilton’s equations for the collective coordinates $Q(t)$. Introducing the notation $\partial_{\mu} \equiv \partial / \partial Q^{\mu}$, the Lagrangian in terms of the collective coordinates is given by $$\label{e:VT-2}
L(\,Q,\dot{Q}\,)
=
\pi_\mu(Q) \, \dot{Q}^\mu - H(\,Q\,) \>,$$ where $\pi_\mu(Q)$ is defined by $$\begin{aligned}
\label{e:VT-3}
\pi_\mu(Q)
&=
\frac{{{\rm i}}}{2} {\!\int\!}\dd{x}
\{ \,
{\tilde{\psi}}^{\ast}(x,Q)\,[\, \partial_\mu {\tilde{\psi}}(x,Q) \,]
\\
& \qquad\qquad
-
[\, \partial_\mu {\tilde{\psi}}^{\ast}(x,Q) \,] \, {\tilde{\psi}}(x,Q)
\,\} \>,
\notag\end{aligned}$$ and $H(Q)$ is given by $$\begin{aligned}
\label{e:VT-4}
H(Q)
&=
{\!\int\!}\dd{x}
\Bigl \{ \,
|\partial_x {\tilde{\psi}}(x,Q) |^2
-
\frac{g}{2} \, |{\tilde{\psi}}(x,Q)|^{4}
\\
& \qquad\qquad
+
V_1(x) \, |{\tilde{\psi}}(x,Q)|^2 \,
\Bigr \} \>.
\notag \end{aligned}$$ Similarly, in terms of the collective coordinates, the dissipation functional is given by $$\label{e:VT-4.1}
F[Q,\dot{Q}]
=
w_{\mu}(Q) \, \dot{Q}^{\mu} \>,$$ where $$\begin{aligned}
\label{e:VT-4.2}
w_{\mu}(Q)
&=
{{\rm i}}{\!\int\!}\dd{x} V_2(x) \,
\{ \,
{\tilde{\psi}}^{\ast}(x,Q)\,[\, \partial_\mu {\tilde{\psi}}(x,Q) \,]
\\
& \qquad\qquad
-
[\, \partial_\mu {\tilde{\psi}}^{\ast}(x,Q) \,] \, {\tilde{\psi}}(x,Q)
\,\} \>.
\notag\end{aligned}$$ The generalized Euler-Lagrange equations are $$\label{e:VT-5}
\pdv{L}{Q^\mu}
-
\dv{t} \Bigl ( \pdv{L}{\dot{Q}^\mu} \Bigr )
=
-
\pdv{F}{\dot{Q}^\mu} \>.$$ Setting $v_{\mu}(Q) = \partial_\mu H(Q)$, we find $$\label{e:VT-6}
f_{\mu\nu}(Q) \, \dot{Q}^\nu
=
u_{\mu}(Q)
=
v_{\mu}(Q) - w_{\mu}(Q) \,$$ where $$\label{e:VT-7}
f_{\mu\nu}(Q)
=
\partial_\mu \pi_\nu(Q) - \partial_\nu \pi_\mu(Q)$$ is an antisymmetric $2n \times 2n$ symplectic matrix. If $\det{f(Q)} \ne 0$, we can define an inverse as the contra-variant matrix with upper indices, $$\label{e:VT-8}
f^{\mu\nu}(Q) \, f_{\nu\sigma}(Q) = \delta^\mu_\sigma \>,$$ in which case the equations of motion can be put in the symplectic form: $$\label{e:VT-9}
\dot{Q}^\mu
=
f^{\mu\nu}(Q) \, u_{\nu}(Q) \>.$$ Poisson brackets are defined using $f^{\mu\nu}(Q)$. If $A(Q)$ and $B(Q)$ are functions of $Q$, Poisson brackets are defined by $$\label{e:PB-1}
{\ensuremath{ \lbrace \,A(Q),B(Q)\, \rbrace }}
=
( \partial_\mu A(Q) ) \, f^{\mu\nu}(Q) \, ( \partial_\nu B(Q) ) \>.$$ In particular, $$\label{e:PB-2}
{\ensuremath{ \lbrace \,Q^\mu,Q^\nu\, \rbrace }}
=
f^{\mu\nu}(Q) \>.$$ It is easy to show that $f_{\mu\nu}(x)$ satisfies Bianchi’s identity. This means that definition satisfies Jacobi’s identity, as required for symplectic variables. The rate of energy loss is expressed as $$\label{e:EC-1}
\dv{H(Q)}{t}
=
- v_\mu(Q) \, f^{\mu\nu}(Q) \, w_{\nu}(Q) \>,
\notag$$ since $f^{\mu\nu}(Q)$ is an antisymmetric tensor.
\[s:FourPar\]Four parameter trial wave function
===============================================
Let us first look at the four parameter trial wave function that we have successfully used to study the effect of weak complex external potentials on the exact solution of the NLS equation in the absence of that potential. That is we will choose: $$\label{e:T4-1}
{\tilde{\psi}}(x,t)
=
A_0 \beta(t) \, \sech[\, \beta(t) \, y(x,t) \,] \,
{{\rm e}}^{{{\rm i}}\, {{\tilde \phi}}(x,t)} \>,$$ where $A_0$ is the amplitude of the exact solution in the presence of the external potential and is a funcition of $m,b,g$, and $$\label{e:T4-2}
{{\tilde \phi}}(x,t)
=
-
\theta(t)
+
p(t) \, y(x,t)
+
\phi(x) \>.$$ Here $\phi(x)$ is given by Eq. and we have put $y(x,t) = x - q(t)$. The four variational parameters are labeled by $$\label{e:T-2.1}
Q^{\mu}
=
\qty{\, q(t), p(t), \beta(t) , \theta(t) \,} \>.$$ The derivatives of ${\tilde{\psi}}(x,t)$ with respect to $t$ and $x$ are given by
\[e:T-3\] $$\begin{aligned}
\label{e:T-3a}
&{\tilde{\psi}}_t(x,t)
=
A_0 \,
\{\,
\dot{\beta} \sech( \beta y )
\\
& \qquad\qquad
-
\beta \sech(\beta y) \, \tanh(\beta y) \,
[\, \dot{\beta} y - \dot{q} \beta \,]
\notag\\
& \qquad
{}+
{{\rm i}}\, \beta \sech( \beta \, y \,) \,
[\, - \dot{\theta} + \dot{p} \, y - p \, \dot{q} \,] \,
\} \, {{\rm e}}^{{{\rm i}}\, {{\tilde \phi}}(x,t)} \>,
\notag \\
&{\tilde{\psi}}_x(x,t)
=
A_0 \, \beta \,
\{\,
-
\beta \, \sech(\beta y) \, \tanh(\beta y) \,
\label{e:T-3b} \\
& \qquad
+
{{\rm i}}\, \sech(\beta y) \,
[\, p + (2/3) \, m \, b \, \sech(x) \,] \,
\} \, {{\rm e}}^{{{\rm i}}\, {{\tilde \phi}}(x,t)} \>,
\notag\end{aligned}$$
where we have used . Then the density and current is given by
\[e:T-4\] $$\begin{aligned}
\rho(x,t)
&=
A_0^2 \, \beta^2 \sech^2(\beta y) \>,
\label{e:T-6a} \\
j(x,t)
&=
2 \, \rho(x,t) \,
[\,
p
+
(2/3) \, m \, b \sech(x)\,
] \>.
\label{e:T-6b}\end{aligned}$$
The time dependent mass, $M(t)$ which is a normalization factor, is given by $$\label{e:T-4.1}
M(t)
=
{\!\int\!}\dd{x} \rho(x,t)
=
2 \, A_0^2 \, \beta(t) \>,$$ and the Lagrangian and dissipation function are given by,
\[e:T-7\] $$\begin{aligned}
L
&=
\frac{{{\rm i}}}{2} {\!\int\!}\dd{x}
\qty[\, \psi^{\ast}\,\psi_t - \psi^{\ast}_t\,\psi \,]
-
H[\,\psi,\psi^{\ast}\,] \>,
\label{e:T-7a} \\
H
&=
{\!\int\!}\dd{x}
[\,
\abs{ \psi_x }^2
-
g \, \abs{ \psi }^4 / 2
+
V_1(x) \, \abs{ \psi }^2 \,] \>,
\label{e:T-7b} \\
F
&=
{{\rm i}}{\!\int\!}\dd{x} V_2(x) \,
\qty[\, \psi^{\ast} \, \psi_t - \psi^{\ast}_t \, \psi \,] \>.
\label{e:T-7c}\end{aligned}$$
The generalized Euler-Lagrange equations are
\[e:T-8\] $$\begin{aligned}
\fdv{L}{\psi^{\ast}}
-
\partial_t \fdv{L}{\psi_t^{\ast}}
&=
- \fdv{F}{\psi_t^{\ast}} \>,
\label{e:Av-3a} \\
\fdv{L}{\psi^{\phantom\ast}}
-
\partial_t \fdv{L}{\psi_t^{\phantom\ast}}
&=
- \fdv{F}{\psi_t^{\phantom\ast}} \>.
\label{e:Av-3b} \end{aligned}$$
For the trial wave function of Eq. , we find $$\begin{aligned}
\label{e:T-9}
L_0[Q]
&\equiv
\frac{{{\rm i}}}{2} {\!\int\!}\dd{x}
[\, {\tilde{\psi}}^{\ast}\,{\tilde{\psi}}_t - {\tilde{\psi}}^{\ast}_t\,{\tilde{\psi}}\,]
\\
&=
2 A_0^2 \, \beta \,
(\, \dot{\theta} + p \, \dot{q} \,)
\equiv
\pi_\mu(Q) \, \dot{Q}^\mu \>,
\notag\end{aligned}$$ where $$\label{e:T-10}
\pi_{q} = 2 A_0^2 \, \beta \, p
\qc
\pi_{p} = 0
\qc
\pi_{\beta} = 0
\qc
\pi_{\theta} = 2 A_0^2 \, \beta \>.$$ The only partial derivatives of $\pi_\mu(Q)$ that survive are: $$\label{e:T-11}
\partial_p \pi_{q} = 2 A_0^2 \, \beta
\qc
\partial_{\beta} \pi_{q} = 2 A_0^2 \, p
\qc
\partial_{\beta} \pi_{\theta} = 2 A_0^2 \>.$$ So the symplectic matrix and its inverse are given by $$\begin{aligned}
\label{e:T-12}
f_{\mu\nu}(Q)
&=
2 A_0^2
\begin{pmatrix}
0 & -\beta & -p & 0 \\
\beta & 0 & 0 & 0 \\
p & 0 & 0 & 1 \\
0 & 0 & -1 & 0
\end{pmatrix} \>,
\\
f^{\mu\nu}(Q)
&=
\frac{1}{2 A_0^2 \, \beta}
\begin{pmatrix}
0 & 1 & 0 & 0 \\
-1 & 0 & 0 & p \\
0 & 0 & 0 & -\beta \\
0 & -p & \beta & 0
\end{pmatrix} \>.
\notag\end{aligned}$$ From the Hamiltonian and our choice of trial wave function we find that $$\begin{aligned}
&H(Q)
=
A_0^2 \, \beta \,
\{\,
(2/3) \, \beta^2
+
2 \, p^2
+
(4/3) \, p \, m \, b \, \beta \, I_1(\beta,q)
\notag \\
&
-
[\, b^2 + m^2 - (4/9) \, m^2 b^2 - 1/4 \,] \, \beta \, I_2(\beta,q)
\,\}
\label{e:T-16} \\
& \qquad\qquad
-
(2/3) \, g \, A_0^4 \, \beta^3 \>,
\notag\end{aligned}$$ where $I_1(\beta,q)$ and $I_2(\beta,q)$ are given in Appendix \[s:integrals\]. Then defining $v_{\mu} = \partial_{\mu} H(Q)$, we find
\[e:T-18\] $$\begin{aligned}
v_q
&=
- A_0^2 \, \beta \,
[\,
(4/3) \, p \, m \, b \, \beta \, f_1(\beta,q)
\label{e:T-18a} \\
& \qquad
-
2 \, [\, b^2 + m^2 - (4/9) \, m^2 b^2 - 1/4 \,] \,
\beta \, f_{6}(\beta,q) \,
] \>,
\notag \\
v_p
&=
A_0^2 \, \beta \,
[\, 4 p + (4/3) \, m \, b \, \beta \, I_1(\beta,q) \,] \>,
\label{e:T-18b} \\
v_\beta
&=
A_0^2 \, \beta \,
\{\,
2 \, \beta
+
2 \, p^2 / \beta
\label{e:T-18c} \\
& \qquad
+
(8/3) \, p \, m \, b \,
[\, I_1(\beta,q) - \beta \, f_{10}(\beta,q) \,]
\notag \\
& \qquad
-
2 \, [\, b^2 + m^2 - (4/9) \, m^2 b^2 - 1/4 \,]
\notag \\
& \qquad
\times
[\, I_2(\beta,q) - \beta \, f_7(\beta,q) \,]
-
2 \, g \, A_0^2 \, \beta \,
\} \,
\notag \\
v_\theta
&=
0 \>,
\label{e:T-18d} \end{aligned}$$
where the $f_i(\beta,q)$ are given in Appendix \[s:integrals\]. From , the dissipation function is given by $$\begin{aligned}
\label{e:T-19}
F[Q,\dot{Q}]
=
w_\mu(Q) \, \dot{Q}^\mu \>,\end{aligned}$$ where
\[e:T-20\] $$\begin{aligned}
w_q
&=
- 4 \, m \, b \, A_0^2 \, \beta^2 \, p \, f_1(\beta,q) \>,
\label{e:T-20a} \\
w_p
&=
4 \, m \, b \, A_0^2 \, \beta^2 \, f_2(\beta,q) \>,
\label{e:T-20b} \\
w_\beta
&=
0 \>,
\label{e:T-20c} \\
w_\theta
&=
- 4 \, m \, b \, A_0^2 \, \beta^2 \, f_1(\beta,q) \>.
\label{e:T-20d}\end{aligned}$$
Here $f_1(\beta,q)$ and $f_2(\beta,q)$ are given in Appendix \[s:integrals\]. In terms of the vector $u_{\mu}(Q) = v_{\mu}(Q) - w_{\mu}(Q)$, Hamilton’s equations for the variational parameters are $$\label{e:T-22}
\dot{Q}^\mu
=
f^{\mu\nu}(Q) \, u_\nu(Q) \>,$$ which gives
\[e:T-24\] $$\begin{aligned}
\dot{q}
&=
2 \, p
+
(2/3) \, m \, b \, \beta \, I_1(\beta,q)
-
2 \, m \, b \, \beta \, f_2(\beta,q) \>,
\label{e:T-24a} \\
\dot{p}
&=
(2/3) \, p \, m \, b \, \beta \, f_1(\beta,q)
\label{e:T-24b} \\
& \qquad
-
[\, b^2 + m^2 - (4/9) \, m^2 b^2 - 1/4 \,] \, \beta \, f_6(\beta,q) \,
\notag \\
\dot{\beta}
&=
- 2 \, \beta^2 \, m \, b \, f_1(\beta,q) \>.
\label{e:T-24c}\end{aligned}$$
The equation for $\dot{\theta}$ is not needed for the evolution of the set of equations given in . For $m=0$, the equations reduce to:
\[e:T-25\] $$\begin{aligned}
\dot{q}
&=
2 \, p \>,
\label{e:T-25a} \\
\dot{p}
&=
-
[\, b^2 -1/4 \,] \, \beta \, f_6(\beta,q) \>,
\label{e:T-25b} \\
\dot{\beta}
&=
0 \>.
\label{e:T-25c}\end{aligned}$$
So in this case, $\beta = 1$ and is fixed. This is because the normalization must be conserved. Equations then reduce to: $$\label{e:T-26}
\ddot{q}
+
2 \, [\, b^2 - 1/4 \,] \, f_6(1,q)
=
0 \>.$$
\[s:smallosc\]Small Oscillation equations
-----------------------------------------
Using the expansions found in Appendix A we obtain for the small oscillation equations (we set $q = \delta q$, $p = \delta p$, and $\beta = 1 + \delta \beta$ with $\delta Q^{\mu}$ assumed small),
\[e:SO4-1\] $$\begin{aligned}
\delta \dot{q}
& =
\frac{\pi}{72} \,
\qty( 9 \pi^2 - 64 ) b m \, \delta \beta + 2 \, \delta p \>,
\label{e:SO4-1a} \\
\delta \dot{p}
&=
-
\frac{8}{15}
\qty( \,
b^2
+
m^2
-
(4/9) \, b^2 m^2
-
1/4 \, ) \, \delta q \>,
\label{e:SO4-1b} \\
\delta \dot{\beta}
&=
- \frac{\pi}{2} \, m b \, \delta q \>.
\label{e:SO4-1c}\end{aligned}$$
Thus we obtain for $\ddot{q}$ $$\delta\ddot{q}
+
\omega^2(b,m) \, \delta q
=
0 \>,$$ where $$\begin{aligned}
\omega^2(b,m)
&=
\frac{\pi^2}{144} \,
(\, 9 \pi^2 - 64 \, ) \, b^2 m^2
\\
& \qquad
+
\frac{16}{15} \,
\qty(
b^2
+
m^2
-
(4/9) \, b^2 m^2
-
1/4 \,) \>.
\notag\end{aligned}$$ The period $T = 2 \pi/ \omega(b,m)$ for $m=0$ and $m=1$ is shown in Fig. \[f:fig1\].
\[s:6cc\]Six parameter ansatz
=============================
One expects that when one increases the number of CC’s the accuracy of the variational approximation increases. For the six parameter Ansatz we will introduce a “chirp” term [@PhysRevD.34.3831] $\Lambda(t)$ which is conjugate to the width parameter $\beta(t)$. That is we will assume: $$\label{e:T-1}
{\tilde{\psi}}(x,t)
=
A(t) \, \sech[\, \beta(t) \, y(x,t) \,] \>,
{{\rm e}}^{{{\rm i}}\, {{\tilde \phi}}(x,t)} \,$$ where $$\label{e:T-2}
{{\tilde \phi}}(x,t)
=
-
\theta(t)
+
p(t) \, y(x,t)+ \Lambda(t) y(x,t) ^2
+
\phi(x) \>.$$ Here $\phi(x)$ is given by Eq. and we have put $y(x,t) = x - q(t)$. We find $$\label{e:P-3}
\rho(x,t)
=
| {\tilde{\psi}}(x,t) |^2
=
A^2(t) \sech^2(\beta y) \>,$$ so that the mass becomes $$\label{e:P-4}
M(t)
=
{\!\int\!\mathrm{d}x\,}\rho(x,t)
=
\frac{2 A^2(t)}{\beta(t)} \>.$$ It will be useful to employ $M(t)$ as a collective coordinate rather than $A(t)$. The six time-dependent collective coordinates then are: $$\label{e:P-5}
Q^{\mu}(t)
=
\{\, M(t), \theta(t), q(t), p(t), \beta(t), \Lambda(t) \,\} \>.$$
The parameters $\beta(t)$ and $\Lambda(t)$ are related to the two point correlation functions $G_2 = \langle (x-q(t))^2 \rangle$ and $P_2 = \ev{[x-q(t) ] \hat p + \hat p [x-q(t)] }$ where $$\label{e:evdef}
\ev{ (\cdot) }
=
\int\limits_{-\infty}^{\infty} (\cdot)|\psi(x,t)|^2 \dd{x}
\Big /
\int\limits_{-\infty}^{\infty}|\psi(x,t)|^2 \dd{x} \>.$$ Thus we find $G_2 = \pi^2 / (12 \beta^2)$, and $$\begin{aligned}
\label{e:P2}
P_2
&=
\frac{{{\rm i}}}{2} {\!\int\!}\dd{x}
[x-q(t)] \,
\qty[\, \psi^\ast \psi_x - \psi^\ast_x \psi \,] / M(t)
\\
&=
\frac{\pi^2 \Lambda}{3 \beta^2}
+
\frac{2}{3} b m \frac{I_3(\beta,q) }{M(t)} \>,
\notag\end{aligned}$$ where $I_3$ is given in Appendix \[s:integrals\]. We see that $P_2$ is directly related to $\Lambda$ when the potential is real.
From the formalism given in Sec. \[s:CCs\], the equations of motion for the collective coordinates follow. For the kinetic term in the Lagrangian, we find $$\begin{gathered}
\label{e:P-12}
\pi_{M} = 0
\qc
\pi_{\theta} = M
\qc
\pi_{q} = M p
\qc
\pi_{p} = 0 \,
\\
\pi_{\beta} = 0
\qc
\pi_{\Lambda} = - M \frac{\pi^2}{12 \beta^2} \>,
\notag\end{gathered}$$ and the only non-zero derivatives are then $$\begin{gathered}
\label{e:P-13}
\partial_M \pi_{\theta}
=
1
\qc
\partial_M \pi_{q}
=
p
\qc
\partial_p \pi_{q}
=
M \,
\\
\partial_M \pi_{\Lambda}
=
- \frac{\pi^2}{12 \beta^2}
\qc
\partial_{\beta} \pi_{\Lambda}
=
M \frac{\pi^2}{6 \beta^3} \>.
\notag\end{gathered}$$
The antisymmetric symplectic tensor is then given by $$\label{e:P-15}
f_{\mu\nu}(Q)
=
\begin{pmatrix}
0 & 1 & p & 0 & 0 & -\pi^2/(12 \beta^2) \\
-1 & 0 & 0 & 0 & 0 & 0 \\
- p & 0 & 0 & -M & 0 & 0 \\
0 & 0 & M & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & M \pi^2/(6 \beta^3) \\
\pi^2/(12 \beta^2) & 0 & 0 & 0 & -M \pi^2/(6 \beta^3) & 0
\end{pmatrix} \>.$$ Since $\det{f_{ij}(Q)} = M^4 \pi^4/( 36 \beta^6)$ and is non-zero, the inverse is given by $$\label{e:P-16}
f^{\mu\nu}(Q)
=
\begin{pmatrix}
0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & -p/M & \beta/(2M) & 0 \\
0 & 0 & 0 & 1/M & 0 & 0 \\
0 & p/M & -1/M & 0 & 0 & 0 \\
0 & -\beta/(2M) & 0 & 0 & 0 & -6 \beta^3/(\pi^2 M) \\
0 & 0 & 0 & 0 & 6 \beta^3/(\pi^2 M) & 0
\end{pmatrix} \>.$$ For the dissipation functional, we obtain $$\begin{aligned}
\label{e:P-26}
F(Q,\dot{Q})
&=
2 M m b \beta
{\!\int\!\mathrm{d}y\,}\sech^2( \beta y ) \sech(y+q) \tanh(y+q) \,
[\,
-
\dot{\theta}
+
\dot{p} y
-
p \dot{q}
+
\dot{\Lambda} y^2
-
2 y \Lambda \dot{q} \,
] \>,\end{aligned}$$ which gives $$\begin{gathered}
\label{e:P-27}
w_{M} = 0
\qc
w_{\theta}
=
- 2 M m b \beta \, f_1(\beta,q)
\qc
w_{q}
=
- 2 M \, m b \beta \,
[\, p \, f_1(\beta,q) + 2 \Lambda f_2(\beta,q) \,]
\qc \\
w_{p}
=
2 M \, m b \beta \, f_2(\beta,q)
\qc
w_{\beta}
=
0
\qc
w_{\Lambda}
=
2 M \, m b \beta \, f_3(\beta,q) \>.
\notag\end{gathered}$$ For $H(Q)$, using the 6-parameter Ansatz we now obtain $$\begin{aligned}
\label{e:P-23}
\frac{H(Q)}{M}
&=
p^2
+
\frac{\beta^2}{3}
+
\frac{ \pi^2 \Lambda^2}{3 \beta^2}
+
\frac{2 \beta}{3} \, p b m \, I_1(\beta,q)
+
\frac{4 \beta}{3} \, b m \Lambda \, I_3(\beta,q)
\\
& \qquad\qquad
-
\frac{g M \beta}{6}
-
\frac{\beta}{2} \,
\Bigl [\,
b^2 + m^2 - \frac{1}{4}
- \frac{4}{9}\, b^2 m^2 \,
\Bigr ] \, I_2(\beta,q) \>.
\notag\end{aligned}$$ All the integrals are defined in Appendix \[s:integrals\]. For $v_{\mu}(Q) = \partial_{\mu} H(Q)$ we obtain
\[e:P-24\] $$\begin{aligned}
v_{M}
&=
p^2
+
\frac{\beta^2}{3}
+
\frac{ \pi^2 \Lambda^2}{3 \beta^2}
+
\frac{2 \beta}{3} \, p b m \, I_1(\beta,q)
+
\frac{4 \beta}{3} \, b m \Lambda \, I_3(\beta,q)
\label{e:P-24a} \\
& \qquad\qquad
-
\frac{g M \beta}{3}
-
\Bigl [\,
b^2 + m^2 - \frac{1}{4}
- \frac{4}{9}\, b^2 m^2 \,
\Bigr ] \, \frac{\beta}{2} \, I_2(\beta,q) \>,
\notag \\
v_{\theta}
&=
0 \>,
\label{e:P-24b} \\
v_q
&=
-
\frac{2 \beta}{3} \, M \, p b m \, f_1(\beta,q)
-
\frac{4 \beta}{3} \, M \, b m \Lambda \, f_2(\beta,q)
\label{e:P-24c} \\
& \qquad\qquad\qquad
+
M \,
\Bigl [\,
b^2 + m^2 - \frac{1}{4}
- \frac{4}{9}\, b^2 m^2 \,
\Bigr ] \, \beta \, f_6(\beta,q) \>,
\notag \\
v_p
&=
2 M p
+
\frac{2 M \beta}{3} \, b m \, I_1(\beta,q) \>,
\label{e:P-24d} \\
v_{\beta}
&=
\frac{2 M \beta}{3}
-
\frac{2 M \pi^2 \Lambda^2 }{3 \beta^3}
+
\frac{2 M}{3} \, p b m \, I_1(\beta,q)
+
\frac{4 M}{3} \, b m \Lambda \, I_3(\beta,q)
\label{e:P-24e} \\
& \qquad
-
\frac{g M^2}{6}
-
\frac{M}{2} \,
\Bigl [\,
b^2 + m^2 - \frac{1}{4}
- \frac{4}{9}\, b^2 m^2 \,
\Bigr ] \, I_2(\beta,q)
-
\frac{4 M \beta}{3} \, p b m \, f_{10}(\beta,q)
\notag \\
& \qquad
-
\frac{8 M \beta}{3} \, b m \Lambda \, f_9(\beta,q)
+
\Bigl [\,
b^2 + m^2 - \frac{1}{4}
- \frac{4}{9}\, b^2 m^2 \,
\Bigr ] \, M \beta \, f_7(\beta,q) \>,
\notag \\
v_{\Lambda}
&=
\frac{2 \pi^2 M \Lambda}{3 \beta^2}
+
\frac{4 \beta M}{3} \, b m \, I_3(\beta,q) \>.
\label{e:P-24f}\end{aligned}$$
The symplectic equations of motion are $$\label{e:P-29}
\dot{Q}^{\mu}
=
f^{\mu\nu}(Q) \, u_{\nu}(Q) \>,$$ from which we find:
\[e:P-32\] $$\begin{aligned}
\dot{M}
&=
- 2 M \, m b \beta f_1(\beta,q) \>,
\label{e:P-32a} \\
\dot{\theta}
&=
-
p^2
+
\frac{2}{3} \, \beta^2
-
\frac{5}{12} \, g \beta M
+
\frac{1}{3} \, m b p \beta \, I_1(\beta,q)
+
2 \, m b \beta \Lambda \, I_3(\beta,q)
\label{e:P-32b} \\
& \qquad
+
2 \, m b p \beta \, f_2(\beta,q)
-
\frac{2}{3} \, m b p \beta^2 \, f_{10}(\beta,q)
-
\frac{4}{3} \, m b \beta^2 \Lambda \, f_9(\beta,q)
\notag \\
& \qquad
-
\frac{1}{4} \,
\Bigl [\,
b^2 + m^2 - \frac{1}{4}
- \frac{4}{9}\, b^2 m^2 \,
\Bigr ] \, \beta \,
[\, 3 \, I_2(\beta,q) - 2 \, \beta \, f_7(\beta,q) \,] \,
\notag \\
\dot{q}
&=
2 p
+
\frac{2 \beta}{3} \, m b \, I_1(\beta,q)
-
2 \, m b \beta \, f_2(\beta,q) \>,
\label{e:P-32c} \\
\dot{p}
&=
\frac{2}{3} \, m b \beta \, p \, f_1(\beta,q)
-
\frac{8}{3} \, m b \beta \, \Lambda \, f_2(\beta,q)
-
\Bigl [\,
b^2 + m^2 - \frac{1}{4} - \frac{4}{9}\, m^2 b^2 \,
\Bigr ] \, \beta \, f_6(\beta,q) \,
\label{e:P-32d} \\
\dot{\beta}
&=
-
m b \, \beta^2 f_1(\beta,q)
-
4 \beta \Lambda
-
\frac{8 \beta^4}{\pi^2} \, m b \, I_3(\beta,q)
+
\frac{12 \beta^4}{\pi^2} \, m b \, f_3(\beta,q) \>,
\label{e:P-32e} \\
\dot{\Lambda}
&=
-
4 \Lambda^2
+
\frac{4 \beta^4}{\pi^2}
+
\frac{4}{\pi^2} \, \beta^3 p m b \, I_1(\beta,q)
+
\frac{8}{\pi^2} \, \beta^3 \Lambda m b \, I_3(\beta,q)
\label{e:P-32f} \\
& \qquad
-
\frac{g \beta^3 M}{\pi^2}
-
\frac{6 \beta^3}{\pi^2} \,
\Bigl [\,
b^2 + m^2 - \frac{1}{4} - \frac{4}{9}\, b^2 m^2 \,
\Bigr ] \, f_8(\beta,q)
\notag \\
& \qquad
-
\frac{8 \beta^4}{\pi^2} \, b m \, p \, f_{10}(\beta,q)
-
\frac{16 \beta^4}{\pi^2} \, b m \Lambda \, f_9(\beta,q) \>.
\notag\end{aligned}$$
In Eq. , we use the identity . Here $M(t)$ is a dynamic variable. In order for the variational trial wave function to match the exact solution at $t=0$, the initial conditions are: $$\begin{gathered}
\label{e:P-33}
q_0 = 0
\qc
p_0 = 0
\qc
\beta_0 = 1
\qc
\Lambda_0
=
0
\qc
\theta_0 = - t \,
\\
g M_0
=
\frac{(4 b^2 - 9)(4 m^2 - 9 )}{18} \>.
\notag\end{gathered}$$ As a check, the right-hand-sides of Eqs. vanish \[except for $\dot{\theta}(0) = -1$\] at these initial values, which guarantees that the exact solution is stationary. For non-zero values of $q_0$ and/or $\beta_0$, the values of $p_0$ and $\Lambda_0$ are sometimes fixed by setting $\dot{q}_0 = 0$ and $\dot{\beta}_0=0$, and solving Eqs. , and for $p_0$ and $\Lambda_0$, which gives:
\[e:P-34\] $$\begin{aligned}
p_0
&=
\frac{1}{2} \,
\Bigl [\,
\dot{q}_0
-
\frac{2}{3} m b \beta_0 \, I_1(\beta_0,q_0)
\label{e:P-34a} \\
& \qquad\qquad\qquad
+
2 \, m b \beta_0 \, f_2(\beta_0,q_0) \,
\Bigr ] \>,
\notag \\
\Lambda_0
&=
\frac{1}{4 \beta_0} \,
\Bigl [\,
-
\dot{\beta}_0
-
m b \beta_0^2 f_1(\beta_0,q_0)
\label{e:P-34b} \\
& \qquad
-
\frac{8}{\pi^2} \, m b \beta^4 \, I_3(\beta_0,q_0)
+
\frac{12}{\pi^2} \, m b \beta^4 \, f_3(\beta_0,q_0) \,
\Bigr ] \>.
\notag\end{aligned}$$
When $m=0$, the external potential is real and $\dot M=0$. The stability of the solutions to this equation for arbitrary $\kappa$ and for repulsive and attractive potential $V_1$ as well as positive and negative $g$ was studied using a variety of methods, and the stability properties and small oscillation frequencies for $q,p,\beta,\Lambda$ were determined in Ref. [@r:Dawson:2017td]. For that problem when we set $\kappa=1$ and $m=0$, our equations simplify to $$\begin{aligned}
\label{e:reduce}
\dot{q}
&= 2 \, p \>,
\\
\dot{\beta}
&= - 4 \beta \Lambda \>,
\notag \\
\dot{\Lambda }
&= - 4 \Lambda^2 + \frac{4 \beta^4}{\pi^2} -
\frac{g \beta^3 M}{\pi^2}
-
\frac{6 \beta^3}{\pi^2} \,
\Bigl [\,
b^2 - \frac{1}{4}
\Bigr ] \, f_8(\beta,q)
\notag \end{aligned}$$ which agrees with the results in Ref. [@r:Dawson:2017td] once we use the fact that $f_3[G,q,\gamma]$ in that paper is just $\beta^2 f_8(\beta, q)$ here. At $m=0$ the small oscillation equations for $\beta$ and $q$ decouple. Using the expansions of the integrals found in Appendix A, we find that the small oscillation equations are: $$\begin{aligned}
\label{e:SOmzero}
\delta \dot q &= 2 \delta p \,
\\
\delta \dot p &= - \frac{8}{15} (b^2-1/4) \, \delta q \,
\notag\end{aligned}$$ so that $$\begin{gathered}
\label{e:SOmzeroOmega}
\delta \ddot q + \omega_q^2 \delta q = 0 \>,
\\
\omega_q^2 = \frac{16}{15} (b^2-1/4) \>.
\notag\end{gathered}$$ This agrees with the result from the 4-parameter Ansatz. However, we get a *different* frequency for the $\beta$ oscillation, $$\begin{aligned}
\label{e:betaosc}
\delta \dot \beta
&=
- 4 \delta \Lambda \>,
\\
\delta \dot \Lambda
&=
\Bigl [\,
\frac{4 b^2}{15} + \frac{4}{\pi^2} - \frac{1}{15} \,
\Bigr ] \, \delta \beta \>,
\notag\end{aligned}$$ so that $$\begin{gathered}
\label{e:betaomega}
\delta \ddot \beta + \omega_\beta^2 \, \delta \beta = 0 \>,
\\
\omega_\beta^2
=
4 \, \Bigl [\,
\frac{4 b^2}{15} + \frac{4}{\pi^2} - \frac{1}{15}\,
\Bigr ] \>.
\notag\end{gathered}$$ Plots of $\omega_q^2$ and $\omega_\beta^2$ for $m=0$ are shown in Fig. \[f:fig2a\].
\[s:LinearRes\]Linear response results for the six CC approximation
===================================================================
We linearize the set of equations given in by expanding the equations about the exact solutions, $Q^{\mu} = Q_0^{\mu} + \delta Q^{\mu}$ keeping only the first order terms. Note that $Q_0^{\mu}$ are given in Eqs. . Using the expansions of Appendix \[ss:expand\], we find
\[e:E-2\] $$\begin{aligned}
\delta \dot{M}
&=
- \frac{\pi}{2} \, m b \, M_0 \, \delta q \>,
\label{e:E-2a} \\
\delta \dot{\theta}
&=
-
\frac{5}{12} \, g \delta M
+
\frac{7\pi}{18} \, m b \, \delta p
\label{e:E-2b} \\
& \quad
+
\frac{1}{3} \,
\Bigl [\,
1
+
\frac{2 \pi^2}{15}
-
\Bigl (\, \frac{1}{2} + \frac{\pi^2}{30} \, \Bigr ) \, g M_0 \,
\Bigr ] \, \delta \beta \>,
\notag \\
\delta \dot{q}
&=
\frac{\pi}{72} \,
\qty(\, 9 \pi^2 - 64 \,) m b \, \delta \beta + 2 \, \delta p \>,
\label{e:E-2c} \\
\delta \dot{p}
&=
\frac{4}{15} \, [\, g M_0 - 4 \,] \, \delta q
-
\frac{4 \pi}{9} \, m b \, \delta \Lambda \>,
\label{e:E-2d} \\
\delta \dot{\beta}
&=
\Bigl [\, \frac{\pi}{2} - \frac{20}{3 \pi} \, \Bigr ] \,
m b \, \delta q
-
4 \, \delta \Lambda \>,
\label{e:E-2e} \\
\delta \dot{\Lambda}
&=
\frac{2 b m}{3\pi} \delta p
-
\frac{ g }{\pi^2} \delta M
\label{e:E-2f} \\
& \qquad
+
\frac{2}{15}
\Bigl [ \,
- g M_0 + \frac{30}{\pi^2} + 4 \,
\Bigr ] \, \delta \beta \>,
\notag\end{aligned}$$
where we have used the relation, $$\label{e:P-35}
b^2 + m^2 - \frac{4}{9} \, b^2 m^2 - \frac{1}{4}
=
2 - \frac{1}{2} \, g M_0 \>.$$
Equations are written as $$\label{e:E-3}
\delta \dot{Q}^{\mu}
=
M^{\mu}{}_{\nu}(Q_0) \, \delta Q^{\nu} \>,$$ from which we find: $$\begin{gathered}
\label{e:E-4}
\delta \ddot{Q}^{\mu}
+
W^{\mu}{}_{\nu}(Q_0) \, \delta Q^{\nu}
=
0
\\
W^{\mu}{}_{\nu}(Q_0)
=
-
M^{\mu}{}_{\sigma}(Q_0) M^{\sigma}{}_{\nu}(Q_0) \>.
\notag\end{gathered}$$ Here $W^{\mu}{}_{\nu}(Q_0)$ is Hermitian. The square of the linearized oscillation frequencies $\omega^2$ are given by the eigenvalues of $W^{\mu}{}_{\nu}(Q_0)$. One can show that the matrix $W^{\mu}{}_{\nu}(Q_0)$ can be split into two blocks, one of them coupling $(\delta q, \delta \Lambda, \delta \theta)$, the other coupling $(\delta p,\delta \beta,\delta M)$. Both of these blocks give identical eigenvalues, a zero eigenvalue and two non-zero eigenvalues. For example, using Eqs. , we find
\[e:eigen-1\] $$\begin{aligned}
\delta \ddot{q} - \qty[\, A \, \delta q + B \, \delta \Lambda \,] &= 0 \>,
\label{e:eigen-1a} \\
\delta \ddot{\Lambda} - \qty[\, D \, \delta q + E \, \delta \Lambda \,] &= 0 \>,
\label{e:eigen-1b}\end{aligned}$$
where
\[e:eigen-2\] $$\begin{aligned}
A
&=
\frac{8}{15} \, \qty( g M_0 - 4 )
\label{e:eigen-2a} \\
& \qquad\qquad
+
\frac{(9 \pi^2 - 64)(3 \pi^2 - 40) \, b^2 m^2}{432} \>,
\notag \\
B
&=
\qty( 16 - 3 \pi^2 ) \, \frac{\pi \, b m}{6} \>,
\label{e:eigen-2b} \\
D
&=
b m \,
\Bigl \{ \,
\frac{g M_0}{2 \pi}
+
\frac{2( 3 \pi^2 - 40 )}{3 \pi^3}
\label{e:eigen-2c} \\
& \qquad\qquad
+
\frac{(g M_0 - 4)(16 - \pi^2 )}{15 \pi} \,
\Bigr \} \>,
\notag \\
E
&=
-
\frac{16}{\pi^2} - \frac{8}{27} \, m^2 b^2 + \frac{8}{15}\, ( g M_0 - 4 ) \>,
\label{e:eigen-2d}\end{aligned}$$
from which we find $$\label{e:eigen-3}
\omega^2
=
\frac{1}{2} \,
\qty[\,
- \qty(A + E) \pm \sqrt{ \qty(A - E)^2 + 4 \, B D } \, ] \>.$$ Although these two frequencies increase together when $m=0$ as a function of $b$, once we get near $m=1$ they start repelling each other and the dependence of the lower frequency has a maximum as a function of $b$ instead of monotonically increasing. This is shown in Fig. \[f:fig2\]. Note that when $m=0$ and $b^2< 1/4$, the potential becomes repulsive, which leads to $\omega^2 <0$ and thus to a translational instability. This was studied in detail in Ref. [@r:Dawson:2017td].
\[s:Numerical\]Computational Strategy
=====================================
In our previous sections we were able to develop a six parameter variational approach to the time evolution of slightly perturbed solutions of the NLS equation in an external complex potential. We were able to get an explicit analytic expression as a function of $m, b$, of two oscillation frequencies that affect the response of the solution to small perturbations. So the first question we would like to answer is how does this analytic result compare to the actual response found by numerically solving the NLS equation. The second question we want to answer is the domain of applicability of the variational approach in terms of predicting the actual time evolution of the low order moments of the solution. This has two parts: (i) for fixed $m,b$ how long is the approximation valid and (ii) as we increase the size of the complex part of the potential, by say varying $m$ for fixed $b$, when does this approach start losing its validity. In our approximation for all $b,m$ that correspond to an attractive potential, there is no translational instability. So we would like to see in our numerical simulations, that for the case $m=1$ (and $\kappa=1$), the translational instability that arises due to mixing of the solution we are considering with the first excited state in the potential occurs at times much later than the domain of applicability of the six CC method. For that case when $0.56 < b < 1.37$ a late time translational instability was found.
To study numerically the evolution of Eqs. , we have used a homemade code using a Crank-Nicolson scheme [@ref:numrec]. In Ref. [@PhysRevE.94.032213] we have shown that the Crank-Nicolson scheme is a reliable method for successfully solving Eq. in the presence of a complex potential. For the sake of comparison with the analytical calculations, the initial soliton shape $\psi(x,0)$ in our simulations is given by Eqs. and at $t=0$. The complex soliton shape in the transverse spatial domain $x$ was represented in a regular grid with mesh size $\Delta x=2\times 10^{-6}$ and free boundary conditions were imposed. The mesh size was chosen to be much smaller than the initial soliton width parameter $1/\beta(0)=1$, so that very small variations of the soliton position could be accurately measured by using a center of mass definition, [*i.e.*]{} $q=\expval{x}$, where the expectation value is defined in Eq. . The soliton width $W(t)$ is the square root of the normalized second moment $G_2 = \pi^2/(12 \beta^2(t))$. The soliton width parameter $1/\beta(t)$ in the simulations was calculated by using the expression $1/\beta(t) = \sqrt{G_2(t)/G_2(0)}$. The other CC’s measured in the simulations were the amplitude $A(t)=\max_{x \in {\mathbb{R}}} \sqrt{\rho(x,t)}$ and the mass $M(t)$ given by Eq. .
\[s:results\]Comparison of collective variable theories with simulations
========================================================================
Our potential is symmetric in $b \leftrightarrow m$. When either $b$ or $m=0$ the potential is real and the small oscillation equations for $q, p$ $\beta, \Lambda$ decouple giving rise to separate oscillation frequencies in that regime. Once the imaginary part turns on, we expect that these two oscillation frequencies appear to a certain degree in all the collective coordinates. Note that in the collective coordinate approach the mass is related to the height and the CC parameter $\beta$ and is not an independent parameter, namely $M(t) = 2 A^2(t)/\beta(t)$. First let us choose $g=1, \kappa=1, m=0$ and $b=1 $ to see how well our CC approximation works when compared with numerical simulations when the potential is real. For our simulations we choose the parameters $g = 1, \kappa = 1, q_0=0.001, \beta = 1.001$ and all other parameters those of the exact solution. The small oscillation theory for this case predicts separate oscillation frequencies for $q$ and $\beta$, namely $T_q= 7.025$, $\omega_q^2 = 0.800$ and $T_\beta=4.038$, $\omega_{\beta}^2=2.421$. These frequencies are located on the two branches in Fig. \[f:fig2a\], and agree with the six CC approximation. The simulation results are represented by the black data points. This is seen in both the six CC approximation and the numerical simulation.
Since the perturbation is so small, we subtract the initial value of $1/\beta_0=1$ from $1/\beta$ to show the oscillation in the numerical simulations. We see that for $q(t)$, both the amplitude as well as period of oscillation are well reproduced by the six CC theory. This is shown in Fig. \[f:fig3\].
For the width parameter $1/\beta$, the oscillation period is $4.00$ which agrees well with the linear response result $4.038$, but not so well with the simulation result. Here the spectrum consists of several peaks around the frequency of $1.132$, which corresponds to the period $5.55$. Moreover, the soliton amplitude $A(t)$ has the period $3.85$, which is rather close to the above value of $4.00$.
For our simulations with a complex potential we choose the parameters $g = 1, \kappa = 1, m = 1$, and three values for $b$. First we choose $ b = 0.1$ so that the imaginary part of the potential is small, and the dissipation is weak. Next we choose $b = 0.5$ which is located in the lower stability regime $0 < b < 0.56$; and $b = 1.45$ is located in the upper stability regime $1.37 < b < 1.5$.
The exact solution Eq. is stationary and is obtained by the CC Ansatz Eq. with the initial conditions (ICs) $q_0 = 0, p_0 = 0, \beta_0 = 1, \Lambda_0 = 0, \theta_0 = 0$, and $g M_0 = (4 b^2-9)(4 m^2-9)/18$, see Eqs. . In order to test the stability of the exact solution, we choose ICs that are slightly different from the above values. This excites intrinsic oscillations of the soliton which are seen in the time evolution of the CCs, which is obtained by solving the six CC equations, Eqs. , by a Mathematica program. These oscillations are compared with the oscillations which are observed in the simulations, i.e. in the numerical solution of the NLS equation. In particular, the frequencies, periods, and amplitudes of the oscillations are compared.
For the case $b = 0.1$ the four CC and six CC results are nearly identical and agree very well with the simulation results in Fig. \[f:fig4\].
The periods of the oscillations are $T_{4CC} = T_{6CC} = 7.14$, compared to $T_{\text{sim}} = 7.69$. This means that the error in the CC theories is only 7%.
For the case $b = 0.5$ the six CC result is much better than the four CC result and agrees rather well with the simulation shown in Fig. \[f:fig5\]. The periods are $T_{4CC} = 5.26$, $T_{6CC} = 6.25$ and $T_{\text{sim}} = 6.67$, the error is 6%.
For the case $b = 1.45$ the four CC result poorly fits the numerical result. The six CC result is very anharmonic and the oscillation amplitudes do not agree well with the simulations as seen in Fig. \[f:fig6\]. Nevertheless, the periods $T_{6CC} = 8.33$ and $T _{\text{sim}} = 7.69$ agree within an error of 8%. Interestingly, the spectra exhibit a second frequency which is obtained also in the linear response theory. Fig. \[f:fig2b\] shows the two frequencies for all values of $b$. However, the simulations show only one frequency.
So far we have always taken $q_0 = 0.001$, and the other ICs as in the exact solution. Choosing a finite value for $p_0$ gives very similar results, because the $q$ and $p$ oscillations are related, see the relations below Eq. . Let us now consider $b=1.45$ and finite values for $\Lambda_0$ which will also affect the width $1/\beta$ because their oscillations are related. Choosing a very small, negative value $\Lambda_0 = -0.00005$, and increasing this value by steps, we find that the anharmonicity of the CCs gradually decreases. For $\Lambda_0 = -0.00025$ the oscillations are nearly harmonic and the periods are the same as in Fig. \[f:fig6\]
For $\Lambda_0 = +0.00025$ the periods are again the same as in Fig. \[f:fig6\]. However, the spectrum of $M(t)$ exhibits a second peak at $T_2 = 2.38$ which is stronger than the first peak at $T_1 = 8.33$. This second peak belongs to the upper branch in Fig. \[f:fig2b\] which was obtained by our linear response theory. However, this peak is not seen in the simulations.
\[s:conclusions\]Conclusions
============================
In this paper we investigated the domain of applicability of a four and six collective coordinate approximation to study the response of the nodeless solution of the NLS equation in the presence of a complex potential to small perturbations. This type of approximation had been used in the past to study the response of exact solutions of the NLS equation when in the presence of weak harmonic complex external potentials. In this paper we instead considered a ${\mathcal{PT}}$-symmetric potential where we could vary the strength of the complex part of the potential from zero to its maximum allowed value. Using a small oscillation approximation to the CC equations we were able to obtain analytic expressions for the two frequencies of small oscillation found in our six CC approximation. These frequencies were quite close to those that were found in the numerical simulations of the discretized PDEs when we perturbed the initial conditions of the exact solution. This was true for all allowed values of the parameter product $bm$ which governed the strength of the imaginary part of the potential. We found that as we increased $bm$, the four CC approximation quickly broke down. The six CC approximation was quite a reasonable approximation even at $bm=1/2$, but at the maximum value we studied $bm=1.45$, it tracked accurately the position of the solitary wave for less than 1/4 of a period and then began to differ from the numerical solution.
F.C. would like to thank the Santa Fe Institute and the Center for Nonlinear Studies at Los Alamos National Laboratory for their hospitality. F.G.M. and N.R.Q. acknowledge financial support from the Ministerio de Economía y Competitividad (Spain) through FIS2014-54497-P. F.G.M. also acknowledges financial support from the Plan Propio of Universidad de Seville and is grateful for the hospitality of the Mathematical Institute of the University of Seville (IMUS) and of the Theoretical Division and Center for Nonlinear Studies at Los Alamos National Laboratory. N.R.Q. also acknowledges financial support from the Junta de Andalucia (Spain) under Projects No. FQM207 and the Excellent Grant P11-FQM-7276. E.A. gratefully acknowledges support from the Fondo Nacional de Desarrollo Cientifico y tecnologico (FONDECYT) project No. 1141223 and from the Programa Iniciativa Cientfica Milenio (ICM) Grant No. 130001. A.K. is grateful to Indian National Science Academy (INSA) for awarding him INSA Senior Scientist position at Savitribai Phule Pune University, Pune, India. B.M. and J.F.D. would like to thank the Santa Fe Institute for their hospitality. B.M. acknowledges support from the National Science Foundation through its employee IR/D program. The work of A.S. was supported by the U.S. Department of Energy.
\[s:integrals\]Definition of integrals
======================================
We note that
\[e:I-1\] $$\begin{aligned}
\dv{}{z} \sech^2(z)
&=
- 2 \sech^2(z) \, \tanh(z)
=
- 2 \sech^3(z) \, \sinh(z) \>,
\label{e:I-1a} \\
\dv{}{z} \tanh(z)
&=
\sech^2(z) \>.
\label{e:I-1b}\end{aligned}$$
Some useful integrals are the following:
\[e:I-2\] $$\begin{aligned}
{\!\int\!}\dd{z} \sech^2(z)
&=
2 \>,
\label{e:I-2a} \\
{\!\int\!}\dd{z} \sech^3(z)
&=
\frac{\pi}{2} \>,
\label{e:I-2b} \\
{\!\int\!}\dd{z} \sech^4(z)
&=
\frac{4}{3} \>,
\label{e:I-2c} \\
{\!\int\!}\dd{z} z^2 \sech^2(z)
&=
\frac{\pi^2}{6} \>,
\label{e:I-2d} \\
{\!\int\!}\dd{z} \sech^2(z) \tanh^2(z)
&=
\frac{2}{3} \>.
\label{e:I-2e}\end{aligned}$$
We define:
\[e:I-3\] $$\begin{aligned}
I_1(\beta,q)
&=
{\!\int\!\mathrm{d}x\,}\sech^2( \beta y ) \sech(x)
=
{\!\int\!\mathrm{d}y\,}\sech^2( \beta y ) \sech(y+q) \>,
\label{e:I-3a} \\
I_2(\beta,q)
&=
{\!\int\!\mathrm{d}x\,}\sech^2( \beta y ) \sech^2(x)
=
{\!\int\!\mathrm{d}y\,}\sech^2( \beta y ) \sech^2(y+q) \>,
\label{e:I-3b} \\
I_3(\beta,q)
&=
{\!\int\!\mathrm{d}x\,}y \sech^2( \beta y ) \sech(x)
=
{\!\int\!\mathrm{d}y\,}y \sech^2( \beta y ) \sech(y+q) \>.
\label{e:I-3c}\end{aligned}$$
Also, we define:
\[e:I-4\] $$\begin{aligned}
f_1(\beta,q)
&=
{\!\int\!}\dd{y} \sech^2(\beta y) \, \sech(y + q) \, \tanh(y + q) \>,
\label{e:I-4a} \\
f_2(\beta,q)
&=
{\!\int\!}\dd{y} y \, \sech^2(\beta y) \, \sech(y + q) \, \tanh(y + q) \>,
\label{e:I-4b} \\
f_3(\beta,q)
&=
{\!\int\!}\dd{y} y^2 \, \sech^2(\beta y) \, \sech(y + q) \, \tanh(y + q) \>,
\label{e:I-4c} \\
f_4(\beta,q)
&=
{\!\int\!}\dd{y} \sech^3(\beta y) \, \sech(y + q) \, \tanh(y + q) \>,
\label{e:I-4d} \\
f_5(\beta,q)
&=
{\!\int\!}\dd{y} y \, \sech^3(\beta y) \, \sech(y + q) \, \tanh(y + q) \>,
\label{e:I-4e} \\
f_6(\beta,q)
&=
{\!\int\!}\dd{y}
\sech^2(\beta y) \,
\sech^2(y + q) \, \tanh(y + q) \>,
\label{e:I-4f} \\
f_7(\beta,q)
&=
{\!\int\!}\dd{y} y \,
\sech^2(\beta y) \, \tanh(\beta y) \,
\sech^2(y + q) \>,
\label{e:I-4g} \\
f_8(\beta,q)
&=
{\!\int\!}\dd{y} y \,
\sech^2(\beta y) \, \sech^2(y + q) \, \tanh(y+q) \>,
\label{e:I-4h} \\
f_9(\beta,q)
&=
{\!\int\!}\dd{y} y^2 \,
\sech^2(\beta y) \, \tanh(\beta y) \, \sech(y + q) \>,
\label{e:I-4i} \\
f_{10}(\beta,q)
&=
{\!\int\!}\dd{y} y \,
\sech^2(\beta y) \, \tanh(\beta y) \, \sech(y + q) \>.
\label{e:I-4j}\end{aligned}$$
Partial derivatives of $I_1(\beta,q)$ are given by
\[e:I-5\] $$\begin{aligned}
\pdv{I_1(\beta,q)}{q}
&=
- {\!\int\!}\dd{y}
\sech^2(\beta y) \, \sech(y + q) \, \tanh(y + q)
=
- f_1(\beta,q) \>,
\label{e:I-5a} \\
\pdv{I_1(\beta,q)}{\beta}
&=
- 2 {\!\int\!}\dd{y} y \,
\sech^2(\beta y) \, \tanh(\beta y) \, \sech(y + q)
=
- 2 f_{10}(\beta,q) \>.
\label{e:I-5b} \end{aligned}$$
Partial derivatives of $I_2(\beta,q)$ are given by
\[e:I-6\] $$\begin{aligned}
\pdv{I_2(\beta,q)}{q}
&=
- 2 {\!\int\!}\dd{y}
\sech^2(\beta y) \, \sech^2(y + q) \, \tanh(y + q)
=
- 2 f_6(\beta,q) \>,
\label{e:I-6a} \\
\pdv{I_2(\beta,q)}{\beta}
&=
- 2 {\!\int\!}\dd{y} y \,
\sech^2(\beta y) \, \tanh(\beta y) \, \sech^2(y + q)
=
- 2 f_7(\beta,q) \>.
\label{e:I-6b} \end{aligned}$$
Partial derivatives of $I_3(\beta,q)$ are given by
\[e:I-7\] $$\begin{aligned}
\pdv{I_3(\beta,q)}{q}
&=
- {\!\int\!}\dd{y} y \,
\sech^2(\beta y) \, \sech(y + q) \, \tanh(y + q)
=
- f_2(\beta,q) \>,
\label{e:I-7a} \\
\pdv{I_3(\beta,q)}{\beta}
&=
- 2 {\!\int\!}\dd{y} y^2 \,
\sech^2(\beta y) \, \tanh(\beta y) \, \sech(y + q)
=
- 2 f_9(\beta,q) \>.
\label{e:I-7b} \end{aligned}$$
A useful identity is obtained by integration of $f_7(\beta,q)$ by parts. Using $$\label{e:ID-2}
\pdv{y} \, \sech^2(\beta y)
=
- 2 \beta \, \sech^2(\beta y) \tanh(\beta y) \>,$$ we find $$\begin{aligned}
\label{e:ID-3}
- 2 \beta f_7(\beta,q)
&=
{\!\int\!}y \, \sech^2(y + q) \,
\dd{ \qty{ \sech^2(\beta y) } }
\\
&=
-
{\!\int\!}\sech^2(\beta y) \, \dd{ \qty{ y \, \sech^2(y + q) \, } }
\notag \\
&=
-
{\!\int\!}\dd{y} \sech^2(\beta y) \sech^2(y + q)
+
2 {\!\int\!}\dd{y} \sech^ 2(\beta y) \sech^2(y + q) \tanh(y + q)
\notag \\
&=
- I_2(\beta,q) + 2 \, f_8(\beta,q) \>.
\notag\end{aligned}$$ That is, $$\label{e:ID-4}
I_2(\beta,q) - 2 \beta f_7(\beta,q)
=
2 \, f_8(\beta,q) \>.$$ We use this identity in the $\dot{\Lambda}$ equation, . Next, we now consider the expansion of the integrals and find to first order:
\[e:E-1\] $$\begin{aligned}
I_1(1+\delta\beta,\delta q)
&=
\frac{\pi}{2} - \frac{\pi}{3} \, \delta \beta \>,
\label{e:E-1a} \\
I_2(1+\delta\beta,\delta q)
&=
\frac{4}{3} - \frac{2}{3} \, \delta \beta \>,
\label{e:E-1b} \\
I_3(1+\delta\beta,\delta q)
&=
- \frac{\pi}{6} \, \delta q \>,
\label{e:E-1c} \\
f_1(1+\delta\beta,\delta q)
&=
\frac{\pi}{4} \, \delta q \>,
\label{e:E-1d} \\
f_2(1+\delta\beta,\delta q)
&=
\frac{\pi}{6} + \frac{\pi}{48} (16 - 3 \pi^2) \, \delta \beta \>,
\label{e:E-1e} \\
f_3(1+\delta\beta,\delta q)
&=
\frac{\pi}{48} ( -32 + 3 \pi^2 ) \, \delta q \>,
\label{e:E-1f} \\
f_6(1+\delta\beta,\delta q)
&=
\frac{8}{15} \, \delta q \>,
\label{e:E-1g} \\
f_7(1+\delta\beta,\delta q)
&=
\frac{1}{3} + \frac{2}{45} ( -15 + \pi^2 ) \, \delta \beta \>,
\label{e:E-1h} \\
f_8(1+\delta\beta,\delta q)
&=
\frac{1}{3} - \frac{2 \pi^2}{45} \, \delta \beta \>,
\label{e:E-1i} \\
f_9(1+\delta\beta,\delta q)
&=
\frac{\pi}{96} ( 16 - 3 \pi^2 ) \, \delta q \>,
\label{e:E-1j} \\
f_{10}(1+\delta\beta,\delta q)
&=
\frac{\pi}{6} + \frac{\pi}{32} ( - 16 + \pi^2 ) \, \delta \beta \>.
\label{e:E-1k}\end{aligned}$$
\[s:TW\]Generalized traveling wave method
=========================================
This method was named and used in a paper by Quintero, Mertens and Bishop [@PhysRevE.82.016606]. We will show here that it is an alternative way to obtain Eq. for the rate of change of the collective coordinates. The authors substitute the trial wave function directly into [Schr[ö]{}dinger]{}’s equation. This gives
\[e:TW-1\] $$\begin{aligned}
&{{\rm i}}\, \dot{Q}^{\nu} \partial_{\nu} {\tilde{\psi}}^{\phantom\ast}(x,Q)
+
{\tilde{\psi}}_{xx}^{\phantom\ast}(x,Q)
+
g \, |{\tilde{\psi}}(x,Q)|^{2\kappa} \, {\tilde{\psi}}^{\phantom\ast}(x,Q)
\notag \\
&=
[\, V_1(x) + {{\rm i}}V_2(x) \, ] \, {\tilde{\psi}}^{\phantom\ast}(x,Q) \>,
\label{e:TW-1a} \\
- &{{\rm i}}\, \dot{Q}^{\nu} \partial_{\nu} {\tilde{\psi}}^{\ast}(x,Q)
+
{\tilde{\psi}}_{xx}^{\phantom\ast}(x,Q)
+
g \, |{\tilde{\psi}}(x,Q)|^{2\kappa} \, {\tilde{\psi}}^{\ast}(x,Q)
\notag \\
&=
[\, V_1(x) - {{\rm i}}V_2(x) \, ] \, {\tilde{\psi}}^{\ast}(x,Q) \>.
\label{e:TW-1b} \end{aligned}$$
Multiply by $\partial_{\mu} {\tilde{\psi}}^{\ast}(x,Q)$ and by $\partial_{\mu} {\tilde{\psi}}(x,Q)$ and add them to give $$\begin{aligned}
\label{e:TW-2}
&{{\rm i}}\,
\{\,
[\partial_{\mu} {\tilde{\psi}}^{\ast}] \,
[\partial_{\nu} {\tilde{\psi}}]
-
[\partial_{\nu} {\tilde{\psi}}^{\ast}] \,
[\partial_{\mu} {\tilde{\psi}}] \,
\} \, \dot{Q}^{\nu}
+
[ \partial_{\mu} {\tilde{\psi}}^{\ast}] \,
{\tilde{\psi}}_{xx}
\\
& \!\!\!\!
+
[\partial_{\mu} {\tilde{\psi}}] \,
{\tilde{\psi}}_{xx}^{\ast}
+
\{\,
g \, |{\tilde{\psi}}|^{2\kappa}
-
V_1(x) \,
\} \,
\{\,
[\partial_{\mu} {\tilde{\psi}}^{\ast}] \, {\tilde{\psi}}+
[\partial_{\mu} {\tilde{\psi}}] \, {\tilde{\psi}}^{\ast} \,
\}
\notag \\
& \quad
=
{{\rm i}}\, V_2(x) \,
\{\,
[\partial_{\mu} {\tilde{\psi}}^{\ast}] \, {\tilde{\psi}}-
[\partial_{\mu} {\tilde{\psi}}] \, {\tilde{\psi}}^{\ast} \,
\} \>.
\notag\end{aligned}$$ Integrating over $x$ and the second term by parts gives $$\label{e:TW-3}
I_{\mu\nu}(Q) \, \dot{Q}^\nu
=
\partial_\mu H(Q) + R_{\mu}(Q) \>,$$ where
\[e:TW-4\] $$\begin{aligned}
I_{\mu\nu}(Q)
&=
{{\rm i}}\!{\!\int\!}\dd{x} \!
\bigl \{
[\, \partial_\mu {\tilde{\psi}}^{\ast} \,]\,
[\, \partial_\nu {\tilde{\psi}}\,]
-
[\, \partial_\nu {\tilde{\psi}}^{\ast} \,]\,
[\, \partial_\mu {\tilde{\psi}}\,]
\bigr \} ,
\label{e:TW-4a} \\
H(Q)
&=
\!{\!\int\!}\dd{x} \!
\bigl \{
|\partial_x {\tilde{\psi}}|^2
-
\frac{g \, |{\tilde{\psi}}|^{2\kappa+2}}{\kappa + 1}
+
V_1(x) |{\tilde{\psi}}|^2
\bigr \} ,
\label{e:TW-4b} \\
R_{\mu}(Q)
&=
{{\rm i}}{\!\int\!}\dd{x}
V_2(x) \,
\bigl \{ \,
[\, \partial_\mu {\tilde{\psi}}^{\ast} \,] \, {\tilde{\psi}}-
{\tilde{\psi}}^{\ast} \, [\, \partial_\mu {\tilde{\psi}}\,] \,
\bigr \} \>.
\label{e:TW-4c}\end{aligned}$$
Here we have interchanged $\mu \leftrightarrow \nu$ in the definition of $I_{\mu\nu}(Q)$ from their Eq. (6) [@PhysRevE.82.016606]. So we see that $R_{\mu}(Q) \equiv - w_{\mu}(Q)$ and we find that
$$\begin{aligned}
\label{e:TW-5}
f_{\mu\nu}(Q)
&=
\partial_\mu \pi_\nu(Q) - \partial_\nu \pi_\mu(Q)
\\
&=
\frac{{{\rm i}}}{2} {\!\int\!}\dd{x}
\{ \,
[\, \partial_\mu {\tilde{\psi}}^{\ast} \, ]\,[\, \partial_\nu {\tilde{\psi}}\, ]
+
{\tilde{\psi}}^{\ast}\,[\, \partial_\mu \partial_\nu {\tilde{\psi}}\, ]
-
[\, \partial_\mu \partial_\nu {\tilde{\psi}}^{\ast} \,] \, {\tilde{\psi}}-
[\, \partial_\nu {\tilde{\psi}}^{\ast} \,] \, [\, \partial_\mu {\tilde{\psi}}\,]
-
[\, \partial_\nu {\tilde{\psi}}^{\ast} \, ]\,[\, \partial_\mu {\tilde{\psi}}\, ]
-
{\tilde{\psi}}^{\ast}\,[\, \partial_\nu \partial_\mu {\tilde{\psi}}\, ]
\notag \\
& \qquad\qquad
+
[\, \partial_\nu \partial_\mu {\tilde{\psi}}^{\ast} \,] \, {\tilde{\psi}}+
[\, \partial_\mu {\tilde{\psi}}^{\ast} \,] \, [\, \partial_\nu {\tilde{\psi}}\,]
\,\} \>,
\notag \\
&=
{{\rm i}}{\!\int\!}\dd{x}
\{ \,
[\, \partial_\mu {\tilde{\psi}}^{\ast} \, ]\,[\, \partial_\nu {\tilde{\psi}}\, ]
-
[\, \partial_\nu {\tilde{\psi}}^{\ast} \,] \, [\, \partial_\mu {\tilde{\psi}}\,]
\,\}
=
I_{\mu\nu}(Q) \>.
\notag\end{aligned}$$
In the notation used in the variational method, Eq. becomes $$\label{e:TW-6}
f_{\mu\nu}(Q) \, \dot{Q}^\nu
=
u_{\mu}(Q) - w_{\mu}(Q)
=
v_{\mu}(Q) \>.$$ So the generalized traveling wave approximation is identical to the variational method. The authors of Ref. [@PhysRevE.82.016606] proved this in another way in Sec. III of their paper for a simpler dissipative system.
|
---
abstract: 'We find a possibility of a weak universality of spin-glass phase transitions in three-dimensional $\pm J$ models. The Ising, the XY and the Heisenberg models seem to undergo finite-temperature phase transitions with a ratio of the critical exponents $\gamma/\nu \sim 2.4$. Evaluated critical exponents may explain corresponding experimental results. The analyses are based upon nonequilibrium relaxation from a paramagnetic state and finite-time scaling.'
address: |
Department of Applied Physics, Tohoku University,\
Aoba-yama 05, Sendai, Miyagi, 980-8579, Japan
author:
- 'Tota Nakamura, Shin-ichi Endoh and Takeo Yamamoto'
title: |
Weak universality of spin-glass transitions\
in three-dimensional $\pm J$ models
---
Introduction
============
A spin-glass (SG) phenomenon has been attracting great interest both theoretically and experimentally [@sgreview]. Applications now cover a wide range of interdisciplinary fields of statistical physics and informational physics, as treated in this special issue. However, many subjects are not well understood, in spite of efforts made over almost thirty years. One of these subjects is whether or not the SG transition of real materials can be explained by a simple random-bond spin model.
Spins of many SG materials are well-approximated by the Heisenberg spins. The simplest theoretical model is the Heisenberg model with random nearest-neighbour interactions. However, numerical studies suggest that there is no finite-temperature SG transition in this model [@mcmillan; @olive]. Kawamura [@chirality1; @chirality2] proposed the chirality mechanism in order to solve this discrepancy. The chiral-glass (CG) transition occurs without the SG order. A small but finite random anisotropy in the real materials mixes the chirality degrees of freedom and the spin degrees of freedom. This anisotropy effect induces the SG transition observed in the real materials. The scenario is based upon results that the SG transition does not occur in the isotropic model. However, Matsubara et al. [@matsubara1; @matsubara2; @matsubara3] recalculated the domain-wall excess energy and the SG susceptibility, from which they suggested that the finite-temperature SG transition does possibly occur. Methods are quite similar to the previous ones [@mcmillan; @olive]. Subtle differences in the analyses of the obtained data drew an opposite conclusion.
The spin-glass problem is one of the most difficult subjects in computational physics. It can be a tough bench-mark test for a new numerical method. It may be applied to other complex systems, if successful in the spin-glass investigations. The difficulty is caused by serious slow dynamics. It requires many Monte Carlo steps to reach the equilibrium states. An observed quantity at each step has a strong correlation even after the equilibration. The system sizes which can be treated in the simulations are accordingly limited to very small ones, e.g., mostly a linear size is twenty or less in three dimensions. Size effects are generally stronger in the continuous spin systems because the spins are soft and the boundary effect propagates faster. Frustration and randomness also yield a considerable size effect. The system sizes treated previously in the studies of the Heisenberg SG models are too small to extrapolate to the thermodynamic limit. This is our motivation for reexamining the SG transition using the nonequilibrium relaxation (NER) method[@ner1; @ner2; @ner3; @ner4; @huse; @blundell].
![A schematic diagram to approach the thermodynamic limit.[]{data-label="fig:gainen"}](gainen-black.eps){width="7cm"}
The difficulty mentioned above can be overcome by using the NER method. This method takes an opposite approach to the thermodynamic limit. schematically shows a comparison between the conventional equilibrium simulational method and the NER method. In the conventional method one takes the infinite time limit first by achieving the equilibrium states in finite sizes. The thermodynamic limit is taken by the finite-size scaling analysis of the obtained data. In the NER method we take the infinite size limit first by dealing with a very large system within a finite time range before the finite-size effect appears. Then, the [*finite-time scaling analysis*]{} [@timescaling1; @timescaling2] is performed to obtain the thermodynamic properties. The cost of a simulation is in the same order of $L^{d+z}$ for both methods. However, a coefficient factor in the NER method is much smaller than that in the conventional method. An observation time length in the NER method is sufficient if we can observe a beginning of a final relaxation to the equilibrium states (equilibrium relaxation). On the other hand, it is necessary to wait until the end of the equilibrium relaxation in the conventional method. The latter time scale is typically $10 \sim 10^2$ times longer than the former one in the spin glass models. (For example, $\chi_\mathrm{sg}$ of $L=17$ in (a) or $\chi_\mathrm{sg}$ at $T=0.56$ in (a).) Therefore, the NER method has an advantage over the conventional method by this factor. We use the residual computational time to enlarge the system size and to increase statistical accuracy.
By using the NER method we have made it clear that the SG transition occurs in the Heisenberg model at the same finite temperature as the CG transition occurs [@totasg1]. The estimated critical exponent $\gamma$ is consistent with the corresponding experimental result [@heisenexp]. The chirality mechanism is not necessary to explain the spin-glass experiments since the chirality trivially freezes if the spin freezes. However, one may question the use and the validity of the NER method in the spin-glass phenomenon. Therefore, we have corroborated our method by studying the Ising SG model. Many numerical investigations [@ogielski; @bhatt; @kawashimayoung; @palassini; @maricampbell] yield consistent results on the existence of the SG transition, the critical temperature and the critical exponents. They are also consistent with the corresponding experimental results [@isingsgexp]. The NER method yields consistent results for a small number of simulations as discussed in .
In this procedure we have found a possibility of a weak universality: a critical exponent divided by $\nu$, for example $\gamma/\nu$, is common among models in a weak universality class. A ratio of the critical exponents $\gamma/z\nu$ appearing in a finite-time scaling analysis is found to be consistent between the Heisenberg model and the Ising model. We have verified that a ratio $\gamma/\nu$ is also consistent by evaluating the dynamic exponent $z$ alone. The analysis is expanded to the XY SG model and the value is also found to be consistent. These findings are quite surprising. We must reconsider the role of the spin dimensions and the distribution of the randomness in the SG phase transition.
This paper is organised as follows: In the model and the method are explained. Descriptions of the procedure of the NER method and the finite-time scaling are given. In the results on the Ising model, the Heisenberg model and the XY model are shown. Then, the possibility of a weak universality is discussed. is devoted to a summary.
Model and Method - Nonequilibrium relaxation {#sec:model}
============================================
A model treated in this paper is the nearest-neighbour $\pm J$ random-bond model, $${\cal H}=\sum_{\langle i,j \rangle} J_{ij}
\mbox{\boldmath $S$}_i \cdot \mbox{\boldmath $S$}_j.$$ A linear size of a lattice is denoted by $L$. Skewed periodic boundary conditions are imposed, i.e., total numbers of spins $N=L\times L\times (L+1)$. An interaction $J_{ij}$ takes two values of $+J$ and $-J$ with the same probability. The temperature $T$ is scaled by $J$.
Spins are updated by a single-spin-flip algorithm. The Metropolis (M) update is used in all models, whereas the heat-bath (H) update [@olive] is used in the Heisenberg model. Physical quantities observed in our simulations are the SG susceptibility $\chi_\mathrm{sg}$, the CG susceptibility $\chi_\mathrm{cg}$ and the Binder parameter in regard to the spin-glass transition $g_\mathrm{sg}$. These quantities are calculated through the overlap between real replicas.
First, we rewrite the thermal average by an arithmetic mean over thermally equilibrium ensembles labelled by $\alpha$ as $$\langle \mbox{\boldmath $S$}_i \cdot \mbox{\boldmath $S$}_j \rangle
= \frac{1}{m}\sum_{\alpha=1}^{m}
\mbox{\boldmath $S$}_i^{(\alpha)}
\cdot
\mbox{\boldmath $S$}_j^{(\alpha)}
= \frac{1}{m}\sum_{\alpha=1}^{m}
\sum_{\mu}^{x,y,z}
S_{i,\mu}^{(\alpha)} S_{j,\mu}^{(\alpha)}.
\label{eq:thermalreplica}$$ The bracket $\langle \cdots \rangle$ denotes the thermal average and $m$ denotes a number of ensembles. The index $\mu$ stands for three components of spins: $x, y$ and $z$. This expression is substituted into the definition of the SG susceptibility: $$\chi_\mathrm{sg} =
\frac{1}{N}\sum_{i,j}\left[
\langle \mbox{\boldmath $S$}_i \cdot \mbox{\boldmath $S$}_j
\rangle^2
\right]_\mathrm{c}
= N \left[
\frac{1}{m^2}\sum_{\alpha, \beta}^{m}\sum_{\mu,\nu} ^{x,y,z}
(q^{\alpha\beta}_{\mu,\nu})^2
\right
] _\mathrm{c},$$ where $q_{\mu, \nu}^{\alpha \beta}
\equiv (1/N)\sum_i S_{i, \mu}^{(\alpha)} S_{i, \nu}^{(\beta)}$ is an overlap between the $\mu$ component of a spin $i$ on an ensemble $\alpha$: $S_{i, \mu}^{(\alpha)}$ and the $\nu$ component of the spin on an ensemble $\beta$: $S_{i, \nu}^{(\beta)}$. The bracket $[\cdots ]_{\rm c}$ denotes the configurational average.
Here, we introduce the following real replicas. Each real replica takes the same random bond configuration and the different paramagnetic initial spin state. They are updated in parallel with different random number sequences. This procedure corresponds to quenching from an infinite temperature. The thermal ensembles are realized by these real replicas which approach different equilibrium states. Therefore, we replace the thermal average by the average over these real replicas as equation . The indices $\alpha$ and $\beta$ now represent real replicas. We do not take into consideration a constant term which arises from the overlap between the same replica $\alpha=\beta$ and use the following expressions in the simulations. $$\begin{aligned}
\chi_\mathrm{sg}&=& N \left [
\frac{2}{m(m-1)}\sum_{\alpha > \beta}^{m}\sum_{\mu,\nu} ^{x,y,z}
(q^{\alpha\beta}_{\mu,\nu})^2 \right ] _\mathrm{c},
\\
\chi_{\rm cg}&=&
\frac{1}{3N} \left[\frac{2}{m(m-1)}\sum_{\alpha > \beta}^m \left(
\sum_{i, \phi}
C_{i, \phi}^{(\alpha)}
C_{i, \phi}^{(\beta)}\right)^2\right]_{\rm c}
,
\\
g_\mathrm{sg}&=&\frac{1}{2}\left(
A-B\frac{\displaystyle \sum_{\mu,\nu,\delta,\rho}\left[
\frac{2}{m(m-1)}\sum_{\alpha > \beta}^m
(q^{\alpha\beta}_{\mu,\nu})^2 (q^{\alpha\beta}_{\delta,\rho})^2
\right]_{\rm c}
}
{\displaystyle \left(\sum_{\mu,\nu}\left[
\frac{2}{m(m-1)}\sum_{\alpha > \beta}^m
(q^{\alpha\beta}_{\mu,\nu})^2
\right]_{\rm c}\right)^2}
\right).\end{aligned}$$ A number of replicas $m$ controls the precision of the thermal average. It is better to take a large value. We prepare eight or nine replicas for each bond configuration in this paper. The scalar chirality is defined by three neighbouring spins as $
C_{i, \phi}^{(\alpha)}=
\mbox{\boldmath $S$}_{i+\hat{\mbox{\boldmath $e$}}_{\phi}}^{(\alpha)}
\cdot
(
\mbox{\boldmath $S$}_{i}^{(\alpha)}
\times
\mbox{\boldmath $S$}_{i-\hat{\mbox{\boldmath $e$}}_{\phi}}^{(\alpha)}
),
$ where $\hat{\mbox{\boldmath $e$}}_{\phi}$ denotes a unit lattice vector along the $\phi$ axis. In the XY model we calculate the vector chirality, which is defined by $
C_{i,\phi}^{(\alpha)}=(1/2\sqrt{2})(
J_{ij}
\mbox{\boldmath $S$}_{i}^{(\alpha)} \times \mbox{\boldmath $S$}_{j}^{(\alpha)}
+J_{jk}
\mbox{\boldmath $S$}_{j}^{(\alpha)} \times \mbox{\boldmath $S$}_{k}^{(\alpha)}
+J_{kl}
\mbox{\boldmath $S$}_{k}^{(\alpha)} \times \mbox{\boldmath $S$}_{l}^{(\alpha)}
+J_{li}
\mbox{\boldmath $S$}_{l}^{(\alpha)} \times \mbox{\boldmath $S$}_{i}^{(\alpha)}
)|_z.
$ Indices $i,j,k,l$ denote four sites forming a square plaquette in the $\phi$ direction from the $i$ site. Constants in a definition of $g_\mathrm{sg}$ are: $A=3, B=1$ for the Ising model, $A=6, B=4$ for the XY model and $A=11, B=9$ for the Heisenberg model.
![A schematic flow of our simulation. Solid bonds and broken bonds in the lattice depict ferromagnetic bonds and antiferromagnetic bonds.[]{data-label="fig:method"}](method2.eps){width="15cm"}
shows a schematic diagram of the simulation procedure. We calculate a physical quantity at each time step $t$ and obtain a relaxation function. Another simulation starts by changing a random bond configuration, initial spin states and a random number sequence. Then, another relaxation function is obtained. Finally, we take an average of data at each step over these different Monte Carlo runs. It should be noted that the average is over independent data. It guarantees an absence of systematic error due to correlations of the observed quantity, which we usually encounter in the conventional Monte Carlo time average. The obtained raw relaxation function is utilised by the following finite-time scaling analysis.
The most important point in the NER method is to exclude the finite-size effect from the raw relaxation function. The method is based upon taking the infinite-size limit first. If a relaxation function includes a finite-size effect, it exhibits converging behaviour because every finite system has a definite equilibrium state. This behaviour misleads us into thinking that the temperature is in the paramagnetic phase even though it is the critical temperature. Therefore, the critical temperature is always underestimated if the size is insufficient. We check the size effect by changing the lattice sizes and always confirm a time range in which the size can be considered as infinity.
The SG susceptibility is expected to diverge at the critical temperature ($T_\mathrm{sg}$) as $\chi_\mathrm{sg}(t)\sim t^{\gamma/z\nu}$[@huse]. We obtain $T_\mathrm{sg}$, $\gamma$ and $z\nu$ by the finite-time scaling analysis on the relaxation functions of $\chi_\mathrm{sg}(t)$ in the paramagnetic phase.[@totasg1] Since the initial spin configuration is completely random, $\chi_\mathrm{sg}(t=0)\sim 1$. We start a set of simulations at a temperature $T$ that is obviously in the paramagnetic phase. The relaxation function $\chi_\mathrm{sg}(t)$ at this temperature increases with $t$ but soon converges to a finite value. As the temperature is lowered to approach the critical temperature, the relaxation function tends to show diverging behaviour. Since the temperature is still in the paramagnetic phase, the relaxation finally converges to a finite value after a correlation time $\tau(T)$. The spin-glass correlation increases with time and reaches the correlation length $\xi(T)$ after this correlation time. Two quantities relate with each other by $z$ as $\tau(T)\sim \xi^z(T)$. Therefore, the correlation time should diverge at $T_\mathrm{sg}$ as $$\tau(T)\sim (T-T_\mathrm{sg})^{-z\nu}.
\label{eq:taufit}$$ The correlation time can be estimated by scaling the raw relaxation function. We obtain $\gamma/z\nu$ and $\tau(T)$ so that the scaled functions $\chi_\mathrm{sg}(t) t^{-\gamma/z\nu}$ at all temperatures plotted against $t/\tau(T)$ fall onto a single curve. Then, the critical temperature and the exponent $z\nu$ are estimated by the least-squares fitting with the equation . Since a ratio $\gamma/z\nu$ is already estimated by the scaling, $\gamma$ is obtained.
The NER of the Binder parameter $g_\mathrm{sg}(t)$ is calculated at the obtained $T_\mathrm{sg}$. Since quantity is related to the fourth-order cumulant, many bond samples are necessary to obtain meaningful data. The number of bond configurations to obtain the results in this paper is summarised in . The Binder parameter is expected to diverge at $T_\mathrm{sg}$ as $g_\mathrm{sg}(t)\times L^d \sim t^{d/z}$[@blundell], by which $z$ is independently obtained. Then, $\nu$ is estimated from a value of $z\nu$ obtained by the $\tau$-fitting explained above. All exponents are now estimated by the scaling relation. It is possible to compare the critical exponents with the experimental results.
The last procedure of our method is to corroborate the results by observing the NER of $\chi_\mathrm{sg}$ at the obtained $T_\mathrm{sg}$. It should diverge as $t^{\gamma/z\nu}$ with the same exponent obtained by finite-time scaling. If the exponents are inconsistent, the scaling analysis is misled by an insufficient time range or by the finite-size effect. In the Ising model we perform another check at $T_\mathrm{sg}$ by observing the NER of the distribution function of the replica overlap, $P(q,t)$. The finite-time scaling plot of $P(q,t)$ should ride on a single scaling function with the same exponent obtained by finite-time scaling of $\chi_\mathrm{sg}$. This is a direct interpretation of the finite-size scaling of $P(q,L)$ [@bhatt] by $t\propto L^z$.
Results {#sec:results}
=======
Numbers of bond configurations to obtain data at the critical temperature are summarised in . The numbers at other temperatures are mostly in the same order. For each bond configuration, we prepared eight replicas for the XY and the Heisenberg model and nine replicas for the Ising model.
-------------------- --------------- ------ -------- -------- ---------------- -------- --------------- -------- ---------------
Step
Model Size $10^3$ $10^4$ $5 \cdot 10^4$ $10^5$ $5\cdot 10^5$ $10^6$ $4\cdot 10^6$
$\chi_\mathrm{sg}$ Ising 49 $\to$ $\to$ $\to$ 393 $\to$ $\to$ 88
XY 39 $\to$ $\to$ $\to$ 5246 120
Heisenberg(M) 59 $\to$ $\to$ $\to$ $\to$ $\to$ 104
Heisenberg(H) 89 $\to$ 58 $\to$ 22
$g_\mathrm{sg}$ Ising 39 255480 85480 18576 12626 $\to$ 1830 172
XY 19 $\to$ $\to$ $\to$ 7803
Heisenberg(H) 39 43114 18316 7038
-------------------- --------------- ------ -------- -------- ---------------- -------- --------------- -------- ---------------
: Numbers of bond configurations to obtain data of $\chi_\mathrm{sg}$ and $g_\mathrm{sg}$ at $T_\mathrm{sg}$ in this paper. Indices (M) and (H) in the Heisenberg model denote update algorithms: (M) for the Metropolis and (H) for the heat-bath. Arrows mean that the number is same as to the right.
\[tab:samplist\]
Ising model
-----------
![(a) The NER of $\chi_\mathrm{sg}$ of the Ising model at high temperatures. (b) The finite-time scaling plot for a choice of $\gamma/z\nu=0.39$. We obtain $\tau(T)$ and $\gamma/z\nu$ so that this scaling plot is good. The scaling is also possible for $\gamma/z\nu=0.38\sim 0.40$. (c) The least-squares fitting of $\tau(T)$ supposing $\tau(T)\propto |T-T_\mathrm{sg}|^{-z\nu}$.[]{data-label="fig:isingsg"}](isingsgnama.eps "fig:"){width="7.5cm"} ![(a) The NER of $\chi_\mathrm{sg}$ of the Ising model at high temperatures. (b) The finite-time scaling plot for a choice of $\gamma/z\nu=0.39$. We obtain $\tau(T)$ and $\gamma/z\nu$ so that this scaling plot is good. The scaling is also possible for $\gamma/z\nu=0.38\sim 0.40$. (c) The least-squares fitting of $\tau(T)$ supposing $\tau(T)\propto |T-T_\mathrm{sg}|^{-z\nu}$.[]{data-label="fig:isingsg"}](isingsgscale.eps "fig:"){width="7.5cm"}
![(a) The NER of $\chi_\mathrm{sg}$ of the Ising model at high temperatures. (b) The finite-time scaling plot for a choice of $\gamma/z\nu=0.39$. We obtain $\tau(T)$ and $\gamma/z\nu$ so that this scaling plot is good. The scaling is also possible for $\gamma/z\nu=0.38\sim 0.40$. (c) The least-squares fitting of $\tau(T)$ supposing $\tau(T)\propto |T-T_\mathrm{sg}|^{-z\nu}$.[]{data-label="fig:isingsg"}](isingsgtau.eps){width="7.5cm"}
shows an analysis to determine the critical temperature and the exponent. The simulation is performed just to check that our method gives results consistent with previous investigations [@ogielski; @bhatt; @kawashimayoung; @palassini; @maricampbell]. Therefore, the system size is very small ($L=19$) and the time range is very short. Finite-size effects are found to appear for $t> 5000$ by comparing with results of $L=29$. Only data before this time are used in the scaling analysis. (b) is an example of finite-time scaling. A choice of $\gamma/z\nu$ is possible for $\gamma/z\nu=0.38\sim 0.40$. A set of the correlation time at each temperature is estimated for each choice of this exponent. Then, the critical temperature is obtained as summarised in . As the exponent increases, $T_\mathrm{sg}$ decreases. We ignore a result of $\gamma/z\nu=0.400$ which deviates a lot from the others. Our estimates are $$T_\mathrm{sg}=1.17(4), \gamma/z\nu=0.3875(75), z\nu=9.3(12), \gamma=3.7(5).$$
$\gamma/z\nu$ $T_\mathrm{sg}$ $z\nu$ $\gamma$ $\chi^2$
--------------- ----------------- ----------- ---------- ----------
0.400 1.05 12.63(18) 5.1(1) 1.40
0.395 1.13 10.07(37) 4.0(2) 0.66
0.390 1.20 8.46(28) 3.3(1) 0.54
0.385 1.22 8.67( 9) 3.3(0) 1.49
0.380 1.21 9.82( 9) 3.7(0) 1.84
: A list of the critical temperature and an exponent $z\nu$ obtained by the finite-time scaling analysis in the Ising model. A ratio of exponents $\gamma/z\nu$ denotes the possible value in the finite-time scaling. The least-squares fitting errors are denoted by $\chi^2$.
\[tab:isingtsg\]
The results of the finite-time scaling analysis are checked by the raw NER data at the obtained critical temperature. (a) shows relaxation data of $\chi_\mathrm{sg}$ and $g_\mathrm{sg} \times L^d$. The SG susceptibility diverges algebraically with an exponent $\gamma/z\nu=0.38$ that is consistent with the scaling result $\gamma/z\nu=0.3875(75)$. The critical relaxation process begins around $t\sim 100$ and seems to continue to infinity. The Binder parameter also shows diverging behaviour with $t^{d/z}$, from which we obtain the dynamic exponent $z=6.2(2)$. Then, an exponent $\nu$ is estimated as $\nu=1.5(3)$. A ratio of the critical exponents $\gamma/\nu=2.4(1)$. The obtained results are consistent with previous numerical investigations [@ogielski; @bhatt; @kawashimayoung; @palassini; @maricampbell] and the corresponding experimental results [@isingsgexp] as summarised in . Since the lattice size and the time range are insufficient, the final numerical results have large error bars. As discussed in the previous section the critical temperature may be underestimated by using a small lattice. We plan to estimate them with high accuracy by large-scale NER analyses.
A time evolution of the distribution function of the overlap $P(q, t)$ at $T=T_\mathrm{sg}=1.17$ is shown in (b). The system size $L=17$. It exhibits a single Gaussian form with a peak at $q=0$ before the size effect of $\chi_\mathrm{sg}$ appears at $t=10^5$ as shown in (a). As the time increases, the width of the distribution grows in accordance with the divergence of the spin-glass susceptibility. It is possible to scale $P(q, t)/t^{\gamma/2z\nu}$ plotted versus $qt^{\gamma/2z\nu}$ for various time steps from $t=10$ to $t=10^4$ ( (c)). The critical exponent $\gamma/z\nu$ is also consistent with the finite-time scaling of $\chi_\mathrm{sg}$. The scaled data deviate a little for $t=10$ because the time is just before the relaxation of $\chi_\mathrm{sg}$ reaches the critical relaxation region as shown in (a). The distribution changes its shape to having two peaks at $\pm q_\mathrm{eq}$ after the finite-size effect appears. The shape is flat at this crossover time.
![(a) The NER of $\chi_\mathrm{sg}$ and $g_\mathrm{sg}\times L^d$ of the Ising model at $T_\mathrm{sg}=1.17$. Two lines, $t^{\gamma/z\nu}$ with $\gamma/z\nu=0.38$ and $t^{d/z}$ with $z=6.2$, are guides for eyes. (b) The NER of the distribution function $P(q,t)$ at $T_\mathrm{sg}$ for $L=17$. The shape changes from the single-peaked to the double-peaked when the size effect of $\chi_\mathrm{sg}$ appears at $t=10^5$. (c) The finite-time scaling plot of $P(q,t)$. (d) A three-dimensional plot of $P(q,t)$.[]{data-label="fig:pofq"}](namat117.eps "fig:"){width="7.5cm"} ![(a) The NER of $\chi_\mathrm{sg}$ and $g_\mathrm{sg}\times L^d$ of the Ising model at $T_\mathrm{sg}=1.17$. Two lines, $t^{\gamma/z\nu}$ with $\gamma/z\nu=0.38$ and $t^{d/z}$ with $z=6.2$, are guides for eyes. (b) The NER of the distribution function $P(q,t)$ at $T_\mathrm{sg}$ for $L=17$. The shape changes from the single-peaked to the double-peaked when the size effect of $\chi_\mathrm{sg}$ appears at $t=10^5$. (c) The finite-time scaling plot of $P(q,t)$. (d) A three-dimensional plot of $P(q,t)$.[]{data-label="fig:pofq"}](L17t117.eps "fig:"){width="7.5cm"}
![(a) The NER of $\chi_\mathrm{sg}$ and $g_\mathrm{sg}\times L^d$ of the Ising model at $T_\mathrm{sg}=1.17$. Two lines, $t^{\gamma/z\nu}$ with $\gamma/z\nu=0.38$ and $t^{d/z}$ with $z=6.2$, are guides for eyes. (b) The NER of the distribution function $P(q,t)$ at $T_\mathrm{sg}$ for $L=17$. The shape changes from the single-peaked to the double-peaked when the size effect of $\chi_\mathrm{sg}$ appears at $t=10^5$. (c) The finite-time scaling plot of $P(q,t)$. (d) A three-dimensional plot of $P(q,t)$.[]{data-label="fig:pofq"}](L17t117scale.eps "fig:"){width="7.5cm"} ![(a) The NER of $\chi_\mathrm{sg}$ and $g_\mathrm{sg}\times L^d$ of the Ising model at $T_\mathrm{sg}=1.17$. Two lines, $t^{\gamma/z\nu}$ with $\gamma/z\nu=0.38$ and $t^{d/z}$ with $z=6.2$, are guides for eyes. (b) The NER of the distribution function $P(q,t)$ at $T_\mathrm{sg}$ for $L=17$. The shape changes from the single-peaked to the double-peaked when the size effect of $\chi_\mathrm{sg}$ appears at $t=10^5$. (c) The finite-time scaling plot of $P(q,t)$. (d) A three-dimensional plot of $P(q,t)$.[]{data-label="fig:pofq"}](L17t117.3D.eps "fig:"){width="7.5cm"}
It is found that the NER function knows the critical phenomenon from its very early time steps: $t=10\sim 100$. The NER method is now clearly shown to be applicable to the spin glass phenomenon.
Heisenberg model {#sec:result.heisen}
----------------
We apply the same analysis performed in the Ising model to the Heisenberg model. Finite-time scaling results have already been shown briefly in Ref. [@totasg1] and the detailed analysis will be reported elsewhere. The system size is $L=59$ and the time scale is 70000 Monte Carlo steps. Typical numbers of bond configurations are same as given in . The finite-time scaling results are $$T_\mathrm{sg}=0.20(2), \gamma/z\nu=0.39(5), z\nu=4.8(10), \gamma=1.9(5).$$ These results are checked by the raw NER at $T=0.21$ as shown in . The SG susceptibility diverges algebraically with an exponent $\gamma/z\nu=0.38$, which is consistent with the result of the finite-time scaling. We performed simulations of both Metropolis update and heat-bath update. The Metropolis result denoted by (M) and the heat-bath result denoted by (H) exhibit the same critical behaviour, while the amplitudes are different by a factor of 3.5. NER behaviours are independent from the update algorithm. The consistency supports the criticality of the SG order at this temperature. It is noted that the critical divergence begins at a very early time: $t\sim 100$.
The Binder parameter exhibits a critical divergence $t^{d/z}$ with $z=6.2(5)$. The value is consistent with that in the Ising model. Since $z\nu$ is obtained by finite-time scaling, $\nu$ is estimated as $\nu=0.8(2)$. A ratio of the critical exponent $\gamma/\nu= 2.3(3)$. The results are compared with an experimental result [@heisenexp] in . They are not inconsistent.
![The NER of $\chi_\mathrm{sg}$ and $g_\mathrm{sg}\times L^d$ of the Heisenberg model at $T_\mathrm{sg}=0.21$. Lattice sizes are $L=39$ for $g_\mathrm{sg}\times L^d$, $L=89$ for $\chi_\mathrm{sg}$(H) and $L=59$ for $\chi_\mathrm{sg}$(M). Indices (H) and (M) denote the heat-bath updated and the Metropolis updated, respectively. Lines, $t^{\gamma/z\nu}$ with $\gamma/z\nu=0.38$ and $t^{d/z}$ with $z=6.2$, are guides for eyes.[]{data-label="fig:heisen"}](namat021.eps){width="7.5cm"}
XY model
--------
It has been considered that there is no SG transition in this model [@xynosg1; @xynosg2]. Only the CG transition with respect to the vector chirality is considered to occur [@xychiral1; @xychiral2; @xychiral3]. However, a possibility of the SG transition has recently been identified by several investigations [@xyoccurs1; @xyoccurs2; @xyoccurs3]. We applied the NER analysis on this model and our result supports the latter conclusion: the SG transition occurs.
Our finite-time scaling analysis on the XY model is not yet conclusive in regard to whether the SG transition and the CG transition occur at the same temperature or not. A system size ($L=39$), Monte Carlo steps ($10^5$) and a temperature range used in the scaling analysis are insufficient to extract a conclusion. However, as we increase the size and the steps, both critical temperatures seem to approach each other: $T_\mathrm{sg}$ increase from low and $T_\mathrm{cg}$ decrease from high. Therefore, we consider that both transitions occur simultaneously. Investigations are now being carried out and the details will be reported elsewhere. What has now been made clear is that both transitions occur in a temperature range of $ 0.4 < T < 0.46$. In this paper we do not examine the issue of simultaneous transition but focus on the existence of the SG transition.
\(a) shows raw NER plots of $\chi_\mathrm{sg}$ and $\chi_\mathrm{cg}$ near and above the critical temperature. There is no difference in $\chi_\mathrm{sg}$ between $T=0.43$ and $T=0.46$ within the present time steps. They exhibit a critical divergence with the same exponent and amplitude. The SG transition is considered to occur near $T=0.43$. From the slope we obtain an exponent $\gamma/z\nu=0.35$. Note that this value is a little smaller than that of the Ising model and the Heisenberg model ($\gamma/z\nu \sim 0.38$). The NER of the Binder parameter is shown in (b). It exhibits a critical divergence with an exponent $d/z$ with $z=6.8(5)$, which is also a little larger than that of the other models. However, we obtain a ratio of the critical exponents $\gamma/\nu=2.4(2)$, which is consistent with the other models.
![(a) The NER of $\chi_\mathrm{sg}$ in the XY model above the critical temperature $T_\mathrm{sg}\sim 0.43$. Lattice size is $L=39$. A line $t^{\gamma/z\nu}$ with $\gamma/z\nu=0.35$ is a guide for eyes. (b) The NER of the Binder parameter multiplied by $L^d$. A line $t^{d/z}$ with $z=6.8$ is a guide for eyes.[]{data-label="fig:xy"}](newfig6a.eps "fig:"){width="7.5cm"} ![(a) The NER of $\chi_\mathrm{sg}$ in the XY model above the critical temperature $T_\mathrm{sg}\sim 0.43$. Lattice size is $L=39$. A line $t^{\gamma/z\nu}$ with $\gamma/z\nu=0.35$ is a guide for eyes. (b) The NER of the Binder parameter multiplied by $L^d$. A line $t^{d/z}$ with $z=6.8$ is a guide for eyes.[]{data-label="fig:xy"}](binderxy.eps "fig:"){width="7.5cm"}
Weak universality
-----------------
The SG transition occurs in all models as shown in the preceding subsections. A ratio of the critical exponents $\gamma/\nu$ takes a common value around 2.4. Therefore, there is a possibility of weak universality among these transitions. Not only a value of $\gamma/\nu$ but the NER functions themselves suggest that the transitions are qualitatively equivalent.
\(a) shows the NER functions of $\chi_\mathrm{sg}$ at the critical temperature for all the models treated in this paper. The data of the Heisenberg model with the Metropolis update are multiplied by a factor 3.5 in order to compare with a result of the Ising model and that of the Heisenberg model with the heat-bath update. These three NER functions are not distinguishable. If we take into account a correction-to-scaling term, the relaxation functions can be fitted from the first few steps (Bold lines in (a)) by an expression [@maricampbell]: $$At^{\gamma/z\nu}[1-B t^{-w/z}].$$ Here, exponents of the leading term are set $\gamma/\nu=2.356$ and $z=6.2$. The correction-to-scaling exponent $w=3$. Coefficient constants are $A=7.6$ and $B=0.7$. The same expression also fits the NER function of the XY model, but with the dynamic exponent $z=6.8$ and a constant $A=3.3$. The NER functions of the Binder parameter are shown in (b). If we multiply the result of the Ising model by a factor 2.2, it is indistinguishable from the curve of the Heisenberg model.
![(a) NER plots of the $\chi_\mathrm{sg}$ at $T_\mathrm{sg}$. Correction-to-scaling fittings are depicted by bold lines with $\gamma/\nu=2.356$, $w=3$, $z=6.2$ for the Ising/Heisenberg model and $z=6.8$ for the XY model. NER functions except the XY model are indistinguishable. The data of the Heisenberg model with the Metropolis update are multiplied by 3.5. (b) NER plots of the Binder parameter multiplied by $L^d$. The relaxation function of the Ising model multiplied by 2.2 coincides with that of the Heisenberg model with the heat-bath update.[]{data-label="fig:weakuniv"}](campbell.eps "fig:"){width="7.5cm"} ![(a) NER plots of the $\chi_\mathrm{sg}$ at $T_\mathrm{sg}$. Correction-to-scaling fittings are depicted by bold lines with $\gamma/\nu=2.356$, $w=3$, $z=6.2$ for the Ising/Heisenberg model and $z=6.8$ for the XY model. NER functions except the XY model are indistinguishable. The data of the Heisenberg model with the Metropolis update are multiplied by 3.5. (b) NER plots of the Binder parameter multiplied by $L^d$. The relaxation function of the Ising model multiplied by 2.2 coincides with that of the Heisenberg model with the heat-bath update.[]{data-label="fig:weakuniv"}](bindersg.eps "fig:"){width="7.5cm"}
Summary {#sec:summary}
=======
By applying the nonequilibrium relaxation method it has been made clear that the $\pm J$ models in three dimensions undergo finite-temperature spin-glass transitions. There is a possibility that these models belong to the same weak universality class with a ratio of the critical exponents $\gamma/\nu \sim 2.4$. We compare our results with other numerical results and the experimental results in . They agree well within the numerical errors. Since the error bars are rather large at present, further efforts to improve precision are necessary in order to prove weak universality.
The spin-glass transition of the Heisenberg model is found to be very similar to that of the Ising model. The relaxation functions of $\chi_\mathrm{sg}$ and $g_\mathrm{sg}$ and values of a ratio of the exponents $\gamma/\nu$ and the dynamic exponent $z$ are consistent between the two models. If one considers that the spin-glass transition occurs in the Ising model, it may be thought that it occurs in the Heisenberg model in the same accuracy. Only the dynamic exponent of the XY model differs from the other models. Spin-glass transition and weak universality in models with Gaussian bond distributions is a problem to be checked in future work. The type of bond distributions may be important.
The NER method has been shown to be particularly effective in the spin-glass study. Critical behaviour is observed from very early time steps even though it takes a very long time to achieve the equilibrium states. What is long is the nonequilibrium relaxation process after a short initial relaxation before the final equilibrium relaxation. This long process is discarded in conventional simulations, while it is utilised in the NER method. This is one reason why the NER method is advantageous in this system. Applications to various complex systems with slow dynamics are fruitful [@totaner1; @totaner2].
$T_\mathrm{sg}$ $\gamma$ $\nu$ $\gamma/\nu$ $z$
------------------------- ----------------- ------------ ------------- -------------- ----------
Ising SG
Present work $1.17(4)$ $3.6(6)$ $1.5(3)$ $2.4(1)$ $6.2(2)$
Ref. [@kawashimayoung] $1.11(4)$ 4.0(8) $1.7(4)$ 2.35(5)
Ref. [@maricampbell] $1.195(15)$ $2.95(30)$ $1.35(10)$ $2.225(25)$ 5.65(15)
Experiment[@isingsgexp] $4.0(3)$ $\sim 1.7 $ $\sim 2.4$
Heisenberg SG
Present work $0.20(2)$ $1.9(5)$ $0.8(2)$ $2.3(3)$ $6.2(5)$
Ref. [@matsubara3] 0.18(1) 2.0(2) 0.97(5) 2.1(1)
Experiment[@heisenexp] $2.3(4)$ 1.25(25) 2.0(7)
XY SG
Present work 0.43(3) $2.4(2)$ $6.8(5)$
: Estimates of $T_\mathrm{sg}$, $\gamma$, $\nu$, $\gamma/\nu$ and $z$ in the $\pm J$ models in three dimensions.
\[tab:explist\]
The authors would like to thank Professor Fumitaka Matsubara for guiding them in the spin-glass study and for their fruitful discussions. The author TN also thanks Professor Nobuyasu Ito and Professor Yasumasa Kanada for providing him with a fast random number generator RNDTIK. Computations were partly done at the Supercomputer Center, ISSP, The University of Tokyo.
References {#references .unnumbered}
==========
[99]{}
For a review, Binder K and Young A P 1986 801\
Mydosh J A 1993 [*Spin Glasses*]{} (Taylor & Francis, London, 1993)\
[*Spin Glasses and Random Fields*]{}, ed. A.P. Young (World Scientific, Singapore, 1997)
McMillan W L 1985 B [**31**]{} 342
Olive J A, Young A P and Sherrington D 1986 B [**34**]{} 6341
Kawamura H 1992 3785 Hukushima K and Kawamura H 2000 E [**61**]{} R1008
Matsubara F, Endoh S and Shirakura T 2000 1927 Endoh S, Matsubara F and Shirakura T 2001 1543 Matsubara F, Shirakura T and Endoh S 2001 B [**64**]{} 092412\
(Matsubara F, Shirakura T and Endoh S 2000 [*Preprint*]{} cond-mat/0011218)
Sadic A and Binder K 1984 [*J. Stat. Phys.*]{} [**35**]{} 517 Stauffer D 1992 [*Physica*]{} A [**186**]{} 197 Ito N 1993 [*Physica*]{} A [**196**]{} 591 Ito N and Ozeki Y 1999 [*Int. J. Mod. Phys.*]{} [**10**]{} 1495 Huse D A 1989 B [**40**]{} 304
Blundell R E, Humayun K and Bray A J 1992 L733
Ozeki Y and Ito N 2001 B [**64**]{} 024416 Ozeki Y, Ogawa K and Ito N 2003 E [**67**]{} 026702
Nakamura T and Endoh S 2002 2113
Vincent E and Hammann J 1987 2659
Ogielski A T 1985 B [**32**]{} 7384 Bhatt R N and Young A P 1985 924 Kawashima N and Young A P 1996 B [**53**]{} R484 Palassini M and Caracciolo S 1999 5128 Mari P O and Campbell I A 2002 B [**65**]{} 184409
Gunnarsson K, Svedlindh P, Nordblad P, Lundgren L, Aruga H and Ito A 1991 B [**43**]{} 8199
Morris B W, Colborne S G, Moore M A, Bray A J and Canisius J 1986 1157 Jain S and Young A P 1986 3913 Kawamura H and Tanemura M 1987 B [**36**]{} 7177 Kawamura H and Tanemura M 1991 608 Kawamura H and Li M S 2001 187204 Maucourt J and Grempel D R 1998 770 Akino N and Kosterlitz J M 2002 B [**66**]{} 054536 Lee L W and Young A P 2003 227203
Shirahata T and Nakamura T 2002 B [**65**]{} 024402 Nakamura T 2003 789
|
---
abstract: 'We present a novel algorithm for simulating heterogeneous road-agents such as cars, tricycles, bicycles, and pedestrians in dense traffic by computing collision-free navigation. Our approach computes smooth trajectories for each agent by taking into account the dynamic constraints. We describe an efficient optimization-based algorithm for each road-agent based on reciprocal velocity obstacles that takes into account kinematic and dynamic constraints. Our algorithm uses tight fitting shape representations based on medial axis to compute collision-free trajectories in dense traffic situations. We evaluate the performance of our simulation algorithm in real-world dense traffic scenarios and highlight the benefits over prior reciprocal collision avoidance schemes.'
author:
- Yuexin Ma
- Dinesh Manocha
- Wenping Wang
bibliography:
- 'sample-bibliography.bib'
title: 'AutoRVO: Local Navigation with Dynamic Constraints in Dense Heterogeneous Traffic'
---
<ccs2012> <concept> <concept\_id>10010147.10010341.10010349</concept\_id> <concept\_desc>Computing methodologies Simulation types and techniques</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010341.10010349.10010359</concept\_id> <concept\_desc>Computing methodologies Real-time simulation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010341.10010349.10011810</concept\_id> <concept\_desc>Computing methodologies Artificial life</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
|
Ecole Centrale Paris\
Laboratoire EM$^2$C\
**Radiative transfer and atom transport\
Transfert radiatif et transport d’atomes\
Postdoc Report of\
Carsten Henkel\
Mars–Juillet 1997 (preprint date: May 2005)**
**Foreword to the preprint version**
This report summarizes a post-doctoral training period of four months that I spent, eight years ago, in the group working on Propagation and Scattering of Electromagnetic Waves that is led by Jean-Jacques Greffet (Ecole Centrale Paris). The report deals with classical and quantum descriptions of particles that interact with smooth random potentials, for example ultracold atoms in the dipole potential of an optical speckle pattern. In addition, a discussion of the link between Radiative Transfer theory and the underlying coherence theory of wave optics is presented. The Radiative Transfer Equation is shown to emerge as a limiting case of the Bethe-Salpeter Equation, and some next-order corrections are discussed.
I decided to put this material on the arxiv because of a recent explosion of interest in the transport of ultracold atoms in the multiple scattering regime. Also a number of colleagues have benefited from the parts dealing with Radiative Transfer. I hope that these pages may be useful for other interested scientists as well.
The report is written in French which may cause difficulties, although only basic vocabulary is used, and the technical terms are mainly identical to the English ones (some English equivalents are given in footnotes, if needed). Just keep in mind that the French *diffusion* means ‘scattering’, while *diffusion spatiale* is used for ‘diffusion’. In case you really get stuck, please do not hesitate to contact me at
carsten.henkel@physik.uni-potsdam.de
C. Henkel, Potsdam, 03 May 2005.
**Remerciements**
Je tiens d’abord à remercier Jean-Jacques Greffet de m’avoir accueilli au Laboratoire EM2C de l’Ecole Centrale Paris, ainsi d’avoir pu jouir d’excellentes conditions de travail.
Merci également aux autres membres de son groupe d’électro-magnétisme : aux thésards Olivier Calvo, Laurent Roux, Jean-Baptiste Thibaud, à Rémi Bussac, Pedro Valle et Anne Sentenac. Merci aussi aux étudiants du groupe des combustionnistes, et plus particulièrement à Metta et Katja, Sébastien Pax(ion), Olivier Delabroy, Manuel et Ulrich.
Merci à Gilbert pour l’assistance avec le réseau d’ordinateurs. Merci à Caroline et Stéphanie pour votre aide avec les démarches administratives.
C’est un temps agréable que j’ai passé avec vous.
Introduction {#introduction .unnumbered}
============
Le problème {#le-problème .unnumbered}
-----------
Le présent rapport résume les quelques idées que j’ai pu explorer au sujet du transport, que ce soit de lumière ou d’atomes. Ce sujet est né d’une question apparue vers la fin de ma thèse :
> lorsque des atomes de basse température traversent un champ lumineux de tavelures, qu’arrive-t-il à leur distribution de vitesse ? Et qu’en est-il de la cohérence spatiale des ondes de [de Broglie]{} associée aux atomes ?
Une telle question a un intérêt immédiat pour ce que l’on appelle l’optique atomique, qu’il s’agisse de l’imagerie d’atomes ou de l’interférométrie. Le groupe d’optique atomique à l’Institut d’Optique à Orsay a pu observer dans l’expérience du miroir à atomes [@Landragin96b] que la traversée du champ de tavelures élargit la distribution des vitesses transverses, perpendiculaires à la vitesse moyenne des atomes. Dans une telle expérience, des champs de tavelures apparaissent à cause de faisceaux lumineux parasites ou bien par diffusion sur une surface rugueuse. L’élargissement de la vitesse transverse dépend en particulier de la quantité de lumière parasite. Dans le cas le moins favorable, il empêchait d’observer la diffraction d’atomes par un réseau en réflexion parce que la largeur angulaire du faisceau atomique devenait trop importante.
D’un point de vue théorique, le mouvement d’une onde de matière dans un champ de tavelures se résume à un problème de propagation d’une onde dans un milieu aléatoire où l’onde subit des multiples collisions élastiques. Sous cet élairage, l’on conçoit aisément qu’il y a un grand nombre de situations physiques où apparaît un problème analogue :
- la diffusion de la lumière dans un milieu fortement diffusant (atmosphères stellaire ou terrestre, liquides, peintures, tissus biologiques) ;
- le transport des électrons dans un solide réel (avec des défauts et des vibrations thermiques), avec des applications pour la conductivité ;
- la diffusion des neutrons dans les réacteurs à fission ;
- la propagation des ondes sismiques à travers le globe terrestre ;
- et d’autres exemples encore (arrêt d’un faisceaux de particules radioactives dans une matrice solide, mouvement de boules dans un terrain rugueux, marche au hasard d’une particule, mouvement d’atomes dans un champ laser résonnant ...)
Il est donc naturel d’explorer les analogies avec ces problèmes pour trouver des méthodes qui permettent de répondre aux questions posées.
Approches théoriques {#approches-théoriques .unnumbered}
--------------------
Une caractéristique commune de ces méthodes théoriques est qu’elles sont des [*approches statistiques*]{} : au lieu de décrire le mouvement d’une particule (ou d’une onde) donnée dans un milieu aléatoire donnée, l’on se résigne à calculer un [*mouvement moyen*]{}, que l’on espère typique pour une mesure portant sur un grand nombre de particules (ondes). L’on construit ainsi une [*théorie de transport*]{} pour les particules (ondes). La quantité dont on calcule la valeur moyenne dépend de la situation expérimentale que l’on envisage : une fonction d’onde moyenne si celle-ci interfère avec un oscillateur local, c’est-à-dire une onde de référence avec une phase fixée ; une fonction de cohérence de l’onde si l’on étudie une figure d’interférence moyennée sur un grand nombre de réalisations du système ; ou encore une fonction de corrélation d’ordre quatrième (corrélations d’intensité), lorsque l’on s’intéresse par exemple aux propriétés de la figure d’interférence créée par une seule réalisation de l’expérience (cette situation s’applique à l’interférence entre deux condensats de [Bose–Einstein]{} ou encore aux tavelures lumineuses d’un faisceau laser cohérent).
Les exemples que l’on vient de donner s’inscrivent essentiellement dans un cadre statistique [*ondulatoire*]{}, où les objets de base sont des ondes (ou champs), comme les ondes de matière ou lumineuses. Il existe également des problèmes où ce sont des [*particules*]{} qui sont diffusées dans un milieu aléatoire. Dans ce cas, il n’a pas de sens de parler d’ondes ou d’interférence : la description naturelle des moyennes du système passe simplement par une densité spatiale ou une distribution des vitesses. L’on est habitué à une telle approche par la théorie cinétique des gaz dont le point de départ est l’équation de [Boltzmann]{} : cette équation décrit comment la fonction de distribution d’un ensemble de particules évolue en présence d’une force extérieure d’une part, et des collisions entre particules d’autre part. Un autre exemple, un peu moins habituel, est celui du transfert radiatif : l’on considère alors le transport de l’énergie lumineuse dans un milieu diffusant (et/ou absorbant), en décrivant cette énergie par sa distribution en fréquence, en position et direction de propagation (la luminance). L’équation de base de cette théorie ressemble à celle de [Boltzmann]{} et s’appelle celle du transfert radiatif ; elle a été introduite au début du siècle et étudiée abondamment par [Chandrasekhar]{} dans un contexte astrophysique, pour modéliser les atmosphères des étoiles [@Chandrasekhar]. Les ingrédients de cette équation sont les propriétés d’absorption et de diffusion du milieu. Pour des atomes dans un potentiel aléatoire, une telle approche serait qualifiée de classique, en opposition à une approche quantique où l’on manipule des ondes de matière. Pourtant, comme on sait que ce sont ces dernières qui ont une réalité plus fondamentale, l’on peut se poser la question du rapport entre ces deux approches théoriques : comment se justifie l’approche corpusculaire à partir de l’approche ondulatoire ? Ou encore : dans quelle limite peut-on se contenter d’une approche corpusculaire, et où vont se manifester des phénomènes proprement ondulatoires ?
Petit historique {#petit-historique .unnumbered}
----------------
Ces questions ont été poursuivies dans le contexte [**optique**]{} depuis les années 1960 environ. Il s’agit de fonder l’équation du transfert radiatif (description quasi-corpusculaire, absence d’interférences) sur la théorie de la diffusion multiple des ondes dans les milieux aléatoires. C’est maintenant que l’on se rend compte du travail énorme des physiciens soviétiques dans ce domaine [@Kravtsov96]. Il est devenu clair que l’approche du transfert radiatif est valable pour des champs lumineux dont les profils spatiaux varient lentement à l’échelle de leur longueur de cohérence (des champs quasi-uniformes). Cette approche n’est plus valable, par exemple, pour rendre compte de la rétro-diffusion exaltée (“[*coherent backscattering*]{}”), ou encore de la localisation (forte) des ondes dans un milieu désordonné. Dans ce rapport, nous allons nous appuyer abondamment sur l’analogie avec la diffusion de la lumière. En particulier, nous allons élaborer un peu les conditions de validité de l’équation du transfert radiatif.
Dans le domaine du transport des [**électrons**]{} dans les solides, le développement s’est poursuivi de façon plutôt indépendante, semble-t-il. Le travail de [P.-W. Anderson]{} sur la localisation des électrons dans un réseau désordonné (“[*Absence of diffusion in certain random lattices*]{}”, [@Anderson58]) a été d’un impact remarquable parce qu’il a donné un exemple où les interférences ont des conséquences dramatiques pour le transport (une propriété plutôt macroscopique). Depuis, d’autres phénomènes d’origine ondulatoire ont été étudiés, comme par exemple les fluctuations de conductance universelles. Un temps assez long s’est écoulé jusqu’à ce que l’on prenne conscience que la localisation des électrons provient simplement de leur caractère ondulatoire, et que la localisation existe par conséquent également pour les ondes lumineuses (voir [@vanTiggelen96]). C’est dans ce contexte qu’a été prédite et observée la rétro-diffusion exaltée de la lumière, un autre effet au-delà de l’équation standard du transfert radiatif.
En ce qui concerne les [**neutrons**]{}, les recherches intenses dans les réacteurs nucléaires ont rapidement conduit à une théorie du transport par analogie avec le transfert radiatif. Des formalismes très divers ont été utilisés, par physiciens et mathématiciens, pour la justifier à partir d’une approche microscopique [@Overhauser59; @Osborn65; @Riska68; @Bobker79]. La conclusion actuelle semble que l’équation du transfert radiatif est valable aux grandes échelles spatiales et pour des énergies pas trop basses.
Plan du rapport {#plan-du-rapport .unnumbered}
---------------
Ce rapport contient trois chapitres dont le [**premier**]{} présente l’équation du transfert radiatif pour la lumière d’un point de vue phénoménologique. Nous étudions en particulier la limite où le milieu diffuseur favorise la diffusion vers l‘avant (les petits angles de diffusion). Dans ce régime, l’on peut reformuler le transfert radiatif à l’aide d’une équation de [Fokker–Planck]{}.
Nous retrouverons cette approche au [**second**]{} chapitre qui est consacré au point de vue classique du transport d’atomes dans un potentiel aléatoire. Nous montrons que c’est aux échelles spatiales grandes par rapport à la longueur de corrélation du potentiel, que la distribution moyenne des atomes dans l’espace des phases vérifie une équation de [Fokker–Planck]{}. Nous en calculons le tenseur de diffusion des vitesses. Aux temps encore plus longs (pour des atomes très lents), la distribution spatiale atomique est décrite par une équation de diffusion spatiale dont nous déterminons le coefficient de diffusion.
Le [**troisième**]{} chapitre étudie le point de vue ondulatoire : la diffusion multiple des ondes de matière dans un milieu aléatoire. Nous introduisons les concepts statistiques pertinents ainsi que leurs équations d’évolution. Le fil rouge du chapitre est de détailler le passage vers une approche du type transfert radiatif. Le problème est présenté de façon à couvrir aussi bien la diffusion de la lumière que celle des atomes. Nous mettons en évidence l’échelle spatiale minimale au-dessus de laquelle l’équation du transfert radiatif est équivalente aux approches ondulatoires statistiques.
Transfert radiatif {#s:radiatif}
==================
Une petite introduction dans la théorie du transfert radiatif. La lumière y est décrite par la [*luminance*]{}. Nous étudions un peu plus en détail la situation d’une diffusion piquée vers l’avant, où l’équation du transfert radiatif peut être écrite sous la forme d’une équation de [Fokker–Planck]{}. Nous verrons au chapitre suivant qu’une équation identique apparaît pour le transport d’atomes. On illustre de cette façon le passage entre des théories ondulatoire et corpusculaire.
La luminance
------------
La quantité centrale pour décrire la lumière dans le transfert radiatif est la [*luminance*]{} ${\cal I}({\bf r}, {\bf u})$ : elle représente la puissance rayonnée ${\rm d}{\cal F}$ par un élément de surface ${\rm d}\sigma$ situé à la position ${\bf r}$, dans un élément d’angle solide ${\rm d}\Omega$ autour de la direction caractérisée par le vecteur unitaire ${\bf u}$ (voir le chapitre 5.7 de [Mandel]{} et [Wolf]{} [@MandelWolf] et le chapitre 4.3 de [Rytov, Kravtsov]{} et [Tatarskii]{} [@Rytov]): $${\rm d}{\cal F} = {\cal I}({\bf r}, {\bf u}) {\rm d}\sigma \cdot
{\rm d}\Omega
\label{eq:def-luminance}$$ L’équation d’évolution pour la luminance est celle du transfert radiatif: $${\bf u}\cdot\nabla_{r} {\cal I} =
\underbrace{ S({\bf r}, {\bf u}) }_{ \mbox{\scriptsize sources} }
- \underbrace{ \beta({\bf r}) {\cal I} }_{ \mbox{\scriptsize att\'enuation} }
+ \frac{ 1 }{ \ell_{\rm ex}( {\bf r} ) }
\int\limits_{4\pi} \underbrace{ {\rm d}\Omega' \,
p({\bf u}, {\bf u}'; {\bf r}) {\cal I}({\bf r}, {\bf u}')
}_{ \mbox{\scriptsize re-diffusion} }
\label{eq:e-t-r}$$ Cette équation traduit les processus suivants:
- l’énergie lumineuse est transportée dans la direction ${\bf u}$ le long des rayons géométriques ;
- la lumière peut être émise par une source $S({\bf r}, {\bf u})$ dans la direction ${\bf u}$ ;
- elle peut être atténuée par de l’absorption ou de la diffusion. Ce processus est caractérisé par un coefficient $\beta({\bf r})$ ;
- la diffusion redistribue les directions de l’intensité lumineuse. Ce processus est décrit par le libre parcours moyen[^1] $\ell_{\rm ex}( {\bf r} )$ (égal à la section efficace totale par unité de volume) et la fonction de phase $p({\bf u}, {\bf u}'; {\bf r})$ que l’on peut interpréter comme une section efficace différentielle de diffusion pour un processus où un rayon venant d’une direction ${\bf u}'$ est diffusé dans la direction ${\bf u}$.
Hypothèses pour le transport radiatif
-------------------------------------
1. L’on décrit le milieu à une [*échelle spatiale grande devant la longueur d’onde optique*]{} (pour pouvoir définir une puissance rayonnée dans une direction par unité de surface). Ceci implique que les coefficients $\beta({\bf r})$ et $f({\bf u}, {\bf u}'; {\bf r})$ et bien sûr la luminance ${\cal I}({\bf r}, {\bf u})$ elle-même varient lentement, en fonction de la position ${\bf r}$, à l’échelle de la longueur d’onde optique.
2. La diffusion de la lumière est supposée [*élastique*]{}: c’est seulement la direction de la lumière, mais non pas sa fréquence qui est modifiée par le milieu. Dans cette hypothèse, l’on peut formuler une équation du transport radiatif pour chaque fréquence lumineuse séparément. Il est cependant immédiat de rendre compte de la diffusion inélastique en introduisant une section efficace inélastique et une dépendance de la luminance de la fréquence.
Relation à une équation de
---------------------------
Afin de comparer l’équation du transport radiatif (\[eq:e-t-r\]) au transport d’atomes, une petite modification: nous allons en déduire une équation de type [Fokker–Planck]{} que nous retrouverons également pour les atomes. A cette fin, nous supposons que la re-diffusion de la lumière s’effectue surtout autour de la direction vers l’avant, c’est-à-dire que dans la fonction de phase $f({\bf u}, {\bf u}'; {\bf r})$, le vecteur diffusé ${\bf u}$ pointe dans une direction voisine du vecteur incident ${\bf u}'$. Une telle situation correspond, par exemple, à la diffusion par des inhomogénéités d’indice, lorsque celles-ci sont beaucoup plus grandes que la longueur d’onde. Ce cas se présente fréquemment dans la diffusion de la lumière par des tissus biologiques.
Dans la limite d’une diffusion piquée vers l’avant, nous pouvons développer la luminance ${\cal I}({\bf r}, {\bf u}')$ au troisième terme dans (\[eq:e-t-r\]) en fonction de ${\bf u}'$. Ceci faisant, l’on obtient l’expression suivante pour le terme de re-diffusion : $$\mbox{re-diffusion} \to
\frac{ 1 }{ \ell_{\rm ex}( {\bf r} ) }
\, {\cal I}({\bf r}, {\bf u})
+ \sum_{\alpha} F_\alpha({\bf r}) \partial_{u_\alpha}
{\cal I}({\bf r}, {\bf u})
+ \sum_{\alpha\beta} D_{\alpha\beta}({\bf r})
\partial_{u_\alpha} \partial_{u_\beta}
{\cal I}({\bf r}, {\bf u})
\label{eq:vers-FP}$$ où $u_{\alpha,\beta}$ désignent les directions perpendiculaires au vecteur ${\bf u}$. (Ce sont seulement ces directions-ci qui interviennent dans la dérivée, étant donné que ${\bf u}$ est un vecteur unitaire.) Dans (\[eq:vers-FP\]), $F_\alpha({\bf r})$ est la force moyenne et $D_{\alpha\beta}({\bf r})$ le tenseur de diffusion. Ils sont donnés par $$\begin{aligned}
F_\alpha({\bf r}) &=& \frac{ 1 }{ \ell_{\rm ex}( {\bf r} ) }
\int\limits_{4\pi} \! {\rm d}\Omega' \,
(u'_\alpha - u_\alpha) \, p({\bf u}, {\bf u}'; {\bf r})
\label{eq:def-force}\\
D_{\alpha\beta}({\bf r}) &=& \frac{ 1 }{ 2 \ell_{\rm ex}( {\bf r} ) }
\int\limits_{4\pi} \! {\rm d}\Omega' \,
(u'_\alpha - u_\alpha) (u'_\beta - u_\beta)
\, p({\bf u}, {\bf u}'; {\bf r})
\label{eq:def-coeff-diffusion}\end{aligned}$$ Notons qu’une telle formulation du transfert radiatif a été étudiée en détail par [G. C. Pomraning]{} [@Pomraning95], dans le contexte de l’arrêt d’un faisceau de particules à l’intérieur d’un solide diffusant. Le groupe de [J. M. Luck]{} l’a également utilisée pour trouver des solutions analytiques dans une géométrie planaire [@Luck96].
Lorsque le milieu diffuseur est statistiquement isotrope (la fonction de phase ne dépend que de $\cos\theta \equiv {\bf u}\cdot{\bf u}'$), la force $F_\alpha( {\bf r} )$ s’annule et le tenseur $D_{\alpha\beta}( {\bf r} )$ est proportionnel au [Kronecker]{} $\delta_{\alpha\beta}$: $$\mbox{milieu isotrope}:\quad
D_{\alpha\beta}( {\bf r} )
= \frac{\delta_{\alpha\beta}}{4\ell_{\rm ex}( {\bf r} )}
\langle \sin^2\theta \rangle_p
\approx
\frac{\delta_{\alpha\beta}}{4\ell_{\rm ex}( {\bf r} )}
\langle \theta^2 \rangle_p
\approx
\frac{\delta_{\alpha\beta}}{2\ell^*_{\rm trans}( {\bf r} )}$$ où $\ell^*_{\rm trans} = \ell_{\rm ex}
/ ( 1 - \langle \cos\theta \rangle )$ est le libre parcours moyen de transport. Nous constatons donc que les éléments du tenseur de diffusion correspondent à la valeur quadratique moyenne de la déviation angulaire par unité de longueur des rayons. Ceci appelle à une interprétation physique de l’équation de [Fokker–Planck]{} (\[eq:vers-FP\]).
Mouvement diffusif des directions des rayons
--------------------------------------------
Introduisons les angles $\theta, \phi$ pour le vecteur unitaire ${\bf u}$ et supposons que le milieu diffusant est sans sources, non-absorbant et isotrope. Le tenseur de diffusion s’écrit alors $D_{\alpha\beta} =
D \,\delta_{\alpha\beta}$ et l’équation de [F.–P.]{} pour la luminance devient $${\bf u} \cdot \nabla_r {\cal I}({\bf r}, \theta, \phi) =
D \left[
\frac{ 1 }{ \sin \theta }
\frac{ \partial }{ \partial \theta }
\left( \sin\theta \frac{ \partial }{ \partial \theta } \right)
+
\frac{ 1 }{ \sin^2\theta } \frac{ \partial^2 }{ \partial\phi^2 }
\right]
{\cal I}({\bf r}, \theta, \phi)
\label{eq:FP-sphere}$$ Les dérivées angulaires correspondent au Laplacien sur la sphère. Leurs fonctions propres sont donc les polynômes de [Legendre]{} $P_n(\cos\theta) {\rm e}^{ i m \phi}$. Pour une luminance qui est initialement collimatée autour de la direction $\theta = 0$ (l’axe $Oz$), [Frisch]{} donne alors la solution explicite suivante ([@Frisch66], éq. (19.20)) $$\begin{aligned}
&& {\cal I}(z, \theta, \phi) = \frac{ 1 }{ 4\pi }
\sum\limits_{n = 0}^{\infty} ( 2 n + 1 )
P_n(\cos\theta) \, {\rm e}^{ - n ( n + 1 ) D z }
\label{eq:solution-Frisch}\\
&& {\cal I}(z = 0, \theta, \phi) = \delta(\theta)
\nonumber\end{aligned}$$ On en déduit que les structures angulaires fines (les harmoniques sphériques supérieurs avec $n$ grands) sont amorties plus rapidement que les variations angulaires molles. Dans la limite $z \to \infty$ d’un milieu profond, il ne subsiste plus qu’une distribution angulaire isotrope ($n = 0$). La distance caractéristique $L$ pour l’isotropisation de la luminance est donnée par l’inverse du coefficient de diffusion $$L \sim \frac{ 1 }{ D } = \frac{ \ell^*_{\rm trans} }{ 2 }
\label{eq:c-est-isotrope}$$ Pour une fonction de phase très piquée vers l’avant, cette distance (le libre parcours moyen de transport) est beaucoup plus grande que la longueur d’atténuation ($\sim \ell_{\rm ex}$) de la partie collimatée du faisceau lumineux. La solution explicite (\[eq:solution-Frisch\]) permet donc d’apprécier la transition entre un transport balistique (distribution angulaire piquée) et un transport diffusif (distribution angulaire isotrope). Ce qui lui manque, c’est de prendre en compte le profil spatial de l’intensité incidente que l’on suppose en effet uniforme.
A titre d’exemple, considérons l’évolution de la directivité de la luminance ; la solution (\[eq:solution-Frisch\]) donne $$\langle \cos\theta \rangle = {\rm e}^{ - z / \ell^*_{\rm trans} }
\label{eq:cos-theta-decroit}$$ Pour des distances faibles par rapport à $\ell^*_{\rm trans}$, on trouve donc un élargissement diffusif $$\langle \theta^2 \rangle \simeq 2 z / \ell^*_{\rm trans},
\label{eq:c-est-diffusif}$$ caractéristique pour une marche au hasard des directions de la lumière sur la sphère des vecteurs unitaires. D’autre part, aux grandes distances $ z \gg \ell^*_{\rm trans}$, l’on a $\langle \cos\theta \rangle = 0 $: la distribution angulaire est devenue isotrope.
##### Remarque.
L’analogie entre cette formulation du transfert radiatif et le mouvement d’un ensemble de particules classique a fait naître l’idée de simuler la diffusion de la lumière par des techniques Monte Carlo [@Feld94]. L’idée est de représenter la luminance comme la somme sur un grand nombre de trajectoires qui ont subi des diffusions dans le milieu. Une telle approche ressemble à la formulation de la mécanique quantique en termes d’une intégrale de chemins, et l’on peut profiter de cette analogie pour faire un calcul analytique approché de l’intégrale de trajectoires, en retenant, par exemple, seulement la trajectoire la plus probable dans la somme. Voir le papier de [Perelman]{} [*et al.*]{} [@Feld94] pour davantage de références.
Conclusions
-----------
Dans la limite des faibles angles de diffusion $\langle \theta^2 \rangle \ll 1$, l’équation du transfert radiatif se transforme d’une équation intégro-différentielle en une équation aux dérivées partielles. Elle ressemble à une équation de [Fokker–Planck]{} pour une distribution de probabilité. Une telle description est justifiée lorsque la fonction de phase, en fonction de l’angle de diffusion, a une portée beaucoup plus courte que la luminance en fonction de la direction.
Les directions des rayons effectuent alors un mouvement diffusif sur la sphère des vecteurs unitaires qui est caractérisé par le tenseur de diffusion $D_{\alpha\beta}$. Pour des milieux diffuseurs plus épais que $\sim 1 / D$, la distribution angulaire du rayonnement devient isotrope.
Transport d’atomes — point de vue classique {#s:classique}
===========================================
Il s’agit maintenant de formuler le problème du mouvement classique d’un ensemble d’atomes dans un milieu aléatoire. Avant d’entreprendre cette étude, nous allons préciser un modèle pour le milieu aléatoire: il s’agit d’un [*potentiel aléatoire*]{} tel qu’il apparaît pour le déplacement lumineux dans un champ de tavelures lumineuses. Ensuite, nous proposons une approche classique pour le transport d’atomes : les atomes seront considérés comme des particules ponctuelles qui se déplacent dans le potentiel aléatoire. Au chapitre suivant, nous utiliserons une approche ondulatoire où l’on étudie la propagation des ondes de matière dans le potentiel. Dans les deux cas, nous allons faire le parallèle à la propagation de la lumière.
Modèle pour un potentiel aléatoire
----------------------------------
Nous supposons les atomes placés dans un champ de [*speckles*]{} lumineux. Nous allons nous restreindre à un champ lumineux monochromatique dont la fréquence est désaccordée beaucoup par rapport à une fréquence de résonance atomique. Dans cette situation, le champ électrique ${\bf E}({\bf r})$ (amplitude vectorielle complexe) donne lieu à un [*potentiel lumineux*]{} proportionnel au carré du champ et inversement proportionnel au désaccord en fréquence $\omega_L - \omega_A$. Nous nous limiterons dans un premier temps à un atome avec un seul niveau interne dans l’état fondamental, donc à une transition atomique $J = 0 \to 1$. Le potentiel lumineux $V({\bf r})$ prend alors la forme $$V({\bf r}) = \frac{ d^2 | {\bf E}({\bf r}) |^2
}{ \hbar (\omega_L - \omega_A ) } ,$$ il est donc proportionnel à l’intensité lumineuse. Dans cette expression, $d$ est l’élément de matrice de l’opérateur dipole électrique entre l’état fondamental et l’état excité. Une écriture alternative du potentiel lumineux qui fait directement intervenir l’intensité lumineuse $I({\bf r}) \equiv |{\bf E}({\bf r})|^2$ est $$V({\bf r}) = \frac{ \hbar \Gamma^2 }{ 8 (\omega_L - \omega_A ) }
\frac{ I({\bf r}) }{ I_{sat} }
\label{eq:potentiel-lumineux}$$ où $I_{sat}$ est l’[*intensité de saturation*]{}. On dèduit de cette formule que l’ordre de grandeur du potentiel lumineux est (très grossièrement) donné par la largeur en énergie $\hbar\Gamma$ de l’état excité.
Les tavelures lumineuses ont une structure spatiale erratique dont l’échelle spatiale typique est la longueur d’onde lumineuse $\lambda_L = 2\pi / k_L =
2 \pi c / \omega_L$. A cause de leur complexité, l’on peut tout au plus raisonner en termes de leurs propriétés moyennes ou statistiques. Nous allons supposer que ces propriétés sont entièrement contenues dans la [*fonction de corrélation de l’intensité lumineuse*]{}: $$\langle V({\bf r}) V({\bf r}') \rangle = B^2
\langle I({\bf r}) I({\bf r}') \rangle
\equiv \bar{V}^2 \left[ 1 + g(|{\bf r} - {\bf r}'|) \right]
\label{eq:def-pot-correlation}$$ où $B$ est le coefficient de proportionnalité entre l’intensité et le potentiel lumineux, $\bar{V}$ est la valeur moyenne du potentiel et $g(r)$ sa fonction de corrélation (sans dimension). Nous supposons que les tavelures sont statistiquement homogènes (les corrélations ne dépendent que de la différence ${\bf r} - {\bf r}'$ entre deux positions) et isotropes ($g$ ne dépend que du module du vecteur différence). L’échelle de variation caractéristique de la fonction de corrélation est la longueur d’onde lumineuse: à des positions plus distantes que quelques $\lambda_L$, la fonction de corrélation décroît rapidement vers zéro $$r \gg \lambda_L:\quad
g(r) \to 0.$$ Un exemple particulier de fonction de corrélation pour un champ de tavelures est donné par: $$g(r) = \left(
\frac{ \sin k_L r }{ k_L r }
\right)^2$$ qui correspond à un champ lumineux avec une distribution angulaire isotrope dans tout l’angle solide (voir [Gori]{} [@Gori94] et [Nussenzveig]{} [@Nussenzveig87a]). Un autre exemple est une fonction de corrélation gaussienne: $$g(r) = \exp\!\left( - \frac{ {\bf r}^2 }{ 2 \ell_c^2 } \right)$$ qui correspond au champ lumineux rayonné dans le champ lointain par une source incohérente non ponctuelle dont le profil spatial d’intensité est gaussien.
##### Remarques.
\(i) Dans ce modèle, le milieu aléatoire est décrit par un [*potentiel*]{}, à la différence du mouvement brownien, par exemple, où l’on modélise les chocs que subit une particule immersé dans un fluide par une force aléatoire et dépendante du temps. Dans ce problème apparaît par exemple un amortissement de la vitesse atomique, en conséquence des chocs avec le fluide environnant. Dans notre problème par contre, l’énergie de l’atome est conservée [*a priori*]{}, il peut tout au plus y avoir un transfert d’énergie entre énergies potentielle et cinétique, ou bien entre énergie cinétique orientée (correspondant à une distribution de vitesse collimatée) et diffuse (correspondant à un mouvement diffus de l’ensemble d’atomes).
\(ii) Si nous nous contentons de la fonction de corrélation en deux points pour décrire le potentiel aléatoire, ceci revient à supposer que ce dernier possède une [*statistique gaussienne*]{}. C’est une hypothèse commode pour le calcul, mais qui peut s’avérer insuffisante dans certaines situations expérimentales (diffusion par les particules en suspension liquide, par exemple). [V. M. Finkel’berg]{} discute un potentiel avec une statistique plus générale et donne les formules nécessaires pour calculer les fonctions de corrélation [@Finkelberg67].
Distribution dans l’espace des phases
-------------------------------------
La quantité de base pour décrire le mouvement d’un ensemble d’atomes classiques est sa [*densité de probabilité dans l’espace des phases*]{} $f({\bf r}, {\bf v})$ qui donne le nombre d’atomes ${\rm d}N$ se trouvant autour de la position ${\bf r}$ et ayant une vitesse ${\bf v}$: $${\rm d}N = f({\bf r}, {\bf v}) \, {\rm d}{\bf r} \, {\rm d}{\bf v}
\label{eq:def-f-de-rv}$$ Nous notons l’analogie à la luminance ${\cal I}({\bf r}, {\bf s})$ définie à l’équation (\[eq:def-luminance\]). A la différence de cette dernière, la distribution dans l’espace des phases dépend à la fois de la direction et du module de la vitesse atomique. Ceci traduit le fait qu’en général, les atomes peuvent avoir une vitesse (une énergie cinétique) quelconque. En outre, lors du mouvement dans le potentiel aléatoire, l’énergie cinétique des atomes est modifiée. A la différence de la lumière, la diffusion d’atomes n’est donc pas monochromatique: la fréquence de l’onde de [de Broglie]{} change. Nous pouvons tout au plus espérer de retrouver une conservation d’énergie cinétique en moyenne lorsque le potentiel aléatoire a une valeur moyenne spatialement constante. Cette loi de conservation serait cependant approchée et seulement valable à une échelle spatiale grande par rapport à l’échelle de variation des tavelures (la longueur d’onde lumineuse).
Dérivation d’une équation de Fokker–Planck
------------------------------------------
### Approche heuristique
Nous partons de l’équation de [Liouville]{} pour la distribution $f({\bf r}, {\bf v}; t)$ dans l’espace des phases: $$\partial_t f({\bf r}, {\bf v}; t) + {\bf v}\cdot\nabla_r f
+ \frac{ 1 }{ M } {\bf F}( {\bf r} )\cdot\nabla_v f = 0 ,
\label{eq:Liouville}$$ où ${\bf F}({\bf r}) = - \nabla_r V({\bf r})$ est la force qui agit sur l’atome.[^2] Un moyen simple d’intégrer l’équation (\[eq:Liouville\]) est d’utiliser la conservation du nombre d’atomes le long des trajectoires classiques. Les trajectoires dans l’espace des phases $( {\bf r}(t), {\bf v}(t) )$ sont les solutions des équations du mouvement classiques, et la conservation du nombre de particules s’écrit: $$f({\bf r}(t), {\bf v}(t); t) = \mbox{const.}
\label{eq:nb-const}$$ Cette expression permet de calculer la fonction de distribution à des instants ultérieurs si l’on se la donne à un instant initial.
Utilisons maintenant l’équation (\[eq:nb-const\]) pour deux instants de temps voisins $t$ et $t - \Delta t:$ $$f({\bf r}_0, {\bf v}_0; t) =
f({\bf r}(t - \Delta t), {\bf v}(t - \Delta t); t - \Delta t)
\label{eq:ft-et-ft+dt}$$ de sorte qu’entre $t - \Delta t$ et $t$, les trajectoires classiques sont de la forme (avec $0 \le \tau \le \Delta t$) $$\begin{aligned}
{\bf r}_0 &\simeq& {\bf r}(t - \tau)
+ \tau {\bf v}_0 \nonumber\\
{\bf v}_0 &\simeq& {\bf v}(t - \tau)
+ \frac{ 1 }{ M }
\int_0^\tau\!{\rm d}\tau' {\bf F}[{\bf r}(t - \tau')]
\label{eq:traj-droites}\end{aligned}$$ Nous supposons donc l’intervalle $\Delta t$ suffisamment court et le potentiel aléatoire suffisamment faible pour que les trajectoires ne soient pas fortement courbées sous l’influence du potentiel aléatoire. En admettant en outre que la fonction de distribution $f({\bf r},
{\bf v}; t)$ varie lentement à l’échelle des déplacements (\[eq:traj-droites\]), nous pouvons la développer à l’équation (\[eq:ft-et-ft+dt\]), pour trouver $$\begin{aligned}
&& f({\bf r}_0, {\bf v}_0; t) - f({\bf r}_0, {\bf v}_0; t - \Delta t) =
\nonumber\\
&& - \, \Delta t \, {\bf v}_0 \cdot \nabla_r f(t - \Delta t)
- ( {\bf v}(t - \Delta t) - {\bf v}_0 )
\cdot \nabla_v f(t - \Delta t) \, +
\nonumber\\
&& \qquad + \, \frac 12 \sum_{ij} ( {\bf v}(t - \Delta t) - {\bf v}_0 )_i
( {\bf v}(t - \Delta t) - {\bf v}_0 )_j
\partial_{v_i} \partial_{v_j} f(t - \Delta t) .
\label{eq:Liouville-ordre-2}\end{aligned}$$ Nous allons maintenant prendre la valeur moyenne de cette équation par rapport au potentiel aléatoire (qui apparaît dans la vitesse ${\bf v}(t - \Delta t)$). Pour factoriser la valeur moyenne de la troisième ligne de (\[eq:Liouville-ordre-2\]), nous supposons que le long des trajectoires, la force aléatoire et la fonction de distribution forment un processus [*markovien*]{}: la fonction de distribution à l’instant $t - \Delta t$ n’est alors pas corrélée avec les valeurs de la force pour $t - \tau > t - \Delta t$. Cette hypothèse semble raisonnable si pendant l’intervalle de temps $\Delta t$, l’atome traverse un grand nombre de longueurs de corrélation du potentiel: $$v \Delta t \gg \lambda_L
\label{eq:condition-Markov}$$ Il s’ensuit que cette approche ne peut ni décrire des atomes très lents, ni rendre compte de leur mouvement à une échelle de temps comparable au temps entre deux collisions.
Avec l’hypothèse de [Markov]{}, la moyenne sur les réalisations du milieu aléatoire fait apparaître l’intégrale de la fonction de corrélation de la force dans le dernier terme de l’équation (\[eq:Liouville-ordre-2\]). Celui-ci devient alors $$\frac{ 1 }{ 2 M^2 } \int\limits_0^{\Delta t}
\! {\rm d}\tau'_1 {\rm d}\tau'_2 \sum_{ij} \,
\left\langle F_i[ {\bf r}_0 - \tau'_1 {\bf v}_0 ]
F_j[ {\bf r}_0 - \tau'_2 {\bf v}_0 ] \right\rangle
\partial_{v_i} \partial_{v_j} \langle f \rangle
\label{eq:force-correlation}$$ Par conséquent, le coefficient de diffusion de la vitesse est donné par l’expression $$D_{ij}({\bf v}) =
\lim_{\lambda_L / v \ll \Delta t \atop \Delta t \ll T}
\frac{ 1 }{ 2 M^2 \Delta t} \int\limits_0^{\Delta t}
\! {\rm d}\tau'_1 {\rm d}\tau'_2 \,
G_{ij}[ (\tau'_1 - \tau'_2) {\bf v} ]
\label{eq:coeff-diffusion}$$ où $T \gg \lambda_L / v$ est l’échelle de temps à laquelle l’on regarde la fonction de distribution moyenne (beaucoup plus grande que le temps entre deux collisions), et où $G_{ij}({\bf r})$ est le tenseur de corrélations de la force aléatoire $$G_{ij}({\bf r}) \equiv
\left\langle F_i( {\bf 0} ) F_j( {\bf r} ) \right\rangle.
\label{eq:def-corr-force}$$
Avant de calculer en détail le coefficient de diffusion de la vitesse $D_{ij}({\bf v})$, donnons l’équation de [Fokker–Planck]{} que l’on trouve par cette procédure. Il y apparaît une complication parce que le coefficient de diffusion dépend de la vitesse. Nous suivons l’article de [Hodapp]{} [*et al.*]{} [@Dalibard95] (qui suit le livre de [van Kampen]{} [@vanKampen]) pour la forme de l’équation de [F.-P.]{}: $$\partial_t \langle f({\bf r}, {\bf v}; t) \rangle
+ {\bf v}\cdot\nabla_r \langle f \rangle =
\sum_{ij} \partial_{v_i}
\left\{ - \frac{ F^{d}_i({\bf v}) }{ M } \langle f \rangle
+ D_{ij}({\bf v}) \partial_{v_j} \langle f \rangle
\right\}
+ \frac{ \partial_{v_i} F^{d}_i({\bf v}) }{ M } \langle f \rangle
\label{eq:Fokker-Planck}$$ où la force de dérive $F^{d}_i({\bf v})$ ainsi que le dernier terme proviennent du ré-arrangement des termes sous les dérivées $\partial_{v_i}$. La force de dérive est donnée par $$F^{d}_i({\bf v}) = M \sum_{j} \partial_{v_j} D_{ij}({\bf v}).
\label{eq:def-friction}$$
##### Calcul du coefficient de diffusion.
Nous donnons d’abord la fonction de corrélation de la force aléatoire (\[eq:def-corr-force\]): $$G_{ij}({\bf r})
= \bar{V}^2 \left[
\frac{ 1 }{ r } \frac{ {\rm d}g }{ {\rm d}r }
\left( \frac{ r_i r_j }{ r^2 } - \delta_{ij} \right)
- \frac{ {\rm d}^2g }{ {\rm d}r^2 } \frac{ r_i r_j }{ r^2 }
\right] ,
\qquad r \equiv |{\bf r}|,
\label{eq:resultat-corr-force}$$ résultat que l’on déduit de la définition élémentaire de la force comme le gradient du potentiel, et en échangeant l’ordre de la moyenne statistique et du processus limite pour le gradient. Ce calcul suppose que le potentiel aléatoire admet une dérivée, hypothèse qui semble justifiée pour un champ de [*speckles*]{}.
Pour calculer le coefficient de diffusion (\[eq:coeff-diffusion\]), passons aux variables d’intégration $(\tau'_1 + \tau'_2) / 2,
\, \tau'_2 - \tau'_1$. Pour l’intégration sur la différence, nous utilisons l’équation (\[eq:condition-Markov\]): l’intervalle $\Delta t$ est beaucoup plus long que le temps de corrélation du tenseur de corrélations $G_{ij}[ (\tau'_1 - \tau'_2) {\bf v} ]$. Par conséquent, celui-ci s’annule aux bornes d’intégration $\pm (\tau'_1 + \tau'_2)/2$ que nous pouvons donc remplacer par $\pm\infty$. L’intégrande ne dépend alors plus de la somme, et l’intégrale est proportionnelle à $\Delta t$. Nous trouvons ainsi le résultat: $$\begin{aligned}
D_{ij}({\bf v}) &=& \frac{ 1 }{ 2 M^2 v } \int\limits_{-\infty}^{+\infty}
\! {\rm d}r \, G_{ij}( r {\bf v} / v )
= \frac{ K }{ v } \left( \delta_{ij} - \frac{ v_i v_j }{ v^2 } \right)
\label{eq:resultat-Dij}\\
\mbox{avec}\quad
K &=& \frac{ \bar{V}^2 }{ M^2 }
\int\limits_0^\infty \! \frac{ {\rm d}r }{ r }
\left( - \frac{ {\rm d}g }{ {\rm d}r } \right) > 0.
\label{eq:def-K}\end{aligned}$$ Le deuxième terme dans (\[eq:resultat-corr-force\]) (qui fait intervenir la deuxième dérivée de la fonction de corrélation) ne contribue pas au coefficient de diffusion parce que son intégrale radiale s’annule. Pour les deux fonctions de corrélations proposées ci-dessus, la constante $K$ prend les valeurs suivantes: $$\begin{aligned}
\mbox{lumi\`ere isotrope}: \quad K &=& \frac{ \pi k_L }{ 3 }
\frac{ \bar{V}^2 }{ M^2 } , \\
\mbox{{\em speckles\/} gaussiens}: \quad K &=& \sqrt{ \frac{ \pi }{ 2 } }
\frac{ 1 }{ \ell_c } \frac{ \bar{V}^2 }{ M^2 } .\end{aligned}$$
##### Force de friction.
Du tenseur de diffusion (\[eq:resultat-Dij\]) nous déduisons que la force de dérive ${\bf F}^{d}({\bf v})$ dans l’équation de [F.–P.]{} (\[eq:Fokker-Planck\]) est en fait une force de friction avec $${\bf F}^{d}({\bf v}) = - 2 M K \frac{ {\bf v} }{ v^3 },
\qquad \nabla_v \cdot {\bf F}^{d} \equiv 8 \pi M K \delta({\bf v})
\label{eq:expression-force-d}$$ Puisque notre approche n’est pas valable pour des atomes à vitesse nulle \[voir la condition (\[eq:condition-Markov\])\], nous allons négliger le terme proportionnel à $\nabla_v \cdot {\bf F}^{d} \propto \delta({\bf v}) \langle f \rangle$ dans l’équation de F.–P. (\[eq:Fokker-Planck\]).
### Discussion physique
##### Amortissement de la vitesse moyenne.
On déduit facilement de l’équation de [Fokker–Planck]{} (\[eq:Fokker-Planck\]) l’équation d’évolution pour la vitesse moyenne (après deux intégrations par parties et en négligeant les termes de bord): $$\begin{aligned}
\frac{ {\rm d} \langle v_k \rangle }{ {\rm d}t }
& = &
\frac{ {\rm d} }{ {\rm d}t }
\int \!{\rm d}{\bf r} {\rm d}{\bf v}
\, v_k \langle f({\bf r}, {\bf v}; t) \rangle
\nonumber\\
&= &
\int \!{\rm d}{\bf r} {\rm d}{\bf v}
\left[ \frac{ F^{d}_k({\bf v}) }{ M }
+ \sum_j \partial_{v_j} D_{jk}({\bf v}) \right]
\langle f({\bf r}, {\bf v}; t) \rangle
\label{eq:amortissement-v-general}
\\
& = &
- 4 K \int \!{\rm d}{\bf r} {\rm d}{\bf v}
\, \frac{ v_k }{ v^3 } \langle f({\bf r}, {\bf v}; t) \rangle
\label{eq:amortissement-v}\end{aligned}$$ La valeur moyenne de la vitesse diminue donc sous l’effet de la force de friction et de la diffusion. Ceci se produit à une échelle de temps caractéristique $T \sim v^3 / K$, qui dépend du module de la vitesse initiale. Pour que notre théorie soit valable, il faut que ce temps soit beaucoup plus grand que l’intervalle entre deux collisions. Cette condition se traduit par l’inégalité: $$\frac{ T }{ \lambda_L / v } \sim
\left( \frac{ M v^2 }{ \bar{V} } \right)^2 \gg 1
\label{eq:deux-temps}$$ Il faut donc que le potentiel aléatoire soit en moyenne beaucoup plus faible que l’énergie cinétique des atomes. Dans ce régime, l’amortissement de la vitesse est lente à l’échelle des collisions avec les [*speckles.*]{}
##### Elargissement diffusif des composantes de vitesse transverses.
Nous nous attendons à ce que l’amortissement de la vitesse moyenne s’accompagne d’une isotropisation de la distribution angulaire de la vitesse parce qu’il semble peu probable que la distribution puisse être refroidie par le potentiel aléatoire.
Nous rappelons que le coefficient de diffusion dans l’équation de [F.–P.]{} donne la valeur quadratique moyenne du changement de la vitesse: $$\langle \Delta v_i(t) \Delta v_j(t) \rangle
\simeq 2 D_{ij}[{\bf v}(0)] t.$$ En utilisant le tenseur de diffusion (\[eq:resultat-Dij\]), nous constatons que les composantes parallèles à la vitesse initiale ${\bf v}(0)$ ne subissent aucune diffusion[^3] $\langle \Delta v^2_\Vert( t ) \rangle = 0$. Par contre, les composantes perpendiculaires à ${\bf v}(0)$ s’élargissent de façon diffusive $$\left\langle {\bf v}^2_\perp (t) \right\rangle \simeq
\frac{ 4 K }{ v(0) } t.$$ La distribution des vitesses devient isotrope lorsque les composantes transverses sont du même ordre de grandeur que la vitesse initiale. Nous trouvons que ce processus a lieu sur la même échelle de temps $T \sim K / v^3(0)$ que l’amortissement de la valeur moyenne de la vitesse.
##### Variation de l’énergie cinétique moyenne.
Calculons maintenant une équation d’évolution pour la valeur moyenne de l’énergie cinétique des atomes. L’équation de [F.–P.]{} (\[eq:Fokker-Planck\]) donne, après un calcul similaire à celui de l’éq.(\[eq:amortissement-v-general\]), $$\begin{aligned}
&& \frac{ {\rm d} }{ {\rm d}t }
\left\langle \frac{ M }{ 2 } {\bf v}^2 \right\rangle
= \nonumber\\
&& \quad =
\int {\rm d}{\bf r} {\rm d}{\bf v}
\left[ {\bf v}\cdot{\bf F}^{d}({\bf v}) + M
\sum_{ij} \partial_{v_i} \left( D_{ij}({\bf v}) v_j \right) \right]
\langle f({\bf r}, {\bf v}; t) \rangle
\label{eq:derivee-energie-cinetique}
\\
&& \quad = - 2 K M
\int {\rm d}{\bf r} {\rm d}{\bf v}
\frac{ 1 }{ v }
\langle f({\bf r}, {\bf v}; t) \rangle.
\label{eq:refroidissement?}\end{aligned}$$ L’énergie cinétique moyenne diminue donc sous l’influence de la force de friction \[le premier terme en crochets de (\[eq:derivee-energie-cinetique\])\]. Ceci apparaît également dans l’équation du mouvement pour la vitesse sous l’influence de la friction: $$\begin{aligned}
&& \frac{ {\rm d} }{ {\rm d}t } {\bf v} =
{\bf F}^{d}( {\bf v} ) = - 2 K \frac{ {\bf v} }{ v^3 }
\\
&& \Longrightarrow
{\bf v}(t) = \frac{ {\bf v}(0) }{ v(0) }
\left[ v^3(0) - 6 K t \right]^{1/3},
\quad \left( 0 < t < v^3(0) / 6 K \right)
\label{eq:solution-v-moyenne}\end{aligned}$$ Toutes les vitesses sont donc amorties radialement et deviennent nulles en un [*temps fini*]{} $T_{v(0)} = v^3(0) / 6 K$ de l’ordre de l’échelle de temps caractéristique $T$ trouvée ci-dessus.
Cependant, nous trouvons ici des résultats en contradiction avec l’intuition que le potentiel aléatoire conserve en moyenne l’énergie cinétique. Cette contradiction nous amène à la remarque suivante:
##### Démonstration malhonnête.
D’après [Keller]{} (cité dans [@Frisch66]), une telle démonstration de l’équation de [F.–P.]{} pour la fonction de distribution moyenne est malhonnête. Nous faisons en effet des hypothèses sur la nature markovienne de la force aléatoire qui ne sont justifiées que dans un certain régime de paramètres ou à certaines échelles de vitesse et de position (grande devant la longueur de corrélation). Nous verrons au prochain paragraphe que l’on peut trouver l’équation de [F.–P.]{} par une procédure différente et mieux justifiée d’après [Frisch]{}. Elle aura en outre la vertu de donner une équation de [F.–P.]{} physiquement acceptable qui conserve l’énergie cinétique moyenne.
Démonstrations plus rigoureuses
-------------------------------
### Méthode pédestre
La façon la plus simple d’éviter la contradiction du refroidissement dans un potentiel aléatoire consiste à renverser une partie de l’argumentation précédente pour déduire l’équation de [Fokker–Planck]{}. (C’est ainsi que [Frisch]{} la trouve habituellement.) Définissons d’abord les valeurs moyennes $$\begin{aligned}
F^{d}_i({\bf v}) &:=& M
\lim_{\Delta t \to 0}
\left.\frac{ \left\langle v_i(t + \Delta t) - v_i(t) \right\rangle }{
\Delta t} \right|_{ \mbox{${\bf v}(t) = {\bf v}$} }
\label{eq:force-Frisch}\\
D_{ij}({\bf v}) & := &
\lim_{\Delta t \to 0}
\left.\frac{
\left\langle \Delta v_i(t + \Delta t) \Delta v_j(t + \Delta t) \right\rangle }{
2 \Delta t} \right|_{ \mbox{${\bf v}(t) = {\bf v}$} }
\label{eq:diff-Frisch}\end{aligned}$$ L’équation de [F.–P.]{} est alors donnée par $$\partial_t \langle f({\bf r}, {\bf v}; t) \rangle
+ {\bf v}\cdot\nabla_r \langle f \rangle =
\sum_{ij} \partial_{v_i}
\left\{ - \frac{ F^{d}_i({\bf v}) }{ M } \langle f \rangle
+ D_{ij}({\bf v}) \, \partial_{v_j} \langle f \rangle
\right\} ,
\label{eq:FP-correcte}$$ où la force de dérive ${\bf F}^{d}({\bf v})$ et le tenseur de diffusion $D_{ij}({\bf v})$ sont donnés par (\[eq:force-Frisch\]) et (\[eq:diff-Frisch\]).
Pour calculer la dérive et le tenseur de diffusion dans un potentiel aléatoire faible, nous pouvons intégrer le gradient du potentiel le long d’une trajectoire droite, comme à l’équation (\[eq:traj-droites\]). De cette façon, l’on trouve que
- la force de dérive $F^{d}_i({\bf v})$ [*s’annule*]{} (en moyenne, le potentiel aléatoire ne donne pas de transfert de vitesse), et
- un résultat identique à (\[eq:resultat-Dij\]) pour le tenseur de diffusion.
Il reste cependant vrai que la valeur moyenne de la vitesse est amortie: l’équation (\[eq:amortissement-v-general\]) est encore correcte et la seule différence par rapport au resultat (\[eq:amortissement-v\]) est un facteur numérique qui change (remplacer $4 K$ par $2 K$). Et finalement, pour l’énergie cinétique moyenne, l’équation (\[eq:derivee-energie-cinetique\]) reste vraie (elle ne dépend que de la forme de l’équation de [F.–P.]{}) et donne un résultat nul parce que le tenseur de diffusion est orthogonal à la vitesse, $\sum_j D_{ij}({\bf v}) v_j = 0$. Nous obtenons donc le résultat attendu que [*l’énergie cinétique moyenne n’est pas modifiée par le potentiel aléatoire.*]{} Ce qui se produit c’est bien une redistribution de la vitesse d’une composante collimatée vers une composante diffuse.
### Développement multi-échelles de [Ryzhik]{}, [Papanicolaou]{} et [Keller]{}
##### Idée du développement.
Nous allons présenter ici une méthode alternative pour établir l’équation de [Fokker–Planck]{}. Elle met en relief le fait que le potentiel aléatoire $V({\bf r})$ d’une part, et la fonction de distribution moyenne $\langle f( {\bf r}, {\bf v}; t )
\rangle$ d’autre part, varient sur des échelles spatiales très différentes. Il s’agit alors d’éliminer les quantités avec des variations spatiales rapides et de formuler une équation fermée pour les quantités moyennes avec une variation spatiale lente. [Ryzhik]{}, [Papanicolaou]{} et [Keller]{} [@Keller96] introduisent à cet effet le rapport entre les échelles caractéristiques rapides et lentes, ce qui revient pour notre situation à introduire le paramètre $$\epsilon = \frac{ \lambda_L }{ L }
\label{eq:def-epsilon-Keller}$$ Ils considèrent ensuite un potentiel aléatoire faible proportionnel à $\sqrt{\epsilon}$. Le calcul consiste à effectuer un développement asymptotique pour $\epsilon \ll 1$.
Cette méthode semble d’abord se limiter à une situation où existe une relation entre la force du potentiel, d’une part, et la portée des corrélations, d’autre part. Cependant, l’on peut aussi la voir sous un autre point de vue et choisir indépendamment la force du potentiel et la longueur de corrélation $\lambda_L$: [*l’approche de [Ryzhik]{} fixe alors l’échelle $L$ à laquelle on regarde la fonction de distribution moyenne.*]{} La discussion précédente confirme que ce point de vue est adapté à notre problème: plus précisément, nous pouvons estimer que l’échelle spatiale caractéristique $L$ de la fonction de distribution moyenne est égale à la distance parcourue nécessaire pour que la diffusion ait rendu isotrope la distribution angulaire de la vitesse: $$L \sim v T \sim \frac{ v^4 }{ K }
\sim \lambda_L \left( \frac{ M v^2 }{ \bar{V} } \right)^2
\label{eq:longueur-caracteristique}$$ Cet argument permet d’établir le lien suivant entre les petits paramètres de notre problème : $$\epsilon = \left( \frac{ \bar{V} }{ Mv^2 } \right)^2 ,
\label{eq:def-epsilon}$$ et la condition $\epsilon \ll 1$ est bien celle d’un potentiel aléatoire faible que nous avons rencontrée et utilisée ci-dessus. Lorsque [Ryzhik]{} [*et al.*]{} parlent d’un potentiel aléatoire proportionnel à $\sqrt{\epsilon}$, ce nombre donne donc l’ordre de grandeur de $V( {\bf r} )$ en unités de l’énergie cinétique des atomes (la seule échelle d’énergie dans le problème).
Notons que le développement multi-échelles est à manipuler avec une certaine précaution: il faut identifier d’avance les échelles caractéristiques spatiales ainsi que les énergies typiques du problème et regrouper ces paramètres dans une seule petite quantité $\epsilon$.
##### Développement.
En fonction du petit paramètre $\epsilon$, la force aléatoire est proportionnelle à $$| {\bf F} | \sim \frac{ \bar{V} }{ \lambda_L } \propto
\frac{ \sqrt{\epsilon} }{ \epsilon } = \frac{ 1 }{ \sqrt{\epsilon} }
\label{eq:force-et-epsilon}$$ Par conséquent, [Ryzhik]{} [*et al.*]{} écriraient l’équation de [Liouville]{} (\[eq:Liouville\]) sous la forme $$\partial_t f({\bf r}, {\bf v}; t) + {\bf v}\cdot\nabla_r f
+ \frac{ 1 }{ M \sqrt{ \epsilon} } \tilde{{\bf F}}\cdot\nabla_v f = 0 ,
\label{eq:Liouville-Keller}$$ où $\tilde{{\bf F}}$ est une quantité de l’ordre unité. La fonction de distribution est alors développée en une série de puissances $$\begin{aligned}
f( {\bf r}, {\bf v}; t ) & = &
f^{(0)}( {\bf r}, {\bf v}; t ) \, +
\nonumber\\
&& + \, \sqrt{\epsilon}
f^{(1)}( {\bf r}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {\bf v}; t ) + \epsilon
f^{(2)}( {\bf r}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {\bf v}; t ) + \ldots
\label{eq:developpement-Keller}\end{aligned}$$ où ${\bf r}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}$ dénotent les variables spatiales lentes et rapides. On fait l’hypothèse que le premier terme $f^{(0)}$ est indépendant de la variable rapide ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}$ et que c’est le seul à donner un résultat non nul après moyenne statistique sur le potentiel. Les termes suivants $f^{(1,2)}$ dépendent à la fois de la variable rapide ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}$ et de la variable lente ${\bf r}$.
La différence en échelle entre ces deux variables se traduit par l’écriture suivante de l’opérateur gradient spatial dans (\[eq:Liouville-Keller\]) $$\nabla_r \mapsto \nabla_r + \frac{ 1 }{ \epsilon } \nabla_\rho
\label{eq:gradient-deux-echelles}$$ Nous pouvons maintenant insérer le développement de la fonction de distribution (\[eq:developpement-Keller\]) dans l’équation de [Liouville]{} (\[eq:Liouville-Keller\]) et séparer les différents ordres en $\epsilon$. Les deux ordres les plus bas donnent alors: $$\begin{aligned}
\mbox{ordre ${\cal O}( 1/\sqrt{\epsilon} )$}: \quad
0 & = &
{\bf v}\cdot\nabla_\rho f^{(1)} + \tilde{{\bf F}}\cdot\nabla_v f^{(0)}
\label{eq:exprimer-f1}\\
\mbox{ordre ${\cal O}( 1 )$}: \quad
0 & = &
\left( \partial_t + {\bf v}\cdot\nabla_r \right) f^{(0)}
+ {\bf v}\cdot\nabla_\rho f^{(2)} +
\tilde{{\bf F}}\cdot\nabla_v f^{(1)}
\label{eq:equation-f0}\end{aligned}$$ La première équation permet d’exprimer la partie fluctuante $f^{(1)}$ de la distribution en fonction de la force aléatoire et de la distribution lente $f^{(0)}$. La deuxiéme équation permet de trouver une équation fermée pour la distribution moyenne $\langle f \rangle = \langle f^{(0)} \rangle$ en insérant la solution de la première équation [*dans la valeur moyenne de la deuxième équation*]{}: le terme $f^{(2)}$ en disparaît en effet après la moyenne.
L’équation (\[eq:exprimer-f1\]) peut se résoudre par une transformation de [Fourier]{}, et l’on trouve $$\tilde{ f }^{(1)}( {\bf r}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}, {\bf v}; t ) =
- \frac{ 1 }{ {\rm i} {\bf v} \cdot {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} + \eta }
\int\!{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}
\exp( - {\rm i} {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} \cdot {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} )
\tilde{ {\bf F} }({{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}) \cdot \nabla_v
f^{(0)}( {\bf r}, {\bf v}; t ) ,
\label{eq:solution-f1}$$ où $\eta \to 0^+$ est un paramètre de régularisation. Nous insérons ce résultat en (\[eq:equation-f0\]) et trouvons après la moyenne une équation de [Fokker–Planck]{} $$\left( \partial_t + {\bf v}\cdot\nabla_r \right) \langle f \rangle
= \sum_{ij} \partial_{v_i} D_{ij}( {\bf v} ) \partial_{v_j}
\langle f \rangle ,
\label{eq:FP-Keller}$$ où le tenseur de diffusion est donné par $$D_{ij}({\bf v}) = \frac{ 1 }{ M^2 }
\int\!{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}'
\langle F_i({{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}) \, F_j({{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}') \rangle
\int\!\frac{ {\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} }{ (2\pi)^3 }
\frac{
\exp{ {\rm i} {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}\cdot( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}' ) } }{ {\rm i} {\bf v}\cdot{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} + \eta }
\label{eq:def-Dij-Keller}$$ Comme le potentiel aléatoire est statistiquement homogène, le tenseur de diffusion (\[eq:def-Dij-Keller\]) ne dépend pas de ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}$. En outre, l’intégrale sur ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}$ peut se calculer[^4] et l’on trouve \[en utilisant (\[eq:def-corr-force\])\] $$D_{ij}({\bf v}) = \frac{ 1 }{ M^2 v } \int\limits_{-\infty}^{0}
\! {\rm d}r \, G_{ij}( r {\bf v}/v )
\label{eq:resultat-Dij-Keller}$$ L’on trouve alors un résultat identique à (\[eq:resultat-Dij\]) si la fonction de corrélation $G_{ij}({\bf r})$ ne dépend pas du signe de ${\bf r}$ (c’est l’invariance par parité). En plus, le tenseur de diffusion se trouve à la bonne position dans l’équation de [F.–P.]{} (\[eq:FP-Keller\]) (entre les dérivées $\partial_{v_i}$) de sorte que celle-ci est compatible avec la conservation en moyenne de l’énergie.
##### Remarque.
Nous constatons que le développement multi-échelles conduit de façon générique à une équation de [F.–P.]{} à partir de l’équation de [Liouville]{}. Ce type de description du transport d’atomes dans un potentiel aléatoire apparaît donc comme l’outil standard dans le point de vue classique.
### Raisonnement formel de [Frisch]{}
##### Idée.
Cette approche procède par analogie avec les équations de propagation des ondes en milieu aléatoire, en les formulant d’une façon assez générale. L’on identifie des opérateurs de [Liouville]{} ${\cal L}_0$ et ${\cal L}_1$ dont le deuxième contient la partie aléatoire de l’équation de propagation: $$\partial_t A(t) = {\cal L}_0 A(t) + {\cal L}_1(t) A(t) ,
\label{eq:equation-generale}$$ où $A(t)$ désigne un champ quelconque. Dans l’équation de [Liouville]{} proprement dite (\[eq:Liouville\]), nous avons les correspondances suivantes $$\begin{aligned}
A(t) &\longleftrightarrow& f({\bf r}, {\bf v}; t)
\nonumber\\
{\cal L}_0 &\longleftrightarrow& - {\bf v}\cdot\nabla_r
\nonumber\\
{\cal L}_1(t) &\longleftrightarrow& - \frac{ 1 }{ M } {\bf F}\cdot\nabla_v
\label{eq:analogie-classique}\end{aligned}$$ Pour l’équation de propagation des ondes de [Schrödinger]{}, l’on aurait par exemple $$\begin{aligned}
A(t) &\longleftrightarrow& \psi({\bf r}; t)
\nonumber\\
{\cal L}_0 &\longleftrightarrow&
\frac{ {\rm i}\hbar }{ 2 M } \nabla^2_r
\nonumber\\
{\cal L}_1(t) &\longleftrightarrow&
- \frac{ {\rm i} }{ \hbar } V({\bf r})
\label{eq:analogie-quantique}\end{aligned}$$
[Frisch]{} montre alors que l’on peut écrire des équations séparées pour la valeur moyenne $\langle A(t) \rangle$ et la partie fluctuante $\delta A(t) = A(t) - \langle A(t) \rangle$ du champ: $$\begin{array}{rcl}
\partial_t \langle A(t) \rangle & = & {\cal L}_0 \langle A(t) \rangle
+ \langle {\cal L}_1(t) \delta A(t) \rangle
\\
\partial_t \delta A(t) & = & {\cal L}_0 \delta A(t)
+ \delta \left\{
{\cal L}_1(t) \left[ \left\langle A(t) \right\rangle
+ \delta A(t) \right] \right\},
\end{array}
\label{eq:2eq-Frisch}$$ La deuxième équation peut être résolue par un développement itératif, au moins de façon formelle. Sa solution exprime donc la partie fluctuante du champ en fonction du champ moyen et du Liouvillien aléatoire. En la reportant dans la première équation, l’on trouve une équation pour le champ moyen. Dans le cas de la propagation des ondes, l’équation résultante est l’équation de [Dyson]{} pour le champ moyen $\langle \psi \rangle$.
##### Approximation pour le champ fluctuant.
Il est cependant difficile de résoudre explicitement pour la partie fluctuante du champ. Ce que [Frisch]{} appelle l’[*approximation de régularisation au premier ordre*]{} revient alors à ne retenir que le premier terme de l’itération.[^5] L’on a alors la solution $$\delta A(t) \simeq \int\limits_{t_0}^t \! {\rm d}t' \,
\exp{\left[ {\cal L}_0 (t - t') \right]}
\left\{ {\cal L}_1(t') \langle A(t') \rangle \right\} .
\label{eq:solution-Bourret}$$ Dans l’approche de [Ryzhik]{} [*et al.*]{}, une telle approximation arrive naturellement parce que la partie fluctuante est calculée par un développement perturbatif: à l’ordre le plus bas, c’est le Liouvillien fluctuant et le champ moyen (d’ordre zéro) qui déterminent les fluctuations du champ (voir l’équation (\[eq:exprimer-f1\])).
Avec cette approximation, l’équation d’évolution pour le champ moyen devient $$\partial_t \langle A(t) \rangle = {\cal L}_0 \langle A(t) \rangle
+ \int\limits_{t_0}^t \! {\rm d}t' \,
\left\langle {\cal L}_1(t)
\exp{\left[ {\cal L}_0 (t - t') \right]}
{\cal L}_1(t') \right\rangle
\langle A(t') \rangle .
\label{eq:Dyson-Frisch}$$ où nous voyons apparaître sous l’intégrale la fonction de corrélation du Liouvillien fluctuant, prise pour les instants $t$ et $t'$. Entre ces instants, l’exponentielle $\exp{\left[ {\cal L}_0 (t - t') \right]}$ décrit le mouvement libre du système sous la seule influence du Liouvillien régulier.
##### Application à notre problème.
En utilisant les correspondances (\[eq:analogie-classique\]), le deuxième terme à l’équation (\[eq:Dyson-Frisch\]) pour le champ moyen s’écrit $$\sum_{ij} \partial_{v_i}
\int\limits_{t_0}^t \! {\rm d}t' \,
\left\langle F_i({\bf r})
F_j( {\bf r} + (t' - t){\bf v} ) \right\rangle
\partial_{v_j}
\langle f({\bf r} + (t' - t){\bf v}, {\bf v}; t') \rangle .
\label{eq:eqdiff-Frisch}$$ où ${\bf r} + (t' - t){\bf v}$ décrit le vol libre aux instants $t'$ antérieurs à $t$. Nous constatons que l’équation pour la fonction de distribution moyenne [*n’est pas équivalente*]{} à l’équation de [F.–P.]{} (\[eq:FP-Keller\]) parce qu’elle fait intervenir une intégrale sur $\langle f \rangle$ à des instants antérieurs. C’est seulement lorsque nous invoquons une [*hypothèse supplémentaire*]{} que nous retrouvons la théorie précédente: cette l’hypothèse étant que la portée de la fonction de corrélation de la force aléatoire (en fonction de $(t' - t){\bf v}$) est beaucoup plus courte que l’échelle spatiale caractéristique pour la fonction de distribution moyenne. Dans cette situation (que nous pourrions appeler markovienne) l’on peut sortir la fonction de distribution moyenne de l’intégrale sur $t'$. En remplaçant ensuite la borne inférieure $t_0$ de l’intégrale par $-\infty$, l’on trouve l’équation de [Fokker–Planck]{} (\[eq:FP-Keller\]).
Notons que l’approche de [Frisch]{} donne elle aussi un tenseur de diffusion à la bonne position (entre les dérivées $\partial_{v_i}$).
Conclusions
-----------
Dans le point de vue classique, le transport d’atomes dans un potentiel aléatoire faible est décrit de façon naturelle par une équation de [Fokker–Planck]{}, et non pas par une équation de transfert radiatif. L’équation de [F.–P.]{} fait seulement intervenir la fonction de distribution au même instant (elle est locale dans le temps) lorsque les fluctuations du potentiel aléatoire ont une échelle spatiale beaucoup plus courte que la fonction de distribution moyenne. Le transport est alors décrit par une équation linéaire aux dérivées partielles. Son tenseur de diffusion est relié au tenseur de corrélation de la force aléatoire, il dépend de la vitesse des atomes. L’équation de [F.–P]{} permet de montrer que dans le potentiel aléatoire, la valeur moyenne de la vitesse décroît parce que la distribution angulaire des vitesses devient isotrope. Le temps caractéristique pour ce processus diffusif est plus long que le temps nécessaire pour traverser une longueur de corrélation du potentiel aléatoire. A cette échelle de temps, la distribution angulaire d’un faisceau d’atomes initialement bien collimaté, s’élargit pour devenir isotrope, tout en conservant son énergie cinétique moyenne.
Nous avons constaté que le tenseur de diffusion dépend de la vitesse. Il faut alors faire attention à sa position dans l’équation de [F.–P.]{} Il y a plusieurs méthodes pour trouver la bonne équation: un retour à la signification statistique de l’équation de [Fokker–Planck]{} est déjà fructueux (notre méthode pédestre) ; des approches plus formelles sont soit le développement à plusieurs échelles (méthode de [Ryzhik, Papanicolaou]{} et [Keller]{}), soit l’analogie à l’approximation de [Bourret]{} de la théorie de propagation des ondes en milieu aléatoire (méthode de [Frisch]{}).
Finalement, une remarque sur les approches du type [F.–P.]{} et transfert radiatif : nous pouvons attribuer leur différence aux descriptions classique ou ondulatoire du mouvement. En effet, la théorie classique part de l’équation de [Liouville]{} où interviennent des dérivées de la fonction de distribution dans l’espace des phases. L’équation de [F.–P.]{} hérite alors de cette même structure. Dans l’approche ondulatoire, la diffusion par le potentiel aléatoire est décrite par des intégrales de diffusion (voir l’équation du transfert radiatif (\[eq:e-t-r\])) parce que pour la diffusion d’une onde, tout potentiel, aussi faible soit-il, peut donner lieu à des grands angles de diffusion avec une probabilité non nulle (bien que faible). Dans une approche classique, par contre, la vitesse change toujours de façon continue, et ce changement tend vers zéro lorsque le potentiel est faible. Les deux descriptions se rejoignent si la longueur d’onde est beaucoup plus petite que la taille des inhomogénéités du potentiel : la diffusion est alors piquée vers l’avant, et l’on peut passer du transfert radiatif à l’équation de [F.–P.]{} (voir au chapitre \[s:radiatif\]).
Exemples de transport classique
-------------------------------
Pour finir ce chapitre sur le transport classique d’atomes, nous donnons quelques exemples où l’on peut trouver des solutions explicites à l’équation de [Fokker–Planck]{}. Il s’agit d’une part de l’élargissement angulaire de la distribution des vitesses et d’autre part de la diffusion spatiale.
Notre point de départ est l’équation de [F.–P]{} suivante $$\left( \partial_t + {\bf v}\cdot\nabla_r \right) f( {\bf r}, {\bf v} ; t )
= \frac{ K }{ v^3 } \triangle_v^{(T)} f
\label{eq:rappel-FP}$$ où la fonction de distribution moyennée dans l’espace des phases est notée $f$, pour simplifier, et où $\triangle_v^{(T)}$ est l’opérateur différentiel Laplacien transverse. En fonction des angles $\theta, \phi$ du vecteur ${\bf v}$, il est donné par l’expression en crochets de l’équation (\[eq:FP-sphere\]). La constante $K$ dans (\[eq:rappel-FP\]) est proportionnelle à $\bar{V}^2$, elle est donnée par (\[eq:def-K\]).
### Diffusion des vitesses
Etudions d’abord l’évolution d’un jet d’atomes collimaté qui traverse un champ de tavelures. Dans un premier temps, supposons que les tavalures occupent une tranche d’épaisseur $Z$ et que le jet a un profil spatial uniforme (voir la figure \[fig:slab\]).
En nous plaçant dans le régime stationnaire, la fonction de distribution ne dépend que de la distance $z$ et de la vitesse ${\bf v}$. Nous notons également que dans l’équation de [Fokker–Planck]{} (\[eq:rappel-FP\]), le module de la vitesse $v$ intervient seulement comme paramètre. Nous pouvons donc raisonner pour une classe de vitesses donnée. A l’entrée du champ de [*speckles*]{}, la fonction de distribution est de la forme $$f( z = 0; {\bf v} ) = f_i(v) \delta(\theta)
= \frac{ f_i(v) }{ 4\pi }
\sum_{n = 0}^{\infty} (2n+1) P_n(\cos\theta)
\label{eq:f-init}$$ où $P_n(\cos\theta)$ sont les polynômes de [Legendre]{} qui permettent de représenter la fonction $\delta$ angulaire (voir aussi (\[eq:solution-Frisch\]) ; $\theta$ dénote l’angle par rapport à la direction initiale du faisceau). En introduisant les coefficients $f_n(z)$ pour le développement en harmoniques sphériques de la fonction de distribution, nous trouvons l’équation suivante : $$\sum_n \cos\theta \frac{ {\rm d}f_n }{ {\rm d}z } ( 2 n + 1 )
P_n(\cos\theta) =
-\frac{ K }{ v^4 }
\sum_n n ( n + 1 ) f_n( z ) ( 2 n + 1 )
P_n(\cos\theta)
\label{eq:FP-Legendre}$$ Lorsque l’on projette cette équation sur un polynôme de [Legendre]{} $P_m(\cos\theta)$, le $\cos\theta$ au membre gauche introduit un couplage entre les polynômes voisins $P_{m\pm1}(\cos\theta)$. L’on trouve alors un système linéaire infini d’équations différentielles ordinaires que l’on peut résoudre, par exemple, avec des techniques de diagonalisation de matrices. Nous allons nous contenter d’une discussion qualitative du régime où le champ de [*speckles*]{} a une épaisseur faible. Plus précisément, nous supposons que la distribution angulaire reste assez bien collimatée de sorte que nous pouvons remplacer le $\cos\theta$ dans (\[eq:FP-Legendre\]) par l’unité. Dans cette approximation, les différentes harmoniques sphériques se découplent, et l’on retrouve la solution (\[eq:solution-Frisch\]) donnée par [Frisch]{} $$f_n( Z ) = {\rm e}^{ - K n (n+1) Z / v^4 } f_i(v)
\label{eq:solution-Frisch-2}$$ Comme nous l’avons vu pour le transfert radiatif piqué vers l’avant (chap. \[s:radiatif\]), les harmoniques élevés (les structures angulaires $\Delta\theta$ fines) disparaissent sur une épaisseur caractéristique donnée par $$\frac{ 1 }{ \Delta\theta^2 } \frac{ K Z }{ v^4 } \sim 1
\Longrightarrow
\Delta\theta \sim \left( \frac{ K Z }{ v^4 } \right)^{1/2}
\label{eq:delta-theta}$$ La solution (\[eq:solution-Frisch-2\]) est valable tant que cette largeur angulaire reste faible. L’équation (\[eq:delta-theta\]) permet de traduire ce critère par une épaisseur maximale du milieu aléatoire : celle-ci est évidemment beaucoup plus grande que la longueur de corrélation (la longueur d’onde optique).
Nous passons maintenant à un deuxième exemple où le potentiel lumineux est branché pendant un temps $T$ fini. Prenons encore un jet d’atomes collimaté à l’instant $t=0$ et spatialement uniforme. Nous supposons également que le potentiel lumineux (lorsqu’il est branché) remplit tout l’espace. La fonction de distribution ne dépend alors que de $t$ et ${\bf v}$. Nous introduisons les coefficients $f_n( t )$ d’un développement en polynômes de [Legendre]{}, et (\[eq:FP-Legendre\]) devient $$\sum_n \frac{ {\rm d}f_n }{ {\rm d}t } ( 2 n + 1 )
P_n(\cos\theta) =
- \frac{ K }{ v^3 }
\sum_n n ( n + 1 ) f_n( z ) ( 2 n + 1 )
P_n(\cos\theta)
\label{eq:FP-Legendre-2}$$ Dans cette équation, les harmoniques sphériques se découplent exactement, et nous trouvons une solution formellement identique à (\[eq:solution-Frisch-2\]) : $$f_n( T ) = {\rm e}^{ - K n (n+1) T / v^3 } f_i(v)
\label{eq:solution-Frisch-2b}$$ De nouveau, nous identifions un temps d’interaction caractéristique $T_{\rm iso} \sim v^3 / K$ au bout duquel la distribution angulaire devient isotrope. Si l’on sait résoudre des détails angulaires $\Delta\theta$ plus fins, l’effet du potentiel aléatoire se fait voir déjà à des temps plus courts, de l’ordre de $\Delta\theta^2 \times T_{\rm iso}$.
Notons finalement que le temps caractéristique $T_{\rm iso}$ dépend à la fois de la vitesse des atomes et du potentiel aléatoire (qui apparaît dans la constante $K$). L’on peut donc faire varier ces deux paramètres pour explorer une large gamme de temps d’interaction effectifs, même si pour des raisons expérimentales (effet de la gravité, taille de la chambre à vide), le temps réel est limité.
### Diffusion spatiale
En dernière application du transport d’atomes, nous allons étudier le régime des temps très longs, où la distribution des vitesses est déjà devenue isotrope. Les atomes continuent cependant d’évoluer à cause d’un gradient spatial dans la densité atomique. Dans ce régime, la distribution spatiale $F({\bf r}; t)$ des atomes vérifie une équation de la diffusion que nous allons déterminer maintenant.
A cet effet, nous écrivons la distribution dans l’espace des phases comme la somme d’une partie isotrope $F({\bf r}; t)$ et d’un terme dipolaire ${\bf v}\cdot{\bf j}({\bf r}; t)$ $$f({\bf r}, {\bf v}; t) = F({\bf r}; t) + {\bf v}\cdot{\bf j}({\bf r}; t)
\label{eq:approx-P1}$$ (Cette approximation est appelée $P_1$parce qu’elle se contente de $P_{0,1}(\cos\theta)$ dans le développement en harmoniques sphériques.) En projetant l’équation de [Fokker–Planck]{} sur les harmoniques sphériques d’ordre $n=0,1$, l’on trouve $$\begin{aligned}
\partial_t F( {\bf r} ; t)
+ \int\frac{ {\rm d}\Omega_v }{ 4\pi }
\nabla_r\cdot{\bf v} [ {\bf v}\cdot{\bf j}( {\bf r}; t ) ]
& = & 0
\label{eq:FP-P0}\\
\partial_t \,{\bf j}( {\bf r}; t )
+ \nabla_r F( {\bf r}; t )
& = &
\frac{ 2 K }{ v^3 } {\bf j}( {\bf r}; t )
\label{eq:FP-P1}\end{aligned}$$ En nous plaçant à des temps longs par rapport au temps caractéristique d’évolution $T_{\rm iso}$ de la distribution angulaire, nous pouvons nous contenter de la solution stationnaire de l’équation (\[eq:FP-P1\]) : elle relie le flux diffusif ${\bf j}( {\bf r} ; t )$ au gradient de la densité spatiale (loi de [Fick]{}) $${\bf j}( {\bf r}; t ) = - \frac{ v^3 }{ 2 K } \nabla_r F( {\bf r}; t )
\label{eq:Fick}$$ En reportant ce résultat à l’équation (\[eq:FP-P1\]) et en calculant l’intégrale angulaire, nous trouvons alors une équation de diffusion spatiale $$\partial_t F( {\bf r}; t ) - \frac{ v^5 }{ 6 K }
\nabla_r^2 F( {\bf r}; t ) = 0
\label{eq:diffusion-spatiale}$$ Le coefficient de diffusion (spatiale) est donc donné par $$D_{\rm sp} = \frac{ v^5 }{ 6 K } .
\label{eq:estimation-Dsp}$$
##### Discussion.
Une situation physique intéressante correspond à l’expansion d’un nuage initialement confiné et plongé dans un potentiel aléatoire. Après un temps $\Delta t$, les atomes ont diffusé à travers une distance quadratique moyenne $$\Delta{\bf r}^2 \simeq 6 D_{\rm sp} \Delta t
= \frac{ v^5 \Delta t }{ K }
\sim v^2 T_{\rm iso} \Delta t
\ll ( v \Delta t )^2
\label{eq:c-est-pas-rapide}$$ A première vue, il semble que les atomes diffusent très rapidement (dépendance en $v^5$). Mais la deuxième formulation dans (\[eq:c-est-pas-rapide\]) nous rappelle que, lorsque le temps $\Delta t$ est plus long que le temps caractéristique $T_{\rm iso}$ de l’évolution angulaire, la diffusion est au contraire beaucoup plus lente que le vol libre.
Il reste à remarquer que pour un temps d’expérience donné, ce sont les atomes les plus lents qui ont le tempsd’entrer dans le régime diffusif (leur $T_{\rm iso} \propto v^3$ est compris dans la durée de l’expérience). Les atomes plus rapides suivent par contre un transport balistique avec $\Delta {\bf r}^2
\simeq ({\bf v} \Delta t)^2$, ils n’ont en outre pas encore oubliéleur distribution angulaire initiale. C’est ainsi qu’ils ont parcouru la distance la plus grande (et que leur distribution en position permet de remonter à la distribution des vitesses initiale). Nous constatons donc qu’il est possible d’observer simultanément plusieurs comportement du transport, pour chacune des classes de vitesses (radiales) des atomes. Le comportement diffusif apparaît seulement pour les classes les plus lentes, il va se manifester, à la fin de l’expérience, au centre de la distribution spatiale.
Transport d’atomes — point de vue ondulatoire {#s:quantique}
=============================================
Un bref aperçu d’approches ondulatoires pour modéliser le transport d’atomes dans un milieu aléatoire. Un grand nombre de résultats sont bien connus dans le contexte optique (diffusion multiple de la lumière). Nous montrerons en particulier comment l’on obtient une équation du transfert radiatif à partir de la description ondulatoire. C’est un exemple particulier du passage du microscopique vers le macroscopique.
Introduction
------------
Nous nous plaçons maintenant dans un contexte ondulatoire où les atomes sont décrits, au niveau le plus fondamental de la théorie, par une fonction d’onde $\psi( {\bf r}, t)$. Son équation d’évolution est l’équation de [ Schrödinger]{} : $${\rm i} \hbar \partial_t \psi( {\bf r}, t ) +
\frac{ \hbar^2 }{ 2 M } \nabla^2 \psi = V( {\bf r} ) \, \psi
\label{eq:Schroedinger-ici}$$ où $V( {\bf r} )$ est le potentiel aléatoire. Nous nous intéressons à des [*propriétés moyennes*]{} des atomes lorsque l’on ignore presque tout du potentiel aléatoire, sauf sa valeur moyenne (que nous supposons égale à zéro, avec une redéfinition convenable de l’énergie[^6]) et sa fonction de corrélation $\langle V({\bf r}_1 ) \, V({\bf r}_2 )\rangle$. Les quantités intéressantes dans le contexte de cette approche statistique sont, d’une part, la [*valeur moyenne*]{} $\langle \psi( {\bf r}) \rangle$ de la fonction d’onde et, d’autre part, sa [*fonction de corrélation*]{} $\langle \psi({\bf r}_1 ) \, \psi^*({\bf r}_2 )\rangle$ que nous appelerons également la fonction de cohérence. Notons que la première de ces quantités n’est pas forcément nulle lorsque l’on se trouve au voisinage d’une source cohérente d’atomes. La fonction d’onde moyenne décrit alors la partie cohérente de l’onde atomique dont la phase est suffisamment bien définie pour qu’elle puisse interférer avec une onde de référence. La fonction de cohérence, quant à elle, décrit la distribution en position et en impulsion de l’ensemble atomique, comme nous verrons dans la représentation de [Wigner]{}. Elle décrit également le contraste des franges lorsque l’on fait interférer les ondes rayonnées par deux points du même ensemble (en opposition à l’interférence avec une onde de référence dont il était question pour la fonction d’onde moyenne).
Dans la suite de cette introduction, nous allons fixer quelques notations. Quant au potentiel aléatoire, nous introduisons sa densité spectrale (spatiale) $$S( {\bf q} ) = \int\!{\rm d}{\bf r} \,
\langle V({\bf r} + {\bf r}_0 ) \, V({\bf r}_0 ) \rangle
\exp{(- {\rm i} {\bf q}\!\cdot\!{\bf r} ) }
\label{eq:def-densite-spectrale}$$ qui est la transformée de [Fourier]{} de la fonction de corrélation du potentiel. Dans cette définition, nous supposons que le potentiel aléatoire est (statistiquement) homogène, c’est-à-dire que sa fonction de corrélation en deux points ${\bf r}_{1,2}$ ne dépend que de la différence ${\bf r}_2 - {\bf r}_1$. La portée de la densité spectrale (\[eq:def-densite-spectrale\]) en fonction du vecteur d’onde ${\bf q}$ est de l’ordre de l’inverse $1 / \ell_c$ de la longueur de corrélation.
Quant à l’ensemble atomique, introduisons sa distribution de [Wigner]{} : $$f( {\bf r}, {\bf k} ) =
\int\!{\rm d}{\bf s} \,
\langle \psi({\bf r} + {{\textstyle\frac12}}{\bf s}) \,
\psi^*({\bf r} - {{\textstyle\frac12}}{\bf s} ) \rangle
\exp{ (- {\rm i} {\bf k}\cdot{\bf s} ) }
\label{eq:def-Wigner}$$ Cette distribution a des propriétés similaires à une densité de probabilité dans l’espace des phases :
- elle dépend à la fois de la position ${\bf r}$ et du vecteur d’onde ${\bf k}$ ;
- son intégrale sur les positions (sur les vecteurs d’onde) donne la distribution en vecteur d’onde (en position) des atomes ;
- le flux moyen atomique peut s’exprimer comme la valeur moyenne du vecteur d’onde : $${\bf j}( {\bf r} ) = \frac{ \hbar }{ M }
\mbox{Im } \langle \psi^*( {\bf r} )
\nabla \psi( {\bf r} ) \rangle
= \int\! {\rm d}{\bf k} \frac{ \hbar {\bf k} }{ M }
f( {\bf r}, {\bf k} ) ;
\label{eq:flux-Wigner}$$
- notons cependant que la distribution de [Wigner]{} n’est pas nécessairement positive, elle ne peut donc pas en toutes circonstances être interprétée comme une densité de probabilité.
Nous allons souvent considérer des atomes avec une énergie $E$ fixée. L’équation de [Schrödinger]{} stationnaire peut alors s’écrire de la façon suivante : $$\begin{aligned}
\setlength{\arraycolsep}{0.0em}
\nabla^2 \psi( {\bf r} ) + k_0^2 \left[ 1 + \mu( {\bf r} ) \right]
\psi( {\bf r} ) & = & 0 ;
\label{eq:Schroedinger-stationnaire}\\
E = \hbar^2 k_0^2 / 2 M , \qquad
\mu( {\bf r} ) & = & - V( {\bf r} ) / E .
\label{eq:def-mu}\end{aligned}$$ Cette forme de l’équation met en évidence l’analogie à la lumière (dans l’approximation scalaire), où $\mu( {\bf r})$ est la déviation de la constante diélectrique de l’unité. L’équation de Schrödinger stationnaire sera notre point de départ pour la théorie de la diffusion multiple des ondes de matière. Une quantité importante dans ce contexte sera la fonction de corrélation de la constante diélectrique, donc du potentiel aléatoire renormalisé : $$\begin{aligned}
\setlength{\arraycolsep}{0.0em}
\langle \mu( {\bf r}_1 ) \, \mu( {\bf r}_2 ) \rangle =
E^{-2} \langle V( {\bf r}_1 ) \, V( {\bf r}_2 ) \rangle & = &
\epsilon^2 g( {\bf r}_2 - {\bf r}_1 ),
\\
\epsilon & = & \langle V^2 \rangle^{1/2} / E
\label{eq:def-epsilon-ici}\end{aligned}$$ Le nombre $\epsilon$ donne donc l’ordre de grandeur du potentiel aléatoire par rapport à l’énergie des atomes. Au chapitre précédent, nous avons seulement considéré le cas $\epsilon \ll 1$. Nous ferons de même ici, notre théorie ne s’applique donc pas au cas d’un potentiel aléatoire qui piègerait les atomes dans ses vallées.
Point de vue de [Ryzhik]{} [*et al.*]{} {#s:Keller-Wigner}
---------------------------------------
Dans l’article de [Ryzhik]{}, [Papanicolaou]{} et [Keller]{} [@Keller96], la diffusion des ondes scalaires par un potentiel aléatoire est étudiée en guise d’introduction. Nous nous contenterons ici d’esquisser leur démarche.
### Equation de transport à grande échelle
Ils cherchent une équation d’évolution pour la distribution de [Wigner]{} à des échelles spatiales grandes par rapport à la portée des corrélations du potentiel aléatoire, d’une part, et à la longueur d’onde atomique, d’autre part. En utilisant un développement multi-échelles, ils parviennent à démontrer que la distribution de [Wigner]{} vérifie une équation du type [*transfert radiatif*]{} qu’ils écrivent de la façon suivante : $$\begin{aligned}
\left( \partial_t + \frac{ \hbar{\bf k} }{ M } \cdot
\nabla_r \right) f( {\bf r}, {\bf k} ) & = &
\frac{ 4 \pi M }{ \hbar^3 }
\int\!\frac{ {\rm d}{\bf k}' }{ (2\pi)^3 }
S( {\bf k}' - {\bf k} )
\,\delta( {\bf k}'^2 - {\bf k}^2 )
\times
\nonumber\\
&& \quad \times
\left[ f({\bf r}, {\bf k}' ) - f({\bf r}, {\bf k} ) \right]
\label{eq:transfert-Keller}\end{aligned}$$ Le dernier terme dans cette équation décrit la diffusion d’une onde atomique du vecteur d’onde initial ${\bf k}$ vers le vecteur d’onde final ${\bf k}'$. L’efficacité de ce processus est proportionnelle à la densité spectrale du potentiel $S({\bf k}' - {\bf k})$ au transfert de vecteur d’onde, comme c’est le cas dans l’approximation de [Born]{}. L’équation du transfert radiatif (\[eq:transfert-Keller\]) implique donc les valeurs suivantes pour la section efficace de diffusion[^7] $\sigma_{\rm{tot}}$ et la fonction de phase[^8] $p( {\bf n}', {\bf n} ; k_0 )$ $$\begin{aligned}
\sigma_{\rm{tot}}( k_0 ) & = & \frac{ k M }{ \pi \hbar^3 }
\int\!\frac{ {\rm d}^2{\bf n}' }{ 4\pi }
S( k_0 ({\bf n}' - {\bf n}) )
\label{eq:section-efficace-Keller}\\
p( {\bf n}', {\bf n} ; k_0 ) & = &
\frac{ k_0 M }{ 4 \pi^2 \hbar^3 \sigma_{\rm{tot}} }
S( k_0 ({\bf n}' - {\bf n}) )
\label{eq:fn-phase-Keller}\end{aligned}$$
### Discussion
Tout d’abord, l’équation de transport (\[eq:transfert-Keller\]) est un résultat utile pour fixer le comportement à grande distance de la distribution de [Wigner]{}. D’une façon plus conceptuelle, le travail de [Ryzhik]{} [*et al.*]{} fournit donc une démonstration alternative de l’équation de transfert radiatif pour les ondes scalaires : celle-ci est valable à des échelles spatiales grandes par rapport à la longueur d’onde et la longueur de corrélation. On en a vu d’autres démonstrations par [Rytov]{} ([@Rytov], § 4.3) et [Barabanenkov]{} et [Finkel’berg]{} [@Barabanenkov67].
Ensuite, il ne faut pas oublier que (\[eq:transfert-Keller\]) contient implicitement l’hypothèse que le potentiel aléatoire est faible ; en effet, nous avons constaté que la diffusion par le potentiel est décrite dans l’approximation de [Born]{} (la section efficace est proportionnelle au carré de la transformée de [Fourier]{} du potentiel). Nous retrouvons là la proposition de [Luck]{} [@Luck93; @Luck96] qui dit que l’équation de [Bethe–Salpeter]{} est équivalente à l’équation du transfert radiatif lorsque l’on tient compte des diagrammes en échelle([*ladder approximation*]{}) ce qui revient à une approximation de diffusion simple et donc de [Born]{}.
Finalement, nous remarquons que l’équation de transfert (\[eq:transfert-Keller\]) se simplifie lorsque l’on se place dans le régime semi-classique. En utilisant la fonction de phase (\[eq:fn-phase-Keller\]), nous constatons que la valeur maximale de l’angle de diffusion $\theta$ est donnée par $$k_0 | {\bf n}' - {\bf n} |_{\max}
= 2 k_0 \sin(\theta_{\max}/2)
\simeq \frac{ 1 }{ \ell_c }
\quad \Longrightarrow \quad
2\sin(\theta_{\max}/2) \simeq \frac{ \lambdabar_{dB} }{ \ell_c }
\ll 1
\label{eq:limite-sc-ici}$$ Dans le régime semi-classique, la diffusion des atomes se produit donc de préférence vers l’avant, et nous pouvons utiliser la formulation de [Fokker–Planck]{} de l’équation de transfert radiatif introduite au chapitre \[s:radiatif\]. Nous avons vérifié que l’on trouve une équation de [Fokker–Planck]{} identique à celle du chapitre \[s:classique\] où nous nous sommes placé d’emblée dans une description classique.[^9]
### Conclusion
A des échelles spatiales grandes par rapport à la longueur de corrélation du potentiel aléatoire et pour un potentiel aléatoire faible, la distribution de Wigner des atomes vérifie une équation de transport du type transfert radiatif. La fonction de phase dans cette théorie s’exprime en fonction de la densité spectrale du potentiel aléatoire, comme c’est le cas dans l’approximation de [Born]{} (de diffusion simple). Nous retrouvons exactement l’image classique du transport lorsque la longueur d’onde atomique est petite par rapport à la longueur de corrélation du potentiel ; la distribution de [Wigner]{} vérifie alors une équation de [Fokker–Planck]{}.
Diffusion multiple des ondes {#s:diff-mult}
----------------------------
Introduisons maintenant un formalisme ondulatoire plus général qui est également en mesure de décrire la diffusion multiple des ondes atomiques. Nous allons présenter les objets et les équations de base d’une telle théorie, à savoir
- la fonction de [Green]{} moyenne, qui permet de calculer la fonction d’onde moyenne et qui vérifie l’équation de [Dyson]{},
- la fonction de cohérence atomique et son équation d’évolution, l’équation de [Bethe–Salpeter]{} (B.–S.).
Au paragraphe \[s:vers-TR\] suivant, nous étudierons de façon générale le lien entre l’équation de B.–S. et la théorie du transfert radiatif. Nous nous servons à cet effet de la représentation de [Wigner]{}.
Dans ce paragraphe, nous nous plaçons dans une situation stationnaire, l’énergie des atomes est donc fixée par le vecteur d’onde atomique $k_0$ (valeur dans le potentiel moyen).
### Fonction de [Green]{} moyenne
La quantité centrale pour décrire la fonction d’onde atomique est la fonction de [Green]{} $G( {\bf r}, {\bf r}')$ pour l’équation de [Schrödinger]{} stationnaire (\[eq:Schroedinger-stationnaire\]) : $$\nabla_r^2 G( {\bf r}, {\bf r}') +
k_0^2 \left[ 1 + \mu( {\bf r} ) \right]
G( {\bf r}, {\bf r}')
= \delta( {\bf r} - {\bf r}')
\label{eq:def-Green}$$ Cet objet décrit la fonction d’onde rayonnée par une source ponctuelle (de fréquence $E/\hbar$) située à la position ${\bf r}'$ dans le milieu aléatoire. Pour une source spatialement étendue, nous trouvons la fonction d’onde en sommant les champs rayonnés par tous les points source.
#### Equation de [Dyson]{}
La fonction de [Green]{} (\[eq:def-Green\]) dépend du potentiel aléatoire en chaque point, et il est impossible dans la pratique de la calculer explicitement. L’objet intéressant est donc la [*fonction de [Green]{} moyenne*]{}, la moyenne étant prise sur les configurations du potentiel aléatoire. Nous utiliserons la notation $$\overline{G}( {\bf r} - {\bf r}' )
= \langle G( {\bf r}, {\bf r}') \rangle
\label{eq:Green-moyenne}$$ pour cette fonction de [Green]{}. Elle décrit donc la valeur moyenne de la fonction d’onde rayonnée par une source ponctuelle. En écrivant (\[eq:Green-moyenne\]), nous avons supposé que le milieu aléatoire est statistiquement homogène : la fonction de [Green]{} moyenne ne dépend alors que la distance ${\bf r} -
{\bf r}'$ entre le point d’observation et le point source.
Il s’agit maintenant de trouver une équation fermée pour la fonction de [Green]{} moyenne. Il est utile à cet effet de transformer l’équation de [Schrödinger]{} (\[eq:def-Green\]) en une équation intégrale : $$G( {\bf r}, {\bf r}' ) = G_0( {\bf r} - {\bf r}' )
+ k_0^2 \int\!{\rm d}{\bf r}_1
G_0( {\bf r} - {\bf r}_1 ) \mu( {\bf r}_1 )
G( {\bf r}_1, {\bf r}' ) ,
\label{eq:Green-integrale}$$ où nous avons utilisé la fonction de [Green]{} $G_0( {\bf r} - {\bf r}' )$ pour l’espace libre : $$G_0( {\bf r} - {\bf r}' ) = -
\frac{ \exp{ {\rm i} k_0 | {\bf r} - {\bf r}' |} }{
4 \pi | {\bf r} - {\bf r}' | }
\label{eq:Green-libre}$$ L’approximation de [Born]{} consiste à résoudre (\[eq:Green-integrale\]) en remplaçant au membre droit, $G$ par la solution en espace libre $G_0$. On constate que cette procédure est le premier terme d’une solution par itération. En allant au-delà de l’approximation de [Born]{}, une telle solution génère des produits avec de plus en plus de facteurs $\mu( {\bf r} )$. L’on peut alors prendre la valeur moyenne de cette série, pour faire apparaître les fonctions de corrélation du potentiel aléatoire. En toute généralité, l’on aura affaire à des corrélations d’ordre arbitrairement élevé, comme par exemple $$\langle \mu( {\bf r}_1 ) \mu( {\bf r}_2 ) \mu( {\bf r}_3 ) \rangle$$ Pour simplifier la théorie, nous allons supposer que le potentiel aléatoire à une statistique gaussienne ; c’est-à-dire que les fonctions de corrélations d’ordre supérieur à deux peuvent s’exprimer au moyen de la fonction de corrélation à deux points. Toutes les corrélations à un nombre de points impair s’annulent alors, et pour les corrélation à quatre points, par exemple, nous avons $$\begin{aligned}
\lefteqn{
\langle \mu( {\bf r}_1 ) \mu( {\bf r}_2 ) \mu( {\bf r}_3 ) \mu( {\bf r}_4
\rangle = \langle \mu( {\bf r}_1 ) \mu( {\bf r}_2 ) \rangle \langle
\mu( {\bf r}_3 ) \mu( {\bf r}_4 \rangle \, + }
\nonumber\\
&& + \,
\langle \mu( {\bf r}_1 ) \mu( {\bf r}_3 ) \rangle \langle
\mu( {\bf r}_2 ) \mu( {\bf r}_4 \rangle +
\langle \mu( {\bf r}_1 ) \mu( {\bf r}_4 ) \rangle \langle
\mu( {\bf r}_2 ) \mu( {\bf r}_3 \rangle .
\nonumber\end{aligned}$$
La solution itérative reste quand même complexe et contient une infinité de termes. Pour les organiser, l’on peut se servir d’une méthode de diagrammes. Cette approche est exposée dans les articles de [Frisch]{} [@Frisch66] et dans les livres de Rytov [*et al.*]{} ([@Rytov], § 4.1) et de Ping [Sheng]{} ([@Sheng95], §§ 4.3, 4.4). En ré-organisant la série itérative, l’on trouve l’équation suivante pour la fonction de [Green]{} moyenne, l’équation de [Dyson]{} : $$\overline{G}( {\bf r} - {\bf r}' ) = G_0( {\bf r} - {\bf r}' )
+ \int\!{\rm d}{\bf r}_1 {\rm d}{\bf r}_2
G_0( {\bf r} - {\bf r}_1 ) m( {\bf r}_1 - {\bf r}_2)
\overline{G}( {\bf r}_2 - {\bf r}' ) ,
\label{eq:Dyson}$$ où $m( {\bf r}_1 - {\bf r}_2)$ est appelé l’opérateur de masse. Son développement itératif ne contient que des diagrammes irréductibles[^10] et s’écrit : $$\begin{aligned}
\lefteqn{m( {\bf r}_1 - {\bf r}_2) =
\epsilon^2 k_0^4 g( {\bf r}_1 - {\bf r}_2)
G_0( {\bf r}_1 - {\bf r}_2 ) \, + }
\label{eq:Bourret}\\
&& + \, \epsilon^4 k_0^8
\int\!{\rm d}{\bf r}_3 {\rm d}{\bf r}_4
G_0( {\bf r}_1 - {\bf r}_3 )
G_0( {\bf r}_3 - {\bf r}_4 ) G_0( {\bf r}_4 - {\bf r}_2 )
\times
\label{eq:operateur-masse}\\
&& \quad \times
\left[
g( {\bf r}_1 - {\bf r}_4) g( {\bf r}_2 - {\bf r}_3)
+ g( {\bf r}_1 - {\bf r}_2) g( {\bf r}_4 - {\bf r}_3)
\right] \, + \ldots
\nonumber\end{aligned}$$ Nous notons que l’opérateur de masse ne dépend que de la différence des positions ${\bf r}_1 - {\bf r}_2$, ceci étant dû à l’homogénéité statistique.
Afin d’interpréter l’opérateur de masse, revenons à une formulation différentielle de l’équation de [Dyson]{}, en appliquant l’opérateur $\nabla_r^2 + k_0^2$ à (\[eq:Dyson\]) : $$\left( \nabla_r^2 + k_0^2 \right)
\overline{G}( {\bf r} - {\bf r}' ) =
\delta( {\bf r} - {\bf r}' )
+ \int\!{\rm d}{\bf r}_1 m( {\bf r} - {\bf r}_1)
\overline{G}( {\bf r}_1 - {\bf r}' ) ,
\label{eq:integro-diff}$$ Ceci est en fait une équation intégro-différentielle qui exprime le champ moyen rayonnée dans le milieu aléatoire comme la somme d’un champ en espace libre (la fonction $\delta$), plus une correction qui dépend de façon non-locale de la valeur du champ autour de la source (le deuxième terme). L’opérateur de masse traduit donc précisément la rétro-action du milieu diffusant sur la propagation du champ moyen.
Nous notons finalement que l’on peut résoudre (\[eq:integro-diff\]) par une transformée de [Fourier]{} puisque le milieu est homogène. L’intégrale du deuxième terme est en effet un produit de convolution, et nous avons donc pour les transformées de [Fourier]{} $$\begin{aligned}
( k_0^2 - {\bf k}^2 ) \overline{G}({\bf k})
& = & 1 + m({\bf k}) \overline{G}({\bf k})
\label{eq:integro-diff-Fourier}\\
\mbox{d'o\`u} : \quad
\overline{G}({\bf k}) & = & \frac{ 1 }{
k_0^2 - {\bf k}^2 - m({\bf k}) }
\label{eq:TF-<G>}\end{aligned}$$ Nous en déduisons la relation de dispersion des ondes de matière dans le milieu désordonneé : $${\bf k}^2 + m({\bf k}) = k_0^2
\label{eq:dispersion}$$ Cette équation définit le vecteur d’onde effectif $k_{\rm{eff}}$ de la fonction d’onde moyenne dans le milieu. Il diffère du vecteur d’onde dans le vide $k_0$ pour deux raisons : d’une part, la diffusion par le milieu introduit de l’atténuation pour le champ cohérent, ce qui se traduit par une partie imaginaire du vecteur d’onde, et d’autre part, la diffusion modifie la phase de propagation des ondes.[^11]
### Approximation de [Bourret]{}
Tout ce que nous venons d’exposer ne serait en fait que des manipulations formelles d’équations s’il n’y avait pas un moyen de calculer l’opérateur de masse $m({\bf r}_1 -
{\bf r}_2)$. Or, le développement (\[eq:operateur-masse\]) permet de le faire en principe (il contient néanmoins une infinité de termes), et en particulier dans une limite perturbative, lorsque l’on suppose $\epsilon \ll 1$. A l’ordre le plus bas, l’on trouve donc l’expression assez maniable de la première ligne (\[eq:Bourret\]) $$m( {\bf r}_1 - {\bf r}_2) \approx \epsilon^2 k_0^4 \,
g( {\bf r}_1 - {\bf r}_2)
G_0( {\bf r}_1 - {\bf r}_2 ) ,
\label{eq:Bourret-ici}$$ que l’on appelle, suivant les auteurs, l’approximation de [Bourret]{} [@Frisch66], l’approximation de régularisation du premier ordre [@Frisch66], la “[*ladder approximation*]{}” [@Rytov]. Elle doit également avoir un lien avec la “[*random phase approximation*]{}” (RPA) utilisée dans la physique des solides.
Nous remarquons que l’approximation de [Bourret]{} (\[eq:Bourret-ici\]) ressemble, au premier abord, à l’approximation de [Born]{} pour l’opérateur de masse. Qu’a-t-on alors gagné par rapport au choix de faire d’emblée l’approximation de [Born]{} pour la fonction de [Green]{} ? La réponse se trouve par exemple dans la formule (\[eq:dispersion\]) qui permet de calculer les nouveaux modes dans le milieu aléatoire, alors qu’avec l’approximation de [Born]{}, la fonction de [Green]{} donne seulement une amplitude de diffusion. Da façon plus profonde, le formalisme de l’équation de [Dyson]{} permet, dans l’approximation de [Bourret]{}, de trouver une fonction de [Green]{} moyenne qui, elle, est déjà une resommation partielle d’une infinité de diagrammes.
Pour illustrer la puissance de l’approximation de [Bourret]{} pour la diffusion multiple des ondes, citons le résultat pour le vecteur d’onde effectif dans un milieu faiblement diffuseur, donné par [Rytov]{} ([@Rytov], éq.(4.61)) : $$k_{\rm{eff}} = k_0 + \frac{ \pi }{ 4 } \epsilon^2 k_0^2
\int\limits_0^\infty \! \frac{ k' {\rm d}k' }{ (2\pi)^3 }
g(k') \log\!\left( \frac{ 2 k_0 + k' }{ 2 k_0 - k' } \right)^2
+ \, {\rm i} \frac{ \pi^2 }{ 2 } \epsilon^2 k_0^2
\int\limits_0^{2k_0} \! \frac{ k' {\rm d}k' }{ (2\pi)^3 }
g(k')
\label{eq:k-eff-Bourret}$$ où la fonction de corrélation est supposée isotrope (sa transformée de [Fourier]{} $g(k')$ ne dépend alors que du module $k'$ du vecteur d’onde). On constate que le vecteur d’onde dans le milieu contient une partie imaginaire positive (atténuation de l’onde cohérente par la diffusion), et une partie réelle un peu plus grande que dans le vide.
Pour donner un ordre de grandeur de ces effets, prenons un potentiel aléatoire avec une corrélation gaussienne. La densité spectrale est alors gaussienne également. La transformée de [Fourier]{} $m({\bf k})$ de l’opérateur de masse est alors donnée par $$\begin{aligned}
m({\bf k}) & = & \frac{ {\rm i} \sqrt{\pi} }{ 2 \sqrt{2} }
\frac{ (\epsilon k_0^2 \ell_c)^2 }{ k \ell_c }
\begin{array}[t]{l}
\left[
{\rm e}^{- (k_0 + k)^2 \ell_c^2 / 2 }
{\rm{erfc}}[-{\rm i} (k_0 + k) \ell_c / \sqrt{2} ] \right. \\
\left. - \,
{\rm e}^{- (k_0 - k)^2 \ell_c^2 / 2 }
{\rm{erfc}}[-{\rm i} (k_0 - k) \ell_c / \sqrt{2} ]
\right]
\end{array}
\label{eq:m(k)-gaussien}\\
&& {\rm{erfc}}(x) = \frac{ 2 }{ \sqrt{\pi} }
\int_x^\infty \! {\rm e}^{- t^2 } \, {\rm d}t
\label{eq:def-erfc}\end{aligned}$$ Cette fonction est représentée sur la figure \[fig:masse\].
Quant à la relation de dispersion (\[eq:dispersion\]), l’on trouve le vecteur d’onde effectif suivant (dans la limite semi-classique $k_0 \ell_c \gg 1$) $$k_{\rm{eff}} \approx k_0 \left( 1 +
\frac{ \epsilon^2 }{ 8 } \right)
+ {\rm i} k_0 \frac{ \pi^2 }{ 2 (2\pi)^{3/2} }
\epsilon^2 k_0 \ell_c .
\label{eq:k-eff-ordre}$$ [Rytov]{} n’oublie pas de nous rappeler que ce résultat n’est valable que dans l’approximation de [Bourret]{}. En comparant aux corrections d’ordre suivant, [Rytov]{} trouve la condition de validité suivante $$\mbox{\sc Bourret}: \quad
\left( \epsilon k_0 \ell_c \right)^2 \ll 1 .
\label{eq:validite-Bourret}$$ Rappelons que dans les méthodes classiques du chapitre \[s:classique\], la condition de validité d’une approche de [Fokker–Planck]{} était que le potentiel soit faible par rapport à l’énergie cinétique des atomes, ce qui revient dans la notation de ce chapitre à la condition $\epsilon^2 \ll 1$ \[voir (\[eq:def-epsilon-Keller\]) et (\[eq:def-epsilon\])\]. L’approximation de [Bourret]{} (\[eq:validite-Bourret\]) impose donc une limite plus stricte au potentiel moyen qui l’est d’autant plus que l’on se place dans le régime semi-classique où $k_0 \ell_c \gg 1$.
##### Remarque.
L’approximation de [Bourret]{} peut être généralisée à un milieu aléatoire dont la statistique n’est pas gaussienne. Les fonctions de corrélations peuvent alors être écrites comme une somme sur des corrélations élémentaires $g_k( {\bf r}_1, \ldots,
{\bf r}_k )$ pour un groupe de $k$ points (développement en essaims[^12]). [Finkel’berg]{} montre alors que pour calculer l’opérateur de masse à l’ordre le plus bas, il suffit de retenir les diagrammes irréductibles qui ne contiennent qu’un seul groupe de points [@Finkelberg67]. Il appelle cette approximation la “[*single-group approximation*]{}”. Son avantage est qu’elle permet également de tenir compte de la diffusion par des agglomérats de plus que deux particules. [Finkel’berg]{} donne la condition de validité suivante pour cette approximation : $$\mbox{{\em single group\/} :} \quad
\left| \frac{ {\rm d} m }{ {\rm d} k^2 } \right| \ll 1.
\label{eq:Finkelberg}$$ Il montre en particulier que cette condition est suffisante pour que la diffusion soit faible (libre parcours moyen beaucoup plus grand que la longueur d’onde).
### Equation de [Bethe–Salpeter]{}
Pour finir ce paragraphe, il ne reste qu’à présenter l’équation d’évolution pour la fonction de corrélation de la fonction d’onde atomique. Nous allons à cet effet considérer non plus la fonction de [Green]{}, mais l’onde rayonnée par une source $\varrho({\bf r})$ quelconque $$\psi({\bf r}) = \int\!{\rm d}{\bf r}' \, G( {\bf r}, {\bf r}')
\, \varrho({\bf r}')
\label{eq:champ-rayonne}$$ L’onde moyenne $\langle\psi({\bf r})\rangle$ s’exprime alors par une équation analogue où intervient la fonction de [Green]{} moyenne $\overline{G}( {\bf r} - {\bf r}')$. En ce qui concerne la fonction de cohérence des atomes, elle s’exprime par une corrélation entre deux fonctions de [Green]{} : $$\langle \psi({\bf r}_1) \psi^*({\bf r}_2) \rangle =
\int\!{\rm d}{\bf r}'_1 \, {\rm d}{\bf r}'_2 \,
\langle G( {\bf r}_1, {\bf r}'_1 )
G^*( {\bf r}_2, {\bf r}'_2 ) \rangle
\,\varrho( {\bf r}'_1 ) \varrho^*( {\bf r}'_2 )
\label{eq:def-corr-psi}$$ Cette fonction de corrélation entre deux fonctions de [Green]{} est l’objet de l’équation de [Bethe–Salpeter]{}. Elle se déduit de l’équation de [Schrödinger]{} (\[eq:Schroedinger-stationnaire\]) de façon analogue à l’équation de [Dyson]{} (\[eq:Dyson\]) (voir [@Frisch66] et § 4.3 de [@Sheng95] pour plus de détails). L’équation de [Bethe–Salpeter]{} permet d’écrire l’équation fermée suivante pour la fonction de cohérence du champ atomique : $$\begin{aligned}
\langle \psi({\bf r}_1) \psi^*({\bf r}_2) \rangle & = &
\langle \psi({\bf r}_1) \rangle
\langle \psi^*({\bf r}_2) \rangle +
\int\!{\rm d}{\bf r}'_1 \, {\rm d}{\bf r}'_2 \,
{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 \, {\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 \,
\label{eq:Bethe-Salpeter}\\
&& \quad
\overline{G}( {\bf r}_1 - {\bf r}'_1)
\overline{G}^*( {\bf r}_2 - {\bf r}'_2) \,
K( {\bf r}'_1, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 | {\bf r}'_2 , {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 ) \,
\langle \psi({{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1) \psi^*({{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2) \rangle
\nonumber\end{aligned}$$ Dans cette équation apparaît la fonction $K( {\bf r}_1, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 | {\bf r}_2 , {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 )$ que [Frisch]{} appelle l’opérateur d’intensité. Chez d’autres auteurs, elle porte le nom de vertex irréductible [@Tsang85; @Sheng95]. D’un point de vue physique, cet objet décrit la corrélation aux positions ${\bf r}_{1,2}$ entre les champs rayonnés par deux sources situées à ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_{1,2}$. Son développement perturbatif (en diagrammes) contient les premiers termes suivants : $$\begin{aligned}
&& K( {\bf r}_1, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 | {\bf r}_2, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 )
= \epsilon^2 k_0^4\, g( {\bf r}_1 - {\bf r}_2 ) \,
\delta( {\bf r}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 ) \,
\delta( {\bf r}_2 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 )
+
\label{eq:K-Bourret}\\
&& \quad +\,
\epsilon^4 k_0^6 \, g( {\bf r}_1 - {\bf r}_2 ) \,
g( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 )
\left[
\delta( {\bf r}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 ) \,
G_0( {\bf r}_2 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 ) + \{ \mbox{$1 \leftrightarrow 2$} \}
\right]
+
\nonumber\\
&& \quad +\,
\epsilon^4 k_0^8 \, g( {\bf r}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 ) \,
g( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 - {\bf r}_2 )
G_0( {\bf r}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 ) \,
G_0( {\bf r}_2 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 )
+
\nonumber\\
&& \quad +\,
\epsilon^4 k_0^8 \, g( {\bf r}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 ) \,
\delta( {\bf r}_2 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 )
\int\!{\rm d}{\bf r}_1'\,
G_0( {\bf r}_1 - {\bf r}_1' ) \,
g( {\bf r}_1' - {\bf r}_2 ) \,
G_0( {\bf r}_1' - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 )
+
\nonumber\\
&& \quad \quad + \, \{ \mbox{$1 \leftrightarrow 2$} \} + \ldots
\label{eq:operateur-K}\end{aligned}$$ L’opérateur d’intensité est donc un objet plutôt encombrant. Son approximation de [Bourret]{} est donnée par la première ligne, l’éq. (\[eq:K-Bourret\]), où l’on constate qu’il revient essentiellement à la fonction de corrélation du potentiel si ce dernier est faible.
Notons encore la propriété générale suivante de l’opérateur (\[eq:operateur-K\]), elle aussi due à l’homogénéité du milieu diffuseur : il est invariant par une translation globale de ses quatre arguments $$K( {\bf r}_1 + \Delta{\bf r}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 + \Delta{\bf r} |
{\bf r}_2 + \Delta{\bf r} , {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 + \Delta{\bf r} )
=
K( {\bf r}_1, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 | {\bf r}_2 , {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 )
\label{eq:invariance-K}$$ Par conséquent, il ne peut dépendre que des différences des positions, comme on le constate aussi dans le développement (\[eq:operateur-K\]). Ceci se traduit par la propriété suivante pour la transformée de [Fourier]{} de l’opérateur d’intensité : $$\begin{aligned}
&& K( {\bf k}_1, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 | {\bf k}_2, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2 )
=
\int\!{\rm d}{\bf r}_1\, {\rm d}{\bf r}_2\,
{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1\, {\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2\,
\nonumber\\
&&\qquad
K( {\bf r}_1, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 | {\bf r}_2 , {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 )
\exp{\rm i}\left(
- {\bf k}_1\!\cdot\!{\bf r}_1 + {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1\!\cdot\!{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1
+ {\bf k}_2\!\cdot\!{\bf r}_2 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2\!\cdot\!{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2
\right)
\label{eq:def-TF-K}\\[0.3\jot]
&& = (2\pi)^3 \,
\delta\!\left(
{\bf k}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 - ({\bf k}_2 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2) \right)
\tilde{K}( {\bf k}_1, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 | {\bf k}_2, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2 )
\label{eq:TF-K}\end{aligned}$$ Les signes des vecteurs d’onde ont été choisis tels que la T.-F. de l’opérateur d’intensité admet l’interprétation suivante : il décrit la corrélation entre les amplitudes pour les deux processus de diffusion $${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 \to {\bf k}_1 \quad \mbox{et} \quad
{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2 \to {\bf k}_2$$ La fonction $\delta$ dans (\[eq:TF-K\]) exprime alors que ces deux processus ne sont corrélés que lorsqu’ils font intervenir le même transfert de vecteur d’onde : $$K \ne 0 \quad \Longleftrightarrow \quad
{\bf k}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 = {\bf k}_2 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2
\label{eq:diffusions-correlees}$$ Cette propriété se comprend aisément dans l’approximation de [Born]{} : l’amplitude de diffusion pour le processus ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 \to {\bf k}_1$ est proportionnelle à la composante de [Fourier]{} $\mu({\bf k}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1)$ du potentiel aléatoire, et les composantes de [Fourier]{} du potentiel aléatoire sont $\delta$-corrélées lorsque le potentiel est statistiquement homogène : $$\langle \mu({\bf q}_1) \mu^*({\bf q}_2) \rangle = \delta({\bf q}_1 -
{\bf q}_2) (2\pi)^3 \epsilon^2 g({\bf q}_1) .
\label{eq:Vq1-Vq2}$$
Dans la pratique, il n’est pas possible de résoudre l’équation de [Bethe–Salpeter]{} (\[eq:Bethe-Salpeter\]), même dans l’approximation de [Bourret]{}. Au paragraphe suivant, nous retraçons une reformulation de l’éq. de B.–S. qui permet de retrouver l’approche du transfert radiatif.
Lien à l’équation du transfert radiatif {#s:vers-TR}
---------------------------------------
L’intérêt de ce paragraphe est de préciser le domaine de validité de l’équation du transfert radiatif, en la déduisant du formalisme microscopique de la diffusion multiple des ondes. Une présentation semblable a été donnée par [Barabanenkov]{} et [Finkel’berg]{} [@Barabanenkov67] et par [Rytov]{} [*et al.*]{} ([@Rytov], § 4.3) pour des ondes scalaires. Pour des ondes électromagnétiques, une généralisation de l’équation du transfert radiatif (portant sur le vecteur de [Stokes]{}) est déjà formulée dans le livre de [Chandrasekhar]{}. Une justification microscopique en a été donnée par [K. M. Watson]{} [@Watson69] dans une situation de diffusion faible, et J.-M. [Tualle]{} a fourni une présentation plus générale [@TualleT]. Une bonne référence est également l’article de revue de [Lagendijk]{} et [van Tiggelen]{} [@vanTiggelen96].
Nous verrons que l’équation du transfert radiatif (ETR) est valable à grande échelle spatiale (ce qui n’est guère étonnant, vu le résultat de [Ryzhik]{} [*et al.*]{}), mais aussi quelle que soit la force du potentiel diffuseur. Ce dernier point semble ne pas avoir été soulevé par d’autres auteurs parce que l’on se place dans la plupart des cas dans le régime de diffusion faible, en utilisant l’approximation de [Bourret]{}. [Luck]{}, par exemple, dit dans ses papiers [@Luck93; @Luck96], au tournant d’une phrase : l’équation de [Bethe–Salpeter]{}, dans l’approximation de [Bourret]{}, est équivalente à l’équation du transfert radiatif. Nous allons par contre rester assez général et ne pas faire l’approximation de [Bourret]{}. Ainsi le formalisme permet-il de voir à quel endroit l’équation du transfert radiatif est moins générale que celle de B.–S.
Nous allons seulement faire l’hypothèse suivante (qui ne semble pas très restrictive)
- le milieu est statistiquement homogène. La fonction de [Green]{} moyenne $\overline{G}$, qui intervient dans l’équation de B.–S. (\[eq:Bethe-Salpeter\]), admet alors le développement de [Fourier]{} (\[eq:TF-<G>\])[^13] $$\begin{aligned}
\overline{G}( {\bf r} - {\bf r}' )
& = &\int\!\dbar{\bf k} \frac{ \exp{\rm i}{\bf k}\cdot(
{\bf r} - {\bf r}' ) }{ k_0^2 - {\bf k}^2 - m({\bf k}) }
\label{eq:TF-<G>-ici}\end{aligned}$$
où $m( {\bf k} )$ est l’opérateur de masse.
### [Bethe–Salpeter]{} dans [Wigner]{}
La première étape consiste à exprimer l’équation de B.–S. (\[eq:Bethe-Salpeter\]) en représentation de [Wigner]{}. En introduisant les transformées de [Fourier]{} des fonctions de [Green]{} moyennées et de l’opérateur d’intensité, l’on trouve l’équation suivante : $$f( {\bf r}, {\bf k} ) = f_{\rm{coh}}( {\bf r}, {\bf k} )
+ f_{\rm{diff}}( {\bf r}, {\bf k} )
\label{eq:BS-0}$$
Calculons d’abord le premier terme qui correspond à la transformée de [Wigner]{} du produit des champs moyens. Nous exprimons le champ moyen à l’aide de la fonction de [Green]{} moyennée ([*cf.*]{} éq. \[eq:champ-rayonne\]) et insérons ensuite la décomposition de [Fourier]{} (\[eq:TF-<G>-ici\]) de cette dernière : $$\begin{aligned}
\lefteqn{
f_{\rm{coh}}( {\bf r}, {\bf k} ) =
}
\nonumber\\
&& =
\int\!{\rm d}{\bf s} \, {\rm e}^{- {\rm i} {\bf k}\!\cdot\!{\bf s} }
\int\!{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 \, {\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 \,
\overline{G}( {\bf r} + {{\textstyle\frac12}}{\bf s} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 ) \,
\overline{G}^*( {\bf r} - {{\textstyle\frac12}}{\bf s} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 ) \,
\varrho( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_1 ) \, \varrho^*( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_2 )
\nonumber\\
&& =
\int\!{\rm d}{\bf s} \, \dbar{\bf k}_1 \, \dbar{\bf k}_2 \,
{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}\,
\overline{G}( {\bf k}_1 ) \, \overline{G}^*( {\bf k}_2 ) \,
f_\varrho[ {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {{\textstyle\frac12}}({\bf k}_1 + {\bf k}_2 ) ] \times
\nonumber\\
&& \quad \times
\exp{\rm i}\left[
- {\bf k}\!\cdot\!{\bf s}
+ {{\textstyle\frac12}}({\bf k}_1 + {\bf k}_2)\!\cdot\!{\bf s} +
({\bf k}_1 - {\bf k}_2)\!\cdot\! ({\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}})
\right]
\label{eq:coh-1}\end{aligned}$$ Nous avons effectué une des deux intégrations sur les positions de la source $\varrho({{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}})$ pour faire apparaître sa transformée de [Wigner]{} $f_\varrho[ {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {{\textstyle\frac12}}({\bf k}_1 + {\bf k}_2 ) ]$. L’intégration sur ${\bf s}$ est maintenant immédiate et l’équation (\[eq:coh-1\]) devient (${\bf q} = {\bf k}_1 - {\bf k}_2$, ${\bf k} = {{\textstyle\frac12}}({\bf k}_1 + {\bf k}_2 )$) : $$f_{\rm{coh}}( {\bf r}, {\bf k} ) =
\int\!
{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}\,
\dbar{\bf q} \,
\tilde{A}( {\bf q} ; {\bf k} ) \,
f_\varrho( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {\bf k} ) \,
\exp{\rm i} {\bf q}\!\cdot\!({\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}})
\label{eq:coh-2}$$ avec l’abbréviation suivante pour le produit des transformées de [Fourier]{} des fonctions de [Green]{} moyennées : $$\tilde{A}( {\bf q} ; {\bf k} ) =
\overline{G}( {\bf k} + {{\textstyle\frac12}}{\bf q} )
\overline{G}^*( {\bf k} - {{\textstyle\frac12}}{\bf q} )
\label{eq:def-A}$$ Nous constatons donc que le terme cohérent relie la luminance observée à celle de la source au même vecteur d’onde ${\bf k}$. Ecrivons ce résultat encore sous la forme intégrale d’une équation de transfert radiatif. Le terme cohérent s’exprime comme une intégrale sur les points sources, pondérés avec une fonction d’atténuation (la transformée de [sc Fourier]{} de $\tilde{A}({\bf q} ; {\bf k} )$) : $$\begin{aligned}
f_{\rm{coh}}( {\bf r}, {\bf k} ) & = &
\int\!{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} \,
A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ; {\bf k} ) f_\varrho( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {\bf k} ) ,
\label{eq:coh-fin}\\
A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ; {\bf k} ) & = &
\int\!\dbar{\bf q} \tilde{A}( {\bf q} ; {\bf k} )
\exp{ {\rm i} {\bf q}\!\cdot\! ({\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}) }
\label{eq:def-att}\end{aligned}$$
Pour le deuxième terme $f_{\rm{diff}}$ dans (\[eq:BS-0\]), nous obtenons d’abord, par une procédure similaire, l’expression suivante qui ressemble à (\[eq:coh-1\]) : $$\begin{aligned}
&& f_{\rm{diff}}( {\bf r}, {\bf k}) =
\int\!{\rm d}{\bf s}\, \dbar{\bf k}_1 \, \dbar{\bf k}_2 \,
{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} \, \dbar{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 \, \dbar{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2 \,
\nonumber\\
&& \quad
K( {\bf k}_1, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 | {\bf k}_2, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2 ) \,
f[ {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {{\textstyle\frac12}}({{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 + {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2)] \,
\overline{G}( {\bf k}_1 ) \, \overline{G}^*( {\bf k}_2 ) \times
\nonumber\\
&& \quad \times
\exp{\rm i}\left[
- {\bf k}\!\cdot\!{\bf s}
+ {{\textstyle\frac12}}({\bf k}_1 + {\bf k}_2)\!\cdot\!{\bf s}
+ ({\bf k}_1 - {\bf k}_2)\!\cdot\!{\bf r}
- ({{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2) \!\cdot\!{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}
\right]
\label{eq:BS-1}\end{aligned}$$ Les intégrations sur ${\bf s}$ et sur la demi-somme ${\bf k} = {{\textstyle\frac12}}( {\bf k}_1 + {\bf k}_2 )$ des vecteurs d’onde s’effectuent de la même façon qu’en (\[eq:coh-1\]) pour faire apparaître la fonction $\tilde{A}( {\bf q} ; {\bf k})$ (\[eq:def-A\]). Pour l’intégration sur ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_{1,2}$, passons à la mi-somme et la différence, et utilisons l’invariance par translation (\[eq:TF-K\]) de l’opérateur d’intensité. Nous trouvons ainsi l’expression assez compacte similaire à (\[eq:coh-fin\]) (avec ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} = {{\textstyle\frac12}}({{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 + {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2 )$) : $$f_{\rm{diff}}( {\bf r}, {\bf k} ) =
\int\!{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} \, \dbar{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} \,
p_A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ; {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
f( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
\label{eq:diff-fin}$$ où intervient une fonction d’atténuation différente qui ressemble plutôt à une fonction de phase (avec ${\bf q} = {\bf k}_1 - {\bf k}_2 = {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_1 -
{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}_2$) : $$\begin{aligned}
&& p_A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ; {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} ) =
\nonumber\\
&& \int\!\dbar{\bf q} \,
\exp[ {\rm i} {\bf q}\!\cdot\! ({\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}) ] \,
\tilde{A}( {\bf q} ; {\bf k} ) \,
\tilde{K}\!\left(
{\bf k} + {{\textstyle\frac12}}{\bf q}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} + {{\textstyle\frac12}}{\bf q} |
{\bf k} - {{\textstyle\frac12}}{\bf q}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} - {{\textstyle\frac12}}{\bf q}
\right)
\label{eq:def-phase}\end{aligned}$$ Ce qui est remarquable, c’est que nous n’avons encore fait aucune approximation pour déduire ce résultat.
### Comparaison à l’équation de transfert
Regardons d’abord ce que l’on a pu faire : l’équation de [Bethe–Salpeter]{} a été transformée en : $$f( {\bf r}, {\bf k} ) =
\int\!{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} \,
A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ; {\bf k} ) f_\varrho( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {\bf k} )
+ \int\!{\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} \, \dbar{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} \,
p_A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ; {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
f( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} ) .
\label{eq:ETR*}$$ La distribution de [Wigner]{} (rappelons qu’elle est équivalente à la luminance pour la lumière) contient un premier terme cohérent (\[eq:coh-fin\])) qui correspond aux champ moyen rayonné par les sources dans le milieu. Le deuxième terme, l’intégrale (\[eq:diff-fin\]), doit alors correspondre aux ondes diffusées par le milieu. Il a bien la structure d’une intégrale de diffusion, où intervient la fonction de [Wigner]{} pour un vecteur d’onde ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}$ différent. En effet, le vecteur d’onde ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}$ peut très bien s’interpréter comme un vecteur d’onde entrantqui est diffusé vers le vecteur d’onde ${\bf k}$, avec une probabilité donnée par l’opérateur d’intensité.
Les intégrales sur ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}$ dans les deux termes de (\[eq:ETR\*\]) ne sont pas si surprenantes qu’elles le semblent au premier abord, parce que nous sommes en train d’étudier une formulation intégrale, qu’il faut comparer à la version intégrale de l’équation du transfert radiatif. En utilisant la forme donnée par [Chandrasekhar]{} ([@Chandrasekhar], Chap. 1, éq.(50)) et en ajoutant la luminance $S( {\bf r}, {\bf n} )$ des sources, la version intégrale de l’ETR s’écrit ($I$ est la luminance) : $$I({\bf r}, {\bf n}) = \int\limits_0^\infty \! {\rm d}s \,
{\rm e}^{- s / \ell} S( {\bf r} - s {\bf n}, {\bf n} )
+ \int\limits_{0}^\infty \!\frac{ {\rm d}s }{ \ell } \,
{\rm e}^{- s / \ell} \int\!{\rm d}{\bf n}' \,
p( {\bf n'}, {\bf n}) I({\bf r} - s{\bf n}, {\bf n}')
\label{eq:ETR-Chandra}$$ où ${\bf r} - s{\bf n}$ est un point sur le rayon de direction ${\bf n}$ à distance $s$ du point d’observation. L’intégrale sur $s$ exprime le fait que la luminance en ${\bf r}$ est le produit de tous les évènements de diffusion survenus aux positions antérieures à ${\bf r}$ sur ce rayon et qui ont amené la lumière dans la direction ${\bf n}$. Par ailleurs, le premier terme dans (\[eq:ETR-Chandra\]) a bien la forme de la partie cohérente du champ, qui est atténuée par la diffusion sur une distance caractéristique égale au libre parcours moyen $\ell$.
A la différence du transfert radiatif (\[eq:ETR-Chandra\]), l’intégrale sur ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}$ dans l’expression exacte (\[eq:ETR\*\]) s’étend [*a priori*]{} sur tout l’espace. La fonction de [Wigner]{} vérifie donc une équation de transport [*non-locale*]{}. Pour retrouver l’ETR standard (\[eq:ETR-Chandra\]) qui, elle, est locale, c’est l’intégrale sur ${\bf q}$ dans la fonction d’atténuation $A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ; {\bf k} )$ (\[eq:def-att\]) qui doit limiter l’intégrale sur ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}$ de sorte qu’elle ne porte plus que sur le rayon arrivant au point d’observation ${\bf r}$ de la direction ${\bf k}$. Nous allons maintenant préciser les conditions pour qu’apparaisse une telle simplification.
### Solution approchée à grandes distances
#### Astuce
Transformons d’abord le produit des deux dénominateurs dans (\[eq:def-A\]) selon la formule suivante (évidente, mais astucieuse…) $$\begin{aligned}
\lefteqn{
\tilde{A}( {\bf q} ; {\bf k} ) =
\overline{G}( {\bf k} + {{\textstyle\frac12}}{\bf q} )
\overline{G}^*( {\bf k} - {{\textstyle\frac12}}{\bf q} )
}
\nonumber\\
&& = \frac{ 1 }{
[ k_0^2 - ({\bf k} + {{\textstyle\frac12}}{\bf q})^2 - m({\bf k} + {{\textstyle\frac12}}{\bf q}) ]
[ k_0^2 - ({\bf k} - {{\textstyle\frac12}}{\bf q})^2 - m^*({\bf k} - {{\textstyle\frac12}}{\bf q}) ] }
\nonumber\\
&& = \frac{
\overline{G}( {\bf k} + {{\textstyle\frac12}}{\bf q} ) -
\overline{G}^*( {\bf k} - {{\textstyle\frac12}}{\bf q} )
}{ 2 {\bf k}\!\cdot\!{\bf q} + m({\bf k} + {{\textstyle\frac12}}{\bf q})
- m^*({\bf k} - {{\textstyle\frac12}}{\bf q}) }
\label{eq:astuce}\end{aligned}$$ Cette expression est encore exacte. Si nous cherchons maintenant le comportement de la distribution de [Wigner]{} à des échelles $|{\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}|$ grandes par rapport à la longueur d’onde $\lambda_{dB}$, l’intégrale (\[eq:def-att\]) montre que l’on peut se contenter des vecteurs d’onde ${\bf q}$ très petits par rapport aux vecteurs d’onde atomiques ${\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}$. L’idée vient alors à l’esprit de faire un développement limité en fonction de ${\bf q}$. Nous allons le faire ici à l’ordre le plus bas ; l’on verra que l’on retrouve alors la forme (\[eq:ETR-Chandra\]) de l’équation du transfert radiatif. Au paragraphe \[s:echelle-min\] suivant, nous irons un ordre plus loin pour obtenir une première correction par rapport à l’équation de transfert.
A l’ordre le plus bas en ${\bf q}$, la formule (\[eq:astuce\]) devient $$\tilde{A}( {\bf q} ; {\bf k} ) \approx
\frac{ - 2 \pi {\rm i}
\delta( k^2 - {\rm{Re}}\, k_{\rm{eff}}^2 ) }{
2 {\bf k}\!\cdot\!{\bf q} - 2 {\rm i} \,\mbox{Im}\, k_{\rm{eff}}^2 } ,
\label{eq:simplification-q-petit}$$ où nous avons utilisé le vecteur d’onde effectif (complexe) pour écrire le pôle du dénominateur : $$m( {\bf k} \pm{{\textstyle\frac12}}{\bf q} ) \approx m( {\bf k} )
\approx k_0^2 - k_{\rm{eff}}^2
\label{eq:approx-masse}$$ Nous avons également approximée la partie imaginaire de la fonction de [Green]{} moyennée (le terme au numérateur dans (\[eq:astuce\])) par une fonction $\delta$. Celle-ci exprime le fait que le module des vecteurs d’onde atomiques finaux ${\bf k}$ (après le processus de diffusion) est fixé par la relation de dispersion dans le milieu (la partie réelle du vecteur d’onde effectif $k_{\rm{eff}}$). Cette approximation est justifiée dans le régime de faible diffusion où le libre parcours moyen est beaucoup plus grand que la longueur d’onde. Une telle situation correspond à l’image intuitive que l’on se fait de l’ETR. La limite opposée correspond à des ondes localisées dans le milieu aléatoire, un régime qui demande une description au-delà de l’ETR standard.
#### Retrouver l’ETR différentielle
En reportant (\[eq:simplification-q-petit\]) dans l’ETR et en négligeant le petit vecteur d’onde ${\bf q}$ dans l’opérateur d’intensité, nous observons que l’ETR (\[eq:ETR\*\]) se résoud facilement par une transformée de [Fourier]{} par rapport à la position ${\bf r}$. Nous notons $\tilde{f}( {\bf q}, {\bf k} )$ la fonction de [Wigner]{} transformée, et elle vérifie l’équation suivante $$\begin{aligned}
&&
\left(
{\rm i}{\bf k}\!\cdot\!{\bf q} + \mbox{Im}\, k_{\rm{eff}}^2
\right) \tilde{f}( {\bf q}, {\bf k} ) =
\nonumber\\
&& \quad
\pi \delta( k^2 - \mbox{Re}\, k_{\rm{eff}}^2 )
\left(
\tilde{f}_{\varrho}( {\bf q}, {\bf k} ) +
\int\dbar{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} \,
\tilde{K}( {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} | {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
\tilde{f}( {\bf q}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
\right)
\label{eq:ETR*-diff}\end{aligned}$$ En revenant dans l’espace directe, nous trouvons donc la forme différentielle de l’ETR, dont le libre parcours moyen et la fonction de phase sont donnés par $$\begin{aligned}
\frac{ 1 }{ \ell } &=& \frac{ \mbox{Im}\,k_{\rm eff}^2 }{ k_p }
= - \frac{ \mbox{Im}\, m(k_p) }{ k_p }
\label{eq:ell-ETR*}\\
p( {\bf u}, {\bf u}' ) & = &
\frac{ 1 }{ ( 4\pi k_p )^2 }
\tilde{K}( k_p {\bf u}, k_p {\bf u}' | k_p {\bf u}, k_p {\bf u}' )
\label{eq:phase-ETR*}\\
k_p & = & ( \mbox{Re}\, k_{\rm eff}^2 )^{1/2}
\label{eq:def-k-p}\end{aligned}$$ Nous avons donc trouvé ici des expressions microscopiques pour les paramètres qui interviennent dans l’ETR. Des formules identiques ont étés écrites par [Barabanenkov]{} et [Finkel’berg]{} (eq. (22) de [@Barabanenkov67]), mais en utilisant une forme approchée pour les opérateurs de masse et d’intensité (approximation de [Bourret]{}). La démarche présentée ici montre que ces expressions restent valables dans un contexte plus général, à condition que les ondes ne soient pas localisées dans le milieu. La seule différence par rapport aux résultats de [Barabanenkov]{} et [Finkel’berg]{} est l’apparition du vecteur d’onde $k_p$ qui décrit l’indice de réfraction du milieu aléatoire. A la limite perturbative, il faut en effet prendre $k_p = k_0$ pour être consistent avec la conservation de l’énergie (l’identité de [Ward]{}, voir [@vanTiggelen96]).
#### Retrouver l’ETR intégrale
Nous pouvons également retrouver la formulation intégrale de l’ETR donnée en (\[eq:ETR-Chandra\]). A cet effet, nous calculons l’intégrale sur ${\bf q}$ dans (\[eq:ETR\*\]) dans l’approximation (\[eq:simplification-q-petit\]) pour la fonction d’atténuation $\tilde{A}( {\bf q}; {\bf k} )$. L’intégrale a d’ailleurs été calculée au chapitre précédent (voir la note \[fn:delta-transverse\] en bas de la page ): $$\begin{aligned}
\lefteqn{
\int\!\dbar{\bf q} \frac{
\exp{\rm i}{\bf q}\!\cdot\!( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}) }{
2 {\bf k}\!\cdot\!{\bf q} - 2 {\rm i} \,\mbox{Im}\, k_{\rm{eff}}^2 }
= }
\label{eq:retrouver-Chandra}\\
&& = \left\{
\begin{array}{l}
( {\rm i} / 2 k )
\delta\!\left( {\bf r}_\perp - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}_\perp \right)
\exp[ - ( \mbox{Im}\, k_{\rm{eff}}^2 / k^2 )
{\bf k}\!\cdot\!({\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}) ] ,
\\
\begin{array}[b]{ll}
&\mbox{lorsque }
{\bf k} \!\cdot\! ( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ) > 0 ;
\\
0 , & \mbox{lorsque }
{\bf k} \!\cdot\! ( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ) < 0 .
\end{array}
\end{array}
\right.
\nonumber\end{aligned}$$ (L’indice $\perp$ fait référence au vecteur ${\bf k}$.) Nous en concluons que l’intégrale sur ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}$ ne porte en fait que sur un demi-rayon qui aboutit au point d’observation ${\bf r}$ en se propageant le long de la direction ${\bf k}$. Le long de ce rayon, l’intensité du champ est atténuée avec une longueur caractéristique $$\ell = \frac{ k }{ \mbox{Im } k_{\rm{eff}}^2 }
\label{eq:def-ell}$$ ce qui correspond bien à l’atténuation du champ cohérent par diffusion. Le terme cohérent s’écrit donc comme une intégrale sur les sources interceptées le long de ce rayon, qui rayonnent dans la direction $\hat{\bf k}$ : $$f_{\rm{coh}}( {\bf r}, {\bf k} ) =
\frac{ \pi }{ k } \,
\delta( k^2 - \mbox{Re}\, k_{\rm{eff}}^2 )
\int\limits_0^\infty\!{\rm d}s' \,
{\rm e}^{ - s' / \ell } \,
f_\varrho( {\bf r} - s' \hat{\bf k}, {\bf k} )
\label{eq:coh-retrouve}$$ Nous retrouvons ici le premier terme de l’équation du transfert radiatif de [Chandrasekhar]{} (\[eq:ETR-Chandra\]). La fonction $\delta$ assure que les seuls vecteurs d’onde de la source qui rayonnent sont ceux qui correspondent à la relation de dispersion dans le milieu aléatoire. (Rappelons encore que la source $\varrho( {\bf r} )$ dans (\[eq:champ-rayonne\]) et (\[eq:coh-retrouve\]) n’a pas la même dimension que le champ, ceci est à l’origine du préfacteur.)
Quant au deuxième terme de (\[eq:ETR-Chandra\]), qui décrit la diffusion, nous remarquons qu’à l’ordre le plus bas, l’opérateur d’intensité dans (\[eq:def-phase\]) ne dépend pas du vecteur d’onde ${\bf q}$ : $$\tilde{K}\!\left(
{\bf k} + {{\textstyle\frac12}}{\bf q}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} + {{\textstyle\frac12}}{\bf q} |
{\bf k} - {{\textstyle\frac12}}{\bf q}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} - {{\textstyle\frac12}}{\bf q}
\right) \approx
\tilde{K}\!\left(
{\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} | {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} \right)
\label{eq:approx-intensite}$$ On peut alors le sortir des intégrales sur ${\bf q}$ et ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}}$ et définir une fonction de phase locale comme suit (attention, sa normalisation est différente de (\[eq:phase-ETR\*\])) $$\begin{aligned}
p_A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ; {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
& = &
A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ; {\bf k} ) \, p( {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
\nonumber\\
p( {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} ) & = & \frac{ \pi }{ k }
\delta( k^2 - {\rm{Re}}\, k_{\rm{eff}}^2 ) \,
\tilde{K}\!\left( {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} | {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} \right) .
\label{eq:fn-phase-generale}\end{aligned}$$ En utilisant le résultat (\[eq:retrouver-Chandra\]) pour la fonction d’atténuation $A( {\bf r} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\rho$}}}{\mbox{\boldmath{$\textstyle\rho$}}}{\mbox{\boldmath{$\scriptstyle\rho$}}}{\mbox{\boldmath{$\scriptscriptstyle\rho$}}}\else\oldbf\rho\fi}} ;
{\bf k})$, le terme de diffusion (\[eq:diff-fin\]) prend donc la forme $$f_{\rm{diff}}( {\bf r}, {\bf k} ) =
\int\limits_0^\infty\! {\rm d}s' \,
{\rm e}^{- s' / \ell} \,
\int\!\dbar{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} \,
p( {\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} ) \,
f( {\bf r} - s' \hat{\bf k}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
\label{eq:diff-retrouve}$$ Puisque la relations de dispersion fixe le module du vecteur d’onde ${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}$, le terme de diffusion est exactement celui donné par [Chandrasekhar]{} en (\[eq:ETR-Chandra\]). (La fonction de phase (\[eq:fn-phase-generale\]) n’est pas normalisée de la même façon que celle de [Chandrasekhar]{}, ceci explique la différence entre les préfacteurs.)
Nous constatons donc qu’aux grandes échelles spatiales, le transport de la distribution de [Wigner]{} des atomes (ou du champ électro-magnétique) est régi par une équation de transport radiatif. Ceci est vrai quelle que soit la force du potentiel aléatoire. Pour un potentiel faible, l’approximation de [Bourret]{} (\[eq:K-Bourret\]) de l’opérateur d’intensité montre que la fonction de phase est proportionnelle à la densité spectrale du potentiel aléatoire $$\begin{aligned}
\mbox{{\sc Bourret}}: &&
\tilde{K}\!\left(
{\bf k} + {{\textstyle\frac12}}{\bf q}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} + {{\textstyle\frac12}}{\bf q} |
{\bf k} - {{\textstyle\frac12}}{\bf q}, {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} - {{\textstyle\frac12}}{\bf q}
\right) =
\nonumber\\
&& \quad = \epsilon^2 k_0^4 \, g( {\bf k} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
= \frac{ 4 M^2 }{ \hbar^4 } S( {\bf k} - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} )
\label{eq:K(k)-Bourret}\end{aligned}$$ Nous retrouvons alors l’approche de [Ryzhik]{} [*et al.*]{}, ainsi que l’idée que s’est faite [Luck]{} de l’équation de transfert. La démonstration que nous venons de donner de cette équation montre cependant qu’elle est également valable pour des potentiels plus forts où l’approximation de [Bourret]{} n’est plus possible.
#### Conclusion
Nous avons constaté qu’aux grandes échelles spatiales, la théorie de la diffusion multiple des ondes prend la forme d’une équation du transfert radiatif. Ce résultat repose seulement sur l’hypothèse d’un milieu invariant par translation et de la diffusion faible (libre parcours moyen beaucoup plus grand que la longueur d’onde). Dans ce régime, la transformée de [Wigner]{} de la fonction de corrélation décrit la luminance du champ, et elle vérifie une ETR (\[eq:ETR\*-diff\]) dont les paramètres macroscopique s’expriment à l’aide des opérateurs de masse et d’intensité.
Nous avons obtenu l’ETR ainsi généralisée par un développement à l’ordre le plus bas par rapport au vecteur d’onde ${\bf q}$, la variable conjuguée au profil spatial en ${\bf r}$ de la luminance. Cette procédure (la limite de [Kubo]{} [@vanTiggelen96]) traduit formellement la limite d’une grande échelle spatiale. Au paragraphe suivant, nous allons pousser le développement jusqu’au second ordre en ${\bf q}$, afin de préciser l’échelle spatiale minimale au-delà de laquelle l’équation du transfert radiatif est justifiée.
### Echelle spatiale minimale[\
]{} pour l’équation du transfer radiatif {#s:echelle-min}
Dans le calcul de la fonction de [Wigner]{} du champ aux grandes distances, nous avons pour l’instant seulement retenu les termes à l’ordre le plus bas en ${\bf q}$. Nous allons ici aller plus loin et analyser les termes de l’ordre suivant pour trouver une limite supérieure $q_{\max}$ à ce développement. Dans l’espace réel, la valeur $q_{\max}$ se traduit par une échelle spatiale minimale $L_{\min} = 1 / q_{\max}$, avec la signification physique la suivante : il faut que la fonction de [Wigner]{} du champ, en fonction de la position ${\bf r}$, varie peu à l’échelle $L_{\min}$ pour que l’équation du transfert radiatif soit valable.
Nous allons nous contenter d’étudier la dépendance de ${\bf q}$ de la fonction d’atténuation $\tilde{A}( {\bf q} ; {\bf k} )$ (\[eq:def-att\]). Quant à la variation de l’opérateur d’intensité $\tilde{K}$ (\[eq:approx-intensite\]) avec ${\bf q}$, l’on en trouvera une discussion dans l’article de revue de [Kravtsov]{} et [Apresyan]{} (§ 6.1 de [@Kravtsov96]) : elle peut être reliée à l’effet de mémoire de la figure de tavelures lorsque l’on fait varier la direction d’incidence du faisceau. Nous notons également que pour un potentiel aléatoire faible (approximation de [Bourret]{}), la formule (\[eq:K(k)-Bourret\]) montre que $\tilde{K}$ est indépendant de ${\bf q}$. Un moyen d’apprécier sa dépendance de ${\bf q}$ est alors de calculer des termes supérieurs (\[eq:operateur-K\]) qui vont au-delà de l’approximation de [Bourret]{}.
Poussons donc le développement de la fonction d’atténuation $\tilde{A}( {\bf q} ; {\bf k} )$ jusqu’à l’ordre quadratique. Nous allons négliger la variation avec ${\bf q}$ de la différence des fonctions de [Green]{} $$\overline{G}( {\bf k} + {{\textstyle\frac12}}{\bf q})
- \overline{G}^*( {\bf k} - {{\textstyle\frac12}}{\bf q})
\approx - 2 \pi {\rm i}
\delta( k^2 - {\rm{Re}}\,k_{\rm{eff}}^2 )$$ et seulement analyser le dénominateur dans (\[eq:astuce\]) qui présente un pôle en ${\bf q}$. A l’ordre quadratique, celui-ci devient : $$\begin{aligned}
\lefteqn{
2 {\bf k}\!\cdot\!{\bf q} + m({\bf k} + {{\textstyle\frac12}}{\bf q})
- m^*({\bf k} - {{\textstyle\frac12}}{\bf q}) \approx
}
\nonumber\\
&&
2 {\bf k}\!\cdot\!{\bf q} - 2 {\rm i} \,{\rm{Im}}\, k_{\rm{eff}}^2
+ {\bf q}\!\cdot\!{\rm{Re}}\,\nabla_k m
+ \frac{ {\rm i} }{ 4 } \sum_{ij} q_i q_j \,{\rm{Im}}\,
\frac{ \partial^2 m }{ \partial k_i \partial k_j }
\label{eq:pole-q2}\end{aligned}$$ Les dérivées de l’opérateur de masse $m( {\bf k})$ sont prises ici pour ${\bf k}$ vérifiant la relation de dispersion. Pour continuer le calcul, nous allons faire l’hypothèse supplémentaire suivante :
- le milieu aléatoire est statistiquement isotrope. La relation de dispersion s’écrit alors $k = k_{\rm{eff}}$ et l’opérateur de masse $m( {\bf k})$ dépend seulement du module $k$ du vecteur d’onde ${\bf k}$.
Par exemple, le milieu à fonction de corrélation gaussienne dont nous avons donné l’opérateur de masse en (\[eq:m(k)-gaussien\]), est isotrope. Par conséquent, les dérivées de l’opérateur de masse de simplifient : $$\begin{aligned}
\nabla_k m & = & \hat{\bf k} \frac{ {\rm d} m }{ {\rm d}k }
\nonumber\\
\frac{ \partial^2 m }{ \partial k_i \partial k_j }
& = &
\delta_{ij} \left( \frac{ 1 }{ k }
\frac{ {\rm d} m }{ {\rm d}k } \right)
+
\hat{k}_i \hat{k}_j \left(
\frac{ {\rm d}^2 m }{ {\rm d}k^2 } -
\frac{ 1 }{ k } \frac{ {\rm d} m }{ {\rm d}k } \right)
\label{eq:dm-dk}\end{aligned}$$ Le dénominateur (\[eq:pole-q2\]) peut alors être écrit sous la forme suivante : $$\begin{aligned}
&&
2 {\bf k}\!\cdot\!{\bf q} + m({\bf k} + {{\textstyle\frac12}}{\bf q})
- m^*({\bf k} - {{\textstyle\frac12}}{\bf q}) \approx
2 {\bf k}'\!\cdot\!{\bf q} -
{\rm i} \gamma
+ {\rm i} \alpha {\bf q}^2
- {\rm i} \beta ( {\bf k}\!\cdot\!{\bf q} )^2
\label{eq:pole-q2-2}\\
&&\qquad {\bf k}' = {\bf k} \left( 1 + \frac{ 1 }{ 2 k }
{\rm{Re}}\, \frac{ {\rm d} m }{ {\rm d}k } \right)
\label{eq:k-prime}\\
&&\qquad \gamma = - 2 \, {\rm Im}\, m( k_{\rm eff} ) =
2 \,{\rm{Im}}\, k_{\rm{eff}}^2
\label{eq:gamma}\\
&&\qquad \alpha =
{\rm{Im}}\left( \frac{ 1 }{ 4 k } \frac{ {\rm d} m }{ {\rm d}k } \right)
\label{eq:alpha}\\
&&\qquad \beta =
\frac{ 1 }{ 4 k^2 } {\rm{Im}}
\left(
\frac{ 1 }{ k } \frac{ {\rm d} m }{ {\rm d}k }
- \frac{ {\rm d}^2 m }{ {\rm d}k^2 }
\right)
\label{eq:beta}\end{aligned}$$ La dérivée première de l’opérateur de masse change donc simplement le module du vecteur d’onde ${\bf k}
\mapsto {\bf k}'$. Le dérivée seconde ajoute des termes quadratiques en ${\bf q}$.
Avant de continuer le calcul, donnons une estimation jusqu’à quel vecteur d’onde maximal ce développement est justifié. La figure \[fig:masse\] et l’expression (\[eq:m(k)-gaussien\]) montrent que l’opérateur de masse $m( {\bf k})$ varie sur une échelle caractéristique de $1 / \ell_c$ en fonction de $k$. Nous trouvons donc la condition de validité $$| {\bf q} | \lesssim \frac{ 1 }{ \ell_c }
\label{eq:condition-q-petit}$$ Dans l’espace réel, cette condition traduit le fait que dans le formalisme du transfert radiatif, la luminance ne résout pas la structure microscopique (longueur de corrélation) du milieu diffusant.
#### Calcul de la fonction d’atténuation
Nous avons donc l’intégrale suivante à calculer : $$\begin{aligned}
\lefteqn{
\int\!\dbar{\bf q} \,
\frac{ \exp{\rm i} {\bf q}\!\cdot\!{\bf r} }{
2 {\bf k}'\!\cdot\!{\bf q} - {\rm i} \gamma
+ {\rm i} \alpha {\bf q}^2
- {\rm i} \beta ( {\bf k}\!\cdot\!{\bf q} )^2 }
} \nonumber\\
&& = \int\!\dbar{\bf Q} \, \dbar q \,
\frac{ \exp{\rm i} \left( {\bf Q}\!\cdot\!{\bf R} + q z \right) }{
2 k' q - {\rm i} \gamma
+ {\rm i} \alpha Q^2
+ {\rm i} ( \alpha - \beta k^2 ) q^2 } .
\label{eq:integrale-q}\end{aligned}$$ Nous avons décomposé les vecteurs ${\bf q}$ et ${\bf r}$ en composantes $\left( {\bf Q}, q \right), \,
\left( {\bf R}, z \right)$ perpendiculaires et parallèles au vecteur ${\bf k}'$. La coordonnée $z$ mesure donc la distance du point d’observation d’un point source situé sur le rayon qui arrive de la direction $\hat{\bf k}$ au point d’observation.
Il se trouve que l’on peut discuter de façon analytique le comportement de l’intégrale (\[eq:integrale-q\]). Nous prenons d’abord l’intégration sur la composante longitudinale $q$, en utilisant le théorème des résidus (nous avons abrégé $\bar{\gamma} =
\gamma - \alpha Q^2, \,
\bar{\alpha} = \alpha - \beta k^2$) : $$\int\!\dbar q \,
\frac{ \exp{\rm i} q z }{
2 k' q - {\rm i} \bar{\gamma} + {\rm i} \bar{\alpha} q^2 }
= \frac{ {\rm{sgn}} \, z }{ \bar{\alpha} ( q_1 - q_2 ) }
\left[
\Theta(z \mbox{Im}\, q_1 )
\, {\rm e}^{ {\rm i} q_1 z }
-
\Theta(z \mbox{Im}\, q_2 )
\, {\rm e}^{ {\rm i} q_2 z }
\right]
\label{eq:solution-residus}$$ où $\Theta(z)$ est la fonction de marche d’escalier et les deux quantités imaginaires $q_{1,2}$ sont les solutions d’une équation quadratique $$\begin{aligned}
&& 2 k' q - {\rm i} \bar{\gamma} + {\rm i} \bar{\alpha} q^2 = 0
\nonumber\\
&& q_{1,2} = \frac{ {\rm i} k' }{ \bar{\alpha} }
\left( 1 \mp \sqrt{ 1 - \frac{ \bar{\gamma} \bar{\alpha} }{
k'^2 } } \right)
\label{eq:q-12}\end{aligned}$$ (Les signes sont choisis tels que $|q_1| < |q_2|$.)
Pour interpréter et simplifier ce résultat, supposons que le potentiel aléatoire est faible (approximation de [Bourret]{}). L’opérateur de masse $m( {\bf k} )$ est alors petit par rapport à $k_0^2$, et le vecteur d’onde effectif $k_{\rm{eff}} = k_{\rm p} +
{\rm i} / 2 \ell$ diffère peu du vecteur d’onde dans le vide. Par conséquent, nous avons $$|\gamma| = | - 2\,{\rm{Im}}\,m( k_{\rm{eff}} ) |
\ll k^2 \simeq k'^2 ,
\qquad
| \alpha |,\, | \bar{\alpha} | \ll 1 ,
\label{eq:gamma-petit}$$ et nous pouvons développer la racine carré dans (\[eq:q-12\]) pour trouver : $$q_1 \approx \frac{ {\rm i} \bar{\gamma} }{ 2 k' }
\qquad
q_2 \approx \frac{ 2 {\rm i} k' }{ \bar{\alpha} }
\label{eq:deux-poles}$$ Le premier terme dans (\[eq:solution-residus\]) qui correspond à la racine $q_1$, représente donc une atténuation exponentielle le long de la direction du vecteur ${\bf k}$ $${\rm{sgn}}\,q_1\,
\Theta( z \mbox{Im}\,q_1 )\, {\rm e}^{ {\rm i} q_1 z } \approx
{\rm{sgn}}\,\bar{\gamma}\,
\Theta( z \bar{\gamma} )\, {\rm e}^{ - z / \ell' } ,
\quad \frac{ 1 }{ \ell' } = \frac{ \gamma }{ 2 k' }
= \frac{ {\rm{Im}}\, k_{\rm{eff}}^2 }{ k' }
\: \mbox{pour ${\bf Q} = {\bf 0}$.}
\label{eq:racine-normale}$$ Le libre parcours moyen qui apparaît ici est modifié par rapport au résultat (\[eq:def-ell\]) parce que $k' \ne k$, à cause du terme proportionnel à $\nabla_k m$ dans (\[eq:k-prime\]).
La deuxième racine $q_2$ donne dans cette approximation $$\Theta(\bar{\alpha} z)\, {\rm e}^{ {\rm i} q_2 z } \approx
\Theta(\bar{\alpha} z)\, {\rm e}^{ - |z| / \ell_2 } ,
\quad \ell_2 = \frac{ | \bar{\alpha} | }{ 2 k' }
\label{eq:racine-bizarre}$$ elle correspond donc à une atténuation très rapide, sur une échelle en dessous de la longueur d’onde dans le milieu. En outre, suivant le signe de la quantité $\bar{\alpha}$, l’expression (\[eq:racine-bizarre\]) est non nulle pour des distances $z$ positives ou negatives, donc pour des sources en amont ou en aval du point d’observation. Dans l’approximation de [Bourret]{}, l’on s’attend à $$\bar{\alpha} =
\frac14 {\rm{Im}}\, \frac{ {\rm d}^2 m }{ {\rm d}k^2 } > 0$$ parce que la partie imaginaire de l’opérateur de masse (reliée au libre parcours moyen) présente un minimum autour de $k \approx k_0$ (voir la figure \[fig:masse\]). L’expression (\[eq:racine-bizarre\]) porte alors sur les positions $z$ positives (des sources en amont du point d’observation).
Il est justifié de négliger cette deuxième contribution à la fonction d’atténuation. Nous cherchons en effet la portée la plus grande en fonction de la distance du point d’observation, et la première racine donne une portée beaucoup plus grande que la deuxième.[^14] Si nous nous plaçons à une échelle spatiale plus grande que la longeur d’onde, nous pouvons donc négliger la contribution de la racine $q_2$. Le résultat de l’intégrale (\[eq:solution-residus\]) sur le vecteur d’onde longitudinal $q$ est alors : $$\int\!\dbar q \,
\frac{ \exp{\rm i} q z }{
2 k' q - {\rm i} \gamma
+ {\rm i} \alpha Q^2
+ {\rm i} \bar{\alpha} q^2 }
=
\frac{ 2 {\rm i} k' \, {\rm{sgn}}\,z \,
\Theta[ z ( \gamma - \alpha Q^2 ) ] }{
4 k'^2 - \bar{\alpha} ( \gamma - \alpha Q^2 ) }
\exp[ - ( \gamma - \alpha Q^2 ) z / 2 k' ]
\label{eq:resultat-integrale-q}$$ où nous avons ré-exprimé $\bar{\gamma}$ en fonction de ${\bf Q}$. Nous notons que la fonction $\Theta$ assure que l’exponentielle est toujours décroissante en fonction de $|z|$.
Il reste maintenant à effectuer l’intégration sur le vecteur d’onde transverse ${\bf Q}$. L’intégrale prend la forme suivante : $$2 {\rm i} k' \, {\rm sgn}\, z
\int\!\dbar{\bf Q}
\frac{
\Theta[ z ( \gamma - \alpha Q^2 ) ] \,
{\rm e}^{{\rm i} {\bf Q}\cdot{\bf R} }
}{ 4 k'^2 - \bar{\alpha} ( \gamma - \alpha Q^2 ) }
\exp[ - ( \gamma - \alpha Q^2 ) z / 2 k' ]
\label{eq:integrale-Q}$$ Il convient de distinguer entre les cas $z > 0$ et $z < 0$ (positions sur le rayon lumineux antérieures et postérieures au point d’observation).
##### Le cas causal $z > 0$.
Le module du vecteur d’onde ${\bf Q}$ est limité à l’intervalle $ 0 \le Q \le ( \gamma / \alpha )^{1/2}$. Nous effectuons l’intégration sur l’angle azimuthal du vecteur ${\bf Q}$. En utilisant une nouvelle variable d’intégration, l’intégrale s’écrit $$\frac{ {\rm i} }{ 8 \pi k' }
\int\limits_0^{\gamma / \alpha } \!
{\rm d}s \,
J_0( \sqrt{ \gamma / \alpha - s } R ) \,
{\rm e}^{ - \alpha s z / 2 k' } ,
\quad s = \gamma / \alpha - Q^2
\label{eq:integrale-causale}$$ Nous avons également négligé le terme proportionnel à $\bar{\alpha} \alpha Q^2$ au dénominateur de (\[eq:integrale-Q\]). Comme $Q^2$ est limité par $\gamma / \alpha$, ce terme est petit par rapport à $k'^2$ en vertu de l’hypothèse (\[eq:gamma-petit\]) que nous avons déjà utilisée ci-dessus (pour calculer $q_{1,2}$).
Il est facile maintenant de déterminer l’échelle caractéristique pour la distance transverse $R$. A partir de l’argument de la fonction de [Bessel]{} $J_0$, nous obtenons : $$\delta R \simeq \sqrt{ \frac{ \alpha }{ \gamma } }
\sim
\sqrt{ \frac{ \ell_c / \ell }{ 2 k' / \ell } }
\simeq \sqrt{ {{\textstyle\frac12}}\ell_c \lambdabar }
\label{eq:echelle-transverse}$$ (Nous avons utilisé l’échelle de variation $1 / \ell_c$ de l’opérateur de masse pour estimer $\alpha \sim
\ell_c (k_0 / \ell) / k_0 = \ell_c / \ell$.) Avant de donner une interprétation physique de ce résultat, étudions le deuxième cas.
##### Le cas acausal $z < 0$.
Le nom acausal provient du fait, rappelons-le, que les positions $z < 0$ se trouvent en amont du point d’observation : elles décrivent donc l’influence sur la luminance du milieu que le rayon n’a pas encore traversé d’un point de vue géométrique.
L’intégrale (\[eq:integrale-Q\]) porte maintenant sur les vecteurs d’onde $ ( \gamma / \alpha )^{1/2} \le
Q \le \infty$. Avec une changement de variables similaire, elle prend la forme $$- \frac{ {\rm i} }{ 8 \pi k' }
\int\limits_0^{\infty} \!
{\rm d}s \,
J_0( \sqrt{ \gamma / \alpha + s } R ) \,
{\rm e}^{ - \alpha s | z | / 2 k' },
\quad s = Q^2 - \gamma / \alpha
\label{eq:integrale-acausale}$$ Nous avons encore négligé la contribution en $Q^2$ dans le dénominateur. Sur les distances qui contribuent à l’intégrale ($\alpha s |z| / 2 k' \sim 1$), elle est négligeable si $z \gg \bar{\alpha} / 2 k'$. Puisque cette distance est bien en-dessous de la longueur d’onde, cette approximation est justifiée aux grandes échelles qui nous intéressent ici.
Pour l’échelle transverse $\delta R$, nous trouvons alors deux limites suivant la distance $|z|$ du point d’observation : $$\delta R \simeq
\left\{
\begin{array}{cl}
\sqrt{ \alpha / \gamma } \simeq \sqrt{ {{\textstyle\frac12}}\ell_c \lambdabar }
& \mbox{pour } z \gg \ell ,
\\
\sqrt{ \alpha |z| / 2 k' } \simeq \sqrt{ {{\textstyle\frac12}}\ell_c \lambdabar
( |z| / \ell ) }
& \mbox{pour } z \ll \ell .
\end{array}
\right.
\label{eq:largeur-acausale}$$
#### Discussion
Nous constatons qu’en un point d’observation ${\bf r}$ donné, la luminance est la somme des processus de diffusion qui ont amené de la lumière dans la direction d’observation ${\bf k}$ et [*qui ont eu lieu dans un lobe autour du rayon géométrique de direction ${\bf k}$ qui aboutit à ${\bf r}$*]{}. Ce ne sont plus des processus de diffusion qui ont eu lieu exactement sur le rayon qui contribuent à la luminance observée : le terme de diffusion de l’équation du transfert radiatif est devenu non-local. La largeur transverse des lobesest de l’ordre de $\delta R \simeq ( \ell_c \lambda )^{1/2}$. En outre, une contribution non nulle de la luminance provient de positions acausales sur le rayon et en aval du point d’observation. La situation est illustrée sur la figure \[fig:lobe\] par des courbes de niveaux.
Nous constatons que la longueur de non-localité transverse est égale à la taille de la zone de [Fresnel]{} lorsque l’on observe une source ponctuelle de longueur d’onde $\lambda$ à la distance $\ell_c$ de la longueur de corrélation. La non-localité de l’ETR que nous mettons ici en évidence est donc reliée à la diffraction dans le champ intermédiaire (diffraction de [Fresnel]{} par rapport à celle de [Fraunhofer]{} dans le champ lointain). En outre, l’échelle $\delta R$ suggère que la déviation par rapport à l’ETR locale est due à la diffusion recurrente entre deux diffuseurs qui sont placés à l’intérieur d’un rayon de corrélation et qui s’éclairent mutuellement par leurs champs proches et intermédiaires. C’est en effet une des prédictions de [Lagendijk]{} et [van Tiggelen]{} pour les phénomènes physiques au-delà de l’ETR standard [@vanTiggelen96].
Notre calcul permet d’obtenir encore un autre résultat curieux qui concerne le comportement à grande distance $|z| \gg \ell$ de la fonction d’atténuation. Dans cette limite, nous pouvons calculer les intégrales sur $s$ (\[eq:integrale-causale\], \[eq:integrale-acausale\]) de façon approchée. Leurs contributions dominantes proviennent en effet de la région $s \to 0$, de sorte que nous obtenons, pour le cas causal par exemple, $$\frac{ {\rm i} }{ 8 \pi k' }
\int\limits_0^{\gamma / \alpha } \!
{\rm d}s \,
J_0( \sqrt{ \gamma / \alpha } R ) \,
{\rm e}^{ - \alpha s z / 2 k' }
=
\frac{ {\rm i} J_0( R / \delta R )
}{ 4 \pi \alpha }
\frac{ 1 }{ z }
\label{eq:decroissance-en-1/z}$$ Si nous pouvons résoudre l’échelle transverse $\delta R$, la fonction d’atténuation est donc de longue portée, avec une décroissance en $1/z$. Nous n’avons pas encore trouvé d’interprétation physique simple de ce résultat. Il est d’autant plus surprenant que l’on trouve un comportement analogue pour le cas acausal.
Il est cependant évident que ce résultat est dû aux vecteurs d’onde transverses $Q^2 \simeq \gamma / \alpha$, qui s’accompagnent d’une décroissance très lente de la fonction d’atténuation en fonction de $|z|$ (voir (\[eq:integrale-Q\])). L’on peut alors revenir à la condition (\[eq:condition-q-petit\]) sur les vecteurs d’onde ($Q \le |q| \lesssim 1 / \ell_c$) et se demander si elle est encore vérifiée pour la situation que nous envisageons ici. En imposant $ ( \gamma / \alpha )^{1/2} \lesssim 1 / \ell_c$ et en utilisant l’estimation pour $\alpha$ donnée ci-dessus, nous trouvons la condition suivante : $$\ell_c \lesssim \lambdabar
\label{eq:lambda-grand}$$ Nos conclusions sont donc seulement valables lorsque la longueur de corrélation est plus petite que la longueur d’onde. Ceci n’est pas très étonnant parce que c’est seulement dans ce régime que la longueur de non-localité $\delta R$ est plus grande que la longueur de corrélation (sur laquelle il faut toujours moyenner dans une théorie statistique de transport). Ce régime apparaît fréquemment dans le domaine optique lors de la diffusion par des très petites particules. Il est par contre plus difficile à réaliser avec les atomes.
Les atomes se trouvent généralement dans le régime opposé où la longueur d’onde est petite par rapport à la longueur de corrélation du milieu aléatoire (régime semi-classique). Dans ce régime, l’on n’a pas le droit d’étendre le domaine d’intégration à des vecteurs d’onde plus grand que $1 / \ell_c$. En limitant les intégrales sur ${\bf Q}$ à cet intervalle, c’est alors la longueur de corrélation qui donne l’échelle spatiale minimale pour l’ETR : $$\mbox{r\'egime semi-classique :}\quad
\delta R \simeq \ell_c
\label{eq:deltaR-semicl}$$ En outre, la fonction d’atténuation ne contient qu’une partie causale, avec une portée longitudinale donnée par le libre parcours moyen $\ell$.
#### Conclusion
L’équation du transfert radiatif est valable lorsque la luminance varie lentement à l’échelle d’une longueur de [Fresnel]{}, définie par la longueur d’onde et la longueur de corrélation. Les déviations par rapport à l’ETR proviennent alors de la diffusion recurrente de la lumière par des diffuseurs qui s’éclairement mutuellement par leurs champs proche et intermédiaire. Cette description suppose des diffuseurs petits par rapport á la longueur d’onde. Dans la limite opposée (régime semi-classique), la longueur minimale pour l’ETR est donnée par la longueur de corrélation.
Nous avons obtenu ce résultat en calculant une correction à l’équation du transfert lorsque l’on se rapproche des échelles spatiales plus courtes. Ce calcul met en outre en évidence une correction au libre parcours moyen qui fait intervenir la dérivée de l’opérateur de masse (le passage de $k$ à $k'$ dans (\[eq:k-prime\])). Cette correction est probablement liée à la vitesse de transport du champ dans le milieu qui n’est donnée ni par la vitesse de groupe ni celle de phase [@vanTiggelen96]. Elle apparaît encore plus nettement dans une approche au transport dépendante du temps, que nous n’avons pas suivie ici.
##### Relation de dispersion non-triviale.
Pour finir, une question d’ordre expérimental : comment est-ce possible que l’opérateur de masse présente une variation importante avec le vecteur d’onde ${\bf k}$ ? L’on aurait alors une vitesse de transport $k'$ très différente de celle de propagation libre, ainsi qu’un libre parcours moyen $\ell'$ fortement corrigé. On peut espérer que tel est le cas dans les solides où la relation de dispersion présente des fortes déviations par rapport à l’espace libre, notamment aux bords de la zone de [Brilluoin]{} (aux bords des bandes permises). Une autre voie peut être ouverte par des particules diffuseurs avec une structure interne, des billes diélectriques avec un noyau creux, par exemple. En effet, la fonction de corrélation de la constante diélectrique $g( r )$ va alors retracer, à courte distance, le profil de l’indice des billes, et ceci se traduira par une variation importante de l’opérateur de masse avec le vecteur d’onde. Ce dernier est en effet donné, dans l’approximation de [Bourret]{} et pour des corrélation isotropes, par $$m({\bf k}) = \frac{ {\rm i} k_0^4 }{ 2 k }
\int\limits_0^\infty \!{\rm d}r \, g( r )
\left( {\rm e}^{ {\rm i} ( k_0 + k ) r } -
{\rm e}^{ {\rm i} ( k_0 - k ) r } \right)$$ ce qui revient essentiellement à la transformée de [Fourier]{} du profil d’indice radial. D’un point de vue expérimental, l’on pourrait donc faire varier la structure interne des billes pour observer un comportement différent du libre parcours moyen. Malheureusement, ceci est une variable qui a également une influence sur d’autres quantités (le contraste diélectrique total, moyenné sur la bille, par exemple). Une astuce peut éviter cet inconvénient : se placer dans une géométrie en deux dimensions où le vecteur d’onde est fixé par la projection sur le plan de symétrie. En choisisant l’angle d’incidence, l’on peut alors faire varier le vecteur d’onde sans changer la fréquence.
Il reste à évaluer quel contraste diélectrique et quelle concentration de diffuseurs sont nécessaires pour que l’effet soit observable. En première approche, l’on peut envisager d’utiliser l’approximation de [Bourret]{} à cet effet.
Réciprocité et rétro-diffusion exaltée
--------------------------------------
Pour finir, nous voudrions étudier la conséquence de la réciprocité de la diffusion pour l’équation de [Bethe–Salpeter]{} dans la présente formulation. Sans en donner une justification précise, la réciprocité exprime que les deux processus de diffusion suivants ont la même amplitude de diffusion : $${{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} \to {\bf k} \quad \mbox{et} \quad
\mbox{$- {\bf k}$} \to - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}}$$ parce qu’ils s’obtiennent l’un de l’autre par un simple renversement du temps. Cette invariance existe pour notre théorie scalaire, elle n’existe plus pour la diffusion électro-magnétique en présence d’un champ magnétique, par exemple. Si les deux processus ont la même amplitude, ceci reste vrai aussi après la moyenne sur le potentiel aléatoire. Par conséquent, l’on s’attend à une intensité augmentée (par interférence constructive) dans ces directions.
En comparant aux arguments de l’opérateur d’intensité dans (\[eq:def-phase\]), nous constatons qu’il correspond à la corrélation entre deux processus réciproques (donc de même amplitude) lorsque nous avons $$\left.
\begin{array}{rcl}
{\bf k} + {{\textstyle\frac12}}{\bf q} & = & - ( {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} - {{\textstyle\frac12}}{\bf q} )
\\[0.5\jot]
{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} + {{\textstyle\frac12}}{\bf q} & = & - ( {\bf k} - {{\textstyle\frac12}}{\bf q} )
\end{array}
\right\} \quad \Longrightarrow \quad
{\bf k} = - {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} .
\label{eq:def-retrodiffusion}$$ C’est donc le cas pour la diffusion vers l’arrière. Dans l’approximation de [Bourret]{}, l’opérateur d’intensité prend la forme (\[eq:K(k)-Bourret\]) et la probabilité relative de diffusion vers l’arrière et vers l’avant est donnée par $$\frac{ \mbox{ arri\`ere } }{ \mbox{ avant } } =
\frac{ S( 2 k ) }{ S( 0 ) }$$ ($S(k)$ est la densité spectrale du potentiel aléatoire, supposée isotrope ici.) Cette probabilité est faible lorsque la longueur d’onde est petite par rapport à la longueur de corrélation. En outre, elle ne présente pas de trace particulière d’un effet d’interférence dans la direction arrière. Ceci est dû au fait que l’approximation de [Bourret]{} ne prend en compte que les processus de simple diffusion. Par contre, si l’on utilise une expression pour l’opérateur d’intensité qui aille au-delà de l’approximation de [Bourret]{}, l’on peut très bien décrire la rétro-diffusion exaltée dans le formalisme de l’équation du transfert radiatif. La bonne nouvelle est qu’il suffit d’inclure une série certes infinie de diagrammes (les [*maximally crossed diagrams*]{}), mais que leur calcul nécessite pas, dans la pratique, d’aller au-delà de l’approximation de [Bourret]{} : l’on s’en sort donc en améliorant le formalisme. Nous renvoyons aux travaux de [Tsang]{} [@Tsang85] ainsi qu’au § 6 du livre de Ping [Sheng]{} [@Sheng95] pour plus de details.
Conclusion et perspectives
--------------------------
L’équation du transfert radiatif ne s’en sort pas mal de cette discussion : sa forme traduit bien le transport de la luminance (la transformée de [Wigner]{} du champ) aux grandes échelles spatiales, même si le potentiel des diffuseurs est fort et que l’approximation de [Born]{} échoue. Il convient cependant de rappeler que l’ETR généralisée a été obtenue pour un milieu statistiquement homogène, donc sans prendre en compte les effets de bords ni les interfaces. Il faut donc modéliser ceux-ci au cas par cas avec des coefficients de réflexion et de transmission, par exemple. En outre, l’ETR est seulement valable lorsque l’on peut parler de la propagation d’ondes dans le milieu : formellement, ceci se traduit par un libre parcours moyen beaucoup plus grand que la longueur d’onde (régime de faible diffusion). L’ETR ne donne donc aucun accès à la localisation des ondes.
Dans la littérature, l’on rencontre souvent des affirmations moins ambitieuses quant au statut de l’ETR : celle-ci ne serait valable que dans l’approximation de [Bourret]{} [@Kravtsov96; @Luck93] ou de “[*single group*]{}”. En effet, l’analyse de [Barabanenkov]{} et [Finkel’berg]{} [@Barabanenkov67] montre que la validité de l’approximation de [Bourret]{} entraîne le régime de faible diffusion : le champ se propage dans le milieu, les diffuseurs se voient dans le champ lointain et la dispersion spatiale (la dépendance de l’opérateur de masse avec le vecteur d’onde) est négligeable. Sous ce point de vue, l’approximation de [Bourret]{} apparaît donc comme une [*condition suffisante*]{} pour l’application de l’ETR. Le libre parcours moyen et la fonction de phase sont alors donnés par les opérateurs de masse et d’intensité que l’on calcule en tenant compte d’un groupe de diffuseurs avec $k = 1,2,...$ particules (caractérisé par une fonction de corrélation à $k$ points). Cette “[*single group approximation*]{}” revient à l’approximation de [Born]{} (les formules de [Ryzhik, Papanicolaou]{} et [Keller]{}) pour la diffusion par un potentiel aléatoire continue. Pour des particules diffuseurs, des corrélations entre particules peuvent donc en principe être décrites par le formalisme, mais il semble difficile dans la pratique de calculer les sections efficaces de diffusion (voir [@Tsang85; @MacKintosh89] pour des exemples).
L’équation du transfert radiatif généralisée (ETR$^*$, éq. \[eq:ETR\*-diff\]) que nous avons trouvée ici ne dépend pas, [*a priori*]{}, de l’approximation de [Bourret]{}. Elle va donc au-delá de l’approximation de la diffusion simple, tout en supposant que la diffusion est faible (ondes non-localisées). Les travaux des Russes indiquent les effets physiques nouveaux qui sont décrits par l’ETR$^*$ :
- la diffusion devient dépendante (diffuseurs s’éclairant dans le champ proche et intermédiaire, diffusion recurrente)
- la relation de dispersion s’élargit (dispersion spatiale : la partie imaginaire de la fonction de [Green]{} moyennée n’est plus une fonction $\delta$)
- il apparaît une vitesse de transport pour l’intensité des ondes.
Nous avons identifié les traces de ces effets en développant la partie propagation de l’équation de [Bethe–Salpeter]{} au-delà de l’approximation habituelle des grandes échelles. Il faut cependant rester prudent sur ces résultats parce que les autres termes dans l’équation de [B.–S.]{} donnent des contributions analogues, la fonction de [Green]{} moyennée et l’opérateur d’intensité. En outre, ces contributions ne sont pas indépendantes parce qu’elles sont reliées entre elles par l’identité de [Ward]{} (une généralisation du théorème optique), qui est valable à toute échelle. La liste des effets physiques ci-dessus semble quand même qualitativement correcte ; [Lagendijk]{} et [van Tiggelen]{} dressent en effet un répertoire analogue [@vanTiggelen96].
Finalement, tout semble indiquer que l’équation du transfert radiatif ne peut pas décrire la localisation (forte) des ondes dans un milieu aléatoire. Notre étude peut tout au plus suggèrer l’image physique suivante : nous avons constaté que l’équation de [Bethe–Salpeter]{} est non-locale, avec une fonction de phase non-positive qui présente des changements de signe. Il est alors concevable que dans l’intégrale sur les processus de diffusion, ceux-ci interfèrent de façon destructive : aux grandes distances, le champ ne se propagerait pas et serait [*localisé*]{} dans le milieu diffusant. Rappelons dans ce contexte que la fonction de [Wigner]{} n’est pas forcément positive, et que ses négativités sont précisément le résultat d’interférences.
Conclusion {#conclusion-3 .unnumbered}
==========
Le transport des ondes {#le-transport-des-ondes .unnumbered}
----------------------
Nous avons étudié dans ce rapport le transport d’une onde (scalaire, électro-magnétique ou de matière) à travers un milieu aléatoire diffusant. L’on peut distinguer trois niveaux de description dans la théorie :
1. le niveau microscopique, où l’on tient compte de l’aspect ondulatoire, en étudiant le champ moyen et la fonction de cohérence. A ce niveau, les effets d’interférence sont inclus dans la théorie, dont les équations de base sont celles de [Dyson]{} et de [Bethe–Salpeter]{}.
2. un niveau mésoscopique, où le champ est décrit par une luminance : elle en donne l’intensité, distribuée en position et selon les directions de propagation. La théorie du transfert radiatif, mais aussi la mécanique classique statistique si situent à ce niveau théorique. Les équations de base sont celle du transfert radiatif, de [Boltzmann]{} ou encore l’équation de [Fokker–Planck]{}[^15]. Le lien à la théorie microscopique passe par l’identification de la luminance avec la fonction de corrélation (de cohérence) du champ en représentation de [Wigner]{}. L’on obtient l’équation du transfert radiatif à partir de celle de [Bethe–Salpeter]{} dans la limite des échelles spatiales grandes par rapport à la longueur de corrélation du milieu (les diffuseurs se voient en champ lointain).
3. Finalement, la diffusion dans le milieu redistribue l’intensité de l’onde du faisceau collimaté incident vers la partie diffuse de la distribution angulaire. Aux grandes distances, la distribution angulaire devient isotrope, et le champ est décrit seulement par son densité locale d’énergie. Sur de telles échelles macroscopiques, le transport de l’onde est gouverné par une équation de la diffusion (spatiale). Dans une expérience avec les atomes où le temps d’interaction est souvent limité, ce sont seulement les atomes les plus lents qui entrent dans le régime du transport diffusif.
Ces différents niveaux théoriques sont représentés sur la figure suivante.
\
[*Représentation schématique des différents niveaux d’approximation pour le transport des ondes dans un milieu aléatoire.*]{}
Perspectives {#perspectives .unnumbered}
------------
Les atomes froids placés dans les tavelures lumineuses réalisent un problème de transport en milieu aléatoire dont on peut facilement varier les paramètres caractéristiques (amplitude du potentiel, longueur de corrélation par rapport à la longueur d’onde (température) des atomes, durée de l’interaction, poids relatif des effets réactifs et dissipatifs). Le transport d’atomes représente donc une sorte de laboratoire où l’on peut étudier des régimes différents du transport. Etant donné la difficulté de refroidir des atomes en dessous de la limite de recul (longueur d’onde atomique plus grande que la longueur de corrélation des tavelures), c’est d’abord le régime mesoscopique du transport que l’on explorera. Ce n’est pas un domaine dépourvu d’intérêt : l’on peut même envisager de modéliser, en présence du champ de pesanteur, un modèle pour la conduction des électrons dans un solide avec des défauts (potentiel aléatoire plus force constante).
La structure interne des atomes offre une autre possibilité intéressante : il est possible d’étudier par exemple le transport d’un ensemble de spins polarisés à travers un champ magnétique aléatoire (statique). Ce problème est pertinent pour l’interférométrie atomique à états internes : un champ magnétique résiduel induit en effet une rotation aléatoire du spin et réduit le contraste des franges d’interférence. D’un point de vue théorique, l’on pourra exploiter l’analogie au transfert radiatif de la lumière polarisée. Le formalisme de [Ryzhik, Papanicolaou]{} et [Keller]{} est également suffisamment général pour trouver rapidement les équations de transport du spin atomique dans l’approximation d’un faible champ résiduel.
Dans le domaine du transfert radiatif proprement dit, les considérations de ce rapport indiquent que l’équation du transfert radiatif est suffisamment générale pour décrire le régime de la diffusion dépendante, à condition de généraliser la section efficace de diffusion à la diffusion par des agglomérats de particules. De la même façon, l’ETR peut incorporer dans une certaine mesure des effets d’interférence comme la rétrodiffusion exaltée. Elle se limite cependant à une description à grande échelle spatiale. A plus courte échelle, le transport est régi par une équation non-locale, et il n’est plus possible de définir une relation de dispersion pour les ondes dans le milieu. L’ETR est également limitée au régime de faible diffusion où le libre parcours moyen est beaucoup plus grand que la longueur d’onde. Le régime opposé correspond à la localisation des ondes pour laquelle il faut revenir à une description microscopique à l’aide de l’équation de [Bethe–Salpeter]{}. Il est étonnant qu’il faille en élaborer une théorie à un niveau si fondamental bien que la localisation se manifeste par l’absence de transport à très grande échelle spatiale (voir le titre de l’article d’[Anderson]{}). Il semble que se trouve là une explication pour la difficulté notoire du problème de la localisation qui reste ouvert même après quarante ans de recherches ardues.
Carsten Henkel\
Châtenay–Malabry, juillet 1997
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[^1]: L’indice ex indique qu’il s’agit de la distance caractéristique pour l’extinction de la partie cohérente du champ lumineux.
[^2]: En écrivant (\[eq:Liouville\]), nous négligeons les interactions entre atomes.
[^3]: Précisons que $\Delta v_\Vert$ dénote la partie fluctuante de la vitesse. C’est la vitesse moyenne qui est amortie par la force de friction ${\bf F}^{d}( {\bf v} )$.
[^4]: \[fn:delta-transverse\]Le résultat est $$\int\!\frac{ {\rm d}{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} }{ (2\pi)^3 }
\frac{ \exp( - {\rm i} {{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} \cdot {\bf r} )
}{ {\rm i} {\bf v}\cdot{{\ifmmode\mathchoice{\mbox{\boldmath{$\displaystyle\kappa$}}}{\mbox{\boldmath{$\textstyle\kappa$}}}{\mbox{\boldmath{$\scriptstyle\kappa$}}}{\mbox{\boldmath{$\scriptscriptstyle\kappa$}}}\else\oldbf\kappa\fi}} + \eta }
= \left\{
\begin{array}{l}
\delta({\bf R}_\perp) / v \quad
\mbox{si ${\bf r}\cdot{\bf v} < 0$,}\\
0 \quad \mbox{sinon.}
\end{array}
\right.$$ où ${\bf R}_\perp$ dénote les composantes de ${\bf r}$ perpendiculaires au vecteur ${\bf v}$.
[^5]: Notons que dans le contexte de la propagation des ondes, cette approximation est celle de [Bourret]{} ou encore la [*ladder approximation*]{} pour l’opérateur de masse. Dans d’autres contextes, on l’appelerait théorie effective à basse énergie ou à grande échelle (physique des particules) ou encore élimination adiabatique des variables rapides (atome piloté par un champ à grand désaccord : réduction des équations de [Bloch]{} optiques à la population de l’état fondamental...)
[^6]: Pour des atomes dans un potentiel aléatoire profond(qui peut devenir plus grand que l’énergie cinétique atomique), cette redéfinition de l’énergie implique que la fonction d’onde moyenne est une onde localisée: son vecteur d’onde $k_0$ est imaginaire parce que $\langle V \rangle
> E_{i}$. Cette situation n’a pas d’analogue lumineux, et il serait intéressant d’en explorer les conséquences d’un point de vue théorique. Mais l’on n’aura pas le temps de le faire ici...
[^7]: Il s’agit en fait du nombre de diffusions par unité de temps. Elle est reliée au libre parcours moyen $\ell_{\rm ex}$ par $\sigma_{tot} = \hbar k_0 / (M \ell_{\rm ex})$.
[^8]: La fonction de phase (\[eq:fn-phase-Keller\]) dépend des deux vecteurs unitaires ${\bf n}'$, ${\bf n}$ ; elle est normalisée telle que $\int\!\mbox{d}^2{\bf n}\,
p( {\bf n}', {\bf n} ; k_0 ) = 1$.
[^9]: Il faut à cet effet exprimer le coefficient de diffusion par la fonction de corrélation du potentiel aléatoire, et utiliser un développement à l’ordre le plus bas en $\lambda_{dB} / \ell_c$ pour retrouver la fonction de corrélation de la force aléatoire.
[^10]: “[*strongly connected*]{}”
[^11]: [Rytov]{} dit que la longueur des rayons augmente par l’effet de la diffusion et par conséquent aussi la partie réelle de $k_{eff}$ ([@Rytov], § 4.2, p. 137).
[^12]: “[*cluster expansion*]{}”
[^13]: Nous nous servons de la notation abrégée $$\mbox{d}\kern-.6em\hbox{\raise1.2ex
\hbox{\kern.25em\vrule width.4em height.3pt depth.1pt}}{\bf k} = \frac{ \mbox{d}{\bf k} }{ (2\pi)^3 } .$$
[^14]: Nous notons aussi que [Rytov]{} a calculé la fonction de [Green]{} moyenne par une technique analogue (éq. (4.54) de [@Rytov]). Il trouve également deux contributions, dont l’une s’atténue à l’échelle du libre parcours moyen et l’autre a une portée de l’ordre de la longueur de corrélation $\ell_c$. Il néglige ensuite cette dernière contribution, en se plaçant à une échelle spatiale plus grande que $\ell_c$.
[^15]: Il dépend du rapport entre la longueur d’onde et la longueur de corrélation si c’est l’ETR ou l’équation de [F.–P]{} qui est bien adaptée au problème. La deuxième correspond au régime des petites longueurs d’onde où la diffusion se produit de préférence aux petits angles.
|
---
abstract: |
Ionic electro-active polymers (E.A.P.) is an active material consisting in a polyelectrolyte (for example Nafion). Such material is usually used as thin film sandwiched between two platinum electrodes. The polymer undergoes large bending motions when an electric field is applied across the thickness. Conversely, a voltage can be detected between both electrodes when the polymer is suddenly bent. The solvent-saturated polymer is fully dissociated, releasing cations of small size. We used a continuous medium approach. The material is modelled by the coexistence of two phases; it can be considered as a porous medium where the deformable solid phase is the polymer backbone with fixed anions; the electrolyte phase is made of a solvent (usually water) with free cations.
The microscale conservation laws of mass, linear momentum and energy and the Maxwell’s equations are first written for each phase. The physical quantities linked to the interfaces are deduced. The use of an average technique applied to the two-phase medium finally leads to an Eulerian formulation of the conservation laws of the complete material. Macroscale equations relative to each phase provides exchanges through the interfaces. An analysis of the balance equations of kinetic, potential and internal energy highlights the phenomena responsible of the conversion of one kind of energy into another, especially the dissipative ones : viscous frictions and Joule effect.
author:
- Mireille Tixier
- Joël Pouget
date: 'Received: date / Accepted: date'
title: 'Conservation laws of an electro-active polymer'
---
Introduction {#intro}
============
Electro-active polymers (EAP) have attracted much attention from scientists and engineers of various disciplines. In particular, researches in the field of biomimetics (for instance, in robotic mechanisms are based on biologically-inspired models) and for the use as artificial muscles (see, for instance, the review of Shahinpoor [@shahinpoor1998-a] and [@shahinpoor1998-b] or [@samatham2007]) and more recently EAPs are excellent candidates for energy harvesting devices [@sodano2004], [@liu2004] and [@liu2005]. Roughly speaking, such polymers have responses to external electric stimulation by displaying a significant shape and size variations. This interesting property offers many promising applications in advanced technologies. In addition, they can be used as actuators or sensors. As actuators the EAPs are characterized by the fact they undergo a large amount of deformation while sustaining large forces. They are often called artificial muscles [@shahinpoor1994], [shahinpoor2000]{} and [@bar-cohen2002]. Electro-active polymers can be divided in several categories according to their process of activation and chemical compositions. Nevertheless, they can be placed in two major categories : electronic and ionic categories. These both categories come in several families [@samatham2007] (among them, ferroelectric polymers, dielectric EAP, electrostrictive paper, electro-viscoelastic elastomers, ionic polymer gels, conductive polymers, etc.). The first category of EAP is the electronic type. Concerning their advantages the E.A.P. can operate in room conditions with rapid response in time; in addition they induce relatively large actuation forces. One of the main disadvantage is that they require high voltage ($150$ MV/m). The second category, the ionic EAPs, with which the present work is concerned, operates with low voltage (few volts) producing large bending displacements. Their drawbacks are more or less slow response and low actuation force. They operate best in humid environment and they can be made as self-contained encapsulated actuators to be used in dry environment.
In the present study the emphasis is placed especially on the ionic polymer metal composite (IPMC) [@nemat2000]. The structure consists of thin ion-exchange membrane of Nafion, Flemion or Aciplex (polyelectrolyte) plated on both faces by conductive electrodes (generally platinum or gold). In short, to explain the mechanism of deformation of an EAP, a thin trip of polymers is placed between thin conductive electrodes. Upon the application of an electric field across a slightly humid EAP, the positive counter ions move towards the negative electrode (cathode), whole negative ions that are fixed (or immobile) to the polymer backbone experience an attractive force from the positive electrode (anode). At the same time, water molecules in the EAP backbone diffuse towards the region of high positive ion concentration (near the negative electrode) to equalize the charge distribution. As a result, the water or solvent concentration in the region near the anode increases and the concentration in the region near the cathode decreases, leading to strain with linear distribution along the step thickness which causes the bending towards the positive anode. Conversely, if the strip of electro-active polymers is suddenly bent, a difference of electric voltage is produced between electrodes [@newbury2002] and [yoon2007]{}.
The theories or models to explain the mechanism of deformation in EAP are yet to emerge. Nevertheless, some heuristic or empiric models are available in the literature. One of the most interesting and comprehensive accounts for chemical mechano-electric effect of the ionic transport coupled to electric field and elastic deformation of the polymer. A micro mechanical model has been developped by Nemat-Nasser [@nemat2000] and [@nemat2002] accounting for coupled ion transport, electric field and elastic deformation to predict the response of the IPMC. The model presented is mostly governed by Gauss equation for the conservation of electric charge, a constitutive equation for ion flux vector and a so-called generalized Darcy’s law for the water molecule velocity. Other models based on linear irreversible thermodynamics have been proposed by Shahinpoor *et al.* [shahinpoor2000]{} and [@degennes2000]. The model considers standard Onsager formulation for the simple description of ion transport (current density) and the flux of the solvent transport. The conjugate forces are the electric field and the gradient of pressure. In different way, Shahinpoor and co-workers propose models for micro-electro-mechanics of ions polymeric gels based on continuum electromechanics [@shahinpoor1994].
The present work focus on a novel approach for electro-active polymers based on thermodynamics of continua. More precisely, we present a detailed approach for such polymer material using the concepts of non-equilibrium thermodynamical processes. The material is then modeled by the coexistence of two phases. The first one is the backbone polymer or the solid phase with fixed anion while the second phase is the solvent containing the free cations. The method consists of computing an average of the different phases over a representative elementary volume containing the phases at the micro scale. The statistical average leads to macro scale quantities defined all over the material. The main difficulty of the method is that we must account for the interfaces which exist between phases for which interfacial quantities must be defined. On using this procedure for different conservation laws of the present multiphase material, we deduce the equation of mass conservation, the electric charge conservation, the conservation of the momentum, the different energy balance equations at the macroscopic scale of the whole material.
The paper is organized as follows. The description of the model and the definition of phases are presented with underlying physics in the next Section. Section 3 is devoted to the equations of conservation of mass. Since the polymer contains electric charges, the electric charge conservation and interface equations are presented in Section 4. The following Section places the emphasis on the linear momentum balance equation where the macroscopic stress tensor is defined. Moreover, the Maxwell’s tensor is placed in evidence due to the action of the electric field on the moving electric charges. The Section 6 presents the energy balance laws, that is, the potential energy, the kinetic energy, the total energy and internal energy balance equations. At last, the discussion is reported in Section 7 and finally conclusions are drawn in Section 8.
Modelling {#sec:1}
=========
As mentioned in the introduction, the system under study is made of a thin membrane of an ionic electro-active polymer saturated with water and coated on both sides with thin metal layers used as electrodes. Water, even in small quantity, causes a quasi-complete dissociation of the polymer and the release of positive ions (cations) in water; negative ions (anions) remain bound to the polymer backbone [@Chabe]. When an electric field perpendicular to the electrodes is applied, cations move towards the negative side, carrying solvent away by an osmosis phenomenon. This solvent displacement leads to a polymer swelling on the negative electrode side and to a compression on the opposite side, resulting in a bending of the strip.
To model this system, we describe the polymer chains as a deformable porous medium; this solid is saturated by an ionic solution composed by water and cations. The whole material is considered as a continuum, which is the superposition of three systems whose velocity fields are different : a deformable solid component made up of polymer backbone negatively charged and fluid trapped in the unconnected porosity (the “solid component”), and a liquid composed of water and cations located in the connected porosity. Anions are bound to the solid component. Quantities relative to the different components will be respectively identified by subscripts $1$, $2$ and $3$ for cations, solvent and solid. Subscript $4$ will refer to the solution, i.e. both components $1$ and $2$. Quantities without subscript refer to the whole material. Solid and solution are separated by an interface (subscript $i$) whose thickness is supposed to be negligible. Components $2$, $3$ and $4$ as well as the global material are assimilated to continua. Modelling of the interface is detailled in the appendix.
Solid and solution are supposed to be incompressible phases. We assume the gravity and the magnetic field are negligible; the only external force acting on the system is the electric force.
To describe this complex dispersed medium, we use a coarse-grained model developed by Nigmatulin [@Nigmatulin79], [@Nigmatulin90], Drew [Drew83]{}, Drew and Passman [@Drew98] and Ishii and Hibiki [@Ishii06] for two-phase mixtures [@Lhuillier03]. We use two scales. The microscopic scale must be small enough so that the corresponding volume only contains a single phase (3 or 4), but large enough to use a continuous medium model. For Nafion completely saturated with water, it is about hundred Angstroms. At the macroscopic scale, the representative elementary volume (R.E.V.) contains phases 3 and 4. It must be large enough so that average quantities relative to the whole material make sense, and small enough so that these quantities can be considered as local. Its characteristic length is about micron [@Colette], [@Gierke] and [@Chabe]. For each phase $3$ and $4$, we define a microscale Heaviside-like function of presence $\chi _{k}\left(
\overrightarrow{r},t\right) $ by
$$\chi _{k}=1\hbox{ when phase }k\hbox{ occupies point }\overrightarrow{r}%
\hbox{ at time t,}\quad \chi _{k}=0\hbox{ otherwise} \label{Presence}$$
$\chi _{k}$ remains unchanged in case of displacement following the interface velocity $\overrightarrow{V_{i}^{0}}$. We obtain
$$\overrightarrow{grad}\chi _{k}=-\overrightarrow{n_{k}}\chi _{i}\qquad \frac{%
\partial \chi _{k}}{\partial t}=\overrightarrow{V_{i}^{0}}\cdot\overrightarrow{%
n_{k}}\chi _{i}\qquad \mbox{for}\;k=3,4 \label{qui}$$
where the Dirac-like function $\chi _{i}=-\overrightarrow{grad}\chi _{k}\cdot%
\overrightarrow{n_{k}}$ (in $m^{-1}$) denotes the function of presence of the interface and $\overrightarrow{n_{k}}$ the outward-pointing unit normal to the interface in the phase $k$.
The quantities related to each phase have significant variation over space and time, as well as the positions of each phase. In order to define macroscale quantities relative to the whole material, we consider a representative elementary volume (R.E.V.) containing the three components and the microscale quantities are statistically averaged over the R.E.V.. This statistical average, denoted by $\left\langle {}\right\rangle $ and obtained by repeating many times the same experiment with the same boundary and initial conditions, is supposed to be equivalent to a volume average (ergodic hypothesis). The average thus defined commutes with the space and time derivatives (Leibniz’ and Gauss’ rules, Drew [@Drew83]; Lhuillier [@Lhuillier03]). On denoting by $\left\langle{}\right\rangle _{k}$ the average over the phase $k$ of a quantity relative to the phase $k$ only, a microscale quantity $g_{k}^{0}$ satisfies
$$g_{k}=\left\langle \chi _{k}g_{k}^{0}\right\rangle =\phi _{k}\left\langle
g_{k}^{0}\right\rangle _{k}$$
where $\phi _{k}=\left\langle \chi _{k}\right\rangle $ is the volume fraction of the phase $k$. The macroscale quantity $g_{k}$ is defined all over the material. In the following, superscript $^{0}$ denotes the microscale quantities of each phase. The macroscale quantities, which are averages defined everywhere, are written without superscript.
Equation of conservation of mass {#sec:2}
================================
In the following, we assume that the polymer is enough hydrated to be completely dissociated. For the water, solution and solid phases, the microscale mass continuity equation can be written as
$$\frac{\partial \rho _{k}^{0}}{\partial t}+div\left( \rho _{k}^{0}%
\overrightarrow{V_{k}^{0}}\right) =0 \label{CMm}$$
where $\overrightarrow{V_{k}^{0}}$ is the local velocity of the phase $k$ and $\rho _{k}^{0}$ its mass density. Phases $2$ and $3$ are incompressible, so we obtain
$$div\left( \overrightarrow{V_{k}^{0}}\right) =0 \label{CMmbis}$$
The different phases do not interpenetrate, thus we can write
$$\overrightarrow{V_{1}^{0}}\chi _{i}=\overrightarrow{V_{2}^{0}}\chi _{i}=%
\overrightarrow{V_{3}^{0}}\chi _{i}=\overrightarrow{V_{4}^{0}}\chi _{i}=%
\overrightarrow{V_{i}^{0}}\chi _{i} \label{CMcl}$$
Using (\[CMm\]) and (\[qui\]) we deduce
$$\frac{\partial \chi _{k}\rho _{k}^{0}}{\partial t}+div\left( \chi _{k}\rho
_{k}^{0}\overrightarrow{V_{k}^{0}}\right) =\rho _{k}^{0}\overrightarrow{%
V_{i}^{0}}\cdot\overrightarrow{n_{k}}\chi _{i}-\rho _{k}^{0}\overrightarrow{%
V_{k}^{0}}\cdot\overrightarrow{n_{k}}\chi _{i} \label{1}$$
For the phase $k$, the mass density relative to the whole material volume and the barycentric velocity are defined respectively by
$$\rho _{k}=\left\langle \chi _{k}\rho _{k}^{0}\right\rangle =\phi _{k}\rho
_{k}^{0}\qquad \qquad \overrightarrow{V_{k}}=\frac{\left\langle \chi
_{k}\rho _{k}^{0}\overrightarrow{V_{k}^{0}}\right\rangle }{\left\langle \chi
_{k}\rho _{k}^{0}\right\rangle }=\overrightarrow{V_{k}^{0}}$$
neglecting the velocities fluctuations on the R.E.V. scale.
$$\rho _{4}^{0}=\rho _{2}^{0}\frac{\phi _{2}}{\phi _{4}}+CM_{1}$$
where $M_{k}$ is the molar mass of the component $k$ and $C$ the cations molar concentration relative to the solution volume. It follows$$\rho _{4}=\rho _{1}+\rho _{2}\qquad \mbox{with}\qquad \rho _{1}=\phi
_{4}CM_{1}$$assuming that the concentration fluctuations are negligible and that the solution is diluted. In the same way the velocity of the solution can be written as
$$\rho _{4}^{0}\overrightarrow{V_{4}^{0}}=CM_{1}\overrightarrow{V_{1}^{0}}%
+\rho _{2}^{0}\frac{\phi _{2}}{\phi _{4}}\overrightarrow{V_{2}^{0}}\qquad
\qquad \rho _{4}\overrightarrow{V_{4}}=\rho _{1}\overrightarrow{V_{1}}+\rho
_{2}\overrightarrow{V_{2}}$$
Averaging over the material R.E.V., we finally obtain$$\frac{\partial \rho _{k}}{\partial t}+div\left( \rho _{k}\overrightarrow{%
V_{k}}\right) =0\qquad \qquad k=1,2,3,4 \label{CMk}$$
The interfaces have no mass. Consequently, we deduce for the complete material
$$\frac{\partial \rho }{\partial t}+div\left( \rho \overrightarrow{V}\right) =0
\label{CM}$$
where $\rho $ and $\overrightarrow{V}$ denote the mass density and the barycentric velocity of the whole material
$$\rho =\sum\limits_{k=3,4}\rho _{k}\qquad \qquad \rho \overrightarrow{V}%
=\sum\limits_{k=3,4}\rho _{k}\overrightarrow{V_{k}}$$
Electric equations {#sec:3}
==================
Electric charge conservation {#sec:31}
----------------------------
The microscale electric charge conservation of the phase $k$ can be written$$div\overrightarrow{I_{k}^{0}}+\frac{\partial \left( \rho
_{k}^{0}Z_{k}^{0}\right) }{\partial t}=0 \label{CCm}$$where $\overrightarrow{I_{k}^{0}}$ denotes the current density vector and $%
Z_{k}^{0}$ the electric charge per unit of mass ($Z_{2}^{0}$ and $Z_{3}^{0}$ are constants).
$$\overrightarrow{I_{3}^{0}}=\rho _{3}^{0}Z_{3}^{0}\overrightarrow{V_{3}^{0}}%
\qquad \qquad \qquad \overrightarrow{I_{4}^{0}}=M_{1}CZ_{1}^{0}%
\overrightarrow{V_{1}^{0}}$$
$$Z_{k}^{0}=\frac{z_{k}F}{M_{k}}\quad \quad \mbox{for}\quad k=1,3\qquad
\quad Z_{2}^{0}=0 \qquad \quad Z_{4}^{0}=\frac{CM_{1}Z_{1}^{0}}{\rho _{4}^{0}}$$
where $z_{k}$ is the number of elementary charges of an ion and $F$ the Faraday’s constant.
Averaging over the R.E.V., we obtain$$div\overrightarrow{I_{k}}+\frac{\partial \rho _{k}Z_{k}}{\partial t}%
=\left\langle -\overrightarrow{i_{k}^{0}}\cdot\overrightarrow{n_{k}}\chi
_{i}\right\rangle \label{CCk}$$in which the macroscale mass charge and current density vector are defined as
$$\rho _{k}Z_{k}=\left\langle \chi _{k}\rho _{k}^{0}Z_{k}^{0}\right\rangle
\qquad \qquad \overrightarrow{I_{k}}=\left\langle \chi _{k}\overrightarrow{%
I_{k}^{0}}\right\rangle$$
with$$\overrightarrow{I_{3}}=\left\langle \chi _{3}\overrightarrow{I_{3}^{0}}%
\right\rangle =\rho _{3}Z_{3}\overrightarrow{V_{3}}\qquad \qquad
\overrightarrow{I_{4}}=\left\langle \chi _{4}\overrightarrow{I_{4}^{0}}%
\right\rangle =\rho _{1}Z_{1}\overrightarrow{V_{1}}$$$\overrightarrow{i_{k}^{0}}=\overrightarrow{I_{k}^{0}}-\rho _{k}^{0}Z_{k}^{0}%
\overrightarrow{V_{k}^{0}}$ denotes the microscale diffusion current in phase $k$. Quantities relative to the interfaces are defined in the appendix. The interface electric charge density per unit surface $Z_{i}$ and the current density vector $\overrightarrow{I_{i}}$ satisfy the following mean condition$$\frac{\partial Z_{i}}{\partial t}+div\overrightarrow{I_{i}}=\left\langle
\overrightarrow{i_{3}^{0}}\cdot\overrightarrow{n_{3}}\chi _{i}+\overrightarrow{%
i_{4}^{0}}\cdot\overrightarrow{n_{4}}\chi _{i}\right\rangle \label{CCi}$$
Adding up equations (\[CCk\]) for the solid, the solution and (\[CCi\]) for the interfaces, it follows for the whole material$$div\overrightarrow{I}+\frac{\partial \rho Z}{\partial t}=0 \label{CC}$$where$$\rho Z=\sum\limits_{3,4}\rho _{k}Z_{k}+Z_{i}\qquad \qquad \overrightarrow{I}%
=\rho _{1}Z_{1}\overrightarrow{V_{1}}+\rho _{3}Z_{3}\overrightarrow{V_{3}}+%
\overrightarrow{I_{i}}$$
Maxwell’s equations {#sec:32}
-------------------
One can reasonably neglect the effects of the magnetic field. The electric fields $\overrightarrow{E_{k}^{0}}$ and the electric displacements $\overrightarrow{D_{k}^{0}}$ of the solid and the solution are governed by the Maxwell’s equations
$$\overrightarrow{rot}\overrightarrow{E_{k}^{0}}=\overrightarrow{0}\qquad
\qquad div\overrightarrow{D_{k}^{0}}=\rho _{k}^{0}Z_{k}^{0} \label{MAXm}$$
The associated boundary conditions can be presented as$$\overrightarrow{n_{3}}\wedge \overrightarrow{E_{3}^{0}}\chi _{i}=-%
\overrightarrow{n_{4}}\wedge \overrightarrow{E_{4}^{0}}\chi _{i}\qquad
\qquad \ \ \overrightarrow{D_{3}^{0}}\cdot\overrightarrow{n_{3}}\chi _{i}+%
\overrightarrow{D_{4}^{0}}\cdot\overrightarrow{n_{4}}\chi _{i}+Z_{i}^{0}\chi
_{i}=0 \label{MAXcl}$$
Averaging equations (\[MAXm\]) over the R.E.V., we derive the following macroscale equations for the solid and the solution $$\overrightarrow{rot}\overrightarrow{E_{k}}=\overrightarrow{0}\qquad \qquad
div\overrightarrow{D_{k}}=\rho _{k}Z_{k}-\left\langle \overrightarrow{%
D_{k}^{0}}\cdot\overrightarrow{n_{k}}\chi _{i}\right\rangle \label{MAXk}$$in which the macroscale electric fields and displacements are defined as $$\overrightarrow{E_{k}}=\frac{\left\langle \chi_{k} \overrightarrow{E_{k}^{0}}%
\right\rangle }{\left\langle \chi_{k} \right\rangle }\qquad \qquad
\overrightarrow{D_{k}}=\left\langle \chi _{k} \overrightarrow{D_{k}^{0}}\right\rangle$$ Electric field is an intensive thermodynamic variable. In principle, it displays spatial and time fluctuations within the R.E.V.. Considering this volume is tiny, we assume that the fluctuations are not relevant; we venture the same hypothesis for the concentration and the velocities of the phases. Furthermore, we suppose that macroscale electric fields are identical in all the phases. Adding up equations (\[MAXk\]) for the solid and the solution, it follows for the whole material $$\overrightarrow{rot}\overrightarrow{E}=\overrightarrow{0}\qquad \qquad div%
\overrightarrow{D}=\rho Z \label{MAX}$$using (\[MAXcl\]). Parameters of the complete material are defined by $$\overrightarrow{E}=\sum\limits_{3,4}\phi _{k}\overrightarrow{E_{k}}=\overrightarrow{E_{k}}
\qquad\qquad \overrightarrow{D}=\sum\limits_{3,4}\overrightarrow{D_{k}}$$
We conclude that the E.A.P. verifies the same Maxwell’s equations and the same law of conservation of charge as an isotropic homogeneous linear dielectric.
Constitutive relations {#sec:33}
----------------------
A reasonable approximation is that solid and solution can be regarded as isotropic linear dielectrics $$\overrightarrow{D_{k}^{0}}=\varepsilon _{k}^{0}\overrightarrow{E_{k}^{0}}
\label{RCm}$$where $\varepsilon _{k}^{0}$ denotes the permittivity of the phase $k$. Average over the R.E.V. gives $$\overrightarrow{D_{k}}=\varepsilon _{k}\overrightarrow{E_{k}} \label{RCk}$$in which :
$$\varepsilon _{k}=\left\langle \chi _{k}\varepsilon _{k}^{0}\right\rangle$$
is the mean permeability of the phase $k$ relative to the total volume.
The constitutive relation of the E.A.P. takes on the following form $$\overrightarrow{D}=\varepsilon \overrightarrow{E} \label{RC}$$where the whole material permittivity is defined by $$\varepsilon =\sum\limits_{k=3,4}\varepsilon _{k}$$
On considering our assumptions, the E.A.P. is equivalent to an isotropic linear dielectric. We however point out that its permittivity a priori varies over time and space because of variations of the volume fractions $%
\phi _{3}$ and $\phi _{4}$.
Linear momentum conservation law {#sec:4}
================================
Particle derivatives and material derivative {#sec:41}
--------------------------------------------
In order to write the remaining balance equations, it is necessary to calculate the variations of the extensive quantities following the material motion. This raises a problem because the different phases do not move with the same velocity : velocities of the solid and the solution are a priori different. For a quantity $g$, we can define particle derivatives following the motion of the solid $(\frac{d_{3}}{dt})$, the solution $(\frac{d_{4}}{dt})$ or the interface $(\frac{d_{i}}{dt})$
$$\frac{d_{k}g}{dt}=\frac{\partial g}{\partial t}+\overrightarrow{grad}g\cdot%
\overrightarrow{V_{k}}$$
Let us consider an extensive quantity of density $g\left( \overrightarrow{r}%
,t\right) $ relative to the whole material. According to the theory developped by O. Coussy [@Coussy95] and implicitly used in [@Biot77] and [@Coussy89], we are able to define a derivative following the motion of the different phases of the medium. We will call it the “material derivative”
$$\frac{D}{Dt}\left( \frac{g}{\rho }\right) =\sum\limits_{k=3,4,i}\frac{\rho
_{k}}{\rho }\frac{d_{k}\left( \frac{g_{k}}{\rho _{k}}\right) }{dt}$$
where $g_{3}$, $g_{4}$ and $g_{i}$ are the densities relative to the total actual volume attached to the solid, the solution and the interface, respectively (for example, if $g$ is the volume density, we set $%
g_{3}=1-\phi $ and $g_{4}=\phi $ where $\phi $ is the porosity)
$$g=g_{3}+g_{4}+g_{i}$$
$\frac{d_{k}}{dt}\left( \frac{g_{k}}{\rho _{k}}\right) $ is the derivative following the motion of the phase $k$ of the mass density associated with the quantity $g_{k}$. Using (\[CMk\]), we derive
$$\rho \frac{D\left( \frac{g}{\rho }\right) }{Dt}=\sum\limits_{k=3,4,i}\frac{%
\partial g_{k}}{\partial t}+div\left( g_{k}\overrightarrow{V_{k}}\right)
\label{DerivMat}$$
for a scalar quantity and
$$\rho \frac{D\left( \frac{\overrightarrow{g}}{\rho }\right) }{Dt}%
=\sum\limits_{k=3,4,i}\frac{\partial \overrightarrow{g_{k}}}{\partial t}%
+\overrightarrow{div}\left( \overrightarrow{g_{k}}\otimes \overrightarrow{V_{k}}\right)
\label{DerivMatVect}$$
for a vector quantity. This derivative must not be confused with the derivative $\frac{d}{dt}$ following the barycentric velocity $%
\overrightarrow{V}$.
Linear momentum balance equation {#sec:42}
--------------------------------
On assuming that the gravity and the magnetic field are negligible, the only applied volume force is the electric one. The microscale momentum balance equation of the phase $k$ is then written as
$$\frac{\partial \rho _{k}^{0}\overrightarrow{V_{k}^{0}}}{\partial t}%
+\overrightarrow{div}\left( \rho _{k}^{0}\overrightarrow{V_{k}^{0}}\otimes \overrightarrow{%
V_{k}^{0}}\right) =\overrightarrow{div}\utilde{\sigma} _{k}^{0}+\rho
_{k}^{0}Z_{k}^{0}\overrightarrow{E_{k}^{0}} \label{CQm}$$
where $\utilde{\sigma} _{k}^{0}$, the microscale stress tensor of the phase $k$, is symmetric. The linear momentum of the interfaces per surface unit is zero (see appendix). On accounting for the assumptions concerning the local velocities, it follows that at the macroscopic scale
$$\frac{\partial \rho _{k}\overrightarrow{V_{k}}}{\partial t}+\overrightarrow{%
div}\left( \rho _{k}\overrightarrow{V_{k}}\otimes \overrightarrow{V_{k}}%
\right) =\overrightarrow{div}\utilde{\sigma} _{k}%
+\rho _{k}Z_{k}\overrightarrow{E_{k}}+\overrightarrow{F_{k}} \label{CQk}$$
where
$$\utilde{\sigma}_{k}=\left\langle \chi _{k}%
\utilde{\sigma}_{k}^{0} \right\rangle \qquad \qquad
\overrightarrow{F_{k}}=\left\langle \utilde{\sigma}
_{k}^{0} \cdot\overrightarrow{n_{k}}\chi _{i}\right\rangle$$
We verify that the macroscale stress tensor of the phase $k$, $%
\utilde{\sigma} _{k}$, is symmetric. $%
\overrightarrow{F_{k}}$ represents the resultant of the mechanical stresses exerted on the phase $k$ by the other phase; it is an interaction force. Concerning the interfaces, we obtain the following mean condition (cf § \[sec:Annexe\]), which expresses the linear momentum conservation law for the interfaces
$$\overrightarrow{F_{3}}+\overrightarrow{F_{4}}=Z_{i}\overrightarrow{E_{i}}
\label{CQcl}$$
The interface momentum is zero, then the volume linear momentum of the whole material is $\rho \overrightarrow{V}$ $=\rho _{3}\overrightarrow{V_{3}}%
+\rho _{4}\overrightarrow{V_{4}}$. On using the definition of the material derivative (\[DerivMatVect\]), we obtain
$$\rho \frac{D\overrightarrow{V}}{Dt}=\overrightarrow{div}\utilde{\sigma }%
+\rho Z\overrightarrow{E} \label{CQ}$$
in which
$$\utilde{\sigma}=\sum\limits_{k=3,4}\utilde{\sigma}_{k}$$
We check that $\utilde{\sigma }$ is a symmetric tensor and that in the absence of any external force ($\overrightarrow{E}=%
\overrightarrow{0}$), the total linear momentum is conserved.
Using Maxwell’s equations (\[MAX\]) and (\[RC\]), (\[CQ\]) becomes
$$\rho \frac{D\overrightarrow{V}}{Dt}=\overrightarrow{div}\left[
\utilde{\sigma }+\varepsilon \left( \overrightarrow{E}\otimes \overrightarrow{E}%
-\frac{E^{2}}{2}\utilde{I}\right) \right] +\frac{E^{2}}{2}\overrightarrow{grad}\varepsilon$$
$\varepsilon \left( \overrightarrow{E}\otimes \overrightarrow{E}-\frac{E^{2}%
}{2}\utilde{I}\right) $ is the Maxwell’s tensor, which is here symmetric. The additional term $\frac{E^{2}}{2}\overrightarrow{grad}\varepsilon $ is produced by the non homogeneous material permittivity.
Energy balance laws {#sec:5}
===================
Potential energy balance equation {#sec:51}
---------------------------------
Solid and solution are supposed to be non-dissipative isotropic linear media. As a consequence the balance equation for the potential energy or Poynting’s theorem can be written in the integral form [@Jackson], [@Maugin]$$\frac{d}{dt}\int_{\Omega }\frac{1}{2}\left( \overrightarrow{E}\cdot\overrightarrow{D}+%
\overrightarrow{B}\cdot\overrightarrow{H}\right) dv=-\oint\nolimits_{\partial
\Omega }\left( \overrightarrow{E}\wedge \overrightarrow{H}\right) \cdot%
\overrightarrow{n}ds-\int_{\Omega }\overrightarrow{E}\cdot\overrightarrow{I}dv$$assuming that no charge goes out of the volume $\Omega $. The left hand side represents the variation of the potential energy attached to the volume $%
\Omega $ following the charge motion. If the charges are mobile, the associated local equation writes for the phase $k$, neglecting the magnetic field
$$\frac{\partial E_{pk}^{0}}{\partial t}+div\left( E_{pk}^{0}\overrightarrow{%
V_{k}^{0}}\right) =-\overrightarrow{E_{k}^{0}}\cdot\overrightarrow{I_{k}^{0}}%
\qquad \qquad k=3,4 \label{Epm}$$
in which
$$E_{pk}^{0}=\frac{1}{2}\overrightarrow{D_{k}^{0}}\cdot\overrightarrow{E_{k}^{0}}%
\qquad \qquad k=3,4$$
is the potential energy per unit of volume of the phase $k$. On taking the statistical average of (\[Epm\]) over the R.E.V., we obtain
$$\frac{\partial E_{pk}}{\partial t}+div\left( E_{pk}\overrightarrow{V_{k}}%
\right) =-\overrightarrow{E_{k}}\cdot\overrightarrow{I_{k}} \label{Epk}$$
where
$$E_{pk}=\left\langle \chi _{k}E_{pk}^{0}\right\rangle =\frac{1}{2}%
\overrightarrow{D_{k}}\cdot\overrightarrow{E_{k}}$$
The mean volume potential energy associated to the interfaces satisfies (see appendix)
$$\frac{\partial E_{pi}}{\partial t}+div\left( E_{pi}\overrightarrow{V_{i}}%
\right) =-\overrightarrow{E_{i}}\cdot\overrightarrow{I_{i}}$$
The potential energy balance equation for the whole material is then
$$\rho \frac{D}{Dt}\left( \frac{E_{p}}{\rho }\right) =-\overrightarrow{E}\cdot%
\overrightarrow{I} \label{Ep}$$
where
$$E_{p}=\sum_{3,4,i}E_{pi}=\frac{1}{2}\overrightarrow{D}\cdot\overrightarrow{E}$$
The production of potential energy in the R.E.V. is equal to the volume power $-\overrightarrow{E}\cdot\overrightarrow{I}$ of the force due to the action of the electric field on the density of electric charges.
Kinetic energy balance equation {#sec:52}
-------------------------------
The microscale kinetic energy balance equation derives from (\[CQm\])
$$\frac{\partial E_{ck}^{0}}{\partial t}+div\left( E_{ck}^{0}\overrightarrow{%
V_{k}^{0}}\right) =div\left( \utilde{\sigma} _{k}^{0}\overrightarrow{V_{k}^{0}}%
\right) -\utilde{\sigma}_{k}^{0}:\utilde{grad}\overrightarrow{V_{k}^{0}}%
+\rho _{k}^{0}Z_{k}^{0}\overrightarrow{E_{k}^{0}}\cdot\overrightarrow{V_{k}^{0}}
\label{Ecm}$$
where the microscale volume kinetic energy of the phase $k$ is
$$E_{ck}^{0}=\frac{1}{2}\rho _{k}^{0}V_{k}^{02}$$
In the same way, (\[CQk\]) is transformed into
$$\frac{\partial E_{ck}}{\partial t}+div\left( E_{ck}\overrightarrow{V_{k}}%
\right) =\overrightarrow{V_{k}}\cdot\overrightarrow{div}\utilde{\sigma}_{k}+\rho _{k}Z_{k}%
\overrightarrow{V_{k}}\cdot\overrightarrow{E_{k}}+%
\overrightarrow{F_{k}}\cdot\overrightarrow{V_{k}} \label{Eck}$$
where
$$E_{ck}=\frac{1}{2}\rho _{k}V_{k}^{2}$$
is the macroscale volume kinetic energy of the phase $k$. The interface kinetic energy is zero (see appendix).
On summing up the equations (\[Eck\]) for phases $3$ and $4$, we arrive at
$$\begin{tabular}{ll}
$\rho \frac{D}{Dt}\left( \frac{E_{c\Sigma }}{\rho }\right) $ & $%
=\sum\limits_{3,4}\left[ \frac{\partial E_{ck}}{\partial t}+div\left( E_{ck}%
\overrightarrow{V_{k}}\right) \right] $ \\
& $=\sum\limits_{3,4}\left[ div\left( \utilde{\sigma}
_{k}\cdot\overrightarrow{V_{k}}\right) -\utilde{\sigma}
_{k}:\utilde{grad}\overrightarrow{V_{k}}\right] +%
\left[ \sum\limits_{3,4}\rho _{k}Z_{k}\overrightarrow{V_{k}}+Z_{i}%
\overrightarrow{V_{i}}\right] \cdot\overrightarrow{E}$%
\end{tabular}
\label{Ecsigma}$$
where $E_{c\Sigma }$ is the sum of the volume kinetic energies of the different phases with respect to the laboratory reference frame
$$E_{c\Sigma }=E_{c3}+E_{c4}$$
$\qquad E_{c\Sigma }$ is distinct from the kinetic energy of the whole material because the phase velocities are different. The total volume kinetic energy $E_{c}$ is defined as
$$E_{c}=\frac{1}{2}\rho V^{2}=\sum\limits_{3,4}\frac{1}{2}\rho _{k}V^{2}$$
From (\[DerivMat\]), we deduce
$$\rho \frac{D}{Dt}\left( \frac{E_{c}}{\rho }\right) =\frac{\partial E_{c}}{%
\partial t}+div\left( E_{c}\overrightarrow{V}\right)$$
Using (\[Ecsigma\]), it follows
$$\begin{tabular}{l}
$\rho \frac{D}{Dt}\left( \frac{E_{c}}{\rho }\right) =\frac{\partial }{%
\partial t}\left( E_{c}-\sum\limits_{3,4}E_{ck}\right) +div\left[
\sum\limits_{3,4}\left( \utilde{\sigma}_{k}\cdot%
\overrightarrow{V_{k}}-E_{ck}\overrightarrow{V_{k}}\right) +E_{c}%
\overrightarrow{V}\right] $ \\
$\qquad \qquad \qquad -\sum\limits_{3,4}\utilde{\sigma}_{k}:\utilde{grad}%
\overrightarrow{V_{k}}+\left( \sum\limits_{3,4}\rho _{k}Z_{k}\overrightarrow{V_{k}}%
+Z_{i}\overrightarrow{V_{i}}\right) \cdot\overrightarrow{E}$%
\end{tabular}
\label{Ec}$$
The last two terms of this equation are source terms. The penultimate one represents the viscous dissipation, that is to say kinetic energy conversion into internal energy. The last term is the electric force volume power, which corresponds to a potential energy conversion into kinetic energy. As for the first two terms, they correspond to the kinetic energy flux, which is both due to the contact forces work $\sum\limits_{3,4}div\left(
\utilde{\sigma}_{k}\cdot\overrightarrow{V_{k}}\right) $ and to the relative velocity of the two phases: the kinetic energy of the phases with respect to the barycentric reference frame becomes indeed part of the internal energy of the whole material.
Total energy conservation law {#sec:53}
-----------------------------
The total energy of the present system is the sum of its internal, potential and kinetic energies. The energy fluxes come from contact forces work and heat conduction. The microscale energy conservation law for the phase $k$ can be written as
$$\frac{\partial E_{k}^{0}}{\partial t}+div\left[ E_{k}^{0}\overrightarrow{%
V_{k}^{0}}-\utilde{\sigma}_{k}^{0}\cdot\overrightarrow{%
V_{k}^{0}}+\overrightarrow{Q_{k}^{0}}\right] =0 \label{Em}$$
where$$E_{k}^{0}=U_{k}^{0}+\frac{1}{2}\rho _{k}^{0}V_{k}^{02}+\frac{1}{2}%
\overrightarrow{E_{k}^{0}}\cdot\overrightarrow{D_{k}^{0}}$$is the total microscale energy of the phase $k$. $\overrightarrow{Q_{k}^{0}}$ denotes the microscale heat flux of the phase $k$ and $U_{k}^{0}$ its microscale internal energy. Average over the R.E.V. leads to
$$\frac{\partial E_{k}}{\partial t}+div\left( E_{k}\overrightarrow{V_{k}}%
\right) -div\left( \utilde{\sigma}_{k}\cdot%
\overrightarrow{V_{k}}\right) +div\overrightarrow{Q_{k}}=\overrightarrow{%
F_{k}}\cdot\overrightarrow{V_{k}}+P_{k} \label{Ek}$$
where
$$E_{k}=\left\langle \chi _{k}E_{k}^{0}\right\rangle
=U_{k}+E_{ck}+E_{pk}\qquad \quad U_{k}=\left\langle \chi
_{k}U_{k}^{0}\right\rangle \qquad \quad \overrightarrow{Q_{k}}=\left\langle
\chi _{k}\overrightarrow{Q_{k}^{0}}\right\rangle$$
and
$$P_{k}=\left\langle -\overrightarrow{Q_{k}^{0}}\cdot\overrightarrow{n_{k}}\chi
_{i}\right\rangle$$
$\overrightarrow{F_{k}}\cdot\overrightarrow{V_{k}}+P_{k}$ represents the energy exchanges between the different phases through the interfaces : contact forces work and heat fluxes. We obtain the following condition for the interfaces (see appendix)
$$\frac{\partial E_{i}}{\partial t}+div\left( E_{i}\overrightarrow{V_{i}}%
\right) =-P_{3}-P_{4}-\overrightarrow{F_{3}}\cdot\overrightarrow{V_{3}}-%
\overrightarrow{F_{4}}\cdot\overrightarrow{V_{4}} \label{Ei}$$
where $E_{i}$ is the total energy density of the interfaces averaged over the R.E.V.. On summing equations (\[Ek\]) for $k=3,4$ and (\[Ei\]), we obtain the conservation law of the total volume energy of the whole material $E$
$$\rho \frac{D}{Dt}\left( \frac{E}{\rho }\right) =div\left( \sum\limits_{k=3,4}%
\utilde{\sigma}_{k}\cdot\overrightarrow{V_{k}}\right)
-div\overrightarrow{Q} \label{E}$$
where
$$E=\sum\limits_{3,4,i}E_{k}=U+E_{c}+E_{p}\quad \quad \quad \quad \quad \quad
\quad \overrightarrow{Q}=\sum\limits_{k=3,4}\overrightarrow{Q_{k}}$$
The source term of this equation is zero, which is the expression of the conservation law of the energy. $\sum\limits_{3,4}\utilde{\sigma}_{k}\cdot%
\overrightarrow{V_{k}}$ and $\overrightarrow{Q}$ represent the volume power of the contact forces and the heat fluxes of the complete medium, respectively.
Internal energy balance equation {#sec:54}
--------------------------------
The internal energy equation is obtained by subtracting kinetic and potential energy equations (\[Ecm\]) and (\[Epm\]) from the total energy conservation law (\[Em\])
$$\frac{\partial U_{k}^{0}}{\partial t}+div\left( U_{k}^{0}\overrightarrow{%
V_{k}^{0}}+\overrightarrow{Q_{k}^{0}}\right) =\utilde{\sigma}_{k}^{0}:\utilde{grad}%
\overrightarrow{V_{k}^{0}}+\left( \overrightarrow{I_{k}^{0}}-\rho
_{k}^{0}Z_{k}^{0}\overrightarrow{V_{k}^{0}}\right) \cdot\overrightarrow{E_{k}^{0}%
} \label{Um}$$
Algebraic manipulations of (\[Ek\]), (\[Eck\]) and (\[Epk\]) lead to
$$\frac{\partial U_{k}}{\partial t}+div\left( U_{k}\overrightarrow{V_{k}}+%
\overrightarrow{Q_{k}}\right) =\utilde{\sigma}_{k}:\utilde{grad}\overrightarrow{V_{k}}%
+ \overrightarrow{i_{k}}\cdot\overrightarrow{E_{k}}-\left\langle \overrightarrow{Q_{k}^{0}}%
\cdot\overrightarrow{n_{k}}\chi _{i}\right\rangle \label{Uk}$$
and for the interfaces (see appendix)
$$\frac{\partial U_{i}}{\partial t}+div\left( U_{i}\overrightarrow{V_{i}}%
\right) =\left\langle \overrightarrow{Q_{3}^{0}}\cdot\overrightarrow{n_{3}}\chi
_{i}+\overrightarrow{Q_{4}^{0}}\cdot\overrightarrow{n_{4}}\chi _{i}\right\rangle
-\overrightarrow{i_{i}}\cdot\overrightarrow{E_{i}}$$
where $U_{i}$ denotes the volume internal energy of interfaces included in the R.E.V..
Let us define $U_{\Sigma }$ as the sum of the volume internal energies of the different phases
$$U_{\Sigma }=U_{3}+U_{4}+U_{i}$$
From (\[DerivMat\]), we derive
$$\rho \frac{D}{Dt}\left( \frac{U_{\Sigma }}{\rho }\right)
=\sum\limits_{3,4}\left( \utilde{\sigma}_{k}:\utilde{grad}\overrightarrow{V_{k}}\right) +%
\overrightarrow{i}\cdot\overrightarrow{E}-div\overrightarrow{Q} \label{Usigma}$$
where $\overrightarrow{i}$ represents the diffusion current, consisting of the diffusion currents of the interfaces and of the cations in the solution
$$\overrightarrow{i}=\overrightarrow{I}-\sum\limits_{k=3,4}\left( \rho
_{k}Z_{k}\overrightarrow{V_{k}}\right) -Z_{i}\overrightarrow{V_{i}}=\rho
_{1}Z_{1}\left( \overrightarrow{V_{1}}-\overrightarrow{V_{4}}\right) +%
\overrightarrow{i_{i}}$$
$U_{\Sigma }$ represents only a part of the internal energy of the whole material; another part comes from the motion of the different phases in the barycentric reference frame. The internal energy of the whole material is defined by
$$U=E-E_{c}-E_{p}=U_{\Sigma }+E_{c\Sigma }-E_{c}$$
One deduces
$$\begin{tabular}{l}
$\rho \frac{D}{Dt}\left( \frac{U}{\rho }\right) =div\left(
\sum\limits_{3,4}E_{ck}\overrightarrow{V_{k}}-E_{c}\overrightarrow{V}\right)
+\frac{\partial }{\partial t}\left( \sum\limits_{3,4}E_{ck}-E_{c}\right) -div%
\overrightarrow{Q}$ \\
$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+\sum\limits_{3,4}\left( \utilde{\sigma}_{k}:%
\utilde{grad}\overrightarrow{V_{k}}\right) +%
\overrightarrow{i}\cdot\overrightarrow{E}$%
\end{tabular}%$$
The first two terms in the right-hand side represent the volume internal energy flux due to the relative velocities of the phases. The fourth one is the volume kinetic energy converted into heat by viscous dissipation. And the last term is the volume heat source by Joule effect in the solution.
Discussion {#sec:6}
==========
The conservation laws obtained for the global material include simplest cases. Assuming that the material is not electrically charged or removing the electric field, we obtain the equations governing a single-phase flow in porous medium [@Coussy95]. In case that the stress tensor is zero and that the velocities of the two phases are identical and uniform, we find the equations of a charged rigid solid subjected to an electric field.
The balance equations of the kinetic, potential, internal and total energies all have the same structure : the energy variation following the motion of one constituent, which is a particle derivative, is the sum of a flux and of source terms. The equations we write are relative to a thermodynamic closed system because of the use of the material derivative. Source terms correspond to conversion of one kind of energy into another one. At the microscopic scale, we obtain the following tables for the phase $%
k$
$$\begin{tabular}{cc}
\hline
& flux \\ \hline
$E_{pk}^{0}$ & \\
$E_{ck}^{0}$ & $div\left( \utilde{\sigma}_{k}^{0}\cdot\overrightarrow{V_{k}^{0}}\right) $ \\
$U_{k}^{0}$ & $-div\overrightarrow{Q_{k}^{0}}$ \\
$E_{k}^{0}$ & $div\left( \utilde{\sigma}_{k}^{0}\cdot%
\overrightarrow{V_{k}^{0}}-\overrightarrow{Q_{k}^{0}}\right) $ \\ \hline
\end{tabular}%$$
and
$$\begin{tabular}{cccc}
\hline
& $E_{c}\longleftrightarrow E_{p}$ & $U\longleftrightarrow E_{p}$ & $%
E_{c}\longleftrightarrow U$ \\ \hline
$E_{pk}^{0}$ & $-\rho _{k}^{0}Z_{k}^{0}\overrightarrow{E_{k}^{0}}\cdot%
\overrightarrow{V_{k}^{0}}$ & $-\left( \overrightarrow{I_{k}^{0}}-\rho
_{k}^{0}Z_{k}^{0}\overrightarrow{V_{k}^{0}}\right)\cdot\overrightarrow{E_{k}^{0}%
}$ & \\
$E_{ck}^{0}$ & $+\rho _{k}^{0}Z_{k}^{0}\overrightarrow{E_{k}^{0}}\cdot%
\overrightarrow{V_{k}^{0}}$ & & $-\utilde{\sigma}
_{k}^{0}:\utilde{grad}\overrightarrow{V_{k}^{0}}$ \\
$U_{k}^{0}$ & & $+\left( \overrightarrow{I_{k}^{0}}-\rho _{k}^{0}Z_{k}^{0}%
\overrightarrow{V_{k}^{0}}\right) \cdot\overrightarrow{E_{k}^{0}}$ & $+%
\utilde{\sigma}_{k}^{0}:\utilde{grad}\overrightarrow{V_{k}^{0}}$ \\
$E_{k}^{0}$ & & & \\ \hline
\end{tabular}%$$
Fluxes can be considered as the rate of variation of the quantity associated with the conduction phenomenon. The flux of kinetic energy is due to the contact force work, and the flux of internal energy to the heat conduction. The total energy flux is then the sum of the two previous ones. We point out that there is no flux for the potential energy. The viscous dissipation $%
\utilde{\sigma}_{k}^{0}:\utilde{grad}\overrightarrow{V_{k}^{0}}$ transforms the kinetic energy into heat, that is to say into internal energy. The work of the electric forces produces two source terms : the first one is the scalar product of the electric field $\overrightarrow{E_{k}^{0}}$ and of the diffusion current $\overrightarrow{I_{k}^{0}}-\rho _{k}^{0}Z_{k}^{0}%
\overrightarrow{V_{k}^{0}}$, which is the electric current measured in the barycentric reference frame. It can be seen as Joule heating, that is as a conversion of potential energy into internal energy. The other part $\rho
_{k}^{0}Z_{k}^{0}\overrightarrow{V_{k}^{0}}\cdot\overrightarrow{E_{k}^{0}}$ results in a motion of the electric charges subject to the electric field; potential energy is thus transformed into kinetic energy. Furthermore, the energy conservation law is consequently satisfied. Accordingly, there is no source term in the balance equation of the total energy.
We can examine in the same way the balance equations for one phase averaged over the R.E.V. That highlights the source terms
$$\begin{tabular}{cccc}
\hline
& $E_{c}\longleftrightarrow E_{p}$ & $U\longleftrightarrow E_{p}$ & $%
E_{c}\longleftrightarrow U$ \\ \hline
$E_{pk}$ & $-\rho _{k}Z_{k}\overrightarrow{V_{k}}\cdot\overrightarrow{E_{k}}$ & $%
-\overrightarrow{i_{k}}\cdot\overrightarrow{E_{k}}$ & \\
$E_{ck}$ & $+\rho _{k}Z_{k}\overrightarrow{V_{k}}\cdot\overrightarrow{E_{k}}$ &
& $-\utilde{\sigma}_{k}:\utilde{grad}\overrightarrow{V_{k}}$ \\
$U_{k}$ & & $+\overrightarrow{i_{k}}\cdot\overrightarrow{E_{k}}$ & $+%
\utilde{\sigma}_{k}:\utilde{grad}\overrightarrow{V_{k}}$ \\ \hline
\end{tabular}%$$
Viscous dissipation and Joule heating transform kinetic energy and potential energy into internal energy, respectively. And conversion of potential energy into kinetic energy is due once more to electric charges motion subject to the effect of the electric field. The other terms of the equations can be presented in the form
$$\begin{tabular}{ccc}
\hline
& flux & interfacial exchanges \\ \hline
$E_{pk}$ & & \\
$E_{ck}$ & $div\left( \utilde{\sigma}_{k}\overrightarrow{V_{k}}\right) $ & $%
+\overrightarrow{F_{k}}\cdot\overrightarrow{V_{k}}$ \\
$U_{k}$ & $-div\overrightarrow{Q_{k}}$ & $-\left\langle \overrightarrow{%
Q_{k}^{0}}\cdot\overrightarrow{n_{k}}\chi _{i}\right\rangle $ \\
$E_{k}$ & $div\left( \utilde{\sigma}_{k}\cdot\overrightarrow{V_{k}}-%
\overrightarrow{Q_{k}}\right) $ & $+\overrightarrow{%
F_{k}}\cdot\overrightarrow{V_{k}}+P_{k}$ \\ \hline
\end{tabular}%$$
where
$$\begin{tabular}{l}
$\overrightarrow{F_{k}}\cdot\overrightarrow{V_{k}}=\left\langle \left(
\utilde{\sigma}_{k}^{0}\cdot\overrightarrow{n_{k}}%
\right) \cdot\overrightarrow{V_{k}^{0}}\chi _{i}\right\rangle $ \\
$P_{k}=\left\langle -\overrightarrow{Q_{k}^{0}}\cdot\overrightarrow{n_{k}}\chi
_{i}\right\rangle $%
\end{tabular}%$$
As before, the flux of internal energy is the heat transfer by conduction, and the flux of kinetic energy is the volume power of the contact forces within the phase. Additional terms arise from this analysis; they represent exchanges between the phases through the interfaces. $\overrightarrow{F_{k}}\cdot%
\overrightarrow{V_{k}}$ is thus the volume power of the interaction forces acting on the phase $k$ and corresponds to a kinetic energy input. $%
-\left\langle \overrightarrow{Q_{k}^{0}}\cdot\overrightarrow{n_{k}}\chi
_{i}\right\rangle $ results from the heat transfer through the interface and modifies the internal energy. The sum of these two terms modifies the total energy of the considered phase.
Concerning the whole E.A.P., we obtain the following decomposition
$$\begin{tabular}{cc}
\hline
& flux \\ \hline
$E_{p}$ & \\
$E_{c}$ & $div\left[ \sum\limits_{3,4}\left( \utilde{%
\sigma}_{k}\cdot\overrightarrow{V_{k}}-E_{ck}\overrightarrow{V_{k}}\right)
+E_{c}\overrightarrow{V}\right] +\frac{\partial }{\partial t}\left(
E_{c}-\sum\limits_{3,4}E_{ck}\right) $ \\
$U$ & $div\left[ \sum\limits_{3,4}E_{ck}\overrightarrow{V_{k}}-E_{c}%
\overrightarrow{V}-\overrightarrow{Q}\right] +\frac{\partial }{\partial t}%
\left( \sum\limits_{3,4}E_{ck}-E_{c}\right) $ \\
$E$ & $div\left( \sum\limits_{3,4}\utilde{\sigma}_{k}%
\cdot\overrightarrow{V_{k}}-\overrightarrow{Q}\right) $ \\ \hline
\end{tabular}%$$
and
$$\begin{tabular}{cccc}
\hline
& $E_{c}\longleftrightarrow E_{p}$ & $U\longleftrightarrow E_{p}$ & $%
E_{c}\longleftrightarrow U$ \\ \hline
$E_{p}$ & $-\left( \sum\limits_{k=3,4}\left( \rho _{k}Z_{k}\overrightarrow{%
V_{k}}\right) +Z_{i}\overrightarrow{V_{i}}\right) \cdot\overrightarrow{E}$ & $-%
\overrightarrow{i}\cdot\overrightarrow{E}$ & \\
$E_{c}$ & $+\left( \sum\limits_{3,4}\rho _{k}Z_{k}\overrightarrow{V_{k}}%
+Z_{i}\overrightarrow{V_{i}}\right) \cdot\overrightarrow{E}$ & & $%
-\sum\limits_{3,4}\utilde{\sigma}_{k}:\utilde{grad}\overrightarrow{V_{k}}$ \\
$U$ & & $+\overrightarrow{i}\cdot\overrightarrow{E}$ & $+\sum\limits_{3,4}%
\left( \utilde{\sigma}_{k}:\utilde{grad}\overrightarrow{V_{k}}\right) $ \\ \hline
\end{tabular}%$$
The energy flux comes from the work of the contact forces in the different phases and from the heat transfer by conduction; the first one is a flux of kinetic energy, the second one is the flux of internal energy. The flux of potential energy is still zero. An additional flux term appears : the kinetic energy of the different phases measured in a barycentric reference frame; this kinetic energy is indeed a part of the internal energy of the global material. The source terms include viscous dissipation, which transforms kinetic energy into heat, and Joule heating, which transforms potential energy into internal energy. This last term is linked to the diffusion current created by the interfacial charges motion and by the cations motion in the solution reference frame. The global motion of the charges under the influence of the electric field turns potential energy on kinetic energy.
Conclusion {#sec:concl}
==========
We have modelled an electroactive, ionic, water-saturated polymer placed in an electric field. The polymer is fully dissociated, releasing cations of small size. This system is depicted as the superposition of two continuous media : a deformable porous medium constituted by the polymer backbone embedded with anions, in which flows an ionic solution composed by water and released cations. We have deduced the microscale conservation laws of each phase : mass continuity equation, linear momentum conservation law, Maxwell’s equations and energy balance laws. Then we derived the physical quantities attached to the interfaces. An average over the R.E.V. of the material has provided one with macroscale conservation laws for each phase first and for the global E.A.P., next. Having the three constituents of the material (solid, solvent and cations) different velocities, we have used for this last step, the material derivative concept in order to obtain an Eulerian formulation of the conservation laws.
We have examined the balance equations of the different energies (kinetic, potential and internal ones), and we have put the emphasis on the phenomena responsible for the conversion of one kind of energy into another : viscous frictions, Joule effect and charge motion under the effect of the electric field. The first two results in dissipation. Moreover, the macroscale equations relative to each phase allow an evaluation of energy exchanges through the interfaces.
Using the linear thermodynamics of the irreversible processes we should now be able to determine the potential of dissipation and to derive the phenomenological equations governing this system. This will be the subject of a forthcoming work.
Appendix : interface modelling {#sec:Annexe}
==============================
In practice, contact area between phases $3$ and $4$ has a certain thickness; extensive physical quantities like mass density, linear momentum and energy continuously vary from one bulk phase to the other one. This complicated reality can be modelled by two uniform bulk phases separated by a discontinuity surface $\Sigma $ whose localization is arbitrary. Let $\Omega $ be a cylinder crossing $\Sigma $, whose bases are parallel to $\Sigma $. We denote by $\Omega _{3}$ and $\Omega _{4}$ the parts of $\Sigma $ respectively included in phases $3$ and $4$.
The continuous quantities relative to the contact zone are identified by a superscript $^{0}$ and no subscript. A microscale quantity per surface unit $%
g_{i}^{0}$ related to the interface is defined by
$$g_{i}^{0}=\lim\limits_{\Sigma \longrightarrow 0}\frac{1}{\Sigma }\left\{
\int_{\Omega }g^{0}dv-\int_{\Omega _{3}}g_{3}^{0}dv-\int_{\Omega
_{4}}g_{4}^{0}dv\right\} \label{Def-i0}$$
where $\Omega _{3}$ and $\Omega _{4}$ are small enough so that $g_{3}^{0}$ and $g_{4}^{0}$ are constant. Its average over the R.E.V. is the volume quantity $g_{i}$ defined by
$$g_{i}=\left\langle g_{i}^{0}\chi _{i}\right\rangle$$
The balance equation of the interfacial quantity $g_{i}^{0}$ is written as (Ishii, [@Ishii06])$$\frac{\partial g_{i}^{0}}{\partial t}+div_{s}\left( g_{i}^{0}\overrightarrow{%
V_{i}^{0}}\right) =\sum\limits_{3,4}\left[ g_{k}^{0}\left( \overrightarrow{%
V_{k}}-\overrightarrow{V_{i}^{0}}\right) \cdot\overrightarrow{n_{k}}+%
\overrightarrow{J_{k}^{0}}\cdot\overrightarrow{n_{k}}\right] -div_{s}%
\overrightarrow{J_{i}^{0}}+\phi _{i}^{0}$$where $div_{s}$ denotes the surface divergence operator. $\overrightarrow{%
J_{i}^{0}}$ is the surface flux of $g_{i}^{0}$, $\overrightarrow{%
J_{k}^{0}}$ the flux of $g_{k}^{0}$ and $\phi _{i}^{0}$ the surface source term.
We arbitrarily fix the interface position in such a way that it has no mass density
$$\rho _{i}^{0}=\lim\limits_{\Sigma \longrightarrow 0}\frac{1}{\Sigma }\left\{
\int_{\Omega }\rho ^{0}dv-\int_{\Omega _{3}}\rho _{3}^{0}dv-\int_{\Omega
_{4}}\rho _{4}^{0}dv\right\} =0 \label{DefI}$$
From (\[CMcl\]), we deduce that the linear momentum and the kinetic energy per surface unit of the interface, respectively denoted $\overrightarrow{%
P_{i}^{0}}$ and $E_{ci}^{0}$, are zero
$$\overrightarrow{P_{i}^{0}}=\overrightarrow{0}\qquad \qquad E_{ci}^{0}=0$$
In the same way, we define the charge per unit surface $Z_{i}^{0}$, the surface current vector $\overrightarrow{I_{i}^{0}}$, the surface diffusion current $\overrightarrow{i_{i}^{0}}$, the surface potential energy $E_{pi}^{0}$, the surface internal energy $U_{i}^{0}$ and the surface total energy $E_{i}^{0}$.
The balance equations of these quantities write$$\frac{\partial Z_{i}^{0}}{\partial t}+div_{s}\left( Z_{i}^{0}\overrightarrow{%
V_{i}^{0}}\right) =\overrightarrow{i_{3}^{0}}\cdot\overrightarrow{n_{3}}+%
\overrightarrow{i_{4}^{0}}\cdot\overrightarrow{n_{4}}-div_{s}\overrightarrow{%
i_{i}^{0}}$$
$$\frac{\partial \overrightarrow{P_{i}^{0}}}{\partial t}+\overrightarrow{div_{s}}%
\left(\overrightarrow{P_{i}^{0}}\otimes \overrightarrow{V_{i}^{0}}\right) =-%
\utilde{\sigma}_{3}^{0}\cdot\overrightarrow{n_{3}}-%
\utilde{\sigma}_{4}^{0}\cdot\overrightarrow{n_{4}}%
+Z_{i}^{0}\overrightarrow{E_{i}^{0}}$$
$$\frac{\partial E_{pi}^{0}}{\partial t}+div_{s}\left( E_{pi}^{0}%
\overrightarrow{V_{i}^{0}}\right) =-\overrightarrow{I_{i}^{0}}\cdot%
\overrightarrow{E_{i}^{0}}$$
$$\frac{\partial E_{i}^{0}}{\partial t}+div_{s}\left( E_{i}^{0}\overrightarrow{%
V_{i}^{0}}\right) =-\left( \utilde{\sigma}_{3}^{0}\cdot%
\overrightarrow{n_{3}}\right) \cdot\overrightarrow{V_{3}^{0}}-\left(
\utilde{\sigma}_{4}^{0}\cdot\overrightarrow{n_{4}}%
\right) \cdot\overrightarrow{V_{4}^{0}}+\overrightarrow{Q_{3}^{0}}\cdot%
\overrightarrow{n_{3}}+\overrightarrow{Q_{4}^{0}}\cdot\overrightarrow{n_{4}}$$
$$\frac{\partial U_{i}^{0}}{\partial t}+div_{s}\left( U_{i}^{0}\overrightarrow{%
V_{i}^{0}}\right) =\overrightarrow{Q_{3}^{0}}\cdot\overrightarrow{n_{3}}+%
\overrightarrow{Q_{4}^{0}}\cdot\overrightarrow{n_{4}}+%
\overrightarrow{i_{i}^{0}}\cdot\overrightarrow{E_{i}^{0}}$$
Averaging over the R.E.V., this leads to the boundary conditions below
$$\frac{\partial Z_{i}}{\partial t}+div\overrightarrow{I_{i}}=\left\langle
\overrightarrow{i_{3}^{0}}\cdot\overrightarrow{n_{3}}\chi _{i}\right\rangle
+\left\langle \overrightarrow{i_{4}^{0}}\cdot\overrightarrow{n_{4}}\chi
_{i}\right\rangle$$
$$\overrightarrow{F_{3}}+\overrightarrow{F_{4}}=Z_{i}\overrightarrow{E_{i}}$$
$$\frac{\partial E_{pi}}{\partial t}+div\left( E_{pi}\overrightarrow{V_{i}}%
\right) =-\overrightarrow{I_{i}}\cdot\overrightarrow{E_{i}}$$
$$\frac{\partial E_{i}}{\partial t}+div_{s}\left( E_{i}\overrightarrow{V_{i}}%
\right) =-P_{3}-P_{4}-\overrightarrow{F_{3}}\cdot\overrightarrow{V_{3}}-%
\overrightarrow{F_{4}}\cdot\overrightarrow{V_{4}}$$
$$\frac{\partial U_{i}}{\partial t}+div\left( U_{i}\overrightarrow{V_{i}}%
\right) =\left\langle \overrightarrow{Q_{3}^{0}}\cdot\overrightarrow{n_{3}}\chi
_{i}+\overrightarrow{Q_{4}^{0}}\cdot\overrightarrow{n_{4}}\chi _{i}\right\rangle
+\overrightarrow{i_{i}}\cdot\overrightarrow{E_{i}}$$
Moreover, we have$$\overrightarrow{I_{i}}=Z_{i}\overrightarrow{V_{i}}+\overrightarrow{i_{i}}$$
Notations {#sec:not}
=========
$k=1,2,3,4,i$ subscripts respectively represent cations, solvent, solid, solution (water and cations) and interface; quantities without subscript refer to the whole material. Superscript $^{0}$ denotes a local quantity; the lack of superscript indicates average quantity at the macroscopic scale. Microscale volume quantities are relative to the volume of the phase, average quantities to the volume of the whole material.
$C$ : cations molar concentration (relative to the liquid phase);
$\overrightarrow{D}$ ($\overrightarrow{D_{k}}$,$\overrightarrow{%
D_{k}^{0}}$) : electric displacement field;
$E$ ($E_{k}$,$E_{k}^{0}$) : total energy density (internal, kinetic and potential);
$\overrightarrow{E}$ ($\overrightarrow{E_{k}}$,$\overrightarrow{%
E_{k}^{0}}$) : electric field;
$E_{c}$ ($E_{c\Sigma }$,$E_{ck}$,$E_{ck}^{0}$) : kinetic energy density;
$E_{p}$ ($E_{pk}$,$E_{pk}^{0}$) : potential energy density;
$F=96487\;C\;mol^{-1}$ : Faraday’s constant ;
$\overrightarrow{F_{k}}$ : resultant of the mechanical stresses exerted on the phase $k$ by the other phase;
$\overrightarrow{I}$ ($\overrightarrow{I_{k}}$,$\overrightarrow{%
I_{k}^{0}}$) : current density vector;
$\overrightarrow{i}\ $($\overrightarrow{i_{k}}$,$\overrightarrow{%
i_{k}^{0}}$) : diffusion current;
$M_{k}$ : molar mass of component $k$;
$\overrightarrow{n_{k}}$ : outward-pointing unit normal of phase $k$;
$P_{k}$ : heat flux through interfaces;
$\overrightarrow{P_{i}^{0}}$ : local surface linear momentum of interface;
$\overrightarrow{Q}$ ($\overrightarrow{Q_{k}}$,$\overrightarrow{%
Q_{k}^{0}}$) : heat flux;
$U$ ($U_{\Sigma }$,$U_{k}$,$U_{k}^{0}$) : internal energy density;
$\overrightarrow{V}$ ($\overrightarrow{V_{k}}$,$\overrightarrow{%
V_{k}^{0}}$) : velocity;
$z_{k}$ : number of elementary charges of a ion $k$;
$Z$ ($Z_{k}$,$Z_{k}^{0}$) : total electric charge per unit of mass;
$Z_{i}$ ($Z_{i}^{0}$) : electric charge density per unit surface;
$\varepsilon $ ($\varepsilon _{k}$,$\varepsilon _{k}^{0}$) : permittivity;
$\rho $ ($\rho _{k}$,$\rho _{k}^{0}$) : mass density;
$\utilde{\sigma }$ ($\utilde{\sigma}_{k}$,$\utilde{\sigma}_{k}^{0}$) : stress tensor;
$\phi _{k}$ : volume fraction of phase $k$;
$\chi _{k}$ : function of presence of phase $k$ ;
The authors would like to thank D. Lhuillier and O. Kavian for their fruitful and stimulating discussions.
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|
---
author:
- |
Edward Meeds\
Informatics Institute\
University of Amsterdam\
Amsterdam, The Netherlands\
`tmeeds@gmail.com`\
Max Welling\
Informatics Institute\
University of Amsterdam\
Amsterdam, The Netherlands\
`welling.max@gmail.com`
title: |
POPE: Post Optimization Posterior Evaluation\
of Likelihood Free Models
---
[**POPE: Post Optimization Posterior Evaluation\
of Likelihood Free Models** ]{}\
Edward Meeds$^{1\ast}$, Michael Chiang$^{2}$, Mary Lee$^{3}$, Olivier Cinquin$^{2}$, John Lowengrub$^{3}$, Max Welling$^{1,4}$\
**[1]{} Informatics Institute, University of Amsterdam, Amsterdam, The Netherlands\
**[2]{} School of Biological Sciences, University of California, Irvine, CA, USA\
**[3]{} Department of Mathematics, University of California, Irvine, CA, USA\
**[4]{} Donald Bren School of Informatics, University of California, Irvine, CA, USA\
$\ast$ E-mail: Corresponding E.W.F.Meeds@uva.nl********
Abstract {#abstract .unnumbered}
========
In many domains, scientists build complex simulators of natural phenomena that encode their hypotheses about the underlying processes. These simulators can be deterministic or stochastic, fast or slow, constrained or unconstrained, and so on. Optimizing the simulators with respect to a set of parameter values is common practice, resulting in a single parameter setting that minimizes an objective subject to constraints. We propose a post optimization posterior analysis that computes and visualizes all the models that can generate equally good or better simulation results, subject to constraints. These [*optimization posteriors*]{} are desirable for a number of reasons among which easy interpretability, automatic parameter sensitivity and correlation analysis and posterior predictive analysis. We develop a new sampling framework based on approximate Bayesian computation (ABC) with one-sided kernels. In collaboration with two groups of scientists we applied POPE to two important biological simulators: a fast and stochastic simulator of stem-cell cycling and a slow and deterministic simulator of tumor growth patterns.
Introduction {#introduction .unnumbered}
============
In science and industry alike, modelers express their expert knowledge by building a simulator of the phenomenon of interest. There is an enormous variety of such simulators, deterministic or stochastic, fast or slow, with or without constraints. For most simulators, e.g. driven by stochastic partial differential equations, it is impossible to write down an expression for the likelihood, which can make it highly challenging to optimize the simulator over its free parameters. This “blind optimization problem" is receiving increasing attention in the machine learning community [@lizotte2008; @osborne2009; @snoek:2012].
However, even if the optimal parameter value $\thetastar$ is found, this leaves the scientist still in the dark with respect to important questions such as: “Which parameters are correlated?"; “Which parameters are robust and which are sensitive?"; “Is my model overfitting, underfitting or just right"? We believe that methods capable of handling these type of questions post optimization are essential to the field of simulation-based modeling. In this paper we propose a new Bayesian framework that allows the scientist to answer these questions by approximating through sampling the posterior distribution of all parameters that may result in equally good or better models. This “Post Optimization Posterior Evaluation" (POPE) is different from standard ABC [@marjoram2003markov; @Wilkinson2013; @sisson:2010] in that standard ABC compares simulator outcomes with observations while POPE reasons about an optimization problem (subject to constraints) without the need for observations. While different philosophically, POPE can be implemented by using one-sided kernels within ABC.
POPE was developed in close collaboration with a number of scientists, and has a number of properties that are beneficial to their work: 1) the posterior distribution over parameters has a clear and interpretable meaning and can be used to suggest alternative parameters to explore, 2) POPE can handle multiple objectives and constraints, 3) unlike most standard optimization methods, POPE can handle simulators with stochastic outputs and complicated input or outputs constraints, 4) POPE can handle multimodal posterior distributions, 5) as part of its computation POPE will generate posterior predictive samples that can be used to evaluate the model fit, and 6) by incorporating Gaussian process surrogate models it can handle expensive simulators.
In this paper we will develop POPE and apply it to two real-world cases: one fast stochastic simulator in the domain of stem cell biology and one slow deterministic simulator developed for cancer research.
Approximate Bayesian Computation {#approximate-bayesian-computation .unnumbered}
================================
The primary goal of Bayesian inference is to draw samples from or learn an approximate model of the following (usually intractable) posterior distribution: $$\pi(\thetav | \ystar_1, \ldots, \ystar_N ) \propto \pi(\thetav) \pi( \ystar_1, \ldots, \ystar_N | \thetav )$$ where $\pi(\thetav)$ is a prior distribution over parameters $\thetav \in {\rm I\!R}^{D}$ and $\pi( \ystar_1, \ldots, \ystar_N | \thetav )$ is the likelihood of $N$ data observations, where $\ystar_n \in {\rm I\!R}^J$. The vector of $J$ values can either be “raw” observations or, more typically, informative statistics of observations. In this paper we consider the case where $N=1$ (though all our methods apply equally to $N>1$) and will henceforth drop the subscripts. The unconventional superscript on $\ystar$ is used to distinguish the observations from the simulator outputs $\y$.
In ABC the likelihood function $\pi( \ystar | \thetav )$ is usually not available as a function but rather as a complex simulation, hence the alternative name for ABC, [*likelihood-free inference*]{}. ABC sampling algorithms treat the simulator as an auxiliary variable generator and discrepancies between the simulator outputs and the observations as proxies for the likelihood value. If we let $\y \simsim \pi( \y | \thetav )$ be a “draw” from the simulator, the likelihood can be written as: $$\pi( \ystar | \thetav ) = \int \lb \y = \ystar \rb \pi( \y | \thetav ) d\y \label{eq:abc_exact_likelihood}$$ where $\lb \cdot \rb = 1$ if the arguments are true, and $0$ otherwise. Equation \[eq:abc\_exact\_likelihood\] implies that we can compute the exact likelihood by integrating over all possible simulation output values. In reality, since this integral requires simulations to match observations exactly, it is only achievable for discrete data. For continuous $\ystar$, $J$ slack variables $\epsvec$ are introduced around $\ystar$. More specifically, an $\epsvec$-kernel function $\pi_{\epsvec}$ is used to measure the discrepancy between simulation results and observations. In practice the likelihood is approximated by a Monte Carlo estimate computed from $S$ draws of the simulator $\ymc \simsim \pi( \y | \thetav )$: $$\pi_{\epsvec}( \ystar | \thetav ) \approx \int \pi_{\epsvec}(\ystar | \y ) \pi( \y | \thetav ) d\y
\approx \frac{1}{S} \sum_{s=1}^S \pi_{\epsvec}(\ystar | \ymc ) \label{eq:abc_mc_approx}$$ This is clearly an *unbiased* estimator of $\pi_{\epsvec}( \ystar | \thetav )$. Common $\pi_{\epsvec}$ functions are the $\epsvec$-tube $\pi_{\epsvec}(\ystar | \y ) \propto \prod_{j}\lb \| \ystarj-\yj \|_1 \leq \epsj \rb$ and the Gaussian kernel $\pi_{\epsvec}(\ystar | \y ) = \prod_{j}\mathcal{N} \lp \ystarj | \yj, \epsj^2 \rp$.
Among the many possible ABC sampling algorithms, Markov chain Monte Carlo (MCMC) ABC is of particular relevance to this work [@marjoram2003markov; @Wilkinson2013; @sisson:2010]. In the Metropolis-Hastings (MH) step the proposal distribution is composed of the product of the proposal for the parameters $\thetav$ and the proposal for the simulator outputs: $$q( \thetavp, \yp | \thetav ) = q( \thetavp | \thetav ) \pi( \yp | \thetavp)$$ i.e. parameters $\thetavp$ are first proposed, then outputs $\yp$ are generated from the simulator with input parameters $\thetavp$.
Using this form of the proposal distribution, and using the Monte Carlo approximation eq \[eq:abc\_mc\_approx\], we arrive at the following Metropolis-Hastings accept-reject probability, $$\alpha = \min \lp 1, \frac{\pi\lp\thetavp\rp \sum_{s=1}^S \pi_{\epsvec}(\ystar | \yps ) q( \thetav | \thetavp )}{\pi\lp\thetav\rp \sum_{s=1}^S \pi_{\epsvec}(\ystar | \ymc ) q( \thetavp | \thetav )} \rp \label{eq:abc_mh_acceptance_with_s}$$ When only the numerator is re-estimated at every iteration (and the denominator is carried over from the previous iteration), then this algorithm corresponds to pseudo-marginal (PM) sampling [@delmoral2008; @andrieu2009pseudo]. PM sampling is asymptotically correct (taking for granted the approximation introduced by the kernel $\pi_{\epsvec}$) but can display very poor mixing properties. By resampling the denominator as well, we improve mixing at the cost of introducing a further approximation. This sampler is known as the marginal sampler [@marjoram2003markov; @sisson:2010]. Even the PM sampler requires $S$ simulations per MCMC move, which may be too expensive for complex simulators. Surrogate modeling—where the history of all simulations are stored in memory and used to build a surrogate of the simulator—may be the only option to make progress in that case.
Post Optimization Posterior Evaluation {#post-optimization-posterior-evaluation .unnumbered}
======================================
In regular ABC the simulator generates output statistics $\y$ that are compared directly with observations $\ystar$. For optimization problems, however, the scientist may interpret $\y_1$ as a cost and $\ystar_1$ as an estimate of the minimum cost. Other simulation statistics $\{\y_j\}$, $j=2..J$ may be constrained, e.g. $\{\y_j \leq \ystar_j \}$. For instance, the cost could be some measure of misfit between simulator outcomes and desirable outcomes while constraints could represent domains within which certain simulation results should lie (constraints can of course also be incorporated into the cost function, but as we will see, it is sometimes beneficial to treat them separately). Our first guess to elucidate some posterior distribution over parameters could be to define a Gibbs distribution $p(\y_1)\propto\exp(-\beta\y_1)$ which we would treat as a likelihood similar to $\pi_{\epsvec}$ and apply ABC, rejecting everything that does not satisfy the constraints. Unfortunately, we do not consider this a satisfactory solution because the posterior does not have a clear interpretation. For instance, simply scaling the arbitrary constant $\beta$ would change the posterior.
A better solution is to define a new type of (one-sided) Heavyside kernel in ABC: $\lb \y_1 \leq \ystar_1 \rb$ which is $1$ when the argument is satisfied and $0$ otherwise. Note that this kernel is applied to both the objective $\y_1$ and the constraints $\{\y_j\}$ alike. The quantity $\ystar_1$ is given by the lowest value of the objective found by some optimization procedure (e.g. grid-search, Bayesian optimization [@snoek:2012], etc). The posterior samples produced by an ABC algorithm that uses this one-sided kernel have a very clean interpretation, namely they represent *the probability that a simulation run at that parameter value will generate an equally good or better (lower) value for the objective while satisfying all the constraints*. This distribution can be used to suggest new regions to explore (e.g. other modes, or regions that are farther away from constraint surfaces), and to visualize dependencies between parameters and their sensitivities.
The posterior described above thus corresponds to $$\pi( \thetav | \ystar ) \propto \pi(\thetav) \int \lb \y \leq \ystar \rb \pi( \y | \thetav ) d\y
\propto \pi(\thetav) \int_{-\infty}^{\ystar} \pi( \y | \thetav ) d\y
\propto \pi(\thetav) \F_{\y | \thetav}(\ystar)$$ where $\F_{\y | \thetav}$ is the cumulative distribution function (CDF) of the conditional probability density function $ \pi( \y | \thetav )$ (or the probability of satisfying the constraint or improving the objective). Since in ABC we cannot compute the likelihood analytically, it is approximated by a Monte Carlo estimate: $$\F_{\y | \thetav}(\ystar) \approx \frac{1}{S} \sum_{s=1}^S \lb \y^{(s)} \leq \ystar \rb ~~~~~~~~~~~~~~~~~~~~~~ \y^{(s)} \simsim \pi( \y | \thetav )$$ Using the one-sided kernel $\lb \y \leq \ystar \rb$ will cause the ABC sampler to get stuck when initialized in a region where $\y > \ystar$ because every proposed sample will get rejected. Even when initialized in a region where $\y \leq \ystar$, this kernel will make it very difficult to move between different “islands" (modes) in parameter space where these conditions hold. This problem is aggravated in high dimensions where $\lb \y \leq \ystar \rb = \prod_j \lb y_j \leq y_j^\star \rb $ and every condition needs to be satisfied for the likelihood to be non-zero. A one-sided $\eps$-tube $\lb \y \leq \ystar + \eps \rb$ adds some relief but suffers the same problem for most useful values of $\eps$.
The solution to this problem is to soften the kernel analogously to the softening of the condition $\lb \y = \ystar \rb$ into $\pi_{\epsvec}(\ystar|\y)$ in generalized ABC [@Wilkinson2013]. If we define $d_j = \yj - \ystarj$, then these soft kernels treat all simulation outputs less than $\ystarj$ with likelihood proportional to $1$ and provide quadratic or linear penalties otherwise. For example, a one-sided Gaussian kernel is defined as $$\begin{aligned}
K_{\eps_j}\lp \y_j;\; \ystar_j \rp = [ d_j \geq 0 ] + [ d_j < 0 ] \exp\lp -\frac{1}{2} \lp\frac{d_j}{\eps_j}\rp^2\rp \end{aligned}$$ and a one-sided exponential kernel (i.e. linear penalty) is defined as $$\begin{aligned}
K_{\eps_j}\lp \y_j;\; \ystar_j \rp = [ d_j \geq 0 ] + [ d_j < 0 ] \exp\lp \frac{d_j}{\eps_j} \rp\end{aligned}$$ By modifying $\epsvec$ we can control the severity of the penalty, allowing us to use annealing schedules that adapt $\epsvec$ during the MCMC run in order to focus the sampling at modes when $\epsvec$ is small.
Up to this point we have only discussed [*one-sided*]{} likelihoods, but there is nothing preventing the likelihoods to incorporate both upper and lower constraints: $$\pi\lp \ystar | \thetav \rp = \int_{\ystar_a}^{\ystar_b} \pi( \y | \thetav ) d\y
= \F_{\y | \thetav}(\ystar_b)-\F_{\y | \thetav}(\ystar_a)$$ The one-sided kernels are easily modified for this, setting the likelihood to $1$ in between the regions, with quadratic or linear penalties outside of the regions.
MODELING THE SIMULATOR RESPONSE {#modeling-the-simulator-response .unnumbered}
===============================
We may want to consider modeling the simulator response $\pi(\y|\thetav)$ if the outcome of the simulator is stochastic or the simulator is expensive to run. In the first case, we can reduce the variance of the Markov chain by learning a *local response model* for every state $\thetav$. For the second case, a [*global response model*]{} (a.k.a. a surrogate model) over the entire $\thetav$-space is more appropriate because it stores and makes use of the entire simulation history to predict responses at new $\thetav$ locations.
Local Response Models {#local-response-models .unnumbered}
---------------------
When the simulator is fast and stochastic, it can be beneficial to the inference procedure to build a local, conditional model of the distribution $\pi(\y|\thetav)$ using $S$ simulator responses in $\yone, \ldots, \yS \overset{\simulator}{\sim} \pi( \y | \thetav)$. The simplest local response model is the *conditional Gaussian*, an approach called [*synthetic likelihood*]{} in ABC [@wood2010statistical]. It computes estimators of the first and second moments of the responses and uses the Gaussian distribution to analytically compute the likelihood (thus providing an alternative to kernel ABC). For our algorithms, this allows the direct computation of the CDF: $$\begin{aligned}
&& \muhattheta = \frac{1}{S} \sum_{s=1}^{S} \y_s ~~~~~~~~~~~
\hat{\Sigma}_{\thetav} = \frac{1}{S-1} \sum_{s=1}^{S} \lp \ys - \muhattheta \rp \lp \ys - \muhattheta \rp^T\\
&&\F_{\y | \thetav}(\ystar; \muhattheta, \hat{\Sigma}_{\thetav}) = \int_{-\infty}^{\ystar} \mathcal{N}\lp \y | \muhattheta, \hat{\Sigma}_{\thetav} \rp d \y\end{aligned}$$ where $\muhattheta$ and $\hat{\Sigma}_{\thetav}$ are computed from the $S$ simulations. In our experiments we often use a factorized model: $\mathcal{N}(\y | \muhattheta,\hat{\Sigma}_{\thetav}) \approx \prod_{j=1}^J \mathcal{N}( y_j | \muhatthetaj, \sigmahatthetaj)$, resulting in a factorized product over CDFs as well. Modeling the response by only the first two moments may be inadequate due to multi-modality, asymmetric noise, etc. For such cases a *conditional KDE* (kernel density estimate) response model can by used. In [@TurnerGenLik2014] this approach is shown to be superior to conditional Gaussians for certain computational psychology models. Note that for Gaussian kernels the conditional KDE is very similar to kernel ABC, but has additional flexibility of adaptively choosing bandwidths (rather than the fixed $\epsvec$ in kernel ABC).
Global Response Models {#global-response-models .unnumbered}
----------------------
For very expensive simulators it is impractical to run simulations at each parameter location in the MCMC run. In these cases it is worth the extra storage and the computational overhead of learning a model of the simulator response surface. For global response models the Metropolis-Hastings diverges from ABC-MCMC in that simulations are only performed if the surrogate is very uncertain. When the surrogate is confident, no simulations are performed.
The natural global extension of the Gaussian conditional model is the Gaussian process (GP). The GP has been used extensively for surrogate modeling [@rasmussen:2003; @kennedyohagan; @lizotte2008; @osborne2009], including more recent applications in accelerating ABC [@Wilkinson2014; @Meeds2014GpsUai]. In [@Wilkinson2014] GPs directly model the log-likelihood in successive waves of inference, each one eliminating regions of low posterior probability. This approach is capable of handling high-dimensional simulator outputs. In [@Meeds2014GpsUai] each dimension of the simulator response is modeled by a GP and explicitly uses the surrogate uncertainty to determine simulation locations (design points). The advantage of this approach is that CDFs can be computed directly from the GPs predictive distributions. A global extension of the conditional KDE is more complicated, but estimators such as the Nadayara-Watson could provide the necessary modeling machinery. We leave these extensions to future work.
$\thetav \gets \thetav_0$ $\yone, \ldots, \yS \overset{\simulator}{\sim} \pi( \y | \thetav)$ $\thetapv \sim q(\thetapv | \thetav )$ $\ypone, \ldots, \ypS \overset{\simulator}{\sim} \pi( \y | \thetavp)$ $\yone, \ldots, \yS \overset{\simulator}{\sim} \pi( \y | \thetav)$ $\alpha \gets \lp 1, \frac{\pi(\thetavp)q(\thetap | \thetavp ) \pi(\thetavp | \ystar, \epsvec)}{ \pi(\thetav) q(\thetapv | \thetap )\pi(\thetav | \ystar, \epsvec)} \rp$ $\thetav \gets \thetavp$ $\yone, \ldots, \yS \gets \ypone, \ldots, \ypS$ Collect $\thetav$ Collection $\thetav$
MCMC for POPE {#sec:adapting .unnumbered}
=============
Algorithm \[algo:kernelpope\] provides the pseudo-code for running kernel ABC POPE (easily modified to accommodate response models by plugging in the appropriate likelihood function). This is simply ABC-MCMC with one-sided kernel likelihoods. There are two possible modes for running POPE: marginal and pseudo-marginal. When running marginal MCMC, the state of the Markov chain only includes $\thetav$, and (as discussed earlier) has the property of improved mixing with the cost of doubling the number of simulations per Metropolis-Hastings step and a less accurate posterior. On the other hand, pseudo-marginal can mix poorly, but uses fewer simulations and is more accurate. Choosing between the two modes is problem specific.
Adaptive POPE {#adaptive-pope .unnumbered}
-------------
In ABC, the choice of $\epsvec$ is crucial to both the MCMC mixing and the precision of the posterior distribution. There is an obvious trade-off between the two as large $\epsvec$ provides better mixing but poorer approximations to the target distribution. It is common in ABC to adapt $\epsvec$ using quantiles of the discrepancies (e.g. in Sequential Monte Carlo ABC [@beaumont2009adaptive]) or using a more complicated approach, for example based on the threshold acceptance curve [@Silk2013]. We propose an online version of the quantile method (see function [*UpdateEpsilons*]{} in Algorithm \[algo:adaptivepope\] (Appendix \[sec:appendixpope\])), setting $\epsvec$ to a quantile of the exponential moving average (EMA) of the discrepancies or some minimum values ${\epsvec^{\textsc{min}}}$, which ever is greater. Minimum values ${\epsvec^{\textsc{min}}}$ are set not only for computational reasons, but also to reflect the scientist’s intuition regarding the relative importance of the constraints. Because $\epsvec$ can fluctuate during the MCMC run, it can explore regions where some constraints are easily satisfied, but others are not, and vice-versa. A quantile parameter ${\beta}$ puts pressure on the chain to keep $\epsvec$ small.
For some problems we may not know certain *objective values* in $\ystar$ before running POPE. For these cases simple adaptive MCMC procedures can estimate $\ystar$ during the MCMC run. For deterministic simulators, $\ystar$ can be updated after each simulation. For stochastic simulators we propose a local averaging procedure based on the EMA of $\y$, similar to the adaptation of $\epsvec$. The intuition behind this is that the best objective value $\ystar$ at $\thetastar$ is the expected value of the simulator response at $\thetastar$. An EMA of the simulation response approximates this expectation and we have found in our experiments with stochastic simulators that it performs well and conveniently fits into the POPE MCMC procedure (i.e. there is no need to set up an entirely different optimization procedure with complicated constraints on the input and outputs since these are already part of POPE). This is function [*UpdateObjectives*]{} in Algorithm \[algo:adaptivepope\] (Appendix \[sec:appendixpope\]).
These are adaptive MCMC algorithms that do not necessarily target the correct posterior distribution. The simplest way to correct this is to simply use a few MCMC runs to set $\epsilon$ or $\ystar$ (if needed) or stop the adaptation altogether after a burnin period, from that point using non-adaptive ABC-MCMC. An alternative to adapting $\epsilon$ is to include $\epsilon$ as part of the state of the Markov chain [@Bortot:2007].
Posterior Analysis of MCMC Results {#posterior-analysis-of-mcmc-results .unnumbered}
----------------------------------
Along with the posterior parameter distribution $p( \thetav\,|\,\ystar)$, which is usually the main distribution of interest in a Bayesian analysis, we will also examine the [*posterior predictive distribution*]{}, denoted as $p(\y | \ystar)$, though perhaps unintuitive, is the distribution of statistics (the predictions) generated by the simulation at the parameters from $p( \thetav\,|\,\ystar)$. Posterior predictive distributions are used in statistics for [*model checking*]{} and [*model improvement*]{} [@gelman], for example, and use the generative model with parameters from the posterior to generate data, then statistics—defined by the statistician and considered important for the problem at hand—from the pseudo, or replicated data, are compared with the statistics from the observations (the real data). One can then examine the bias and variance of the posterior predictive distributions with respect to the observations $\ystar$, or perform Bayesian t-tests (how probable are the observations $\ystar$ under $p(\y | \ystar)$) (see [@gelman], Chapter XXX).
For ABC, the posterior analysis comes naturally, and usually, for free. Using ABC-MCMC algorithms, statistics (judged important a priori by the scientist) are generated at each Metropolis-Hastings step. Simply storing the pairs $\{\y, \thetav\}$ from the MH step is sufficient to produce both $p(\y\,|\,\ystar)$ and $p( \thetav\,|\,\ystar)$. In addition to the posterior predictive, visualizing the input-output posteriors, i.e. a joint $p( \y_j, \thetav_d\,|\,\ystar)$ from the combined posterior predictive and posterior distribution, can lead to additional insight.
CASE 1: STEM-CELL NICHE GEOMETRY IN C. ELEGANS {#case-1-stem-cell-niche-geometry-in-c.-elegans .unnumbered}
==============================================
Minimizing the time it takes to develop an organ or to return to a desired steady state after perturbation is an important performance objective for biological systems [@Lander:2009fr; @Itzkovitz:2012fj]. Control of the cycling speed of stem cells and of the timing of their differentiation is critical to optimize the dynamics of development and regeneration. This control is often exerted in part by stem cell niches. While stem cell niches are known to employ a number of molecular signals to communicate with stem cells [@Li:2005gz], the impact of their geometry on stem cell behavior has received less attention. To begin to address this question, we ask here how niches should be shaped to minimize the amount of time to produce a given number of differentiated cells.
We consider a model organ inspired from the C. *elegans* germ line, which is similar to a number of other systems [@Cinquin:2009ep]. Cells reside within a tube-like structure; one end defined by the niche is closed, while the other is open and allows cells to exit. The set of possible positions that can be assumed by stem cells is constrained by the geometry of the niche; a dividing cell that is surrounded by neighbors pushes away one of its neighbors, which in turn might need to push away one of its own neighbors; cells pushed outside of the niche by one of these chain displacement reactions are forced to leave the cell cycle and differentiate. A simulator we developed tracks cell division and movement, and outputs the time it takes to produce N cells for a given geometry. This geometry is such that rows are defined along the main axis of the organ; each cell row has its own size, comprised between 1 and 400 cells. There are several constraints that are put on the niche geometry to help the model remain realistic: the niche should hold fewer than 400 cells total, row size should monotonically increase along the niche axis, and the geometry should be “well-behaved" (i.e., there should not be large jumps in row size along the axis).
Experimental set-up {#experimental-set-up .unnumbered}
-------------------
We performed several sets of experiments aimed at discovering the effects that realistic niche geometry constraints have on the time to 300 cells. We therefore define a single statistic $\y_1$ to be the time to $N=300$ cells for a niche of $D$ rows; a niche geometry vector $\thetav$ defines the simulator input parameters. In this study we set the number of rows in the niche to $D=8$. To enforce the monotonicity constraints, we define $\thetav_1 = 1+ {{\bf g}}_1$ and $\thetav_d = \thetav_{d-1}+{{\bf g}}_d$, $\forall d > 1$, i.e. we define niche geometries in terms of niche increment parameters ${{\bf g}}_d \geq 0$. With this set-up, we can change the prior constraints and observe the effects on the posterior predictive distribution $p(\y_1 | \ystar_1)$.
There are three sets of constraints on $\thetav$ (and/or ${{\bf g}}$), each with their own kernel epsilon parameter; the constraint ${{\bf g}}_d \geq 0$ is strictly enforced. For all experiments, the first cell row was given a flexible range $\thetav_1 \in \{1,400\}$, thus the first constraint is $K_{\eps_{g_1}}\lp {{\bf g}}_1;\; \tau_{g_1} \rp$, where $\eps_{g_1}=0.1$ and $\tau_{g_1} = 399$. The second set of constraints is on the niche geometry increments $K_{\eps_{g_d}}\lp {{\bf g}}_d;\; \tau_{g_d} \rp $, where $\eps_{g_d} = 0.1$ and $\tau_{g_d}$ is set to $10$ (to capture well-behaved niche increments) or $399$ (essentially removing the constraint on niche increments); see experiment details below. The final constraint on $\thetav$ is on the total niche geometry size $K_{\eps_{\theta}}\lp \sum_{d=1}^D \thetav_d;\; \tau_{\theta} \rp$, where $\eps_{\theta} = 1$ and $\tau_{\theta}$ is set to $400$ or $1500$. For all experiments, a one-sided Gaussian kernel was used. The prior over ${{\bf g}}$ is therefore: $$\begin{aligned}
\pi\lp{{\bf g}}\rp &\propto K_{\eps_{\theta}}\lp \sum_{d=1}^D \thetav_d;\; \tau_{\theta} \rp K_{\eps_{g_1}}\lp {{\bf g}}_1;\; \tau_{g_1} \rp \prod_{d=2}^D K_{\eps_{g_d}}\lp {{\bf g}}_d;\; \tau_{g_d} \rp \end{aligned}$$
The likelihood is a one-sided kernel $\pi\lp \ystar\,|\,\y_1 \rp \propto K_{\eps_y}\lp \y_1;\; \ystar_1 \rp$, where $\eps_y = 0.01$ (except for experiment D, below) and $\ystar_1 = 27.05$. For this problem we did not know $\ystar_1$ a priori, so we ran 5 runs of marginal kernel ABC with $S=1$ and adapted $\ystar$ (see algorithm 2 in appendix). We choose $\ystar_1 = 27.05$, the median value from 5 runs (which produced values $26.99$, $27.03$, $27.05$, $27.07$, $27.28$). Table \[tab:niche\_parameters\] summarizes the parameters and results from these experiments. For all experiments, $5$ marginal ABC-MCMC runs of length $10000$ were run and the first $2000$ samples were discarded as burnin.
Experiment M $\ystar$ $\tau_{g_1}$ $\tau_{g_d}$ $\tau_{\theta}$ $\mean{\y_1}$ $\median{\y_1}$ $\mode{\y_1}$ $P( \y_1 < 27.05)$
------------ ---- ---------- -------------- -------------- ----------------- --------------- ----------------- --------------- --------------------
1 27.05 $399$ $399$ $400$ 27.042 27.037 27.029 0.53
1 27.05 $399$ $10$ $400$ 27.059 27.054 27.076 0.49
1 $\infty$ $399$ $399$ $400$ 27.078 27.081 27.076 0.43
1 $\infty$ $399$ $10$ $400$ 27.298 27.150 27.114 0.32
1 $\infty$ $399$ $399$ $1500$ 30.159 30.184 30.224 0.00
1 27.05 $399$ $399$ $1500$ 27.322 27.227 27.150 0.24
10 27.05 $399$ $399$ $400$ 27.053 27.049 27.043 0.51
10 27.05 $399$ $10$ $400$ 27.056 27.053 27.050 0.47
: []{data-label="tab:niche_parameters"}
Experiment A: realistic constraints on ${{\bf g}}_d$ {#experiment-a-realistic-constraints-on-bf-g_d .unnumbered}
----------------------------------------------------
The first set of experiments compared posterior inference using a $\tau_{g_d}=399$ and $\tau_{g_d}=10$. Figure \[fig:nicheA\] shows the posterior geometries with $\tau_{g_d}=399$ (top row) and with a realistic constraint $\tau_{g_d}=10$ (bottom). Without the realistic constraint, the sizes start smaller (averaging around 5), increase slowly until row 6, then jump to a larger size (over 100) at row 8. With the realistic constraint, the sizes start larger (averaging around 20), and increase steadily until row 8, with no jumps, to an average of about 50. The posterior predictive distributions are very similar for both results, with the probability of $\y_1 < 27.05$ without the constraint being $0.53$ compared to $0.49$ with the constraint, indicating that the constraints do remove some regions of the parameter space with shorter time to 300 cells. The medians and modes of $\y_1 | \ystar_1$ also support this (without: $27.037$/$27.029$, with: $27.054$/$27.076$).
![. Comparison of niche geometry posteriors with $\tau_{g_d} = 400$ (top row) and $\tau_{g_d} = 10$ (bottom row). The left column illustrates the posterior geometries $\thetav$ by plotting circles of radius proportional to their posterior fraction of that size for that row (rounded to integers). The right column is the posterior predictive distribution $p(\y_1 | \ystar_1)$, with shading indicating the probability mass $P(\y_1 < 27.05\,|\,\ystar_1)$. []{data-label="fig:nicheA"}](figure1.pdf){width="0.8\columnwidth"}
Experiment B: removing constraint on time to 300 cells {#experiment-b-removing-constraint-on-time-to-300-cells .unnumbered}
------------------------------------------------------
We next removed the effect of the likelihood term on the posterior by setting $\ystar_1 = \infty$ (which is equivalent to sampling from the prior, with soft boundaries, using MCMC). Results for this experiment are shown in Figure \[fig:nicheB\]. Surprisingly, the posteriors of $\thetav$ have the same form as in experiment A, though with some decreases in $P(\y_1 < 27.05\,|\,\ystar_1)$: from $0.53$ to $0.43$ (for $\tau_{g_d}=399$) and from $0.49$ to $0.32$ (for $\tau_{g_d}=10$). This result clearly shows that there is significant [*prior mass*]{} having $\y_1 < 27.05$.
![[]{data-label="fig:nicheB"}](figure2.pdf){width="0.8\columnwidth"}
Experiment C: increasing threshold on total niche cells {#experiment-c-increasing-threshold-on-total-niche-cells .unnumbered}
-------------------------------------------------------
In this experiment we compare an increase in $\tau_{\theta}$ in an attempt to determine the most important factor for minimizing the time to 300 cells, the likelihood constraint $\ystar_1$ or the constraint on the total size. Results are shown in Figure \[fig:nicheC\]. For both results, $\tau_{\theta} = 1500$, but in the top row, the likelihood constraint is removed (and kept in the bottom row). By increasing the total niche geometry permitted ($\tau_{\theta} = 1500$) and removing the constraint on $\y_1$ (top row), the posterior predictive distribution degrades severely, with no samples satisfying $\y_1 < 27.05$. However, when the constraint on $\y_1$ is reintroduced (bottom row), a $P(\y_1 < 27.05\,|\,\ystar_1)=0.24$ is significant, and the posteriors of $\thetav$ are very similar to experiment A with $\tau_{g_d}=399$.
![[]{data-label="fig:nicheC"}](figure3.pdf){width="0.8\columnwidth"}
The results of experiments A-C demonstrate the relative importance of the input and output constraints on the posterior probability of $\y_1\,|\,\ystar_1$. The most important constraints are $\sum \theta_d$ and $\y_1 < \ystar_1$. Both have similar effects on the posterior predictive distribution. The constraint $\tau_{g_d}$ has little effect on $P(\y_1 < 27.05\,|\,\ystar_1 )$, but does produce significantly different posterior geometries, mainly due to the prior constraints.
Experiment D: replacing statistics with average of replicates {#experiment-d-replacing-statistics-with-average-of-replicates .unnumbered}
-------------------------------------------------------------
One final experiment on niche geometries was performed, aimed at exploring the effect that reducing the simulator noise has on the posterior distributions. To do this, we repeat each simulation $M$ times, using the same parameter setting; i.e. $\y = \frac{1}{M} \sum_{m=1}^M \y^{(m)}$, where $\y^{(m)} \simsim \pi( \y | \thetav)$. The variance of the statistic therefore decreases with $M$. Although, as expected, the posterior predictive distribution contracts around $\y$, we found no significant changes to the posterior $p(\thetav | \ystar)$ when $M=1$ (see Figure \[fig:nicheD\]). This experiment gives evidence that the scientist should instead change the value $\eps$ to control the posterior predictive distribution rather than $M$, which has an $M$-fold increase in computation.\
![[]{data-label="fig:nicheD"}](figure4.pdf){width="0.8\columnwidth"}
![. Effect of $M$, the number of replicates used to compute the output statistic $\y_1$: $M=1$ (top) versus $M=10$ (bottom). The left column correspond to $\tau_{g_d}=399$ and the right column $\tau_{g_d}=10$. []{data-label="fig:nicheDb"}](figure5.pdf){width="0.8\columnwidth"}
Experiments A-D illustrate the usefulness of POPE for exploring the roles constraints play on the optimization posterior. We found that the constraints on the prior over valid regions of $\thetav$ had significant influence on the posterior, and played a similar role to the likelihood term. Using realistic constraints on changes in row sizes had very little detrimental effect on the time to 300 cells, compared to having no realistic constraint. More important was the constraint on total geometry size. We found very little difference in the posteriors when the statistics were averages of simulation replicates versus a single simulation. This makes sense if the simulation noise is taken into account when setting $\epsilon$: when increasing the number of replicates in the average, $\epsilon$ should be decreased (from its setting at $M=1$) to take into account the population mean variance, but this seems unnecessary since the posteriors change little, but the number of simulations increases.
CASE 2: SPOTTED PATTERNS IN COLON CANCER TUMORS {#case-2-spotted-patterns-in-colon-cancer-tumors .unnumbered}
===============================================
A remarkable pattern of spots is visible in the tissue of colon cancer tumors when stained for markers indicating glycolytic activity. It is hypothesized that the spotted regions indicate localized areas of glycolytic cells, whereas surrounding areas are considered oxidative cells. Furthermore it is thought that Wnt signaling (an important cell signaling pathway in development and healing) plays a critical role in reducing glycolytic activity [@Pate2014], thereby resulting in significant changes in spot formation. Experiments blocking Wnt by overexpression of a dominant negative form of lymphoid enhance factor (dnLEF-1) have shown that interfering with the Wnt pathway leads to fewer but larger spots and lighter background staining color than *Mock* tissue (tumors that have not received dnLEF-1 intervention). Based on these findings, a simulator of a mathematical model of reaction-diffusion equations was built that produces spatial and temporal dynamics of a population fraction of oxidative cells and glycolytic cells, as well as the activity of Wnt and a Wnt inhibitor. The Wnt and Wnt inhibitor equations are based on the Gierer-Meinhardt activator-inhibitor model, where Wnt is the activator which produces a factor that inhibits Wnt activity. The goal of these experiments is to provide feedback to the mathematical biologists regarding the characteristics of simulation parameters that produce [*simulated patterns different from Mock patterns*]{}. For this reason, this problem does not have a predefined cost function, but instead uses the observed Mock values as constraints. The simulation produces 1D spatial and temporal patterns (see Figure \[fig:spots\_at\_fifty\] for 2D examples) from which $J=4$ statistics are computed: $\y_1$ the average spot width (based on wave patterns in 1D images); $\y_2$ the number of spots (waves, in 1D); $\y_3$ the average background level; and $\y_4$ the average Wnt level. There are $D=9$ simulator parameters including rates of production and decay for Wnt and Wnt inhibitor, and their diffusion coefficients. These are described in Table \[table:spottheta\]. The $\thetav$ settings in column [*Mock*]{} in Table \[table:spottheta\] generate patterns that were judged similar to the Mock spotting patterns in tissue photographs. Their corresponding statistics $\ystar = \{0.604, 5, 0.807, 5.67\}$ are shown in Table \[table:spotstats\], along with statistics from other $\thetav$ settings $A$ to $E$, described below.
The Mock values $\ystar$ define the constraints on simulator statistics $\y$. More precisely, they constrain the posterior to regions where $\lb \y_1 > \ystar_1\rb$, $\lb \y_2 < \ystar_2\rb$, $\lb \y_3 < \ystar_3 \rb$, and $\lb \y_4 < \ystar_4 \rb$, which correspond to the goal of producing different patterns from Mock. For example, the first constraint states that we want the spot widths from simulation to be greater than $\ystar_1 = 0.604$, the average width of spots for the Mock setting $\thetav$. Similarly, we want fewer than $5$ spots, a background lighter than $0.807$, and a Wnt level less than $5.67$. Further constraints are added to avoid degenerate simulation results; as an example, we set its likelihood to zero when there are no spots detected.
![[]{data-label="fig:spots_at_fifty"}](figure6.pdf){width="\columnwidth"}
Parameter $\thetav$ Description Mock A B C D E
--------------------- ------------------------------------- ----------- ----------- ------------ ---------- ----------- -----------
$\kappa_W > 0$ Rate of nonlinear Wnt production $4$ $0.442$ $0.951$ $2.44$ $0.399$ $0.315$
$\kappa_{W_I} > 0$ Rate of Wnt inhibitor production $1$ $27.4$ $0.484$ $0.161$ $0.486$ $0.188$
$\mu_W \geq 0$ Decay rate of Wnt $2$ $0.642$ $0.179$ $0.791$ $0.545$ $0.936$
$\mu_{W_I} \geq 0$ Decay rate of Wnt inhibitor $4$ $2.36$ $1.30$ $1.10$ $0.569$ $1.064$
$a \geq 0$ Constant of inhibition $10^{-8}$ $0.4006$ $0.416$ $0.0384$ $0.00491$ $0.0284$
$b \geq 0$ Constant of inhibition by $W_I$ $1$ $0.0125$ $7.94$ $20.05$ $0.616$ $0.640$
$S_W \geq 0$ Rate of constitutive Wnt production $1$ $0.00167$ $ 0.00351$ $17.75$ $0.00005$ $0.00009$
$1 \geq D_W > 0$ Diffusion coefficient of Wnt $0.01$ $0.0180$ $0.00322$ $0.0955$ $0.0336$ $0.0810$
$1 \geq N > 0$ Nutrient level $1$ $0.818$ $0.897$ $0.984$ $0.959$ $0.970$
: Simulation parameters $\thetav$ for spotted patterns in colon cancer tumors.[]{data-label="table:spottheta"}
Statistic $\y$ Feasible Region Mock ($\ystar$) A B C D E
----------------- ---------------------------- ----------------- -------- -------- -------- ------- --------
Avg. Spot Width $\y_1 > 0.604$ $0.604$ $1$ $0.65$ $0.65$ $1$ $1.75$
Number of Spots $\y_2 \in \lb 2, 3, 4 \rb$ $5$ $3$ $4$ $2$ $3$ $2$
Avg. Background $\y_3 < 0.807$ $0.807$ $0.77$ $0.75$ $0.70$ $0.6$ $0.70$
Avg. Wnt $\y_4 < 5.67 $ $5.67$ $3.25$ $1.50$ $0.75$ $1$ $2$
: Simulation statistics $\y$ for spotted patterns in colon cancer tumors.[]{data-label="table:spotstats"}
This simulator is deterministic but expensive to evaluate, requiring roughly 30 seconds to complete for the 1D simulator used in our experiments, and 90 seconds for the 2D simulator, used for generating 2D images only. We ran $6$ chains of length $4000$ pseudo-marginal kernel ABC-MCMC with S=1. To initialize the chains, a short rejection sampling procedure was used to select $\thetav_0$ for each random seed. This is necessary as many random configurations of $\thetav$ result in degenerate simulation results (i.e. zero likelihood). Diffuse log-normal prior distributions were placed over $\thetav_1$ to $\thetav_7$ and weak Beta priors put on $D_{W}$ and $N$. At least $100$ initial samples were discarded from each chain; sometimes more if the chain had not yet reached a location where all the constraints were satisfied. In total there were $22257$ samples in the posterior.
Analysis of the posterior predictive distribution revealed distinct distributions when conditioned on $\y_2$, the number of spots. The posterior distribution can therefore be viewed as a mixture of 3 spotting patterns, with $p(\y_2\,|\,\ystar ) = [ 0.505, 0.185, 0.310 ]$, where $\y_2 \in \{2,3,4\} $. The marginal posterior predictive distributions are shown in Figure \[fig:spots\_ppd\_pairwise\] for pairs of statistics, and in Figure \[fig:spots\_ppd\_marginal\] for marginal distributions. To illustrate the role of the spotting patterns, by visual inspection of the posterior predictive distributions displayed in Figure \[fig:spots\_ppd\_pairwise\], we selected statistics labeled $A$ through $E$. Parameters $\thetav$ corresponding to the modes $A$-$E$ were ran in both the 1D and 2D simulator producing images in Figure \[fig:spots\_at\_fifty\], showing the desired shift away from Mock patterns. Spot distributions were also found for $p(\thetav\,|\,\ystar)$, most distinctly for the Wnt and Wnt inhibitor decay rates ($\mu_W$ and $\mu_{W_I}$, respectively), which showed decreasing value for fewer spots, validating the original experimental results that blocking Wnt production by dnLEF-1 overexpression leads to qualitatively different spotting patterns. The marginal posteriors are shown in Figure \[fig:spots\_post\_marginal\], along with the prior, for reference. The strong relationship between $\mu_W$ and $\mu_{W_I}$ is shown in Figure \[fig:spots\_muw\_muwi\]. Subsamples from the posterior are overlaid with markers indicating the number of spots.
![[]{data-label="fig:spots_ppd_pairwise"}](figure7.pdf){width="\columnwidth"}
![[]{data-label="fig:spots_ppd_marginal"}](figure8.pdf){width="\columnwidth"}
![[]{data-label="fig:spots_post_marginal"}](figure9.pdf){width="\columnwidth"}
![[The posterior distribution of $\log \mu_{W}$ versus $\log \mu_{W_I}$. Overlaid are subsamples from the posterior with colored symbols indicating the number of spots its setting produced, showing the strong relationship between these parameters and the number of spots.]{}[]{data-label="fig:spots_muw_muwi"}](mix_contours_2d_mu_w_v_mu_WI_all_spots.pdf)
This case study illustrates the usefulness of POPE for exploratory simulation analysis. As a first attempt at studying this simulator from an ABC perspective, POPE revealed several regions of parameter settings that produce qualitatively different images from Mock. Now experts can examine these various solutions to further develop the simulator or to increase the number of statistics. For example, some of the parameter settings in the posterior seem to be similar to the prior, indicating they have little influence on the posterior. If this does not match the intuition of the experts, the role these parameters have the simulator can be re-evaluated. The $J=4$ statistics may also not be the most informative for the experts; based on our results learning the statistics (using computer vision techniques applied to the images) or modifying the current statistics may improve the ability of the experts to learn more about the spot formation process.
CONCLUSION {#conclusion .unnumbered}
==========
There is considerable excitement in the machine learning community about optimizing objectives that are hard to evaluate, such as those defined by simulators. However, there is almost no work on analyzing such problems “post optimization". We have found that this is exactly what scientists desire in order to study parameter dependencies and sensitivities and to compare different models in terms of their goodness of fit. We propose a post optimization posterior evaluation tool, POPE, by extending likelihood-free (ABC) MCMC samplers with one-sided kernels.
Two case studies conducted in close collaboration with biologists show the usefulness of this new modeling framework. For these studies we applied POPE in an optimization setting (stem-cell niche geometry) and a non-optimization setting (spotting patterns in cancer tissue), showing its usefulness for with [*general*]{} constraint-based likelihoods. These preliminary results offer many avenues for future work. Simulations with the stem-cell model could address whether giving cells some flexibility in the position at which they differentiate allows for more flexibility in the optimal geometry, perhaps allowing that geometry to also satisfy competing performance objectives. Ongoing research with a modified version of the tumor metabolism simulator will include non-constant nutrient levels and various therapeutic regimes, which will improve our understanding of cancer metabolism, and in turn aid the development of new treatments or therapies.
POPE {#sec:appendixpope}
====
$\thetav \gets \thetav_0$ $\thetapv \sim q(\thetapv | \thetav )$ $\ypone, \ldots, \ypS \overset{\simulator}{\sim} \pi( \y | \thetavp)$ $\yone, \ldots, \yS \overset{\simulator}{\sim} \pi( \y | \thetav)$ $\alpha \gets \lp 1, \frac{\pi(\thetavp)q(\thetap | \thetavp ) \pi(\thetavp | \ystar, \epsvec)}{ \pi(\thetav) q(\thetapv | \thetap )\pi(\thetav | \ystar, \epsvec)} \rp$ $\thetav \gets \thetavp$ $\yone, \ldots, \yS \gets \ypone, \ldots, \ypS$ $\ymu \gets {E}\lb \yone, \ldots, \yS \rb$ $\ystar, {\y^{\textsc{ema}}}\gets $ [UpdateObjectives]{}$\lp \ystar, \ymu, {\y^{\textsc{ema}}}, \gamma \rp$ $\epsvec, {\epsvec^{\textsc{ema}}}\gets $ [UpdateEpsilons]{}$\lp \ystar, \ymu, {\epsvec^{\textsc{ema}}}, {\epsvec^{\textsc{min}}}, \delta, {\beta}\rp $ Collect $\thetav, \ymu, \epsvec, \ystar$ Collections $\thetav, \ymu, \epsvec, \ystar$ ${\yj^{\textsc{ema}}}\gets (1-\gamma){\yj^{\textsc{ema}}}+ \gamma \yj$ $\ystarj \gets \min\lp \ystarj, {\yj^{\textsc{ema}}}\rp$ $\ystar, {\y^{\textsc{ema}}}$ $\Delta_j \gets (\ystarj - \yj)\textsc{Heavyside}\lp\ystarj - \yj\rp$ ${\epsj^{\textsc{ema}}}\gets (1-\delta){\yj^{\textsc{ema}}}+ \delta \Delta_j$ $\epsj \gets \max\lp {\epsj^{\textsc{min}}}, {\beta}{\epsj^{\textsc{ema}}}\rp$ $\epsvec, {\epsvec^{\textsc{min}}}$
Spots
=====
![[]{data-label="fig:spots_pp_mock_full"}](figure11.pdf){width="\columnwidth"}
![[]{data-label="fig:spots_pp_thetas_a_full"}](figure12.pdf){width="0.9\columnwidth"}
![[]{data-label="fig:spots_pp_thetas_b_full"}](figure13.pdf){width="0.9\columnwidth"}
![[]{data-label="fig:spots_pp_thetas_c_full"}](figure14.pdf){width="0.9\columnwidth"}
![[]{data-label="fig:spots_pp_thetas_d_full"}](figure15.pdf){width="0.9\columnwidth"}
![[]{data-label="fig:spots_pp_thetas_e_full"}](figure16.pdf){width="0.9\columnwidth"}
|
---
abstract: 'The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. New representations for the distance standard deviation are obtained in terms of Gini’s mean difference and in terms of the moments of spacings of order statistics. Inequalities for the distance variance are derived, proving that the distance standard deviation is bounded above by the classical standard deviation and by Gini’s mean difference. Further, it is shown that the distance standard deviation satisfies the axiomatic properties of a measure of spread. Explicit closed-form expressions for the distance variance are obtained for a broad class of parametric distributions. The asymptotic distribution of the sample distance variance is derived.'
author:
- '[Dominic Edelmann,]{}[^1] [Donald Richards,]{}[^2] [ and Daniel Vogel]{}[^3]'
title: ' **The Distance Standard Deviation** '
---
[[*Key words and phrases*]{}. characteristic function; distance correlation coefficient; distance variance; Gini’s mean difference; measure of spread; dispersive ordering; stochastic ordering; U-statistic; order statistic; sample spacing; asymptotic efficiency.]{}
[[*2010 Mathematics Subject Classification*]{}. Primary: 60E15, 62H20; Secondary: 60E05, 60E10.]{}
Introduction {#sec:intro}
============
In recent years, the topic of distance correlation has been prominent in statistical analyses of dependence between multivariate data sets. The concept of distance correlation was defined in the one-dimensional setting by Feuerverger [@feuerverger1993] and subsequently in the multivariate case by Székely, et al. [@szekely2007; @szekely2009], and those authors applied distance correlation methods to testing independence between random variables and vectors.
Since the appearance of [@szekely2007; @szekely2009], enormous interest in the theory and applications of distance correlation has arisen. We refer to the articles [@rizzo2010; @szekely2013; @szekely2014] on statistical inference; [@fiedler2016a; @fokianos2016; @jentsch2016; @zhou2012] on time series; [@Dueck2014; @Dueck2015; @Dueck2016] on affinely invariant distance correlation and connections with singular integrals; [@lyons2013] on metric spaces; and [@sejdinovic2013] on machine learning. Distance correlation methods have also been applied to assessing familial relationships [@kong2012], and to detecting associations in large astrophysical databases [@martinez2014; @richards2014].
For $z \in \C$, denote by $|z|$ the modulus of $z$. For any positive integer $p$ and $s,x \in \R^p$, we denote by $\langle s,x\rangle$ the standard Euclidean inner product on $\R^p$ and by $\|s\| = \langle s,s \rangle^{1/2}$ the standard Euclidean norm. Further, we define the constant $$c_p = \frac{\pi^{(p+1)/2}}{\Gamma\big((p+1)/2\big)}.$$
For jointly distributed random vectors $X \in \R^p$ and $Y \in \R^q$, let $$f_{X,Y}(s,t) = \E \exp\big(\sqrt{-1}(\langle s,X\rangle + \langle t,Y\rangle) \big),$$ $s \in \R^p$, $t \in \R^q$, be the joint characteristic function of $(X,Y)$ and let $f_X(s) = f_{X,Y}(s,0)$ and $f_Y(t) = f_{X,Y}(0,t)$ be the corresponding marginal characteristic functions. The [*distance covariance*]{} between $X$ and $Y$ is defined as the nonnegative square root of $$\label{eq:dcov}
\V^2(X,Y) = \frac{1}{c_p c_q} \int_{\R^{p+q}}
\big|f_{X,Y}(s,t)-f_X(s)f_Y(t)\big|^2 \, \frac{\dd s {\hskip 1pt}\dd t}{\|s\|^{p+1} \, \|t\|^{q+1}};$$ the [*distance variance*]{} is defined as $$\begin{aligned}
\label{eq:dvar}
\V^2(X) := \V^2(X,X) &= \frac{1}{c_p^2} \int_{\R^{2p}}
\big|f_{X}(s+t)-f_X(s)f_X(t)\big|^2 \, \frac{\dd s {\hskip 1pt} \dd t}{\|s\|^{p+1} \, \|t\|^{p+1}};
\end{aligned}$$ and we define the [*distance standard deviation*]{} $\V(X)$ as the nonnegative square root of $\V^2(X)$. The [*distance correlation coefficient*]{} is defined as $$\label{eq:dcor}
\mathcal{R}(X,Y) = \frac{\V(X,Y)}{\sqrt{\V(X) \V(Y)}}$$ as long as $\V(X), \V(Y) \neq 0$, and $\mathcal{R}(X,Y)$ is defined to be zero otherwise.
The distance correlation coefficient, unlike the Pearson correlation coefficient, characterizes independence: $\mathcal{R}(X,Y)= 0$ if and only if $X$ and $Y$ are mutually independent. Moreover, $0 \leq \mathcal{R}(X,Y) \leq 1$; and for one-dimensional random variables $X,Y \in \R$, $\mathcal{R}(X,Y)=1$ if and only if $Y$ is a linear function of $X$. The empirical distance correlation possesses a remarkably simple expression ([@feuerverger1993], [@szekely2007 Theorem 1]), and efficient algorithms for computing it are now available [@Huo2015].
The objective of this paper is to study the distance standard deviation $\V(X)$. Since distance standard deviation terms appear in the denominator of the distance correlation coefficient (\[eq:dcor\]) then properties of $\V(X)$ are crucial to understanding fully the nature of $\mathcal{R}(X,Y)$. Now that $\mathcal{R}(X,Y)$ has been shown to be superior in some instances to classical measures of correlation or dependence, there arises the issue of whether $\V(X)$ constitutes a measure of spread suitable for situations in which the classical standard deviation cannot be applied.
As $\V(X)$ is possibly a measure of spread, we should compare it to other such measures. Indeed, suppose that $\E (\|X\|^2) < \infty$, and let $X$, $X'$, and $X''$ be independent and identically distributed (i.i.d.); then, by [@szekely2007 Remark 3], $$\label{rep:dvar2}
\V^2(X) = \E (\|X-X'\|^2) + (\E \|X-X'\|)^2 - 2 \E (\|X-X'\| \cdot \|X-X''\|),$$ The second term on the right-hand side of (\[rep:dvar2\]) is reminiscent of the Gini mean difference [@gerstenberger2015; @yitzhaki2013], which is defined for real-valued random variables $Y$ as $$\label{GMD}
\Delta(Y) := \E |Y-Y'|,$$ where $Y$ and $Y'$ are i.i.d. Furthermore, if $X \in \R$ then one-half the first summand in (\[rep:dvar2\]) equals $\sigma^2(X)$, the variance of $X$: $$\frac12 \E (|X-X'|^2) = \frac12 \E (X^2 - 2XX' + X'^2) = E(X^2) - E(X)E(X') \equiv \sigma^2(X).
$$
Let $X$ and $Y$ be real-valued random variables with cumulative distribution functions $F$ and $G$, respectively. Further, let $F^{-1}$ and $G^{-1}$ be the right-continuous inverses of $F$ and $G$, respectively. Following [@Shaked1994 Definition 2.B.1], we say that $X$ is [*smaller than*]{} $Y$ [*in the dispersive ordering*]{}, denoted by $X \leq_{\hskip 1pt \rm{disp}} Y$, if for all $0 < \alpha \leq \beta < 1$, $$\label{eq:dispersive}
F^{-1}(\beta) - F^{-1}(\alpha) \leq G^{-1}(\beta) - G^{-1}(\alpha).$$ According to [[@Bickel2012]]{}, a [*measure of spread*]{} is a functional $\tau(X)$ satisfying the axioms:
1. $\tau(X) \geq 0$,
2. $\tau(a+bX) = |b| \, \tau (X)$ for all $a,b \in \R$, and
3. $\tau(X) \leq \tau(Y)$ if $X \leq_{\hskip 1pt \text{disp}} Y$.
The distance standard deviation $\V(X)$ obviously satisfies (C1). Moreover, Székely, et al. [@szekely2007 Theorem 4] prove that:
1. If $\V(X) = 0$ then $X= \E[X]$, amost surely,
2. $\V(a+bX) = |b| \, \V(X)$ for all $a,b \in \R$, and
3. $\V(X+Y) \leq \V(X) + \V(Y)$ if $X$ and $Y$ are independent.
In particular, $\V(X)$ satisfies the dilation property (C2). In Section \[sec:furtherproperties\], we will show that $\V(X)$ satisfies condition (C3), proving that $\V(X)$ is a measure of spread in the sense of [@Bickel2012]. However, we will also derive some stark differences between $\V(X)$, on the one hand, and the standard deviation and Gini’s mean difference, on the other hand.
The paper is organized as follows. In Section \[sec:ineq\], we derive inequalities between the summands in the distance variance representation (\[rep:dvar2\]). For real-valued random variables, we will prove that $\V(X)$ is bounded above by Gini’s mean difference and by the classical standard deviation. In Section \[sec:altrep\], we show that the representation (\[rep:dvar2\]) can be simplified further, revealing relationships between $\V(X)$ and the moments of spacings of order statistics. Section \[sec:distr\] provides closed-form expressions for the distance variance for numerous parametric distributions. In Section \[sec:furtherproperties\], we show that $\V(X)$ is a measure of spread in the sense of [@Bickel2012]; moreover, we point out some important differences between $\V(X)$, the standard deviation, and Gini’s mean difference. Section \[sec:estimator\] studies the properties of the sample distance variance.
Inequalities between the distance variance, the variance, and Gini’s mean difference {#sec:ineq}
====================================================================================
The integral representation in equation (\[eq:dvar\]) of the distance variance $\V^2(X)$ generally is not suitable for practical purposes. Székely, et al. [@szekely2007; @szekely2009] derived an alternative representation; they show that if the random vector $X \in \R^p$ satisfies $\E \|X\|^2 < \infty$ and if $X$, $X'$, and $X''$ are i.i.d. then $$\label{eq:dvartwo}
\V^2(X) = T_1(X) +T_2(X) -2 \, T_3(X),$$ where $$\label{T1T2}
\begin{aligned}
T_1(X) &= \E (\|X-X'\|^2), \\
T_2(X) &= (\E\|X-X'\|)^2, \\
\end{aligned}$$ and $$\begin{aligned}
\label{T3}
T_3(X) &= \E \big(\|X-X'\| \cdot\|X-X''\|\big),
\end{aligned}$$ Corresponding to the representation (\[eq:dvartwo\]), a sample version of $\V^2(X)$ then is given by $$\label{eq:sampledvar}
\V_n^2(\bX) = T_{1,n}(\bX) + T_{2,n}(\bX) -2 \, T_{3,n}(\bX),$$ where $$\label{T1T2sample}
\begin{aligned}
T_{1,n}(\bX)&=\frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \|X_i-X_j\|^2, \\
T_{2,n}(\bX) &= \Big(\frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \|X_i-X_j\| \Big)^2, \\
\end{aligned}$$ and $$\begin{aligned}
\label{T3sample}
T_{3,n}(\bX) &= \frac{1}{n^3} \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^n \|X_i-X_j\| \cdot \|X_i-X_k\|.
\end{aligned}$$ We remark that the version (\[eq:sampledvar\]) is biased; indeed, throughout the paper, we work with biased sample versions to avoid dealing with numerous complicated, but unessential, constants in the ensuing results. In any case, an unbiased sample version can be defined in a similar fashion; see, e.g., [@szekely2013]).
In the following we will study inequalities between the summands showing up in equations (\[eq:dvartwo\]) and (\[eq:sampledvar\]). In the one-dimensional case, these inequalities will lead to crucial results concerning the relationships between the distance standard deviation, Gini’s mean difference and the standard deviation.
[\[lem:Tineq\]]{} Let $X=(X^{(1)},\ldots,X^{(p)})^t \in \R^p$ be a random vector. Moreover let $\bX=(X_1,\ldots,X_n)$ denote a random sample from $X$ and let $T_1(X)$, $T_2(X)$, $T_3(X)$, and $T_{1,n}(\bX)$, $T_{2,n}(\bX)$, $T_{3,n}(\bX)$ be defined as in equations (\[eq:dvartwo\])-(\[T3sample\]). Then $$\label{Tineqs1and2}
T_{2,n}(\bX) \leq T_{3,n}(\bX) \leq T_{1,n}(\bX), \quad \quad \quad T_{1,n}(\bX) \leq 2 T_{3,n}(\bX).$$ Further, if $\E\|X\|^2 < \infty$ then $$\label{Tineqs3and4}
T_2(X) \leq T_3(X) \leq T_1(X), \quad \quad \quad T_1(X) \leq 2 T_3(X).$$
<span style="font-variant:small-caps;">Proof.</span>
First note that $$\begin{aligned}
T_{3,n}(\bX) &= \frac{1}{n^3} \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^n \|X_i-X_j\| \cdot \|X_i-X_k\| \\ &= \frac{1}{n^3} \sum_{i=1}^n \Big( \sum_{j=1}^n \|X_i-X_j\| \Big) ^2.
\end{aligned}$$ By the Cauchy-Schwarz inequality, $(\sum_{i=1}^n a_i)^2 \leq n \sum_{i=1}^n a_i^2$ for all $a_1,\ldots,a_n \allowbreak \in \R$; applying this inequality to the sums which define $T_{1,n}$, $T_{2,n}$ and $T_{3,n}$, we obtain $$\begin{aligned}
T_{2,n}(\bX)&= \frac{1}{n^4} \Big( \sum_{i=1}^n \sum_{j=1}^n \|X_i-X_j\| \Big)^2\\ &\leq \frac{n}{n^4} \sum_{i=1}^n \Big(\sum_{j=1}^n \|X_i-X_j\| \Big)^2 = T_{3,n}(\bX)
\end{aligned}$$ and $$\begin{aligned}
T_{3,n}(\bX)&=\frac{1}{n^3} \sum_{i=1}^n \Big( \sum_{j=1}^n \|X_i-X_j\| \Big) ^2\\ &\leq \frac{n}{n^3} \sum_{i=1}^n \sum_{j=1}^n \|X_i-X_j\|^2 = T_{1,n}(\bX).
\end{aligned}$$
The second assertion in (\[Tineqs1and2\]) follows by the triangle inequality: $$\begin{aligned}
T_{1,n}(\bX) &= \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \|X_i-X_j\|^2 \\
&= \frac{1}{n^3} \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^n \|X_i - X_j\| \cdot \|X_i-X_k+X_k-X_j\| \\
& \leq \frac{1}{n^3} \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^n \|X_i - X_j\| \, \Big(\|X_i-X_k\| + \|X_k-X_j\| \Big) \\
&= 2 \, T_{3,n}(\bX).
\end{aligned}$$
The corresponding inequalities (\[Tineqs3and4\]) for the population measures follow from the strong consistency of the respective sample measures. Alternatively they can be derived by applying Jensen’s inequality and the triangle inequality, respectively. $\qed$
Using the inequalities in Lemma \[lem:Tineq\], we can derive upper bounds for the distance variance in terms of the variance of the components $X^{(1)},\ldots,X^{(p)}$ and the Gini mean difference of the vector $X$.
[\[th:dimpineq\]]{} Let $X = (X^{(1)},\ldots,X^{(p)})^t \in \R^p$ be a random vector with $\E\|X\|< \infty$, and let $X' = (X^{'(1)},\ldots,X^{'(p)})^t$ denote an i.i.d. copy of $X$. Then $$\V^2(X) \leq \sum_{i=1}^p \sigma^2(X^{(i)}),$$ and $$\V^2(X) \leq (\E\|X-X'\|)^2.$$
<span style="font-variant:small-caps;">Proof.</span>
To prove the first assertion, we note that $$\begin{aligned}
\V^2(X) & = \lim_{n \to \infty} \big(T_{1,n}(\bX)+T_{2,n}(\bX)-2 T_{3,n}(\bX) \big) \\
& \leq \lim_{n \to \infty} T_{2,n}(\bX) \\
& = (\E \|X-X'\|)^2,
\end{aligned}$$ where the inequality follows by Lemma $\ref{lem:Tineq}$.
To extablish the second inequality we can assume, without loss of generality, that $\E\|X\|^2 < \infty$. Then $$\begin{aligned}
T_1(X)&= \E \|X-X'\|^2 \\
&= \E \sum_{i=1}^p (X^{(i)}-X'^{(i)})^2 \\
&= \sum_{i=1}^p \E \Big[(X^{(i)}-\E X^{(i)}) + (\E X^{(i)}-X'^{(i)}) \Big]^2 \\
&= 2 \, \sum_{i=1}^p \sigma^2(X^{(i)}).
\end{aligned}$$ Applying Lemma \[lem:Tineq\] yields $$\begin{aligned}
\V^2(X) &= T_1(X)+T_2(X)-2 T_3(X) \\
&\leq T_1(X)-T_3(X) \\
&\leq \tfrac12 \, T_1(X) \\
&= \sum_{i=1}^p \sigma^2(X^{(i)}).
\end{aligned}$$ The proof now is complete. $\qed$
In the one-dimensional case, Theorem \[th:dimpineq\] implies that the distance variance is bounded above by the variance and the squared Gini mean difference.
\[cor:vargmdineq\] Let $X$ be a real-valued random variable with $\E\|X\| < \infty$. Then, $$\V^2(X) \leq \sigma^2(X), \quad \quad \quad \V^2(X) \leq \Delta^2(X).$$
Let us note further that for $X \in \R$, the inequality $T_2(X) \leq T_1(X)$ can be sharpened.
\[prop:T2T1ineq\] Let $X$ be a real-valued random variable with $\E(|X|^2) < \infty$. Then, $$T_2(X) \leq \tfrac{2}{3} \, T_1(X).$$
<span style="font-variant:small-caps;">Proof.</span>
By [@yitzhaki2013 p. 25], $$\label{prop:T2T1ineq1}
1 \geq [\Cor(X,F(X))]^2 = \frac{\Cov^2(X,F(X))}{\sigma^2(X) \,\sigma^2(F(X))}.$$ By [@yitzhaki2003 equation (2.3)], $\Cov(X,F(X))=\Delta(X)/4$; also, since $F(X)$ is uniformly distributed on the interval $[0,1]$ then $\Var(F(X)) = 1/12$. By the definition of the Gini mean difference (\[GMD\]) and by (\[T1T2\]), $\Delta^2(X) =T_2(X)$ and $\sigma^2(X) = T_1(X)/2$. Therefore, it follows from (\[prop:T2T1ineq1\]) that $$1 \ge \frac{12}{16} \, \frac{\Delta^2(X)}{{\sigma^2(X)}} = \frac{3 \,T_2(X)}{2 \, T_1(X)},$$ and the proof now is complete. $\qed$
Interestingly, Gini’s mean difference and the distance standard deviation coincide for distributions whose mass is concentrated on two points.
\[th:bernoulli\] Let $X$ be Bernoulli distributed with parameter $p$. Then $$\V^2(X) = \Delta^2(X) = 4 p^2 (1-p)^2.$$ Conversely, if $X$ is a non-trivial random variable for which $\V^2(X) = \Delta^2(X)$ then the distribution of $X$ is concentrated on two points.
<span style="font-variant:small-caps;">Proof.</span>
It is straightforward from (\[eq:dvartwo\]) to verify that, for a Bernoulli distributed random variable $X$, $\Delta(X) = 2 \, \sigma^2(X) = 2 \, T_3(X) = 2 \, p (1-p)$. Hence, by (\[eq:dvartwo\]), $$\V^2(X) = 2 \, \sigma^2(X)+\Delta^2(X) - 2 \, T_3(X) = 4 \, p^2 (1-p)^2.$$
Conversely, if $X$ is a non-trivial random variable for which $\V^2(X) = \Delta^2(X)$ then the conclusion that the distribution of $X$ is concentrated on two points follows from Theorem \[th:repdiff\]. $\qed$
For the Bernoulli distribution with $p=\tfrac{1}{2}$, Theorem \[th:bernoulli\] implies immediately that $\V^2(X)$, $\sigma^2(X)$, and $\Delta^2(X)$ attain the same value, namely, $1/4$. Hence, applying Corollary \[cor:vargmdineq\] and the dilation property $\V(a X) = |a| \V(X)$ in (C2), we obtain
[\[cor:dvarmax\]]{} Let $\mathcal{X}$ denote the set of all real-valued random variables and let $c>0$. Then $$\max_{X \in \mathcal{X}} \{\V^2(X): \sigma^2(X)=c\} = \max_{X \in \mathcal{X}}\{\V^2(X): \Delta^2(X) = c\} = c,$$ and both maxima are attained by $Z = 2 \, c^{1/2} \, Y$, where $Y$ is Bernoulli distributed with parameter $p = \tfrac12$.
This result answers a question raised by Gabor Székely (private communication, November 23, 2015).
We remark, that the second implication of Theorem \[th:dimpineq\] as well as Theorem \[th:bernoulli\] also follow directly from the result for the generalized distance variance in [@lyons2013 Proposition 2.3]. However, the proof presented here provides a different and more elementary approach to these findings.
New representations for the distance variance
=============================================
[\[sec:altrep\]]{}
The representation of $\V$ given in (\[eq:dvartwo\]), although more applicable than the expression given in equation (\[eq:dvar\]), still has the drawback that it is undefined for random vectors with infinite second moments. This problem can be circumvented by considering the representation $$\label{eq:dvarthree}
\V^2(X) = \Delta^2(X) + W(X),$$ where $$W(X) = \E \Big[\|X-X'\| \cdot \big(\|X-X'\| - 2 \, \|X-X''\| \big) \Big].$$ In the one-dimensional case, the representation (\[eq:dvarthree\]) can be further simplified using the concept of order statistics.
\[th:repdiff\] Let $X$ be a real-valued random variable with $\E|X| < \infty$, and let $X$, $X'$, and $X''$ be i.i.d. copies of $X$. If $X_{1:3} \leq X_{2:3} \leq X_{3:3}$ are the order statistics of the triple $(X,X',X'')$ then $$\begin{aligned}
\V^2(X) &= \Delta^2(X)- \tfrac43 \, \E[(X_{2:3}-X_{1:3})\,(X_{3:3}-X_{2:3})] \label{eq:dvarfour}\\
&= \Delta^2(X)- 8 \, \E[(X-X')_{+}\,(X''-X)_+], \label{eq:dvarfour2}
\end{aligned}$$ where $t_+ = \max(t,0)$, $t \in \R$.
<span style="font-variant:small-caps;">Proof.</span>
We first prove the theorem for the case in which $X$ is continuous. In this case, we apply the Law of Total Expectation and use the independence of the ranks and the order statistics [@Vandervaart2000 Lemma 13.1] to obtain $$\begin{aligned}
& W(X)\\ &=\E \Big[|X-X'| \, \big(|X-X'| - 2 \, |X-X''| \big) \Big] \\
&= \sum_{\substack{k,k',k''=1 \\ k,k',k'' \text{are pair-} \\ \text{wise distinct}}}^3 \hspace{-12pt} \E \Big[|X-X'| \big(|X-X'| - 2 |X-X''| \big) \Big| (r_X,r_{X'},r_{X''})=(k,k',k'') \Big] \\
& \quad \quad \quad \quad \quad \quad \quad \times \, \P \big((r_X,r_{X'},r_{X''})=(k,k',k'') \big)
. \displaybreak \\
\end{aligned}$$ Using the symmetry of $X$, $X'$, and $X''$, it follows that $$\begin{aligned}
W(X)&= \frac16 \sum_{\substack{k,k',k''=1 \\ k,k',k'' \text{are pair-} \\ \text{wise distinct}}}^3 \hspace{-12pt} \E \Big[|X_{k:3}-X_{k':3}| \, \big(|X_{k:3}-X_{k':3}| - 2 \, |X_{k:3}-X_{k'':3}| \big) \Big] \\
&= \frac16 \sum_{\substack{k,k',k''=1 \\ k,k',k'' \text{are pair-} \\ \text{wise distinct}}}^3 \hspace{-12pt} \E \Big[|X_{k:3}-X_{k':3}|^2 \Big] - \E \Big[|X_{k:3}-X_{k':3}| \cdot |X_{k:3}-X_{k':3}| \Big].
\end{aligned}$$ Evaluating the first summand in the latter equation yields $$\begin{aligned}
\frac16 &\sum_{\substack{k,k',k''=1 \\ k,k',k'' \text{are pair-} \\ \text{wise distinct}}}^3 \hspace{-12pt} \E \big[|X_{k:3}-X_{k':3}|^2 \big] \\
&= \frac13 \, \Big( \E \big[(X_{1:3}-X_{2:3})^2 \big] + \E \big[(X_{1:3}-X_{3:3})^2 \big] + \E \big[(X_{2:3}-X_{3:3})^2 \big] \Big).
\end{aligned}$$ Proceeding analogously with the second summand and simplifying the outcome, we obtain $$W(X) = - \frac43 \, \E\big[(X_{2:3}-X_{1:3})\,(X_{3:3}-X_{2:3})\big].$$ This proves (\[eq:dvarfour\]) in the continuous case.
For the case of general random variables, we now apply the method of quantile transformations. Let $U$ be uniformly distributed on the interval $[0,1]$ and let $U$, $U'$, and $U''$ be i.i.d.. Further, let $F$ denote the cumulative distribution function of $X$. With $F^{-1}(p) = \inf\{x: F(x) \geq p\}$ denoting the right-continuous inverse of $F$, we define $\tilde{X} = F^{-1}(\tilde{U})$, $\tilde{X'} = F^{-1}(\tilde{U'})$, and $\tilde{X''} = F^{-1} (\tilde{U''})$. By [@Vandervaart2000 Theorem 21.1], the random variables $\tilde{X}$, $\tilde{X'}$, and $\tilde{X''}$ are i.i.d. copies of $X$ and [ $$\begin{aligned}
&W(X)\\ &=\E \Big[|\tilde{X}-\tilde{X'}| \cdot \big(|\tilde{X}-\tilde{X'}| - 2 \, |\tilde{X}-\tilde{X''}| \big) \Big] \\
&= \sum_{\substack{k,k',k''=1 \\ k,k',k'' \text{are pair-} \\ \text{wise distinct}}}^3 \hspace{-12pt} \E \Big[|\tilde{X}-\tilde{X'}| \cdot \big(|\tilde{X}-\tilde{X'}| - 2 \, |\tilde{X}-\tilde{X''}| \big) \Big| (r_U,r_{U'},r_{U''})=(k,k',k'') \Big] \\
& \quad \quad \quad \quad \quad \quad \quad \times \, \P \big((r_U,r_{U'},r_{U''})=(k,k',k'') \big) \\ \displaybreak \\
&= \frac16 \sum_{\substack{k,k',k''=1 \\ k,k',k'' \text{are pair-} \\ \text{wise distinct}}}^3 \hspace{-12pt} \E \Big[|X_{k:3}-X_{k':3}| \cdot \big(|X_{k:3}-X_{k':3}| - 2 \, |X_{k:3}-X_{k'':3}| \big) \Big] \\
&= - \frac43 \, \E[(X_{2:3}-X_{1:3})\,(X_{3:3}-X_{2:3})].
\end{aligned}$$]{} The second representation for $W(X)$, (\[eq:dvarfour2\]), now follows by a combinatorial symmetry argument from the first representation. $\qed$
In the continuous case with finite second moment, equation (\[eq:dvarfour2\]) is equivalent to $${\label{eq:gmdvar}}
\E ( |X-X'| \cdot |X''-X'| ) = \sigma^2(X) + 4 J(X),$$ where $$J(X) = \int_{x= -\infty}^\infty \int_{y =- \infty}^{x} \int_{z=x}^\infty (x-y) \, (z-x) f(z) \, f(y) f(x) \dd z \dd y \dd x .$$ Formula (\[eq:gmdvar\]) is essentially the key result in the classical paper by Lomnicki [@lomnicki1952], who also gave a simple expression for the variance of the empirical Gini mean difference, $$\label{eq:DeltahatnX}
\widehat{\Delta}_n(\bX) = \frac{2}{n\,(n-1)} \sum_{1\leq i < j \leq n} |X_i - X_j|.$$ Indeed, it is shown in [@lomnicki1952] that $${\label{eq:gmdvartwo}}
\Var\big(\widehat{\Delta}_n(\bX) \big) = \frac{1}{n \, (n-1)} \big(4 \,(n-1) \, \sigma^2(X)+ 16 \, (n-2) J(X)- 2 \, (2 n-3) \Delta^2(X) \big).$$
We note two consequences of Theorem \[th:repdiff\] and equation (\[eq:gmdvartwo\]). First, Theorem \[th:repdiff\] implies that the decomposition (\[eq:gmdvartwo\]) holds in an analogous way for the non-continuous case. Second, for distributions with finite second moment, calculating the distance variance yields the variance of $\widehat{\Delta}_n$ and [*vice versa*]{}. These considerations imply that the asymptotic variance $ASV(\widehat{\Delta}(\bX)) = \lim_{n \to \infty} n \, \Var(\big(\widehat{\Delta}(\bX) \big) )$ can be expressed alternatively as $${\label{eq:ASV}}
ASV(\widehat{\Delta}(\bX)) = 4 \, \sigma^2(X) - 2 \, \V^2 (X) - 2 \, \Delta^2(X).$$
For a random sample $X_1,\ldots,X_n$ of real-valued random variables, the difference between successive order statistics, $D_{i:n} := X_{i+1:n} - X_{i:n}$, $i=1,\ldots,n-1$, is called the $i$th [*spacing of*]{} $\bX = (X_1,\ldots,X_n)$. Jones and Balakrishnan [@Jones2002] (see also [@yitzhaki2003; @yitzhaki2013]) studied closed-form expression for the moments of spacings and showed that $$\label{eq:varrep}
\sigma^2(X) = \E(D_{1:2}^2) = 2 \operatornamewithlimits\iint\limits_{-\infty<x<y<\infty} F(x) (1-F(y)) \dd x \, \dd y$$ and $$\label{eq:gmdrep}
\Delta(X) = \E(D_{1:2}) = 2 \int_{-\infty}^{\infty} F(x) (1-F(x)) \dd x.$$ By applying results in [@Jones2002], we obtain an analogous representation for the distance variance.
\[th:repre.spacing\] Let $X$ be a real-valued variable with $\E(|X|) < \infty$ and let $X$, $X'$, $X''$, and $X'''$ be i.i.d. Then, $$\begin{aligned}
\V^2(X)&= 8 \operatornamewithlimits\iint\limits_{-\infty<x<y<\infty} F^2(x) (1-F(y))^2 \dd x \, \dd y \label{eq:dvarrepcum}\\
&= \frac{2}{3} \E [(X_{3:4}-X_{2:4})^2]\label{eq:dvarrepcum2},
\end{aligned}$$ where $X_{1:4} \leq X_{2:4} \leq X_{3:4} \leq X_{4:4}$ denote the order statistics of $(X,X',X'',X''')$.
<span style="font-variant:small-caps;">Proof.</span>
By equation (\[eq:gmdrep\]), we obtain $$\begin{aligned}
\Delta^2(X) &= \Big[ 2 \int_{- \infty}^{\infty} F(x) \, (1-F(x)) \dd x \Big]^2 \\
&= 4 \,\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} F(x) \, [1-F(x)] \, F(y) \, [1-F(y)] \, \dd x \, \dd y \\
&= 8 \, \operatornamewithlimits\iint\limits_{-\infty<x<y<\infty} F(x) \, [1-F(x)] \, F(y) \, [1-F(y)] \, \dd x \, \dd y.
\end{aligned}$$ Moreover, by [@Jones2002 equation (3.5)] $$\begin{aligned}
& \E[(X_{2:3}-X_{1:3})\,(X_{3:3}-X_{2:3})] \\ & \quad \quad \quad \quad \quad \quad = 8 \, \operatornamewithlimits\iint\limits_{-\infty<x<y<\infty} F(x) \, [F(y)-F(x)] \, [1-F(y)] \, \dd x \, \dd y. \\
\end{aligned}$$ Hence, $$\begin{aligned}
\V^2(X) &= \Delta^2(X)- \tfrac43 \, \E[(X_{(2)}-X_{(1)})\,(X_{(3)}-X_{(2)})] \\
&= 8 \, \operatornamewithlimits\iint\limits_{-\infty<x<y<\infty} [F(x)]^2 \, [1-F(y)]^2 \, \dd x \, \dd y,
\end{aligned}$$ which proves (\[eq:dvarrepcum\]).
Finally, the formula (\[eq:dvarrepcum2\]) follows from (\[eq:dvarrepcum\]) and from [@Jones2002 equation (3.4)]. $\qed$
Theorem \[th:repre.spacing\] now yields for the distance variance a new sample version which is distinct from $\V_n^2(\bX)$, as follows.
\[cor:newsampledistvar\] Let $X$ be a real-valued variable with $\E(|X|) < \infty$ and let $\bX=(X_1,\ldots,X_n)$ be a random sample from $X$. Then, a strongly consistent sample version for $\V^2(X)$ is $$\label{eq:newsampledvar}
\U_n^2(\bX) = {n \choose 2}^{-2} \sum_{i,j=1}^{n-1} \big(\min(i,j)\big)^2 \big(n-\max(i,j)\big)^2 D_{i:n} D_{j:n} ,$$ where $D_{k:n}= X_{{k+1}:n} - X_{k:n}$ denotes the $k$th sample spacing of $X$, $1 \le k \le n-1$.
<span style="font-variant:small-caps;">Proof.</span>
Let $h: \R^4 \mapsto \R$ be the symmetric kernel defined by $$h(X_1,\ldots,X_4) = \frac{2}{3} (X_{3:4}-X_{2:4})^2,$$ where $X_{1:4} \leq X_{2:4} \leq X_{3:4} \leq X_{4:4}$ are the order statistics of $X_1,\ldots,X_4$. By Theorem \[th:repre.spacing\], we have $\E [h(X_1,\ldots,X_4)] < \infty$. Hence, by Hoeffding [@Hoeffding1961], $$\widehat{\U}_n^2(\bX) = \frac{2}{3} {n \choose 4}^{-1} \sum_{1\leq i_1 < i_2 < i_3 < i_4 \leq n} h(X_{i_1},\ldots,X_{i_4})$$ is a strongly consistent estimator for $\V^2(X)$. Using a straightforward combinatorial calculation, we obtain $$\widehat{\U}_n^2(\bX) = \frac{2}{3}{n \choose 4}^{-1} \sum_{1 \leq i < j \leq n} (i-1) \, (n-j) (X_{j:n}-X_{i:n})^2.$$ On inserting the definition of the spacings, the latter equation reduces to $$\begin{aligned}
\widehat{\U}_n^2(\bX) &= \frac{2}{3} {n \choose 4}^{-1} \sum_{1 \leq i < j \leq n} (i-1) \, (n-j) \, (D_{i:n} + \cdots + D_{j-1:n})^2 \\
&\equiv \frac{2}{3}{n \choose 4}^{-1} \sum_{1 \leq i < j \leq n} (i-1) \, (n-j) \, \sum_{k,l=i}^{j-1} D_{k:n} D_{l:n}.
\end{aligned}$$ Interchanging the above summations, we obtain $$\begin{aligned}
\widehat{\U}_n^2(\bX) &= \frac{2}{3}{n \choose 4}^{-1} \sum_{k,l=1}^{n-1} D_{k:n} D_{l:n} \sum_{i=1}^{\min(k,l)} \sum_{j = \max(k,l)+1}^n (i-1) \, (n-j) \\
&= \frac{1}{6} {n \choose 4}^{-1} \sum_{k,l=1}^{n-1} D_{k:n} D_{l:n} \min(k,l) \, \big(\min(k,l)-1\big) \\
& \qquad\qquad\qquad\qquad\qquad \times \big(n-\max(k,l)\big) \, \big(n-\max(k,l)-1\big),
\end{aligned}$$ where the latter equality follows from the fact that $\sum_{i=1}^k i = k (k-1)/2$. Since $$\frac{1}{6} {n \choose 4}^{-1} = \frac{4}{n \, (n-1) \, (n-2) \, (n-3)},$$ then we deduce that $\U_n^2(\bX) = \widehat{\U}_n^2(\bX)+o(1)$. This completes the proof. $\qed$
Denoting the vector of spacings by $D = (D_{1:n},\ldots,D_{n-1:n})$, we can write the quadratic form in (\[eq:newsampledvar\]) as $$\U_n^2(\bX) = D^t \, V \, D,$$ where the $(i,j)$th element of the matrix $V$ is $$V_{i,j} = {n \choose 2}^{-2} \, \big(\min(i,j)\big)^2 \, \big(n-\max(i,j)\big)^2$$ Both the squared sample Gini mean difference and the sample variance $$\widehat{\sigma}_n^2(\bX) := \frac{1}{n \, (n-1)} \sum_{i=1}^n (X_i - \overline{X}_n)^2$$ can also be expressed as quadratic forms in the spacings vector $D$; specifically, $$\widehat{\Delta}_n^2(\bX) = D^t \, G \, D, \quad \quad \widehat{\sigma}_n^2(\bX) = D^t \, S \, D,$$ where the elements of $G$ and $S$ are given by $$G_{i,j} = {n \choose 2}^{-2} \, i\, j \, (n-i) \ (n-j)$$ and $$S_{i,j} = \frac{1}{2} {n \choose 2}^{-1} \, \min(i,j) \, (n-\max(i,j)).$$
![Illustration of (from left to right) the sample distance variance $\U^2_n$, the squared sample Gini mean difference $\widehat{\Delta}^2$, and the sample variance $\widehat{\sigma}^2$ [*via*]{} their respective quadratic form matrices $V$, $G$, and $S$ for sample size $n=1,000$. The coordinate $(i,j)$ corresponds to the $(i,j)$th entry of the corresponding matrix, and the size of the corresponding matrix element is specified [*via*]{} color code (see legend).[]{data-label="figure1"}](dispersionges.png){width="100.00000%"}
Hence, comparing $\U^2_n$, $\Delta_n^2$, and $\sigma_n^2$ is equivalent to comparing the matrices $V$, $G$ and $S$. We use this fact to graphically illustrate differing features of $\V$, $\Delta$, and $\sigma$ by plotting the values of the underlying matrices; see Figure \[figure1\].
Moreover, these quadratic form representations lead to the rediscovery of results from Section \[sec:ineq\]. For example, since $V$ and $G$ have the same diagonal entries then it follows that $\V$ and $\Delta$ are equal for Bernoulli-distributed random variables. Also, if $n$ is even then the elements $V_{n/2,n/2}$, $G_{n/2,n/2}$, and $S_{n/2,n/2}$ all coincide, representing the fact that the underlying measures coincide for the Bernoulli distribution with $p=\tfrac12$. Finally, since $V_{ij} \leq G_{ij}$ and $V_{ij} \leq S_{ij}$ for all $i,j$ then we obtain an alternative proof of Corollary \[cor:vargmdineq\].
It is also remarkable that $V$ is twice the second Hadamard power of $S$ and that $V$ and $S$ both are positive definite, while $G$ is positive semidefinite with rank $1$. Finally, we mention that there are numerous other statistics which can be written as quadratic forms or square-roots of quadratic forms in the spacings, e.g., the Greenwood statistic, the range, and the interquartile range.
Closed form expressions for the distance variance of some well-known distributions
==================================================================================
[\[sec:distr\]]{}
Exploiting the different representations of the distance variance derived in the preceding sections, we can now state the distance variance of many well-known distributions. In the following result, we use the standard notation ${}_1F_1$ and ${}_2F_1$ for the classical confluent and Gaussian hypergeometric functions.
\[th:distri\]
1. Let $X$ be Bernoulli distributed with parameter $p$. Then $
\V^2(X) = 4\,p^2\,(1-p)^2.
$
2. Let $X$ be normally distributed with mean $\mu$ and variance $\sigma^2$. Then $$\V^2(X) = 4\Big( \frac{1-\sqrt{3}}{\pi} + \frac{1}{3} \Big) \sigma^2.$$
3. Let $X$ be uniformly distributed on the interval $[a,b]$. Then $
\V^2(X) = 2(b-a)^2/45.
$
4. Let $X$ be Laplace-distributed with density function, $f_X(x)=(2 \alpha)^{-1}$\
$\exp(-|x-\mu|/\alpha)$, $x \in \R$, $\alpha > 0$, $\mu \in \R$. Then $
\V^2(X) = 7\alpha^2/12.
$
5. Let $X$ be Pareto-distributed with parameters $\alpha > 1$ and $x_m > 0$, and density function $f_X(x) = \alpha x_m^\alpha x^{-(\alpha+1)}$, $x \geq x_m$. Then, $$\V^2(X) = \frac{4 \alpha^2 x_m^2}{(\alpha-1) \, (2 \alpha-1)^2 \, (3 \alpha -2)}$$
6. Let $X$ be exponentially distributed with parameter $\lambda > 0$ and density function $f_X(x)= \lambda \exp(-\lambda \, x)$, $x \geq 0$. Then, $
\V^2(X) = (3 \lambda^2)^{-1}.
$
7. Let $X$ be Gamma-distributed with shape parameter $\alpha > 0$ and scale parameter $1$. Then $$\V^2(X) = 2^{2(2-2\, \alpha)} \sum_{j, k=1}^\infty A_{j,k}(\alpha)^2,$$ where $$\begin{aligned}
&A_{j,k}(\alpha) = 2^{-j-k} \, \left(\frac{(\alpha)_j \, (\alpha)_k}{j! \, k!} \right)^{1/2} \\
& \times \frac{\Gamma(2 \alpha+j+k-1)}{\Gamma(\alpha+j) \, \Gamma(\alpha+k)}\ {}_2F_1\left(-j-k+2,1-\alpha-j;2-2 \alpha-j-k;2 \right).
\end{aligned}$$
8. Let $X$ be Poisson-distributed with parameter $\lambda > 0$. Then $$\V^2(X) = \sum_{j, k=1}^\infty \frac{4^{j+k-1} }{j! \, k!} \, \lambda^{j+k} \, A_{jk}^2,$$ where $$\begin{aligned}
A_{jk} = \frac{1}{(j-1)!} \sum_{l=0}^{\lfloor (j-k)/2\rfloor} \binom{j-k}{2l} (-1)^l (\tfrac12)_l \, (\tfrac12)_{j-l-1} \ {}_1F_1(j-l-\tfrac12;j;-4\lambda).
\end{aligned}$$
9. Let $X$ be negative binomially distributed with parameters $c$ and $\beta$. Then $$\V^2(X) = (1-c)^{4\beta} \sum_{j,k=1}^\infty \frac{(\beta)_j \, (\beta)_k}{j! \, k!} (1+c^2)^{-2\beta-2j} 2^{2k} c^{j+k} A_{jk}^2,$$ where $$\begin{aligned}
A_{jk} &= \sum_{l_1,l_2=0}^{j-k} \binom{j-k}{l_1} \binom{j-k}{l_2} (-c)^{l_1} (-1)^{l_2} (|l_1-l_2|)! \sum_{l=0}^\infty \frac{(\beta+j)-l}{l!} \bigg(\frac{2c}{1+c^2}\bigg)^l \\
& \qquad \times \sum_{m=0}^{|l_1-l_2|} (-2)^m \frac{(m)_{|l_1-l_2|}}{(|l_1-l_2|-m)! \, (2m)!} \frac{2^{k+m-1} \, (\tfrac12)_{k+m-1}}{(k+m-1)!} \\ & \qquad \qquad \times {}_2F_1(-l,k+m-\tfrac12;k+m;2).
\end{aligned}$$
10. Let $X=(X_1,\ldots,X_p)$ be a multivariate normally distributed random vector with mean $\mu=(\mu_1,\ldots,\mu_p)$ and identity covariance matrix $I_p := \diag(1,\ldots,1)$. Then $$\V^2(X) = 4 \pi \, \frac{c_{p-1}^2}{c_{p}^2} \, \left[
\frac{\Gamma(\tfrac12 p) \, \Gamma(\tfrac12 p + 1)}
{\big[\Gamma\big(\tfrac12(p+1)\big)\big]^2}
- 2 \ {}_2F_1 \! \left(-\tfrac12,-\tfrac12;\tfrac12 p;\tfrac14\right) + 1\right].$$
<span style="font-variant:small-caps;">Proof.</span>
1\. See Theorem \[th:bernoulli\].
2\. See the proof of Theorem 7 in [@szekely2007] or [@Dueck2014 p. 14].
3\. and 4. These follow directly from Theorem \[th:repdiff\] and the results in Table 3 in [@gerstenberger2015].
5\. and 6. These results follow directly from the representation (\[eq:dvartwo\]) and [@Zenga2004 equations (4.2) and (4.4)].
7., 8., and 9. See [@Dueck2016 Propositions 5.6, 5.7, and 5.8].
10\. See [@Dueck2014 Corollary 3.3]. $\qed$
By equations (\[eq:gmdvartwo\]) and (\[eq:ASV\]), we can also derive expresssions for the variance and asymptotic variance of the sample Gini mean difference for the distributions 1.- 9. in Theorem \[th:distri\]. To the best of our knowledge, these expressions are novel for the Gamma, Poisson, and negative binomial distributions.
The distance standard deviation as a measure of spread {#sec:furtherproperties}
======================================================
In this section, we show that the distance standard deviation $\V(X)$ satisfies the criteria (C1)-(C3) stated in Section 1 and therefore is an axiomatic measure of spread in the sense of [@Bickel2012]. Moreover, we point out some differences and commonalities between $\V$, $\Delta$ and $\sigma$. First, we state some additional preliminaries about stochastic orders.
[ A random variable $X$ is said to be [*stochastically smaller than*]{} a random variable $Y$, or $X$ is [*smaller than $Y$ in the stochastic ordering*]{}, written $X \leq_{\text{st}} Y$, if $\P(X>u) \leq \P(Y>u)$ for all $u \in \R$. ]{}
A necessary and sufficient condition that $X \leq_{\text{st}} Y$ is that $$\label{eq:stochorder}
\E[\phi(X)] \leq \E[\phi(Y)]$$ for all increasing functions $\phi$ for which these expectations exist.
Another important ordering of random variables is the [*dispersive order*]{}, $\leq_{\hskip 1pt \rm{disp}}$, which was stated earlier at (\[eq:dispersive\]) in the introduction. Bartoszewicz [@Bartoszewicz1986] proved the following result.
\[th:barto\] Let $(X_1,\ldots,X_n)$ and $(Y_1,\ldots,Y_n)$ be random samples from the random variables $X$ and $Y$, respectively, and let $D_j = X_{j+1:n}-X_{j:n}$ and $E_j = Y_{j+1:n}-Y_{j:n}$, $j=1,\ldots,n-1$ denote the corresponding sample spacings. If $X \leq_{\hskip 1pt \rm{disp}} Y$ then $D_{j:n} \leq_{\hskip 1pt \rm{st}} E_{j:n}$ for all $j = 1,\ldots,n-1$.
Applying this result to the representation of the distance variance derived in Theorem \[th:repre.spacing\], we conclude that the distance standard deviation $\V$ is indeed a measure of spread in the sense of [@Bickel2012].
\[th:disp\] If $X \leq_{\hskip 1pt \rm{disp}} Y$ then $\V(X) \leq \V(Y)$.
<span style="font-variant:small-caps;">Proof.</span>
Let us consider i.i.d. replicates $(X,Y)$, $(X',Y')$, $(X'',Y'')$, and $(X''',Y''')$. Moreover, let $X_{1:4} \leq X_{2:4} \leq X_{3:4} \leq X_{4:4}$ and $Y_{1:4}\leq Y_{2:4}\leq Y_{3:4}\leq Y_{4:4}$ denote the respective order statistics. By Proposition \[th:barto\], $$(X_{3:4}-X_{2:4}) \leq_{\hskip 1pt \rm{st}} (Y_{3:4}-Y_{2:4}).$$ Applying equation (\[eq:stochorder\]) and Theorem \[th:repre.spacing\] concludes the proof. $\qed$
Using similar arguments, we can show that the result of Theorem \[th:disp\] holds analogously for the standard deviation and Gini’s mean difference; see also [@Kochar1999].
\[th:logconcave\] The random variable $X$ satisfies the property $$X \leq_{\hskip 1pt \rm{disp}} X+Y \ \text{for any random variable $Y$ which is independent of $X$}$$ if and only if $X$ has a log-concave density.
Applying Theorem \[th:logconcave\], we obtain the following corollary of Theorem \[th:disp\].
[\[cor:logconcave\]]{} Let $X$ be a random variable with a log-concave density. Then $$\V(X+Y) \geq \V(X)$$ for any random variable $Y$ independent of $X$.
In particular if $X$ and $Y$ are independently distributed, continuous, random variables with log-concave densities, then $$\V(X+Y) \geq \max(\V(X),\V(Y)).$$ It is well known, both for the standard deviation and for Gini’s mean difference, that analogous assertions hold without any restrictions on the distributions of $X$ and $Y$. In particular, for any pair of independent random variables $X$ and $Y$ with existing first or second moments, respectively, there holds $$\label{eq:varsum}
\sigma^2(X+Y) = \sigma^2(X)+\sigma^2(Y) \geq \max(\sigma^2(X),\sigma^2(Y)).$$ Also, letting $X'$ and $Y'$ denote i.i.d. copies of $X$ and $Y$, respectively, we have $$\label{eq:gmdsum}
\Delta(X+Y) = \E[\max(|X-X'|,|Y-Y'|)] \geq \max(\Delta(X),\Delta(Y)).$$ However we now show that this property does not hold generally for the distance standard deviation, $\V$, thereby answering a second question raised by Gabor Székely (private communication, November 23, 2015).
Let $X$ be Bernoulli distributed with parameter $p=\tfrac12$ and let $Y$ be uniformly distributed on the interval $[0,1]$ and independent of $X$. Then $\V(X) > \V(X+Y)$.
<span style="font-variant:small-caps;">Proof.</span>
By a straightforward calculation using (\[eq:dvartwo\]), we obtain $$\begin{aligned}
\V^2(X+Y) &= T_1(X+Y)+T_2(X+Y)-2\,T_3(X+Y) \\
&=
\frac{2}{3} + \frac{4}{9} - \frac{14}{15} = \frac{8}{45}.
\end{aligned}$$ However, by Theorem \[th:bernoulli\], $\V^2(X) = 1/4 > \V^2(X+Y)$. $\qed$
Other common properties of the classical standard deviation and Gini’s mean difference concerns differences and sums of independent random variables. From the representations of $\sigma^2(X+Y)$ and $\Delta(X+Y)$ given in (\[eq:varsum\]) and (\[eq:gmdsum\]), we see that $$\Delta(X+Y) = \Delta(X-Y), \quad \quad \quad \sigma(X+Y) = \sigma(X-Y)$$ for any independent random variables $X$ and $Y$ for which these expressions exist. On the other hand, these properties do not hold in general for the distance standard deviation.
Let $X$ and $Y$ be independently Bernoulli distributed with parameter $p \neq \tfrac12$. Then $\V(X+Y) > \V(X-Y)$.
<span style="font-variant:small-caps;">Proof.</span>
By a straightforward calculation using (\[eq:dvartwo\]), we obtain $$\V^2(X+Y) = 8 \, (p-p^2)^2 \, \big(2 \, (p-p^2)^2-6\,(p-p^2)+2 \big)$$ and $$\V^2(X-Y) = 8 \, (p-p^2)^2 \, \big(2 \, (p-p^2)^2-2\,(p-p^2)+1 \big).$$ Hence, $$\V^2(X+Y)-V^2(X-Y) = 8 \, (p-p^2)^2 \, (1-2p)^2,$$ and this difference obviously is positive for $p \neq \tfrac12$. $\qed$
However, an analogous property holds when either of the two variables has a symmetric distribution.
Let $X$ and $Y$ be independent real random variables with $\E|X+Y| < \infty$. Then $\V^2(X+Y) = \V^2(X-Y)$ if either $X$ or $Y$ is symmetric about $\mu$.
<span style="font-variant:small-caps;">Proof.</span>
Since $\V(X-Y)=\V(Y-X)$ then we can assume, without loss of generality, that $Y$ is symmetric. Moreover since $\V^2(X+\mu) = \V^2(X)$ then we can assume that the point of symmetry is at $0$.
By equation (\[eq:dvar\]), $$\begin{aligned}
&\V^2(X-Y)\\ &= \frac{1}{\pi^2} \int_{\R^2}
\big|f_{X-Y}(s+t)-f_{X}(s)f_{-Y}(t)\big|^2 \, \frac{\dd s \dd t}{|s|^2 \, |t|^2} \\
&=\frac{1}{\pi^2} \int_{\R^2}
\big|f_{X}(s+t)\,f_{-Y}(s+t)-f_X(s)f_{-Y}(s)f_X(t)f_{-Y}(t)\big|^2 \, \frac{\dd s \dd t}{|s|^2 \, |t|^2} \\
&= \frac{1}{\pi^2} \int_{\R^2}
\big|f_{X}(s+t)\,f_{Y}(s+t)-f_X(s)f_{Y}(s)f_X(t)f_{Y}(t)\big|^2 \, \frac{\dd s \dd t}{|s|^2 \, |t|^2} \\
&= \V^2(X+Y),
\end{aligned}$$ where the third equality follows from the fact that $Y$ and $-Y$ have the same distribution. $\qed$
The distance standard deviation as an estimator {#sec:estimator}
===============================================
In this section, we investigate the properties of the sample distance variance $\V^2_n(\bX)$ and the sample distance standard deviation $\V_n(\bX)$ as estimators and derive their asymptotic distributions. For these purposes, we employ the representation (\[eq:dvarthree\]), [*viz.*]{}, $$\V^2(X) = W(X) + \Delta^2(X)$$ where $$W(X) = \E \left[\|X-Y\| \left(\|X-Y\| - 2 \, \|X-Z\| \right) \right],$$ and $\Delta(X)$ denotes the population value of Gini’s mean difference. Throughout this section, $X$, $Y$, $Z$ are i.i.d. $p$-variate random vectors with distribution $F$. For a sample of i.i.d. random vectors $\bX =(X_1,\ldots, X_n)^t$, each with distribution $F$, we define the corresponding empirical quantities, $$\Delta_n(\bX) = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \|X_i - X_j\|$$ and $${W}_n(\bX) = \frac{1}{n^3} \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^n \|X_i-X_j\| \left( \|X_i-X_j\| - 2 \|X_i-X_k\| \right).$$ Note that $$W_n(\bX) = T_{1,n}(\bX) - 2 \, T_{3,n}(\bX),$$ cf. (\[eq:sampledvar\]), and $$\V_n^2(\bX) = {W}_n(\bX) + {\Delta}^2_n(\bX).$$ Further, it is straightforward to verify that $$\label{expectedvalueofwn}
\E {W}_n(\bX) = \frac{(n-1)(n-2)}{n^2}W(X).$$ The statistic $\Delta_n(\bX)$ does not wear a hat to distinguish it from $\widehat{\Delta}_n(\bX)$, the unbiased version of the sample Gini difference, defined for univariate observations in (\[eq:DeltahatnX\]) and which we extend to multivariate observations by replacing the absolute value $|\cdot|$ by the Euclidean norm $\|\cdot\|$; thus, $$\Delta_n^2(\bX) = \frac{(n-1)^2}{n^2} \widehat{\Delta}_n^2(\bX).$$ Similar to $\widehat{\Delta}_n(\bX)$, we define $$\begin{aligned}
\widehat{W}_n(\bX) &= \frac{1}{n(n-1)(n-2)} \sum_{\substack{1 \le i,j,k \le n \\ i\neq j, j \neq k, k \neq i}} \| X_i - X_j\|\left( \|X_i-X_j\| - 2 \|X_i- X_k\| \right) \\
&\equiv \frac{n^2}{(n-1)(n-2)} W_n(\bX).
\end{aligned}$$ By (\[expectedvalueofwn\]), $\widehat{W}_n(\bX)$ is an unbiased sample version of $W(X)$.
Also, $\widehat{\V}_n(\bX) = [\widehat{W}_n(\bX) + \widehat{\Delta}_n^2(\bX)]^{1/2}$ is an alternative to $\V_n(\bX)$ as an empirical version of the distance standard deviation $\V(X)$. Although $\widehat{\V}_n(\bX)$ is based on the unbiased estimators $\widehat{W}_n(\bX)$ and $\widehat{\Delta}_n(\bX)$, the estimator $\widehat{\V}_n(\bX)$ itself has a larger bias than $\V_n(\bX)$. The results of Table \[tab:2\] below indicate that $\V_n(\bX)$ is to be preferred over $\widehat{\V}_n(\bX)$ as an estimator of $\V(X)$ because it exhibits smaller finite-sample bias and smaller variance for scenarios considered in our simulations.
Nevertheless, $\V_n^2(\bX)$ and $\widehat{\V}_n^2(\bX)$ have the same asymptotic distribution. In order to establish that result, we define for $x \in \R^p$, $$\begin{array}{lcl}
\psi_1(x) = \E \| x - Y\|^2, & \quad & \psi_2(x) = \E (\|x-Y\| \cdot \|x-Z\|), \\
\psi_3(x) = \E (\|x-Y\| \cdot \| Y-Z\|), & \quad & \psi_4(x) = \E\|x-Y\|,
\end{array}$$ and, with $T_1(X)$, $T_2(X)$, and $T_3(X)$ as defined in (\[T1T2\])-(\[T3\]), we also define $$\label{Mmatrix}
\begin{aligned}
m_{11} &= 4\, \E \big[ \psi_1(X) - \psi_2(X) - 2\psi_3(X) \big]^2 - 4\big(T_1(X) - 3 T_3(X)\big)^2, \\
m_{12} &= 4\, \E \big[ \psi_4(X) \big(\psi_1(X) - \psi_2(X) - 2\psi_3(X)\big) \big] - 4T_2(X)\big(T_1(X) - 3 T_3(X)\big), \\
m_{22} &= 4\, \E \psi_4^2(X) - 4 \big(T_2(X)\big)^2.
\end{aligned}$$ and let $$\gamma = m_{11} + 4 m_{12} \Delta(X) + 4 m_{22} \Delta^2(X).$$ We now provide in the following result the asymptotic distribution of $\V_n^2(\bX)$ and $\widehat{\V}_n^2(\bX)$.
\[th:asymptotics\] Suppose that $\E(\|X\|^4) < \infty$. Then, as $n \to \infty$, $$\label{Vnasymptoticdistn}
\sqrt{n}\big(\V_n^2(\bX) - \V^2(X)\big) \cid N(0,\gamma)$$ and the same result holds for $\widehat{\V}_n^2(\bX)$.
<span style="font-variant:small-caps;">Proof.</span>
Consider the bivariate statistic $\widehat{B}_n(\bX) = \big(\widehat{W}_n(\bX),\widehat{\Delta}_n(\bX)\big)^t$, which has expected value $B(X) = (W(X),\Delta(X))^t$. Define the functions $K, L: \R^p \times \R^p \times \R^p \to \R$ such that $$\begin{aligned}
K(x,y,z) & = \|x\!-\!y\|(\|x\!-\!y\|\!-\!2\|x\!-\!z\|) + \|y\!-\!z\|(\|y\!-\!z\|\!-\!2\|y\!-\!x\|) \\
& \qquad + \|z\!-\!x\|(\|z\!-\!x\|\!-\!2\|z\!-\!y\|)
\end{aligned}$$ and $$L(x,y,z) = (\|x-y\| + \|y-z\| + \|z-x\|),$$ $(x,y,z) \in \R^p \times \R^p \times \R^p$. Then the statistic $\widehat{B}_n(\bX)$ can be written as a U-statistic with the bivariate, permutation-symmetric kernel of order three, $h:\R^p \times \R^p \times \R^p \to \R^2$, where $$h(x,y,z) = \frac13 \begin{pmatrix} K(x,y,z) \\ L(x,y,z)\end{pmatrix},$$ $(x,y,z) \in \R^p \times \R^p \times \R^p$. , where $$K(x,y,z) = \|x\!-\!y\|(\|x\!-\!y\|\!-\!2\|x\!-\!z\|) + \|y\!-\!z\|(\|y\!-\!z\|\!-\!2\|y\!-\!x\|) + \|z\!-\!x\|(\|z\!-\!x\|\!-\!2\|z\!-\!y\|)$$ and $$L(x,y,z) = \frac13 (\|x-y\| + \|y-z\| + \|z-x\|).$$ $$\frac{1}{3}
\begin{pmatrix}
\|x\!-\!y\|(\|x\!-\!y\|\!-\!2\|x\!-\!z\|) + \|y\!-\!z\|(\|y\!-\!z\|\!-\!2\|y\!-\!x\|) + \|z\!-\!x\|(\|z\!-\!x\|\!-\!2\|z\!-\!y\|) \\
\|x-y\| + \|y-z\| + \|z-x\| \\
\end{pmatrix}.$$ Define the function $h_1:\R^p \to \R^2$, where $h_1(x) = \E h(x,Y,Z) - B(X)$; then, $h_1$ is the linear part in the Hoeffding decomposition of the kernel $h$, and we calculate that $$h_1(x) = \frac{2}{3}
\begin{pmatrix}
\psi_1(x) - \psi_2(x) - 2\psi_3(x) - T_1(X) + 3 T_3(X) \\
\psi_4(x) -T_2(X)
\end{pmatrix},$$ $x \in \R^p$. Since $\E(\|X\|)^4 < \infty$ then $\E [(h(X,Y,Z))^2 ] < \infty$; therefore, we deduce from a classical result of Hoeffding [@Hoeffding1948 Theorem 7.1] that $$\sqrt{n} \big(\widehat{B}_n(\bX) - B(X)\big) \cid N_2\big(0, 9 \, \E h_1(\bX) h_1(\bX)^t\big).$$
Denote the symmetric $2 \times 2$ matrix $9 \, \E h_1(\bX) h_1(\bX)^t$ by $M = (m_{ij})_{i,j = 1,2}$, where the elements $m_{11}$, $m_{12}$, and $m_{22}$ are given in (\[Mmatrix\]). Define $g:\R^2 \to \R$ by $g(x,y) = x + y^2$; then $\widehat{\V}_n^2(\bX) = g\big(\widehat{B}_n(\bX)\big)$. Since $\nabla h(x,y) = (1,2y)^t$ then, by applying the Delta Method, we obtain $\sqrt{n}\big(\widehat{\V}_n^2(\bX) - \V^2(X)\big) \cid N(0,\gamma)$.
In the case of $\V_n^2(\bX)$, we need only to apply the formulas $W_n(\bX) = (n-1)(n-2) \widehat{W}_n(\bX)/n^2$ and $\Delta_n^2(\bX) = (n-1)^2 \widehat{\Delta}_n^2(\bX)/n^2$ to deduce that $\V_n^2(\bX) - \widehat{\V}_n^2(\bX) = o(n^{-1})$. Then it follows by the Delta Method that $\V_n^2(\bX)$ has the same asymptotic distribution as $\V_n^2(\bX)$, as given in (\[Vnasymptoticdistn\]). $\qed$
The asymptotic distribution of the sample distance standard deviation $\V_n(\bX)$ now follows from Theorem \[th:asymptotics\] by the Delta Method:
\[cor:asymptotics\] Under the conditions of Theorem \[th:asymptotics\], we have $$\sqrt{n}(\V_n(\bX) - \V(X)) \cid N\big(0, \gamma/4 \V^2(X)\big),$$ and the same result holds for $\widehat{\V}_n(\bX)$.
[c|@ D[.]{}[.]{}[4]{}@ D[.]{}[.]{}[4]{}@ D[.]{}[.]{}[4]{}@ D[.]{}[.]{}[4]{}@ D[.]{}[.]{}[4]{}]{} Distribution, $F$ & & & &\
$N(0,1)$ & 0.784 & 1 & 0.876 & 0.978\
$L(0,1)$ & 0.952 & 0.8 & 1 & 0.964\
$t_5$ & 0.992 & 0.4 & 0.941 & 0.859\
$t_3$ & 0.965 & 0 & 0.681 & 0.524\
\[tab:1\]
In the following, we study the empirical distance standard deviation $\V_n(\bX)$ as an estimator of spread in the univariate case. For any $\sqrt{n}$-consistent and asymptotically normal estimator $s_n(\bX)$, we define its asymptotic variance $ASV(s_n(\bX);F)$ at the distribution $F$ to be the variance of the limiting distribution of $\sqrt{n}(s_n(\bX) - s)$, as $n \to \infty$, where $s_n(\bX)$ is evaluated at an i.i.d. sequence drawn from $F$ and $s$ denotes the corresponding population value of $s_n(\bX)$ at $F$. Any estimators $s_n^{(1)}(\bX)$ and $s_n^{(2)}(\bX)$ which estimate possibly different population values $s_1$ and $s_2$, respectively, at a given distribution $F$, and which obey the dilation property (C2) in Section 1, can be compared efficiency-wise by standardizing them through their respective population values. We define the [*asymptotic relative efficiency of*]{} $s_n^{(1)}(\bX)$ with respect to $s_n^{(2)}(\bX)$ at the population distribution $F$ as $$ARE\big(s_n^{(1)}(\bX),s_n^{(2)}(\bX);F\big) = \frac{ASV(s_n^{(1)}(\bX);F)/s_1^2}{ASV(s_n^{(2)}(\bX);F)/s_2^2} .$$
[cc|@ D[.]{}[.]{}[4]{}@ D[.]{}[.]{}[4]{}@ D[.]{}[.]{}[4]{}@ D[.]{}[.]{}[4]{}@ D[.]{}[.]{}[4]{}]{} Distribution, $F$ & &\
& & & & & &\
$N(0,1)$ & $\E(\V_n)$ & 0.663 & 0.658 & 0.640 & 0.634 & 0.633\
& $\E(\widehat{\V}_n)$ & 0.701 & 0.665 & 0.639 & 0.634 & 0.633\
& $n \Var(\V_n)$ & 0.297 & 0.276 & 0.255 & 0.255 & 0.256\
& $n \Var(\widehat{\V}_n)$ & 0.359 & 0.298 & 0.259 & 0.255 & 0.256\
$L(0,1)$ & $\E(\V_n)$ & 0.888 & 0.861 & 0.790 & 0.767 & 0.764\
& $\E(\widehat{\V}_n)$ & 0.942 & 0.864 & 0.785 & 0.766 & 0.764\
& $n \Var(\V_n)$ & 0.955 & 0.836 & 0.668 & 0.605 & 0.613\
& $n \Var(\widehat{\V}_n)$ & 1.136 & 0.858 & 0.663 & 0.604 & 0.613\
$t_5$ & $\E(\V_n)$ & 0.818 & 0.799 & 0.744 & 0.727 & 0.725\
& $\E(\widehat{\V}_n)$ & 0.866 & 0.804 & 0.741 & 0.727 & 0.725\
& $n \Var(\V_n)$ & 0.761 & 0.632 & 0.474 & 0.432 & 0.424\
& $n \Var(\widehat{\V}_n)$ & 0.931 & 0.655 & 0.471 & 0.432 & 0.424\
$t_3$ & $\E(\V_n)$ & 1.003 & 0.960 & 0.861 & 0.817 & 0.810\
& $\E(\widehat{\V}_n)$ & 1.074 & 0.967 & 0.855 & 0.816 & 0.810\
& $n \Var(\V_n)$ & 5.762 & 2.001 & 1.089 & 0.777 & 0.680\
& $n \Var(\widehat{\V}_n)$ & 8.420 & 2.177 & 1.067 & 0.774 & 0.680\
\[tab:2\]
Even in the univariate case with normally distributed data, the integrals underlying the parameters $m_{11}$, $m_{12}$, and $m_{22}$ in (\[Mmatrix\]) do not admit straightforward analytical expressions. Nevertheless, by means of numerical integration, we can obtain values for the asymptotic variance of $\V_n(\bX)$ for given population distributions and thus deduce properties of the efficiency of $\V_n(\bX)$ in relation to other widely-used estimators of scale.
In Table \[tab:1\], we provide the asymptotic relative efficiency of the distance standard deviation with respect to the respective maximum likelihood estimator at the normal distribution, the Laplace distribution, and $t_\nu$ distributions with $\nu = 5$ and $\nu = 3$. The maximum likelihood estimator of scale in the location-scale family generated by the $N(0,1)$, or standard normal, distribution is the standard deviation. In the Laplace model, the analogous estimator of scale is the mean deviation $\widehat{d}_n(\bX) = n^{-1} \sum_{i=1}^n |X_i - m_n(\bX)|$, where $m_n(\bX)$ denotes the sample median of $\bX$. In the case of the $t_\nu$-distribution, the maximum likelihood estimator of the scale parameter does generally not admit an explicit representation.
In Table \[tab:1\], the asymptotic efficiencies of the distance standard deviation are compared with those of the standard deviation $\widehat{\sigma}_n(\bX)$, the mean deviation $\widehat{d}_n(\bX)$, and Gini’s mean difference $\widehat{\Delta}_n(\bX)$. The asymptotic variance of the maximum likelihood estimator of the scale parameter for the $t_\nu$-distribution is $(\nu + 3)/2\nu$. The population values and asymptotic variances of the other estimators mentioned at the respective distributions are given by Gerstenberger and Vogel [@gerstenberger2015 Tables 2 and 3].
While the distance standard deviation has moderate efficiency at normality, it turns out to be asymptotically very efficient in the case of heavier-tailed populations. For the $t_5$- and $t_3$-distributions, the distance standard deviation outperforms its three competitors considered here and moreover is very close to the respective maximum likelihood estimator.
In Table \[tab:2\], we complement our asymptotic analysis by finite-sample simulations. For sample sizes $n = 5, 10, 50$, and $500$ and the same population distributions as above, the (simulated) expectations and variances (based on 10,000 observations) of the empirical distance standard deviation $\V_n(\bX) = [W_n(\bX) + \Delta_n^2(\bX)]^{1/2}$ and the alternative version $\widehat{\V}_n(\bX) = [\widehat{W}_n(\bX)$ + $\widehat{\Delta}_n^2(\bX)]^{1/2}$ are given along with their respective asymptotic values. The corresponding values for the competing estimators $\widehat{\sigma}_n(\bX)$, $\widehat{d}_n(\bX)$, and $\widehat{\Delta}_n(\bX)$ are also provided by Gerstenberger and Vogel [@gerstenberger2015]. Table \[tab:2\] indicates that $\V_n(\bX)$ indeed is to be preferred over $\widehat{\V}_n(\bX)$ in terms of bias as well as variance.
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[^1]: German Cancer Research Center, Im Neuenheimer Feld 280, 69120 Heidelberg, Germany.
[^2]: Department of Statistics, Pennsylvania State University, University Park, PA 16802, U.S.A.
[^3]: Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, U.K. $^*$Corresponding author; e-mail address: dominic.edelmann@dkfz-heidelberg.de
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author:
- 'Weiwei Zhu, Di Li, Rui Luo, Chenchen Miao, Bing Zhang, Laura Spitler, Duncan Lorimer, Michael Kramer, David Champion, Youling Yue, Andrew Cameron, Marilyn Cruces, Ran Duan, Yi Feng, Jun Han, George Hobbs, Chenhui Niu, Jiarui Niu, Zhichen Pan, Lei Qian, Dai Shi, Ningyu Tang, Pei Wang, Hongfeng Wang, Mao Yuan, Lei Zhang, Xinxin Zhang, Shuyun Cao, Li Feng, Hengqian Gan, Long Gao, Xuedong Gu, Minglei Guo, Qiaoli Hao, Lin Huang, Menglin Huang, Peng Jiang, Chengjin Jin, Hui Li, Qi Li, Qisheng Li, Hongfei Liu, Gaofeng Pan, Bo Peng, Hui Qian, Xiangwei Shi, Jinyuo Song, Liqiang Song, Caihong Sun, Jinghai Sun, Hong Wang, Qiming Wang, Yi Wang, Xiaoyao Xie, Jun Yan, Li Yang, Shimo Yang, Rui Yao, Dongjun Yu, Jinglong Yu, Chengmin Zhang, Haiyan Zhang, Shuxin Zhang, Xiaonian Zheng, Aiying Zhou, Boqin Zhu, Lichun Zhu, Ming Zhu, Wenbai Zhu, Yan Zhu'
bibliography:
- 'myrefs.bib'
title: A Fast Radio Burst discovered in FAST drift scan survey
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The authors thank Shu-Xu Yi, Nan Li, and Zhi-Yuan Ren for discussions and the referee for a careful review and suggestions. This work is supported by National Key R&D Program of China No. 2017YFA0402600, the CAS-MPG LEGACY project and the FAST FRB key science project. WWZ is supported by the CAS Pioneer Hundred Talents Program, the Strategic Priority Research Program of the CAS Grant No. XDB23000000, and by the National Natural Science Foundation of China under grant No. 11690024, 11743002, 11873067. LQ is supported in part by the Youth Innovation Promotion Association of CAS (id. 2018075). YLY is supported by CAS “Light of West China” Program. ZCP is supported by the National Natural Science Funds of China (Grant No. 11703047) and the CAS “Light of West China” Program. DRL is supported by National Science Foundation OIA Award 1458952. This research made use of Astropy,[^1] a community-developed core Python package for Astronomy [@astropy:2013; @astropy:2018]. This work is supported by Chinese Virtual Observatory (China-VO) and Astronomical Big Data Joint Research Center, co-founded by National Astronomical Observatories, Chinese Academy of Sciences and Alibaba Cloud. [*Facilities:*]{}
[^1]: http://www.astropy.org
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abstract: |
We present detailed analyses of the absorption spectrum seen in QSO 2359–1241 (NVSS J235953$-$124148). Keck HIRES data reveal absorption from twenty transitions arising from: , , , , and . HST data show broad absorption lines (BALs) from $\lambda$1857, $\lambda$1549, $\lambda$1397, and $\lambda$1240. Absorption from excited states constrains the temperature of the absorber to $2000\ltorder T \ltorder10,000$ K and puts a lower limit of $10^5$ cm$^{-3}$ on the electron number density. Saturation diagnostics show that the real column densities of and can be determined, allowing to derive meaningful constraints on the ionization equilibrium and abundances in the flow. The ionization parameter is constrained by the iron, helium and magnesium data to $-3.0\ltorder \log(U) \ltorder-2.5$ and the observed column densities can be reproduced without assuming departure from solar abundances. From comparison of the and absorption features we infer that the outflow seen in QSO 2359–1241 is not shielded by a hydrogen ionization front and therefore that the existence of low-ionization species in the outflow (e.g., , , ) does not necessitate the existence of such a front. We find that the velocity width of the absorption systematically increases as a function of ionization and to a lesser extent with abundance. Complementary analyses of the radio and polarization properties of the object are discussed in a companion paper (Brotherton et al. 2000).
[*Subject headings:*]{} quasars: absorption lines
author:
- |
Nahum Arav[^1], Michael S. Brotherton[^2][^3], Robert H. Becker$^{1,3}$, Michael D. Gregg$^{1,3}$,\
Richard L. White[^4], Trevor Price$^{1,3}$, Warren Hack$^{4}$
title: |
THE INTRINSIC ABSORBER IN QSO 2359–1241:\
KECK AND HST OBSERVATIONS
---
=cmr7
=9.0in
2[ergss$^{-1}$cm$^{-2}$]{} 3[[cm]{}\^[-3]{}]{} 2[[cm]{}\^[-2]{}]{} ø5007[\[O[iii]{}\]$\lambda5007$]{} ø[ø]{}
INTRODUCTION
============
The radio source NVSS J235953-124148 (z=0.868), hereafter QSO 2359–1241, is unique among quasars. It shows intrinsic absorption from: and , which appear in less than 1% of optically selected quasars; , which is even less frequent; and from a meta-stable level, which is only seen in two or three other AGNs. We use the term “intrinsic absorption” following the definition given by Hamann et al (1997) and Barlow (1997). Whenever this absorption is significantly blue-shifted with respect to the systemic redshift of the quasar, we interpret it as rising from an outflow connected with the AGN. In § 2 we establish the intrinsic nature of the absorption seen in QSO 2359–1241. Besides its rare absorption features, QSO 2359–1241 has very high intrinsic polarization ($\sim$5%) and is moderately reddened (Brotherton et al. 2000). A low-resolution Keck spectrum of the object is shown in figure 1.
By exhibiting absorption from and QSO 2359–1241 is classified as a low-ionization BALQSO. Low-ionization BALQSOs were studied by Boroson and Meyers (1992), Voit, Weymann and Korista (1993) Wampler, Chugai & Petitjean (1995); Becker et al. (1997) and de Kool et al. (2000), among others. All the low-ionization BALQSOs show absorption from , but only a subset of these show absorption from (Prominent examples include: QSO 0059–2735, Wampler et al. 1995; Arp 102B, Halpern et al 1996; FIRST 0840+3633 and 1556+3517, Becker et al. 1997; QSO 1044+3656, de Kool et al. 2000). An even smaller subset shows in absorption (Arp 102B, QSO 1044+3656). For example, from the six objects studied by Voit, Weymann and Korista (1993) only one (QSO 0059–2735) shows absorption and none show absorption. QSO 2359–1241 shows absorption features from all these ions as well as from . In the discussion we elaborate on the conditions needed for detecting these lines (both physical and observational) and argue that even though the low-ionization features in QSO 2359–1241 are too narrow to be classified as classical BALs, they are definitely part of a BAL outflow because of their association with the much wider high-ionization absorption troughs ( $\lambda$1549 and $\lambda$1397) seen in the ultraviolet spectrum.
The ions detected in absorption and their unique characteristics make QSO 2359–1241 a promising probe for the study of quasar outflows. Features that can be used as diagnostics include:
1\) Appearance of relatively unblended absorption features from both components of the doublet allows us to determine whether the flow completely covers the emission region. A partial covering of the source is taken as a direct evidence for the intrinsic nature of the absorber (Barlow 1997; Hamann et al. 1997; Arav et al. 1999b), that is, an outflow associated with the AGN. It also shows that the apparent column densities extracted from the absorption trough are only lower limits (Arav 1997; Arav et al. 1999b).
2\) Neutral helium absorption lines from the highly meta-stable level 2$^3$S. Very few AGNs show these lines, where known examples are Mrk 231 (Boksenberg et al 1977; Rudy, Stocke & Foltz 1985), NGC 4151 (Anderson 1974) and perhaps 3CR 68.1 (Brotherton et al. 1998). The lines are important diagnostics for the ionization state of the gas. From the ionization equilibrium of this level we can infer lower limits on the He$^+$ column density and therefore lower limits on the column density. The different oscillator strengths of the detected lines allow us to determine the real optical depth and covering factor of the absorbers.
3\) Detection of absorption from excited states of allows for lower limits and sometimes even determination of the number density of electrons ($n_e$) in the gas. In a study of QSO 1044+3656 (de Kool et al. 2000) we were able to determine $n_e$ in the outflow and combined with photo-ionization constraints showed that the outflow is situated about 1000 pc. from the central source.
4\) Appearance of absorption necessitates a low ionization parameter, since it is difficult to shield from ionizing photons. (The ionization parameter $U$ is defined as the ratio of number densities between hydrogen ionizing photons and hydrogen nuclei in all forms.)
To realize this diagnostic potential we observed the optical spectrum of QSO 2359–1241 using the HIRES spectrograph on the Keck telescope. In this paper we concentrate on a detailed analysis of the numerous absorption features seen in our high-resolution ground-based spectroscopic data (§ 2), and in low-resolution HST UV prism data (§ 3). In § 4 we analyze the ionization equilibrium and abundances of the outflow. The discovery of the object, its radio, polarization, and overall optical characteristics are described in a companion paper (Brotherton et al. 2000).
INTRINSIC ABSORPTION IN THE KECK HIRES SPECTRUM
===============================================
Data Acquisition and Reduction
------------------------------
On December 26, 1998 we used the High Resolution Echelle Spectrometer (HIRES, Vogt et al. 1994) on the Keck I 10-m telescope to obtain $4
\times 1500$ second exposures of QSO 2359$-$1241 covering 4330 - 7450Å using a 11 wide slit. The orders overlap up to 6410Å, beyond which small gaps occur between orders. The slit was rotated to the parallactic angle to minimize losses due to differential atmospheric refraction. The observing conditions were excellent with sub-arcsecond seeing and near-photometric skies. The spectra were extracted using routines tailored for HIRES (Barlow 2000). The resolution of the final spectrum is $R=39000$.
A smooth continuum was fit to the spectrum in regions free of absorption or emission features. This procedure is somewhat subjective, particularly in the case of blended emission lines (such as ). While unlikely to create any false absorption features, this normalization introduces some uncertainty in the true continuum level; consequently, equivalent widths of absorption features are somewhat uncertain.
Intrinsic Nature of the Absorption
----------------------------------
Many of the absorption features in the HIRES data arise from a complex intrinsic absorber. The evidence for the absorber being intrinsic is based on: 1) Comparing the absorption features seen in the doublet (the apparent optical depth ratio differs from the nominal value of 1:2). 2) Existence of absorption from a meta-stable state, which requires the density of the absorber to be several orders of magnitude larger than is seen in the ISM and IGM. 3) Existence of absorption from excited levels, which also necessitates high density. 4) Appearance of full fledge BALs ( $\lambda$1549, $\lambda$1397) in the HST FOC spectrum which coincide in velocity with the absorption seen in the HIRES data. Each of these separate pieces of evidence is enough by itself to identify the absorption system as intrinsic. In the rest of this section we analyze the features in the HIRES data that are associated with this intrinsic absorber.
A high resolution spectrum of the absorption feature reveals a rich and complex structure. In figure 2 we show HIRES data for a part of the absorption, the spectral region shown in figure 2 is marked on the insert to figure 1. We have labeled five distinct absorption features ($a-e$) associated with the red doublet component, which are also seen in the blue doublet component. Absorption features $f$ and $g$ are seen in the blue doublet component, but their red counterparts are blended with the ($a-e$) blue complex.
The appearance of the same features in both doublet components allow us to study the covering factor and real optical depth of the outflow. However, the situation in QSO 2359–1241 is not ideal since the red component of feature $g$ falls in the middle of blue component of feature $e$; the red component of feature $f$ contaminates the blue component of feature $b$; and there is a shallower red absorption contribution across the entire blue absorption structure $a-e$. These contaminations do not allow for a clean simultaneous solution for the covering factor and the optical depth (as was done for the BAL in QSO 1603+3002; Arav et al 1999b). However, most of the contaminating absorption is rather shallow and we can still get semi-quantitative results from analyzing the relationship between the same absorption features seen in both doublet components.
To analyze the residual intensities we first need to normalize the data. Since the absorption occur on the blue wing of the broad emission line (BEL), there are two physical choices here. First, the outflow may cover both the continuum source and the BEL region. In this case we model the emission line with two Gaussians and divide the data by the assumed unabsorbed emission, which consists of the BEL plus continuum. Second, the outflow may only cover the continuum source and not the BEL region. In BALQSO 1603+3002 there is strong evidence that the flow does not cover the BELs (Arav et al 1999b). For this case we need to subtract the modeled BEL from the data in order to determine the flux seen by the absorber. We then divide this flux by the continuum to obtain the normalized flux seen by the absorber. We will argue below that the second scenario is more probable and use this normalization in figure \[hires\_mg2\].
Figure \[mg2\_abs\_fits\_combined\] shows a comparison of the absorption features in each doublet component. For direct comparison we use velocity presentation and the two panels show the two normalizations discussed above. We use a thick solid line for the blue component data and a thin line for the red one. If the flow fully covers the QSO’s emission (both continuum source and BEL regions), the expected residual intensity of the absorption features seen in the blue doublet component is: $I_b(v)$\[expected\]$=I^2_r(v)$\[observed\]; where $I_r$ is the residual intensity of the absorption features seen in the red doublet component. We plot this expected blue residual intensity in figure \[mg2\_abs\_fits\_combined\] as a dotted line. In both normalizations it is evident that the depth of features $a, b$ and $c$ in the blue doublet component is smaller than expected by assuming complete coverage. This is a clear indication for partial covering and hence for the intrinsic nature of the absorption (Barlow 1997). We note that partial covering is not restricted only to geometrical coverage. The photons at the bottom of the troughs may also arise from scattering contribution. If we assume that the flow does not cover the BEL region, the top panel in figure \[mg2\_abs\_fits\_combined\] show that the residual intensities for features $b$ and $c$ are identical within the errors in both doublet components. In this case both features are saturated ($\tau_{red}\gtorder3$) and the shape of the absorption profile is completely dependent on the covering factor. If the flow covers the continuum and the BEL equally, then from the bottom panel of figure \[mg2\_abs\_fits\_combined\] we also infer that the blue doublet absorption of features $a, b$ and $c$ is not deep enough if we assume complete coverage. Therefore, we conclude that no matter which normalization we use the flow in features $a, b$ and $c$ show partial coverage and hence demonstrate the intrinsic nature of the absorber.
Which of the two covering scenarios is more probable? In the case of BALQSO 1603+3002 we argued that “in the absence of a physical preference for $\tau_{real}$ values of order unity, values between 2–5 necessitate some fine tuning whereas the range $5-\infty$ is simply much more probable numerically.” This argument suggests that it is more probable that features $b$ and $c$ do not cover the BEL region. However, we notice that the equal covering normalization (bottom panel in Fig. \[mg2\_abs\_fits\_combined\]) shows that feature $d$ might completely cover both the continuum source and the BEL region (expected and observed blue component are equal). This scenario is simpler, since it does not necessitate a partial covering factor, and thus is more appealing. We suggest that the first argument is somewhat stronger and therefore it is more probable that features $b$ and $c$ do not cover the BEL region. However, the evidence is weaker than is seen in BALQSO 1603+3002.
In addition to components $a-g$ there is a high velocity trough at $-5000$ , which can be seen in the insert to figure 1. We do not show the HIRES data since they do not reveal qualitatively different structure than the one seen in the insert. A doublet structure in evident in the feature and the HIRES data confirm that the depth ratio of the two components is 1:1, that is, complete saturation. This 1:1 depth ratio is caused by partial covering factor, which indicates that the absorption system is intrinsic.
Apparent Column Densities
-------------------------
In Table 1 we give the apparent column density measurements of the intrinsic absorption features for all the identified lines. Apparent column density are derived by substituting the apparent optical depths of the troughs (defined as $\tau=-ln(I_r)$, where $I_r$ is the normalized residual intensity seen in the trough) in: $$\rm{N_{ion}} =
\frac{ 3.7679 \times 10^{14}~\rm{cm}^{-2} }{
\lambda_{\rm{o}}\rm{f_{ik}} } \int {\tau\/(v) dv},
\label{eq:column}$$ where $\lambda_{\rm{o}}$ and $\rm{f_{ik}}$ are the transition’s wavelength and oscillator strength, and where the velocity is measured in . We note that the apparent column densities give good estimates for the real column densities only if the absorbing material covers the emission source completely and uniformly and where scattered-photons do not contribute appreciably to the residual intensity in the troughs (see discussions in Korista et al 1992; Arav 1997). Otherwise, apparent column densities are only lower limits on the real ones (Arav et al 1999a, Arav et al 1999b ). We define $gf\lambda$ (where $g$ is the statistical weight, $f$ is the oscillator strength and $\lambda$ is the transition’s wavelengths) as the strength factor of the line and give its value (normalized to that of the strongest transition; data from Verner, Verner & Ferland 1996). For the lines we multiplied the expression in equation (\[eq:column\]) by the ratio of statistical weight of the lower level (summed over all observed states) to that of the specific state. This procedure yields independent estimates for $\rm{N_{\feii}}$ from each transition provided the level populations are in LTE.(see § 2.6). In components $c$ and $d$ are embedded in a wide extension of component $e$, their integration interval only covers the regions where these feature are distinctive, without compensation for the fact that some of the absorption is due to the wing of component $e$. For the other lines the different absorption components are well separated, and therefore the integration over the components is straightforward. For the measurments we corrected for the atmospheric absorption seen in the spectral region around the intrinsic $\lambda3890$ line (especially evident on the red side of component $e$, see Fig. \[hires\_he1\]). The estimated error for each line is derived by performing the integration given in equation (\[eq:column\]) on two absorption free regions in the spectral vicinity of the measured features and taking the average of their absolute values. This procedure takes into account both signal-to-noise and continuum uncertainties. The error is formally derived for component $e$ and can be used as a conservative estimate for the other components since they are narrower. For the error is significantly larger since we had to take the blending of the different components into account.
[llcccccccc]{}\
& & & & & & & & &\
& & & & & & & & &\
& 3970 & 1 & 12.43 & 11.06 & 11.68 & 11.59 & 10.90 & 11.1 & 12.57\
& 3889 & 1 & 13.80 & 12.70 & 13.14 & 13.19 & 12.76 & 12.3 & 14.02\
& 3189 & .36 & 13.85 & 12.29 & 12.22 & 12.94 & 12.49 & 12.8 & 13.90\
& 2946 & .17 & 13.87 & 12.47 & 12.96 & 13.30 & 12.82 & 13.0 & 14.00\
& 2853 & 1 & 11.88 & 10.98 & 11.44 & 11.53 & 10.68 & 10.7 & 12.18\
& 2803 & 1$^d$ & 14.21 & 13.21 & 13.30 & 13.38 & 12.89 & 12.6 & 14.43$^e$\
& 2632$^{*,f}$ & 0.46 & 13.59 & 12.84 & 11.45 & 12.25 & 12.32 & 12.6 & 13.65\
& 2612$^*$ & 0.33 & 13.65 & 13.13 & 13.09 & 11.77 & 12.50 & 12.8 & 13.83\
& 2608$^*$ & 0.22 & 13.83 & 12.35 & 13.02 & 12.87 & 13.10 & 13.1 & 14.00\
& 2600 & 0.76 & 13.66 & 12.83 & 12.76 & 12.91 & 12.14 & 12.5 & 13.83\
& 2587 & 0.17 & 13.92 & 12.56 & 13.47 & 13.30 & 12.88 & 13.1 & 14.16\
& 2405.6$^*$ & 0.43 & 13.63 & 12.37 & 12.74 & 13.00 & 12.01 & 13.0 & 13.61\
& 2396.4$^*$ & 0.68 & 13.62 & 12.11 & 11.42 & 13.08 & 11.80 & 12.8 & 13.73\
& 2382 & 1.00 & 13.61 & 12.67 & 12.71 & 12.87 & 12.75 & 12.6 & 13.81\
$^a$ - Expected absorption strength ratio for lines from the same ion (essentially normalized $gf\lambda$, see text).\
$^b$ - Log$_{10}$ of the column density for each absorption subcomponent (see Figs. 4 and 5).\
$^c$ - Also in units Log$_{10}$ of the column density. The error is roughly appropriate for each individual component, and takes into account both signal-to-noise and continuum uncertainties.\
$^d$ - Although the red doublet component of is only as half as strong as the blue one, we give it strength=1 since we do not report the blue component measurements separately.\
$^e$ - Total column density for includes contributions from component $f$ ($\log(N)=13.41$) and $g$ ($\log(N)=13.06$), which are associated with the blue component of the doublet.\
$^*$ - Transition from excited level.\
$^f$ - A blend of two excited transitions $\lambda$2632.11 and $\lambda$2631.83.\
Figure \[hires\_he1\] shows absorption in , and associated with the intrinsic absorber. All the lines are plotted on the same velocity scale where the wavelength of the transition in the rest frame of the object is at 0 . The spectral segments are plotted in the same normalized flux scale which is shifted for each line for presentation purposes. Absorption features from the He triplet lines are easily recognized as being part of the intrinsic outflow. The column densities for the three lines in each feature are in agreement given the errors, implying that unlike the case the absorption is not saturated. This finding is a prerequisite for determining the ionization equilibrium and abundances in the flow (see § 4), since it gives us the actual column density as opposed to a lower limit available from the apparent column density.
The observed lines all arise from the meta-stable level 2$^3$S. Extensive treatment of this level appears in the literature and the following discussion is largely based on: Macalpine (1976); Rudy, Stocke, & Foltz (1985); Clegg (1987); Oudmaijer, Busfield, & Drew (1997), all of which rely to some extent on Osterbrock’s “Astrophysics of Gaseous Nebulae” (1974). In equilibrium, the population of the 2$^3$S is determined by the balance of arrivals from recombination to all triplet levels versus departures mainly due to collisional transition to other levels. Since under most conditions all recombinations to the triplet levels end up in the 2$^3$S level we obtain:
$${n_{\scriptscriptstyle He^+}}n_e{\alpha_{\scriptscriptstyle T}}={n_{\scriptscriptstyle 2^3S}}\left[A_{21}+n_e(q_{tr}+q_{ci})
+\int_{{\nu_{\scriptscriptstyle 0}}}^{\infty}\frac{a_{\nu}L_{\nu}}{4\pi r^2h\nu}d_{\nu}\right],
\label{eq:he}$$
where ${n_{\scriptscriptstyle He^+}}$ is the number density of singly ionized helium, $n_e$ is the electron number density ${\alpha_{\scriptscriptstyle T}}$ is the total recombination coefficient to all triplet levels, ${n_{\scriptscriptstyle 2^3S}}$ is the number density of neutral helium in the 2$^3$S level, $A_{21}$ is the Einstein A coefficient for the forbidden transition (625 Å) from the 2$^3$S level to the ground level (1$^1$S), $q_{tr}$ is the rate of collisional transfer to all singlet level (which is dominated by collisions to the 2$^1$S and 2$^1$P levels), $q_{ci}$ is the collisional ionization rate which becomes important above 20,000 K (Clegg 1987), $a_{\nu}$ is the photoionization cross section for 2$^3$S, $L_{\nu}/(4\pi r^2h\nu)$ is the flux of ionizing photons ($L_{\nu}$ is the luminosity per unit frequency, $r$ is the distance to the emitting source and $h$ is Planck’s constant), ${\nu_{\scriptscriptstyle 0}}$ is the threshold frequency for ionizing the 2$^3$S level (4.77 eV, 2600 Å).
Equation (\[eq:he\]) can give us a maximum for the ${n_{\scriptscriptstyle He^+}}/{n_{\scriptscriptstyle 2^3S}}$ ratio. Neglecting photoionization, collisional ionization and radiative transition to the ground level (i.e., assuming $n_e$ larger than the critical density of $3\times10^3$ cm$^{-3}$) we obtain: ${n_{\scriptscriptstyle He^+}}/{n_{\scriptscriptstyle 2^3S}}={\alpha_{\scriptscriptstyle T}}/q_{tr}$ Using the values given for ${\alpha_{\scriptscriptstyle T}}$ and $q_{tr}$ given in Clegg (1987) and Osterbrock (1974) we find ${n_{\scriptscriptstyle 2^3S}}/{n_{\scriptscriptstyle He^+}}=6\times10^{-6}$ More generally, Clegg (1987) gives the above ratio as function of $n_e$ and temperature (including radiative transition to the ground level but neglecting photoionization) as: $$\frac{{n_{\scriptscriptstyle 2^3S}}}{{n_{\scriptscriptstyle He^+}}}=\frac{5.8\times10^{-6}T_4^{-1.19}}
{1+3110T_4^{-0.51}n_e^{-1}}
\label{eq:he_ratio}$$ where $T_4$ is the temperature in units of $10^4$ K. Equation (\[eq:he\_ratio\]) is a good approximation for $8,000<T<20,000$, where in our case the temperature might be somewhat lower (see below). We can use these estimates combined with our measurement for the total column density seen in the metastable lines, to set a minimal He$^+$ column density of $\sim 2\times10^{19}$ cm$^{-2}$ in the intrinsic absorber. Assuming solar abundances this estimate yields a minimal column density of $\sim 2\times10^{20}$ cm$^{-2}$.
Many absorption features are detected in the HIRES data. In table 1 we give column density measurements for the most unambiguous detections and in figure \[hires\_fe2\] we plot the absorption associated with five of these on the same velocity scale of the absorption seen in $\lambda$2803 line. The S/N of the features is lower than that of the magnesium and features due to the lower throughput of the HIRES detector at shorter wavelengths. For this reason we concentrate our discussion on feature $e$ which is the strongest one detected in all the lines.
Absorption features from excited levels of are clearly detected. These transitions arise from energy levels 0.05 eV. ($\lambda$2612, $\lambda$2396.4) and 0.08 eV. ($\lambda$2608, $\lambda$2405.6) above ground. Transitions from two slightly higher energy levels are also detected (0.11 and 0.12 eV.), although we do not include them in the table since their detection significance is lower. The $\rm{N_{\feii}}$ values given in table 1 assume LTE population of the levels at the limit $kT\gg\Delta E$ (where $k$ is Boltzmann’s constant, $T$ is the temperature and $\Delta E$ is the energy difference between the levels). That is, the ratio of optical depth of two features is given by: $$\frac{\tau_1}{\tau_2}=\frac{n_1f_1\lambda_1}{n_2f_2\lambda_2}=
\frac{g_1f_1\lambda_1}{g_2f_2\lambda_2},
\label{eq:tau_ratio}$$ where $n_1$ and $n_2$ are the level populations of lower level that give rise to each transition; $\lambda_1$ and $\lambda_2$ are the transitions’ wavelengths, $f_1$ and $f_2$ are the oscillator strengths and $g_1$ and $g_2$ are the statistical weights of the levels. For the last equality we used the assumption of LTE population at the limit $kT\gg\Delta E$. We define $gf\lambda$ as the strength factor of the line and give its value (normalized to that of the strongest transition; data from Verner, Verner & Ferland 1996) in table 1. Six of the lines (from both ground and excited states) give consistent estimates for $N_{\rm{\feii}}$, validating the LTE assumption, and like the case show that the inferred $N_{\rm{\feii}}$ are actual determinations and not lower limits. The two inconsistent $N_{\rm{\feii}}$ estimates are probably due to uncertainties in the oscillator strength of these transitions (see de Kool et al. 2000).
Since the highest energy level we detect (0.12 eV.) is equivalent to a temperature of $\sim1000$ K, we infer that the absorbing gas is at $T\gtorder2000$ K. Lower temperature will cause a significant reduction in the higher level population due to the exponential factor in the Boltzmann equation, and this is not seen in the data. We are also able to constrain the temperature from above due to the non-detection of $\lambda$2563, which arise from a 1.0 eV. level. In order to supress the 1.0 eV. level population below our detection limit we must have $T\ltorder10,000$ K. The detection of excited transitions necessitates the gas to be above the critcal density for these transitions. From this we infer $n_e\gtorder 10^5$ cm$^{-3}$ (see de Kool et al. 2000). Finally, the non detection of $\lambda2524$ necessitate $\log(U)>-5$. Photoionization models (see § 4) show that at smaller values of $U$ we should have detected an appreciable absorption in the line given the inferred column density (see Fig. \[cloudy\_u\]).
and
------
In figure \[hires\_he1\] we identify components $b, c$ and $e$ in $\lambda2853$. A detection of this line is significant since the ionization potentiol of the ion is 7.6 eV. As disscussed in de Kool et al. (2000), this low ionization energy does not allow a hydrogen ionization front to protect the ions from destruction by shielding it from ionizing photon. This is in contrast to the case of with ionization potentiol of 15.0 eV, which is higher than that of hydrogen (13.6 eV). Therefore, a hydrogen ionization front can protect the ions from photodissociation (Voit, Weymann & Korista 1993). Since shielding does not work for a low ionization parameter is needed in order for it to survive in the typical radiation environment produced by the quasar. We discuss this issue further in § 4.
Components $b$ and $c$ are most prominenet in . This gives independant support to the conclusion from the analysis that componets $b$ and $c$ have larger optical depth compared to the other components. In we do not detect components $a$ and $d$. Using the upper limits for $a$ and $d$ we conclude that $\tau_b$ and $\tau_c$ are at list five times larger than $\tau_a$ and $\tau_d$. Our spectral coverage also contains the red component of the doublet ($\lambda$3970). The line is clearly detected in component $e$ and marginally deteced in components $b$ and $c$.
HST FOC SPECTRUM
================
A low-resolution UV spectrum of QSO 2359–1241 was obtained using the Faint Object Camera (FOC) on-board the HST. The data reduction and characteristics are described in Brotherton et al (2000). Figure \[foc\_linear\_shift\] shows part of the FOC spectrum. Two deep and wide absorption features are seen in the spectrum. These are $\lambda$1549 and $\lambda$1397 BALs from the same system. Several additional absorption features are also noticeable. We use the absorption template to determine the relationship between the absorption features seen in the FOC data to those seen in the ground-based observations. To identify FOC absorption features associated with the absorption, we displace the template to the expected wavelength position of a candidate transition. This is done by multiplying the rest wavelength of the template by $\lambda_{c}/{\lambda_{\scriptscriptstyle \mbox{\mgii}}}$, where $\lambda_{c}$ is the wavelength of the candidate transition. The result (shown in figure \[foc\_linear\_shift\]) confirms the existence of absorption features from the resonance lines $\lambda$1857, $\lambda$1549, $\lambda$1397, and $\lambda$1240 associated with the intrinsic absorption. We also note that the steep intensity drop on the red wing of the [Ly$\alpha$]{} BEL can be explained by a [Ly$\alpha$]{} BAL from the same system.
As evident from figure \[foc\_linear\_shift\], there are differences in the shape of the absorption features in the FOC data to the template, especially the feature. Some of these are caused by the much lower resolution of the FOC spectrum. However, the fact that the and absorption features are more extensive than can be extrapolated from the template is not unique. QSO 0059–2735 shows a similar behavior (Weymann et al 1991, Wampler, Chugai & Petitjean 1995), where the and BALs are much wider than the BAL. In table 2 we give apparent column densities for the BALs seen in the FOC spectrum. A comparison with table 1 shows that the apparent column density of is similar to that of , a phenomenon that is also observed in QSO 0059–2735 (Weymann et al. 1991, Wampler, Chugai & Petitjean 1995) and in QSO 1232+1325 (Voit, Weymann & Korista 1993). In contrast, the apparent column density of the and BALs are roughly an order of magnitude larger than the BAL, which is a consequence of the much larger width of the and BALs. Again this is similar to what is observed in QSOs 0059–2735 and 1232+1325. We elaborate further on the relationship between the high and low ionization absorbers in the discussion.
In summary, the FOC spectrum shows the following BALs associated with the intrinsic absorption: $\lambda$1857, $\lambda$1549, $\lambda$1397, and $\lambda$1240. The low resolution of the FOC spectrum does not allow for much analysis of these BALs, other than the fact that the $\lambda$1549, $\lambda$1397 are full fledged BALs with a full-width-half-maximum of $\sim8000$ .
[lcc]{}\
& &\
& &\
& 1857 & 14.0$\pm0.15$\
& 1549 & 15.7$\pm0.1$\
& 1397 & 15.4$\pm0.1$\
& 1240 & 15.2$\pm0.1$\
IONIZATION EQUILIBRIUM AND ABUNDANCES
=====================================
In § 2 we established that the and absorption featured are not saturated, therefore their apparent column density measurements are the actual column densities of the absorber and not just lower limits. The importance of this verification can hardly be overstated. Unless we have a direct evidence that a specific absorption trough in a quasar’s outflow is not saturated, we must assume that it is, since in most cases where saturation diagnostics exist, we find that the troughs are indeed saturated (Arav 1997; Telfer et al. 1998; Arav et al. 1999a; Churchill et al. 1999; Arav et al. 1999b; De Kool et al. 2000). In these cases, using the apparent column densities as real ones undermine the ionization equilibrium and abundances results inferred from ionization models, since the output of these models are the real column densities.
Optically Thin Models
---------------------
We begin by examining photoionization models that are optically thin in the Lyman limit (${\tau_{\scriptscriptstyle LL}}\ll 1$). The output we are most interested in are the ionic column densities ($\rm{N_{ion}}$), These depend strongly on the input ionization parameter ($U$), total hydrogen column density ($\rm{N_H}$) and metalicity. To a lesser extent $\rm{N_{ion}}$ also depend on the shape of the incident continuum and are relatively insensitive to variation of ${n_{\scriptscriptstyle H}}$ in the range $ 10^5$ cm$^{-3}$ (our lower limit based on the troughs) to $ 10^{10}$ cm$^{-3}$ (the estimated density in the broad emission line region). With all other parameters fixed, $\rm{N_{ion}}$ depends linearly on $\rm{N_H}$, the atoms become more ionized with the increase of $U$ and a higher metalicity correlates linearly with higher $\rm{N_{ion}}$ for the metals. In all our models we assume solar metalicity. The elimination of this degree of freedom naturally tightens the constraints we derive for the ionization equilibrium. As we show below, models based on solar metalicity can satisfactorily reproduce the observed $\rm{N_{ion}}$. For the incident spectrum we used AGN continua given in the photoionization package CLOUDY (Ferland et al. 1996): Table AGN, which is essentially the Mathews and Ferland spectrum (Mathews and Ferland 1987) and two variants of a superposition of a black body and various power laws. We found that the result are only moderately dependent on which of the three input continua is used (see § 4.3) and therefore we concentrate on results obtained using the Table AGN continuum.
Figure \[cloudy\_u\] shows the relative $\rm{N_{ion}}$ as a function of $U$ for most of the ions we detect in the HIRES spectrum. The most important constraint arise from comparing the prediction and measurement for the ratio $\rm{N_{\feii}}/\rm{N_{\hei^*}}$ (where $^*$ designate in the 2$^3$S metastable level). When is the dominant iron ion, we expect $\rm{N_{\feii}}/\rm{N_{\hei^*}}$ to be between 30–100 since the solar abundance of iron is $3.2\times10^{-5}$ relative to hydrogen whereas that of the $^*$ is only $\sim6\times10^{-7}$ (based on eq. \[eq:he\_ratio\], assuming most helium is in and using helium abundance of 10% relative to hydrogen; we note that the temperature of the models ranges from 8000 K to 20,000 for $log(U)$ between –6 and –2, respectively). Since the observed $\rm{N_{\hei^*}}$ is somewhat larger than $\rm{N_{\feii}}$ we conclude that cannot be the dominant iron ion. From figure \[cloudy\_u\] we deduce that the observed $\rm{N_{\feii}}/\rm{N_{\hei^*}}$ exclude models with $log(U)<-3.5$. Furthermore, the considerable difference in the slopes of $\rm{N_{\feii}}$ and $\rm{N_{\hei^*}}$ allows for only a narrow range around $log(U)=-3$ for acceptable models.
What about the other observed ions? The magnesium column densities fits very well into the above picture. Comparing the measurements in table 1 to the models in figure \[cloudy\_u\] we note that the observed $\rm{N_{\mgi}}/\rm{N_{\hei^*}}$ can be reproduced by the models only in a narrow range of $log(U)\gtorder-3$. The modeled $\rm{N_{\mgii}}$ are consistent with this picture since the measurements are only lower limits (see § 2.3). For we also get consistent results and the upper limit for $\rm{N_{\fei}}/\rm{N_{\feii}}$ is readily satisfied as long as $log(U)>-5$, while the upper limit for $\rm{N_{\fei}}/\rm{N_{\hei^*}}$ necessitates $log(U)>-3.5$
Models with a Hydrogen Ionization Front
---------------------------------------
Although optically thin models can produce all the observed $\rm{N_{ion}}$ ratios, they have difficulties in explaining some of the measured column densities themselves, especially $\rm{N_{\hei^*}}$. For $log(U)=-3$, figure \[cloudy\_u\] shows that $\rm{N_{Mg}}/\rm{N_{\hei^*}}\simeq250$, or using solar abundances $\rm{N_{H}}/\rm{N_{\hei^*}}\simeq7\times10^6$. Therefore, in order to produce the observed $\rm{N_{\hei^*}}=6\times10^{13}$ cm$^{-2}$ we need a total $\rm{N_{H}}=4\times10^{20}$ cm$^{-2}$. This amount of $\rm{N_{H}}$ produces a strong hydrogen ionization front (i.e., a region where the dominant hydrogen ion shifts from to ) with ${\tau_{\scriptscriptstyle LL}}> 1000$ for $log(U)=-3$, thus invalidating our optically thin assumption. Moreover, models with such a thick hydrogen ionization front cannot produce the observed $\rm{N_{ion}}$ either. For the specific example above, the predicted $\rm{N_{\feii}}/\rm{N_{\hei^*}}$ is more than 100 times the observed one.
Figure \[cloudy\_thick\] illustrates the situation for models with a hydrogen ionization front. Prior to the development of the hydrogen ionization front, the relative column densities are similar to the ones in the optically thin models. We chose to present a model with $log(U)=-3$ since the predicted ratio $\rm{N_{\feii}}/\rm{N_{\hei^*}}$ agreed with the observed one in the optically thin part. Behind the front the situation changes radically. The appearance of lines from the 2$^3$S metastable level requires a significant fraction of in the gas. However, in close proximity to the ionization front we find a ionization front making the dominant helium species. Therefore, the fraction of behind a hydrogen ionization front decreases sharply and with it that of $^*$ (see eq.(\[eq:he\_ratio\])). We note that the increase in the relative fraction of $^*$ in the vicinity of the front is mainly due to the helium ionization transition. For optically thin models with $log(U)=-3$ the more abundant helium ion is (although only by a factor 1.4 compared to ). As the equilibrium shifts towards , at some point becomes most abundant, which causes the abundance peak for the $^*$. As explained above, at a higher column density become the dominant species accompanied by a sharp decline in the fraction and hence in that of $^*$.
For the lines the situation is quite the opposite. A hydrogen ionization front protects the ions from photoionization and thus making it the dominant species. As a result, almost all the iron behind the front is in the form of . In the example given in figure \[cloudy\_thick\] the relative fraction of increases by a factor of 300 across the front, whereas the fraction of and with it that of $^*$ decreases sharply behind the front. Regardless to the $U$ value of the incident spectrum, a strong hydrogen ionization front always causes to be the dominant iron species behind the front. Since even under the most favorable conditions $\rm{N_{Fe}}/\rm{N_{\hei^*}}\gg 10$ (based on eq. \[\[eq:he\_ratio\]\], assuming solar abundances and the allowed temperature range discussed in § 2.6), we conclude that models with a hydrogen ionization front cannot reproduce the observed and $^*$ column densities simultaneously.
Realistic Model Fits
--------------------
From the analysis above we concluded that optically thin models have difficulties in reproducing the observed column densities while models with a hydrogen ionization front fail completely. The obvious thing to try next are models with an intermediate optical depth (${\tau_{\scriptscriptstyle LL}}$ of a few). We aim to find models that can yield the observed column densities of component $e$ (see Figs \[hires\_he1\] and \[hires\_fe2\], and table 1) for which we have the most accurate measurements. In doing so we put most weight on the and results since we have direct evidence that the column densities we measure for these two ions are real ones and not lower limits. It is highly probable that the and are also not saturated (due to similarities with the and features, as well as their moderate depth), however we do not have direct diagnostics to confirm that.
In figure \[ionization\_models\] we show the column density results from three models as well as the observed ones. In order to keep the models simple we used two AGN continua given in the photoionization package CLOUDY (Ferland et al 1996; A full description is given in the “Hazy” document which describes the code and is available at: http://nimbus.pa.uky.edu/cloudy/cloudy\_94.htm). The “Table AGN” model is fully determined by the package and is described in detail in Mathews and Ferland (1987). For the model which is a superposition of a black body “big bump” and various power laws, we used two settings: 1) the default parameter choices given in the Hazy document , T=150,000 K, $\alpha_{ox}=-1.4$, $\alpha_{uv}=-0.5$, $\alpha_{x}=-1$; 2) T=100,000 K, $\alpha_{ox}=-2$, $\alpha_{uv}=-0.5$, $\alpha_{x}=-1$, which gives a somewhat lower temperature across the cloud. The average temperature for each model is given in Table 3, where the difference in temperature across each model is less than 15%. As noted above, the column densities are largely insensitive to variation of ${n_{\scriptscriptstyle H}}$ in the range $10^5$ cm$^{-3}$ to $10^{10}$ cm$^{-3}$. For the presented models we chose ${n_{\scriptscriptstyle H}}$ value close to our -inferred lower limit, since they gave somewhat lower temperatures, in better agreement with the temperature constraints. The Table AGN and big bump models give excellent fits for the and column densities. The models over-predict $\rm{N_{\mgi}}$ by about a factor of three and give a reasonable fit for $\rm{N_{\caii}}$. Although we did not include aluminum in figure \[ionization\_models\] (since we do not have high-resolution data for it), we note that these models also reproduce the apparent $\rm{N_{\aliii}}$ (see table 2) to within a factor of two. The low abundance of aluminum ($[Al/H]_{\odot}=3\times10^{-6})$ coupled with the inferred $\rm{N_{H}}$ in the object, suggests a low level of saturation (if any) in the absorption, thus implying that the apparent $\rm{N_{\aliii}}$ is a reasonable approximation for the actual one.
How does the presence of other flow components affect our ionization analysis? Although the measurements of components $a-d$ are not as accurate, it seems that their column density ratios, and therefore their ionization equilibrium, are similar to that of component $e$. This situation argues in favor of models with with lower ${\tau_{\scriptscriptstyle LL}}$. A moderately strong ${\tau_{\scriptscriptstyle LL}}$ ($\gtorder4$) in a given component will strongly attenuate the incident ionizing spectrum seen by components further from the central source. As a result, their ionization equilibrium will be different. If component $e$ is the furthest away from the source and is the only optically thick one, we still need fine tuning to have similar column density ratios in optically thin and optically thick slabs. The higher ${\tau_{\scriptscriptstyle LL}}(e)$ is, the more difficult it is to explain the similar ionization equilibrium in the other components. Based on these arguments we prefer models that fits the data with the smallest ${\tau_{\scriptscriptstyle LL}}(e)$. This is best achieved by choosing incident continuum that yields low temperature. As evident from equation (\[eq:he\_ratio\]), with all else equal, we get higher concentration of in the 2$^3$S level at lower temperatures. A smaller $\rm{N_{H}}$ is then needed for a given $\rm{N_{\hei^*}}$, which leads to smaller ${\tau_{\scriptscriptstyle LL}}$. Additional support for lower temperature models come from the inferred T$<10,000$ K (see § 2.6).
[lccccc]{}\
& & & & &\
& & & & &\
table agn & 20.2 & -2.7 & $10^6$ & 6.5 & 12\
agn bb 1 & 20.0 & -2.8 & $10^5$ & 3.6 & 10\
agn bb 2 & 19.9 & -2.7 & $10^5$ & 1.5 & 8\
Conditions Needed for Detecting Absorption
-------------------------------------------
In spite of their high S/N, our low-resolution observations do not reveal absorption from $\lambda$2853. The reason for this is the small equivalent width of the features, which in our case can only be detected with a combination of high S/N - high spectral resolution data. As we discuss in § 2 the detection of puts important constraints on photoionization models. It is therefore important to observe BALQSOs at high spectral resolution in order to determine the existance of in the intrinsic absorber.
The ionization models account for the weakness of the features compared to those of . For material optically thin at the Lyman edge, figure \[cloudy\_u\] shows that for a typical AGN spectrum $\rm{N_{\mgii}}/\rm{N_{\mgi}}\gtorder100$ for $-6<\log(U)<-3$; this result also holds up to $\log(U)=-8$. Thus, for $-8<\log(U)<-3$ the dominant magnesium ion in absorption is with showing roughly 1% of it’s column density. That translates to a factor of 30 difference between the optical depths of $\lambda$2796 and $\lambda$2853. If the absorber is highly saturated we may detect absorption even in low-resolution data, as is the case in QSO 1044+3656 (de Kool et al. 2000). Otherwise, only high resolution data can reveal its existance in the deepest parts of the troughs, which is the case in QSO 2359–1241. In order to detect at $\log(U)\gtorder-2$ a hydrogen ionization front is needed (Voit, Weymann & Korista 1993) to keep the relative fraction of ions high enough. But in that case, we should not expect to see absorption features. For example, a model using table AGN continuum, $\log(U)=-2, {N_{\scriptscriptstyle H}}=10^{22}$ cm$^{-2}$ (a combination which produces a thick hydrogen ionization front) and ${n_{\scriptscriptstyle H}}=10^{8}$ cm$^{-3}$, predicts that behind the front 99% of the magnesium is in the form of while only $2\times10^{-4}$ is in . A good observational example is QSO 0059–2735, where is detected, is not (Wampler, Chugai & Petitjean 1995) and a black Lyman limit is observed (Turnshek et al. 1996).
Relationship Between the High and Low Ionization Absorbers
----------------------------------------------------------
What is the relationship between the low-ionization absorption (, , ...) and the high-ionization absorption (, , ...) that are seen in the spectrum of low-ionization BALQSOs? Voit, Weymann & Korista (1993) noted that the low-ionization absorption tends to concentrate at low ejection velocity compared to the high ionization absorption, where the latter normally encompass the velocity range of the former and extends to a much higher velocity. The best examples for that behavior are QSO 0059–2735 and QSO 1232+1325. However, this is not always the case. QSO 0932+5010 shows strong absorption in its high velocity trough (data courtesy of Ray Weymann and Kirk Korista) and in QSO 2359–1241 we observe a high velocity trough at –5000 . As discussed in § 3, our HST UV prism-spectroscopy data show a wide BAL trough in both and (FWHM $\sim8000$ ), which encompasses both the high and low velocity troughs seen in . This relationship strongly suggests that the low-ionization absorption seen in QSO 2359–1241 is indeed physically connected to the BAL flow seen in the UV lines. It appears that a simple picture where the low ionization outflow is confined to low velocity does not hold. A more elaborate model, perhaps including ionization stratification (Arav et al. 1999a), is called for. This picture is strengthen by the observation that in QSO 0059–2735 there is a clear detection of $\lambda1034$ BAL (Turnshek et al. 1996). It is very difficult to construct a single-zone ionization model where significant optical depth arises from both and . We point out that ionization models with large local density gradient give natural explanation for such occurrence (Arav 1996; Arav et al. 1999a) since they allow for material with large variation in ionization parameter to exist at close proximity.
SUMMARY
=======
The spectrum of QSO 2359–1241 contains powerful diagnostics for the state of it’s outflow. In particular we emphasize the importance of the lines from the 2$^3$S meta-stable level. Under optimal conditions, the abundance of this level is roughly 30 times lower than the abundances of iron and magnesium. Detecting a somewhat higher column density of the 2$^3$S meta-stable level than that of indicates that 95–99% of the the iron is in higher ionization stages and allows for tight constraints on the ionization parameter. The three well separated lines also allow for an excellent saturation/covering-factor analysis, since their oscillator strength differ by a factor of six. With the combined constraints available from the and lines we are able to tightly constrain the ionization equilibrium in the flow; reproduce the observed column densities without invoking departure from solar abundances; and exclude a hydrogen ionization front in this outflow.
A strong connection between the low ionization absorber and the BAL phenomenon is evident in the HST FOC data. High ionization BALs (from , and ) are detected in the HST data and these encompass and expand the velocity range seen in the low ionization species. It will be very valuable to obtain better HST spectroscopy of the high ionization lines, as well as ground observations of the $\lambda$1857 absorption. These will allow to study the connection between the high and low ionization absorbers in greater detail, and will yield additional constraints on the low-ionization absorber through the $\lambda$1857 and $\lambda$1670 lines.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We thank Kirk Korista, Martijn de Kool and the referee for numerous valuable suggestions. We acknowledge support from NASA HST grant GO-06350, NSF grant AST-9802791 and STScI. Part of this work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.
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[^1]: Physics Department, University of California, Davis, CA 95616 I: arav@astro.berkeley.edu
[^2]: Kitt Peak National Observatory, 950 North Cherry Avenue, P. O. Box 26732, Tuscon, AZ 85726
[^3]: IGPP LLNL, L-413, P.O. Box 808, Livermore, CA 94550
[^4]: Space Telescope Science Institute, Baltimore, MD 21218
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---
author:
- Andreas Reuter
- Randy Bush
- Italo Cunha
- 'Ethan Katz-Bassett'
- 'Thomas C. Schmidt'
- Matthias Wählisch
bibliography:
- 'rov-measurement-ccr.bib'
title: Towards a Rigorous Methodology for Measuring Adoption of RPKI Route Validation and Filtering
---
[0.8]{}(0.1,0.02) If you cite this paper, please use the CCR reference: A. Reuter, R. Bush, I. Cunha, E. Katz-Bassett, T. C. Schmidt, M. Wählisch. 2018. Towards a Rigorous Methodology for Measuring Adoption of RPKI Route Validation and Filtering. *ACM SIGCOMM Computer Communications Review (CCR)* 48(1) (January 2018), pp. 19–27.
<ccs2012> <concept> <concept\_id>10003033.10003039.10003045.10003046</concept\_id> <concept\_desc>Networks Routing protocols</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003033.10003079.10011704</concept\_id> <concept\_desc>Networks Network measurement</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003033.10003083.10003014.10003015</concept\_id> <concept\_desc>Networks Security protocols</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003033.10003106.10010924</concept\_id> <concept\_desc>Networks Public Internet</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
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abstract: 'We use molecular dynamics simulations to explore the impact of a non-ionic solvent on the structural and capacitive properties of supercapacitors based on an ionic liquid electrolyte and carbon electrodes. The study is focused on two pure ionic liquids, namely 1-butyl-3-methylimidazolium hexafluorophosphate and 1-butyl-3-methylimidazolium tetrafluoroborate, and their 1.5 M solutions in acetonitrile. The electrolytes, represented by coarse-grained models, are enclosed between graphite electrodes. We employ a constant potential methodology which allows us to gain insight into the influence of solvation on the polarization of the electrodes as well as the structural and capacitive properties of the electrolytes at the interface. We show that the interfacial characteristics, different for two distinct pure ionic liquids, become very similar upon mixing with acetonitrile.'
author:
- Céline Merlet
- Mathieu Salanne
- Benjamin Rotenberg
- 'Paul A. Madden'
title: Influence of solvation on the structural and capacitive properties of electrical double layer capacitors
---
double layer capacitance ,solvation ,molecular dynamics simulations ,interfacial structure ,graphite
Introduction {#Intro}
============
Supercapacitors store energy at the electrode/electrolyte interface without involving faradaic reactions; this confers on them characteristics very distinct from batteries. They can be used as high-power generators and can undergo one million charge/discharge cycles without deterioration. Nevertheless, compared to batteries, they suffer from a relatively low energy density. There are four principal components in supercapacitors on which we can act to optimize these systems: the active matter, the current collectors, the separator and the electrolyte. As the energy stored in a supercapacitor is proportional to the capacitance and the square of the operating voltage ($E=\frac{1}{2}CU^2$), the improvements will come by optimizing the electrode morphology, which determines the capacity of the system, and the electrolyte, which sets the maximum voltage by its decomposition limit [@Simon12]. The modifications of these two components will also impact the power which can be delivered by the capacitor, as this property is a function of the maximum voltage and resistance ($P=\frac{U}{4R}$).
Focusing on liquid electrolytes, different fluids have been studied up to date which are suited for distinct applications. Aqueous electrolytes are attractive because of their high ionic conductivities ($>$ 400 mS.cm$^{-1}$), which allow for a higher specific power, but they have relatively narrow electrochemical windows (1.2 V) [@Wang12b]. On the contrary, ionic liquids (ILs) and organic electrolytes exhibit larger electrochemical windows, up to 5 V and 3 V respectively [@Galinski06; @Zhang09; @Balducci07]. Ionic liquids have a number of attractive properties such as low combustibility, high thermal stability and low vapor pressure, which make them *a priori* safe. They are also adaptable thanks to the broad choice of anions and cations that can be combined. From mixtures of different ILs, various operating conditions can be ameliorated, for example the working temperature range can be enlarged [@Lin11]. The major drawback of using ILs in supercapacitors is their low ionic conductivity ($<$ 15 mS.cm$^{-1}$) [@Galinski06].
Consequently, in many experimental studies and applications of supercapacitors, organic electrolytes using acetonitrile (ACN) or propylene carbonate (PC) as solvent are still used instead of pure ILs [@Chmiola06; @Centeno11]. It has been found that adding ACN to ILs enhances greatly the ionic conductivity of the system [@Chaban12; @Wang03]. The structural and dynamic effects of solvation on the bulk properties of electrolytes have been examined by experiments [@Wang03; @Huo07; @Sadeghi11] and simulations [@Chaban12; @Wu05; @Chaban11b] but the interfacial properties of the mixtures are less thoroughly studied, especially from a theoretical standpoint. Recent electrochemical measurements on carbide-derived carbon electrodes immersed in pure 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (\[EMI\]\[TFSI\]) and in a 2 M solution of \[EMI\]\[TFSI\] in ACN display an unexpected peak in the cyclic voltammetry (CV) curve of the solution [@Lin09b]. Some experimental results indicates that this peak is not associated with a faradaic process and a clear explanation is missing. Molecular simulations of the interface between porous carbons and organic electrolytes may be needed to explain this effect, but, even simpler interfaces between, for example, planar electrodes and solutions may contribute to a better understanding.
The Gouy-Chapman theory, which describes highly-diluted solutions at smooth planar interfaces, is not applicable for common electrolytes as typical salt concentrations are 1 M or more. To study these interfaces, there is a need for more complex theories [@Kornyshev07] or molecular simulations which correctly describe ionic correlations. Up-to-date, molecular simulations involving the interface between various carbon structures (carbon nanotubes [@Frolov12; @Yang09], slit pore [@Jiang12b; @Tanaka10], graphite [@Feng10b; @Feng11b; @Shim11; @Shim12]) and organic electrolytes have mainly focused on the liquid side of the interface. In the present study, we investigate the influence of mixing two salts, namely 1-butyl-3-methylimidazolium hexafluorophosphate (\[BMI\]\[PF$_6$\]) and 1-butyl-3-methylimidazolium tetrafluoroborate (\[BMI\]\[BF$_4$\]), with ACN on the electrolyte and electrode properties. We use a comparison between pure ILs and their corresponding 1.5 M organic electrolytes to highlight the modifications of structural and capacitive properties associated with solvation. The use of a constant potential approach to model the electrodes, instead of the constant charge methodology, allows us to gain knowledge about the polarization of the electrodes by the electrolytes and how this is affected by the ACN solvent.
Computational details {#Comput}
=====================
Molecular dynamics simulations are conducted on four different electrolytes surrounded by model graphite electrodes: pure \[BMI\]\[PF$_6$\] and \[BMI\]\[BF$_4$\] and their corresponding 1.5 M solutions with ACN as a solvent. All molecules are represented by a coarse-grained model in which the forces are calculated as the sum of site-site Lennard-Jones potential and coulombic interactions. Parameters for the ions and carbon atoms are the same as in our previous works [@Merlet11; @Merlet12b]: three sites are used to describe the ACN and the cation and the anions are treated as spheres. The two ILs differ by the nature of the anion, the diameter of PF$_6^-$ being larger than the one of BF$_4^-$ by approximately 0.5 . The model for ACN was developed by Edwards *et al.* [@Edwards84]. All the parameters of the force field are recalled in table \[param\] and a schematic representation of the molecules is given in figure \[CGM\].
Each electrode is modelled as three fixed graphene layers. The electrolyte is enclosed between two planar electrodes and two-dimensional periodic boundary conditions are applied, i.e. there is no periodicity in the direction perpendicular to the electrodes. Figure \[snapshot\] shows a snapshot of the simulation cell for the ACN-\[BMI\]\[BF$_4$\] mixture with a salt concentration of 1.5 M. The molecular dynamics simulations were conducted in the NVT ensemble using a time step of 2 fs and a Nosé-Hoover thermostat [@Martyna92] with a time constant of 10 ps. The Ewald summation is done consistently with the two-dimensional periodic boundary conditions [@Reed07; @Gingrich10].
Pure ILs and electrolyte solutions are simulated at 400 K and 298 K respectively. These temperatures were chosen because of the very high viscosities of the pure ILs at room temperature (261.4 mPa.s for \[BMI\]\[PF$_6$\] and 100.2 mPa.s for \[BMI\]\[BF$_4$\] [@Tokuda04]) and the fact that ACN boils at 355 K. Nevertheless, static properties usually depend weakly on the temperature for conditions far from phase transitions. Thus, the qualitative conclusions raised in this article should hold for pure ILs at lower temperatures and the comparison with ACN-based electrolytes is relevant. The sizes of the simulation cells are chosen in order to reproduce the experimental densities of the electrolytes. Table \[cells\] gathers the lengths and number of molecules for all the simulation cells.
Site C1 C2 C3 PF$_6^-$ BF$_4^-$ N C Me
--------------------------------- -------- -------- -------- ---------- ---------- -------- ------- -------
q (e) 0.4374 0.1578 0.1848 -0.78 -0.78 -0.398 0.129 0.269
M (g.mol$^{-1}$) 67.07 15.04 57.12 144.96 86.81 14.01 12.01 15.04
$\sigma_i$ () 4.38 3.41 5.04 5.06 4.51 3.30 3.40 3.60
$\varepsilon_i$ (kJ.mol$^{-1}$) 2.56 0.36 1.83 4.71 3.24 0.42 0.42 1.59
Electrolyte Temperature (K) N$_{\rm ions}$ N$_{\rm ACN}$ $L_z$ (nm)
----------------------- ----------------- ---------------- --------------- ------------
\[PF$_6$\] 400 320 — 12.32
\[BF$_4$\] 400 320 — 11.26
ACN-\[BMI\]\[PF$_6$\] 298 96 896 12.27
ACN-\[BMI\]\[BF$_4$\] 298 96 896 11.89
Following our previous works [@Merlet11; @Merlet12b; @Merlet12], the electrodes are held at constant potential using a method developed by Reed [*et al*]{} [@Reed07] from an original proposal by Siepmann and Sprik [@Siepmann95]. A potential difference $\Delta\Psi^0$ is imposed between the positive ($\Psi^+$) and negative ($\Psi^-$) electrodes such that: $\Psi^+$ = $-\Psi^-$ = $\Delta\Psi^0/2$. Five potential differences between 0.0 V and 2.0 V are explored for the different electrolytes. The simulations are conducted starting with the 0.0 V potential difference and increasing it by steps of 0.5 V to facilitate the equilibration process. When the potential difference is increased, the system is allowed to equilibrate for at least 100 ps before collecting data for 1 ns. The constant potential approach is computationally expensive compared to the constant charge approach used in other works [@Frolov12; @Shim11; @Shim12] but enables the analysis of the polarization of the electrodes by the electrolyte [@Merlet12b; @Merlet12]. Furthermore, this constant potential method may be applied to irregular porous electrodes [@Merlet12].
Results and discussion {#Results}
======================
On the organization of the electrolyte at the interface
-------------------------------------------------------
The impact of the presence of solvent molecules on the structure of the electrolyte at the interface can be probed by computing molecular densities, $\rho_z$, in the direction perpendicular to the graphite electrodes. The molecular densities of the center of mass of the different species are represented in figure \[Density1\] (and \[Density2\]) for \[BMI\]\[PF$_6$\] (and \[BMI\]\[BF$_4$\]) based electrolytes. The calculated quantities for each species are normalized by the appropriate bulk densities to ease the comparison between the various electrolytes.
The first notable fact is the presence of molecular layering for all the species at the interface in both pure ILs and electrolyte solutions. This feature of the density profiles, which is well-known and has been observed for planar electrodes in both experiments [@Atkin11; @Hayes11] and simulations [@Shim12; @Merlet11; @Merlet12b; @Pounds09; @Tazi10; @Vatamanu10b], cannot be recovered by mean-field theories which neglect ionic correlations, molecular sizes and fluctuations. We can underline here again that the classical Gouy-Chapman theory is not suited for studies of concentrated electrolytes.
The second important phenomenon, which is also present for all the electrolytes examined, is the reorganization of the layers upon charging. Alternating layers of ions of opposite charge are associated with the so-called overscreening effect [@Feng11b; @Reed07; @Lanning04; @Bazant11; @Heyes81; @Esnouf88]: The charge in the first adsorbed layer overcompensates that on the electrodes, and, due to correlation between ions, the residual charge is successively overcompensated by the charge in the second adsorbed layer and so on, until the bulk density is reached. At this point, we can underline the first notable consequence of solvation which is to reduce the region where density oscillations are visible from a thickness of around 2 nm for pure ILs to approximately 1 nm for electrolyte solutions. More precisely, in the electrolyte [*s*olutions]{}, the first adsorbed layer of counterions overcompensates the charge on the electrode but the overscreening effect does not go beyond two molecular layers. This reduction of the overscreening can be highlighted by plotting the integral of the charge density of the liquid normalized by the the electrode surface charge as described by Feng *et al.* [@Feng11b]. Our results (not shown here for brevity) are in qualitative agreement with molecular dynamics simulation of the \[BMI\]\[BF$_4$\] and ACN-\[BMI-BF$_4$\] mixtures, near graphite electrodes, for which it is shown that both the intensity and extension of the overscreening effect are decreased in electrolyte solutions in comparison with neat ionic liquids [@Feng11b].
Going further into the analysis of the molecular densities, it appears that the density peaks of the first adsorbed ionic layers are enhanced in \[BMI\]\[PF$_6$\] based electrolytes compared to \[BMI\]\[BF$_4$\] based electrolytes. The positions of the first peaks in the ionic density profiles are not shifted away from the electrode upon addition of a solvent, leading to the conclusion that tightly adsorbed ions exist at the interface. This situation can be visualized in snapshots of the simulations (see figure \[SnapNV\]). In a given electrolyte, the ionic species also show different affinities for the graphite surface, the resulting dissymmetry in the density profiles being larger in the ACN containing mixtures. A common feature is the important variation of the heights and positions of the ionic density peaks when the potential difference is changed.
On the contrary, the molecular density of ACN depends neither on the nature of the anion nor on the potential difference applied. As a consequence, while in pure ILs, the potential difference increase induces mainly a polarization of the layers near the electrodes, in electrolyte solutions, exchanges of ions between different layers occur. One should keep in mind that the ions, in the 1.5 M solutions, represent only 10 % of the molecules. Thus, even if the ionic densities are modified upon charging, the major component at the interface and in the bulk is the solvent which can probably accommodate the charging of the electrode by rearranging only a small number of molecules. On the positive electrode side, this induces a structure where the counter-ions and ACN molecules are located in the same plane (see figure \[SnapNV\]). The cations are simply reoriented and the solvation shell of the ions is slightly distorted in the vicinity of the graphite surface. On the negative electrode side, the positions of the anions are shifted away from the graphite surface but still lie in the same plane as the ACN molecules.
Another notable consequence of the presence of solvent is the stronger expulsion of co-ions from the first adsorbed layer for the 2 V potential difference. Indeed, in pure ILs for $\Delta\Psi$ = 2.0 V, the heights of the co-ion peaks of the first layers are reduced by a factor between 2 and 6 but peaks are still apparent. For the same potential difference in electrolyte solutions, co-ions are almost absent in the first adsorbed layer. This suggests that the removal of co-ions is facilitated in the electrolyte solution.
Another way of looking at these results is to plot the free energy profiles of the various species: $$A(z) - A_{bulk} = -kT\ln(\frac{\rho_z}{\rho_{bulk}}),$$ where $k$ is the Boltzmann constant, $T$ is the temperature of the system and $\rho_z$/$\rho_{bulk}$ is the molecular density at a given position normalized by the bulk density. The free energy profiles for the anions and cations in the \[BMI\]\[BF$_4$\] and ACN-\[BMI\]\[BF$_4$\] electrolytes are shown in figure \[free\_eng\]. For the zero potential difference, all free energy profiles are characterized by a well near the wall at a distance of approximately 0.4 nm. For a 2 V potential difference, the energy barrier to overcome for an ion to go from the bulk to the first adsorbed layer increases for a favorably charged surface and decreases for an unfavorably charged surface. The same observation was made from simulations of 1,3-dimethylimidazolium chloride near graphite walls using positive, negative and neutral probes [@Lynden-Bell12]. In the case of unfavorably charged surfaces, this well is still visible in the solvent-free electrolyte, even if very small in the case of BF$_4^-$, but missing in ACN-\[BMI\]\[BF$_4$\]. The curves for the \[BMI\]\[PF$_6$\] based electrolytes (not shown) lead to the same conclusions. The fact that co-ions are expelled from the first adsorbed layer more easily in electrolyte solutions, in comparison to pure ILs, is linked with the decrease of ion-ion correlations upon addition of a solvent and is consistent with a charging mechanism where ions can be exchanged between different layers at the interface.
The next step in the structural analysis is the calculation of coordination numbers in the bulk and at the interface for various potential differences. The coordination number for each species was estimated as the average number of molecules at a distance smaller than the first minimum in the appropriate bulk radial distribution function. The coordination numbers at the interface were computed as the average coordination number for molecules located in the first adsorbed layer. All the computed values are summarized in table \[coord\].
Electrode Potential Electrolyte $N_{\rm C}({\rm A})$ $N_{\rm A}({\rm C})$ $N_{\rm ACN}({\rm A})$ $N_{\rm ACN}({\rm C})$
--------------------- ----------------------- ---------------------- ---------------------- ------------------------ ------------------------
\[BMI\]\[PF$_6$\] 6.0 6.0 – –
ACN-\[BMI\]\[PF$_6$\] 1.8 1.8 9.3 6.7
\[BMI\]\[BF$_4$\] 6.0 6.0 – –
ACN-\[BMI\]\[BF$_4$\] 1.9 1.9 8.8 6.7
\[BMI\]\[PF$_6$\] 5.0 5.0 – –
ACN-\[BMI\]\[PF$_6$\] 1.6 1.6 7.0 5.0
\[BMI\]\[BF$_4$\] 4.8 4.8 – –
ACN-\[BMI\]\[BF$_4$\] 1.4 1.4 7.2 5.2
\[BMI\]\[PF$_6$\] 5.1 4.0 – –
ACN-\[BMI\]\[PF$_6$\] 2.1 0.9 6.7 5.5
\[BMI\]\[BF$_4$\] 4.9 3.9 – –
ACN-\[BMI\]\[BF$_4$\] 2.0 0.8 6.2 5.1
\[BMI\]\[PF$_6$\] 4.6 5.4 – –
ACN-\[BMI\]\[PF$_6$\] 1.2 2.2 7.7 5.2
\[BMI\]\[BF$_4$\] 4.6 5.5 – –
ACN-\[BMI\]\[BF$_4$\] 0.9 2.1 7.9 5.3
Firstly, we focus on the counter-ion coordination numbers around a given ion. At 0V, we observe a systematic decrease of this coordination number at the interface compared to the bulk, due to the proximity of the carbon atoms of the electrode surface. This reduction is more pronounced in the pure ILs (-1.0 to -1.2 units) compared to the ACN-based electrolytes (-0.2 to -0.5 units), with an increased effect when the anion is BF$_4^-$. As soon as a positive (respectively negative) potential is applied, the anions (cations) coordination sphere tends to diminish further due to the presence of compensating charge at the surface of the electrode. Equally, the cation (anion) interaction with the carbon now needs to be screened, provoking an increase of the coordination number, which can even go above the bulk one in the case of electrolyte solutions.
Secondly, we consider the coordination numbers of ACN around ions for electrolyte solutions which can be referred to as solvation numbers. An interesting feature is the larger decrease of coordinating ACN compared to coordinating ions at the interface. This shows that it is somewhat easier to remove ACN from the coordination shell compared to ions. We note that, upon polarization, very slight changes are observed in the ACN coordination numbers around ions.
We can also see that the PF$_6^-$ ions are more readily desolvated than the BF$_4^-$ ions. This statement is consistent with the Born model which describes the Gibbs energy of the ion-solvent interaction as: $$\Delta G_{IS} = -\frac{z^2e^2N_a}{8\pi\varepsilon_0r_{ion}} \times \left (1-\frac{1}{\varepsilon_r} \right ),$$ where $z$ is the valence of the ion, $e$ is the elementary charge, $N_a$ is Avogadro constant, $\varepsilon_0$ and $\varepsilon_r$ are respectively the permittivity of vacuum and solvent. When the ion size is smaller, the interaction energy increases and the desolvation is more difficult.
On the polarization of the electrodes
-------------------------------------
Our constant potential approach for modelling electrochemical systems allows us to gain insight into the influence of the presence of solvent molecules on the polarization of the electrodes. The charges on the electrode atoms fluctuate during the simulations and it is possible to plot charge distribution functions, i.e. the fraction of carbon atoms that have a given charge. Charge distributions for the studied electrolytes and various potential differences are shown in figure \[histo\] (the analysis is focused on the first graphene layer near the electrolyte). The mean charge and the charge corresponding to the maximum occurrence are detailed in table \[mean\_max\].
Electrolyte Mean charge (e) Most frequent charge (e)
----------------------- ---------------------- -------------------------------------
\[PF$_6$\] $\pm$ 8.02 $10^{-3}$ + 5.97 $10^{-3}$ / - 7.96 $10^{-3}$
\[BF$_4$\] $\pm$ 9.10 $10^{-3}$ + 6.96 $10^{-3}$ / - 7.96 $10^{-3}$
ACN-\[BMI\]\[PF$_6$\] $\pm$ 8.91 $10^{-3}$ + 8.96 $10^{-3}$ / - 8.40 $10^{-3}$
ACN-\[BMI\]\[BF$_4$\] $\pm$ 8.65 $10^{-3}$ + 8.47 $10^{-3}$ / - 8.45 $10^{-3}$
The presence of solvent seems to have two effects on the polarization of the electrodes. The first one is a decrease in the skewness of the charge distributions shapes. When the distribution is less skewed, the charge corresponding to the most frequent charge is closer to the average charge. Secondly, going from pure ILs to electrolyte solutions, the charge distribution functions become very similar when passing from \[BMI\]\[PF$_6$\] to \[BMI\]\[BF$_4$\] and can be superimposed. The difference between mean charges is reduced from 13 % between pure ILs to 3 % between electrolyte solutions.
Independently of the nature of the electrolyte, there is a broadening of the charge distribution functions going from negative potentials to zero potentials to positive potentials. This may be attributed to the asymmetry between anions and cations in the ILs with the smaller anions inducing larger local positive charges on the graphite. When solvent is present, the orientation of the dipolar ACN molecules changes slightly with the sign of the electrode potential, which can induce different polarization at the surface. Our constant potential calculations thus reveal the importance of several factors on the polarization of the electrodes: i) The potential difference applied generates a different environment near the electrode and a broadening/contraction of the distribution curve, ii) In pure ILs, the nature of the anion has an influence on the shape of the curves, iii) The solvation of salts induces a shift and symmetrization of the charge distribution curves, and reduces the impact of the size of the anion.
On the capacitive behavior of the system
----------------------------------------
The presence of solvent has an effect on the molecular densities at the interface, on the polarization of the electrodes and consequently on the capacitive properties of the system. The differential capacitance of each interface depends on the surface charge on the electrode and on the potential drop across the interface. The surface charge is taken as the average total charge on the electrode divided by the surface area of one graphene layer. The potential drop is extracted from the electrostatic potential profile which is a function of the charge density and is described by Poisson’s equation: $$\Psi(z) = \Psi_q(z_0) - \frac{1}{\varepsilon_0}\int_{z_0}^zdz'\int_{-\infty}^{z'}dz''\rho_q(z''),$$ where $z_0$ is a reference point inside the left-hand electrode and thus, $\Psi_q(z_0)$ = $\Psi^+$, $\varepsilon_0$ is the vacuum permittivity and $\rho_q(z)$ is the charge density. The two potential drops, depending on the considered electrode, are defined as follows [@Pounds09; @Tazi10; @Merlet11]: $$\Delta\Psi^{\pm} = \Psi^{\pm} - \Psi_{\rm bulk},$$ and the differential capacitances, for the positive and negative electrodes, result from the differentiation of the surface charge with respect to these potential drops: $$\rm C^{\pm} = \frac{\partial\sigma_{\rm S}}{\partial\Delta\Psi^{\pm}}.$$ Figure \[capa\] gives the surface charge variations as a function of the potential drops for the studied electrolytes. All the functions plotted show linear trends over the range of potentials sampled and the differential capacitances are calculated as the slopes of these functions and gathered in table \[capa\_values\]. We note that the capacitances were all evaluated separately for negative and positive electrodes but in the case of electrolyte solutions, in the light of the errors made in the estimation of the average charges and potential drops (see figure \[capa\]), it would be possible to fit the entire curve by a single linear function.
Electrolyte C$^+$ ($\mu$F.cm$^{-2})$ C$^-$ ($\mu$F.cm$^{-2})$
----------------------- -------------------------- --------------------------
\[PF$_6$\] 3.9 ($\pm$ 0.3) 4.8 ($\pm$ 0.5)
\[BF$_4$\] 3.9 ($\pm$ 0.3) 5.5 ($\pm$ 0.1)
ACN-\[BMI\]\[PF$_6$\] 4.6 ($\pm$ 0.2) 4.6 ($\pm$ 0.2)
ACN-\[BMI\]\[BF$_4$\] 4.8 ($\pm$ 0.2) 4.3 ($\pm$ 0.2)
Looking at the surface charge versus potential drop plots and capacitance values, it clearly appears that the trend is the same as for the polarization of the electrodes: In the solutions, the behavior of the two salts become very similar on both the positive and negative sides, despite the strong asymmetry in size and shape between the ions (this is particularly true for \[BMI\]\[BF$_4$\]). On the positive electrode side, the capacitance is increased when going from pure ILs to solvated ions, and the reverse is observed for the negative electrode side, leading to a mean value of around 4.6 $\mu$F.cm$^{-2}$ for both interfaces in the electrolyte solutions.
This uniformization of the capacitive behaviors upon addition of a solvent is consistent with the molecular density profiles which are dominated by a small dependency on the electrode potential for ACN and with the charge distributions functions which are more gaussian shaped for electrolyte solutions compared to pure ILs. In the presence of ACN in the first adsorbed layer, specific adsorption effects due to the molecular details appear to be wiped off, even if at this concentration highly-diluted theories remain irrelevant.
We would like to point out here that we could expect a lower capacitance value for the diluted electrolytes compared to the pure ionic liquids as the screening efficiency decreases with a decrease of ionic concentration. Our results reveal that this not the case. It presumably reflects the fact that the solvent enables cations and anions to be more readily separated by the application of a potential difference so that the layer compensating the charge on the electrode is narrower consistently with the reduction of the overscreening in electrolyte solutions.
The fact that the capacitance of each interface is nearly constant upon addition of a solvent was also observed in other molecular simulations. Feng *et al.* [@Feng11b] studied the interface between ACN-\[BMI\]\[BF$_4$\] electrolytes and graphite with a mass fraction of ACN ranging between 0 % and 50 % (their highest mass fraction is slighlty smaller than our 63 % mass fraction of ACN for the ACN-\[BMI\]\[BF$_4$\] solution). They show that the capacitance of the system is nearly constant and comprised between 6.5 $\mu$F.cm$^{-2}$ and 7.0 $\mu$F.cm$^{-2}$. With a slightly different electrolyte, ACN-\[EMI\]\[BF$_4$\], Shim *et al.* also reach the conclusion that the capacitance depends only weakly on the presence of the solvent. Interestingly, they do not observe the uniformization of the negative and positive capacitances, and we observe quantitative differences with their results. Our simulation procedure differs a lot from theirs due to the use of a constant potential approach for the electrodes with coarse-grained electrolytes in our case, while these authors have used a constant charge method with all-atom force fields. This observation should be explored in future works. We note that in a completely distinct electrolyte consisting of \[Li\]\[PF$_6$\] and solvent mixtures [@Vatamanu12b; @Xing12b], an asymmetry between negative and positive electrodes was noticed but, in this case, the dissymmetry between the anion and the cation is much more important.
Conclusions {#Conclusion}
===========
In this article we have examined the impact of the presence of non-ionic solvent on the structural, polarization and capacitive properties of the interfaces between planar graphitic electrodes and liquid electrolytes. The main effect of solvation on the structure of the interface is the reduction of the region where ionic/molecular layering is observed near the graphite electrodes. Although the density of ions at the surface is lower in the solutions, this reduction in layering appears to result in a smaller reduction in the capacitance with respect to the pure ILs than might be expected. The molecular density profiles also show that the ACN molecules are only weakly affected by the potential difference applied between the electrodes. The reorganization of the layers upon charging of the electrodes varies when going from pure ILs, in which a polarization of the ionic layers occur, to electrolyte solutions, where a mechanism based on exchange of ions between the different layers is at play. From the coordination numbers at the interface, we conclude that, for the electrolyte solutions, the coordination number of ACN molecules around ions is more affected by the interface than the coordination numbers between ions.
The polarization of the electrode is highly influenced by the type of electrolyte present. Solvent-free electrolytes generates charge distributions functions with skewed shapes and are impacted by the nature of the ions. On the contrary, charge distributions curves for electrolyte solutions have shapes closer to gaussians and do not depend on the size of the anion. In pure ILs, the polarization of the electrode also leads to a larger dissymmetry between positive and negative applied potentials compared to electrolyte solutions because of the asymmetry of the ions.
The effect of solvation extends to the capacitive properties of the system. When a solvent is present in the electrolyte, the size/asymmetry of the ions do not generate different capacitances and the two solutions have similar properties. Moreover, the dissymmetry between positive and negative potentials is attenuated and a general linear trend is observed for the surface charge versus potential drop curves.
This work raises conclusions about equilibrium interfaces between planar graphitic electrodes and free-solvent/electrolyte solutions which cannot be extended straightforwardly to porous electrodes systems. The effect of solvation in supercapacitors including porous electrodes will require further work in order to understand experimental results and design new electrolytes/electrodes. The impact of solvation on dynamic properties and charge/discharge processes should also be investigated to go further into the understanding of supercapacitors.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge the support of the French Agence Nationale de la Recherche (ANR) under grant ANR-2010-BLAN-0933-02 (‘Modeling the Ion Adsorption in Carbon Micropores’). We are grateful for the computing resources on JADE (CINES, French National HPC) obtained through the project x2012096728. This work made use of the facilities of HECToR, the UK’s national high-performance computing service, which is provided by UoE HPCx Ltd at the University of Edinburgh, Cray Inc and NAG Ltd, and funded by the Office of Science and Technology through EPSRC’s High End Computing Programme.
[40]{} natexlab\#1[\#1]{}\[2\][\#2]{} , , , DOI: 10.1021/ar200306b (). , , , () . , , , () . , , () . , , , , , , , () . , , , , , , , , , , () . , , , , , , () . , , , () . , , , , () . , , , , () . , , , () . , , () . , , , , () . , , () . , , , , , , , , () . , () . , , , , () . , , , , () . , , , , () . , , , , , , , , () . , , , , , () . , , , , , () . , , , () . , , , () . , , , , () . , , , () . , , , () . , , , () . , , , () . , , () . , , , , , () . , , () . , , () . , , , , , , , () . , , () . , , , , , , , , () . , , , , , , () . , , , , () . , , , , , , () . , , , () . , , () . , , , () . , , () . , , , () . , , , () . , , , () . , , , , , () .
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abstract: 'An existence of the charge-induced instability is well known for the $^4$He crystal surface in the rough state. Much less is known about charge-induced instability at the $^4$He crystal surface in the smooth well-faceted state below the roughening transition temperature. To meet the lack, we examine here the latter case. As long as the electric field normal to the crystal facet is below the critical value same as for the rough surface, the crystal faceting remains absolutely stable. Above the critical field, unlike absolutely unstable state of the rough surface, the crystal facet crosses over to the metastable state separated from new crushed state with a potential barrier proportional to the square of the linear facet step energy. The onset and development of the instability at the charged crystal facet has much in common with the nucleation kinetics of first-order phase transitions. Depending on the temperature, the electric breaking strength is determined either by thermal activation at high temperatures or by quantum tunneling at sufficiently low temperatures.'
author:
- 'S. N. Burmistrov'
title: 'Charge-induced instability and macroscopic quantum nucleation phenomena at the crystal $^4$He facet'
---
Introduction
============
It is well known that a charged interface between two fluids can develop an electrohydrodynamic instability at sufficiently high density of charges. Such charge-induced instability results from the competition between the electric forces of like charges and forces of surface tension and gravity. Naturally, the liquid phases of helium have become one of physical systems for the theoretical and experimental studies of electrohydrodynamic instabilities [@Lei92], in particular, as softening of the gravitational-capillary wave spectrum [@Gor; @Mim; @Lei79], charge-induced deformations of the interface [@Ebn; @Ike; @Shi], formation of regular array of dimples [@Ebn; @Ike; @Shi], individual multielectron dimples [@Mel81], and hexagonal reconstructuring [@Mel82].
It is of particular interest to compare the onset and development of the electrohydrodynamic instability at the liquid-solid $^4$He interface with that at the interface between two fluids. The first theoretical and experimental studies have shown that a charged-induced instability at the superfluid-solid $^4$He looks roughly like the electrohydrodynamic instability at the free liquid $^4$He surface [@Sav; @Uwa; @Lei95; @Bod]. For the high temperature region where the crystal surface is in the rough state, such similar behavior is expectable since the superfluid-solid interface in the rough state has a very high mobility and interface excitations represent weakly damping crystallization waves whose dispersion [@And; @Kes] is quite similar to that of usual gravitational-capillary waves at the free liquid surface.
To date, no systematic study has been made on the onset and development of electrohydrodynamic instability at the well-faceted and atomically smooth crystal surfaces which may have an infinitely large stiffness and excitation spectrum differing from the usual crystallization wave spectrum. The most striking distinction of the smooth faceted crystal surface from the rough one is the existence of nonanalytic cusplike behavior in the angle dependence for the surface tension, e.g., [@Bal05; @Lan; @Noz]. The origin of the singularity is directly connected with nonzero magnitude of the facet step energy below the roughening transition temperature of about 1.2 K.
In present work we attempt the electrohydrodynamic instability at the smooth faceted surface of a $^4$He crystal in contact with its liquid phase. As we will see below, the close similarity between the rough and smooth states of the crystal surfaces extends until the charge density is below the critical one and the state and shape of the surface are stable. As the charge density increases, the development of the instabilities becomes different in kind. Unlike the rough crystal surface, the faceted surface crosses over to a metastable state and the further development of the instability is accompanied by overcoming some nucleation barrier. The barrier height is proportional to the square of the facet step energy and drastically reduces as the charge density increases. At the sufficiently low temperatures the thermal activation mechanism of overcoming the barrier is replaced with the quantum tunneling through the nucleation barrier. On the whole, the charge-induced reconstructuring of the faceted crystal surface resembles much first-order phase transitions and macroscopic quantum nucleation phenomena in the helium systems [@Tsy; @Ruu; @Bur; @Tan; @Bal02].
For simplicity, we keep in mind the basal plane of hexagonal $^4$He crystal as an example of the crystal facet and neglect any anisotropy in the plane. We also suppose that the temperature is below the roughening transition temperature and the crystal surface is well-defined and faceted.
Hamiltonian. The onset of instability at the crystal facet
===========================================================
Let us assume that the crystal surface is parallel to the $xy$ plane, with vertical position at $z=0$. In order to consider the stability of the surface, we proceed as follows. First, we call $\zeta =\zeta (\bm{r})$ the displacement of the surface from its horizontal position $z=0$ with $\bm{r}=(x,y)$ as a two-dimensional radius-vector. In addition to the surface tension force and the force of gravity due to difference in the densities between the solid and liquid states $\varDelta\rho$, one should involve also the interaction of the charges with electric field $E$ and the direct Coulomb interaction between the charges. Then the total energy $U$ of a charged surface can be written as $$\begin{gathered}
U=\int\! d^2r\biggl(\alpha (\bm{\nu})\sqrt{1+(\nabla\zeta )^2}+ \varDelta\rho\,
g\frac{\zeta ^2}{2} +eEn(\bm{r})\zeta\biggr)\nonumber
\\
+\frac{1}{2}\iint\! d^2r\, d^2r'\,\frac{en(\bm{r})en(\bm{r}')}{|\bm{l}-\bm{l}'|} .
\label{f01}\end{gathered}$$ Here $\bm{l}=(\bm{r},\zeta)$ stands for the three-dimensional coordinate of a point at the surface, $n(\bm{r})$ is the density of electrons with charge $e$, $g$ is the acceleration of gravity, and $\alpha (\bm{\nu})$ is the energy of a unit surface area or surface tension.
Unlike the fluid-fluid interface, the surface tension coefficient $\alpha
(\bm{\nu})$ for the crystal facet depends essentially on the direction of the normal $\bm{\nu}$ to the interface against crystallographic axes. In our simplest case this is a function of angle $\vartheta$ alone between the normal and the crystallographic \[0001\] or $c$-axis of the crystal hcp structure with the geometric relation $\mid\tan\vartheta\mid=\mid\nabla\zeta\mid$.
For the crystal facet tilted by small angle $\vartheta$ from the basal plane, the expansion of surface tension $\alpha (\vartheta )$ usually written, e.g. [@Bal05; @Lan; @Noz], as $$\alpha (\vartheta ) = (\alpha _0+\alpha _1\tan\mid\vartheta\mid +
\ldots)\cos\vartheta , \;\;\;\;\;\; \mid\tan\vartheta\mid =\mid\nabla\zeta\mid ,$$ can be represented for the small angles by a series $$\alpha (\vartheta ) = \alpha (0)+\alpha _1\mid\vartheta\mid+ \alpha
^{\prime\prime}(0)\frac{\vartheta ^2}{2} +\ldots , \;\;\;\;\;\;
\mid\vartheta\mid\ll 1.$$ We intentionally do not write the next terms of expansion, e.g., cubic one due to step-step interaction, since we assume to study only small bending of the crystal surface. The angular behavior has a nonanalytic cusplike behavior at $\vartheta
=0$ due to $\alpha _1=\alpha _1(T)$ representing a ratio of the linear facet step energy $\beta$ to the crystallographic interplane spacing. Below the roughening transition temperature for the basal plane $T_R\sim$1.2 K the facet step energy $\beta=\beta (T)$ is positive and vanishes for temperatures $T>T_R$.
To determine the equilibrium shape of the surface $\zeta (\bm{r})$ and equilibrium charge distribution $n(\bm{r})$, we must minimize the energy functional (\[f01\]) against $\zeta$ and $n$ at a given total surface charge $Q$. This condition can readily be taken into account by augmenting the energy functional with the Lagrange factor $\lambda$ in the form $$-\lambda \int en(\bm{r})\,d^2r .$$ In addition, treating the energy functional, we naturally imply one more obvious condition $n(\bm{r})\geqslant 0$.
In the general form the minimization of the energy functional is a practically unsolvable problem because its expression (\[f01\]) contains not only quadratic terms in $\zeta$ and $n$. Thus, we start first from analyzing small surface bending $\zeta (\bm{r})$ and small gradients $|\nabla\zeta |\ll 1$. The latter implies implicitly that $|\zeta |\ll r$ and we can put approximately $\bm{l}=\bm{r}$ in the denominator of the Coulomb term in Eq. (\[f01\]). Next, we expand the surface tension term in $|\nabla\zeta|$, retain the linear and quadratic terms alone, and arrive at the following expression for the total excess energy $U$ associated with nonzero surface bending $\zeta (\bm{r})$ $$\begin{gathered}
U=\int\! d^2r\biggl(\alpha _1|\nabla\zeta|+\frac{\alpha _0}{2}(\nabla\zeta )^2+
\frac{\varDelta\rho\, g}{2}\zeta ^2 +eEn(\bm{r})\zeta \biggr)\nonumber
\\
+\frac{1}{2}\iint\! d^2r\, d^2r'\,\frac{en(\bm{r})en(\bm{r}')}{|\bm{r}-\bm{r}'|} .\end{gathered}$$ Here we have labeled $$\alpha _0=\alpha (0)+\alpha ^{\prime\prime}(0)$$ as a surface stiffness.
The spatial scale of surface distortion is usually determined by the capillary length $\lambda _0 =(\alpha _0/\varDelta\rho\, g)^{1/2}\approx 1$ mm if one takes $\alpha _0\approx 0.2$ erg/cm$^2$ and $\varDelta\rho\approx 0.018$ g/cm$^3$ for $^4$He [@Bal05]. The electric field $E$ and charge surface density $en$ have the same dimensionality and their typical scale is $(\alpha _0\varDelta\rho\,
g)^{1/4}\approx 400$ V/cm. Accordingly, the typical electron density equals $(\alpha _0\varDelta\rho\, g)^{1/4}/e\approx 2.8\times 10^{9}$ cm$^{-2}$. The number of electrons $\pi\alpha _0^{5/4}(\varDelta\rho\, g)^{-3/4}$ within the circle of radius $\lambda _0$ runs to $10^8$. And lastly, unit of energy corresponds to $\alpha _0^2/(\varDelta\rho\, g)\approx 2.2\times 10^{-3}$ erg.
So, if we measure $\zeta$ and $\bm{r}$ in units of capillary length $\lambda _0$, electric field and charge density in units of $(\alpha _0\varDelta\rho\,
g)^{1/4}$, and energy in units of $\alpha _0^2/(\varDelta\rho\, g)$, the total excess energy $U$ can be expressed in terms of dimensionless units as $$\begin{gathered}
U=\int\! d^2r\biggl(\frac{\alpha _1}{\alpha _0}|\nabla\zeta|+\frac{(\nabla\zeta
)^2}{2}+ \frac{\zeta ^2}{2} + En (\bm{r} )\zeta\biggr)\nonumber
\\
+\frac{1}{2}\iint\! d^2r\, d^2r'\,\frac{n(\bm{r})n(\bm{r}')}{|\bm{r}-\bm{r}'|} .
\label{f06}\end{gathered}$$ As for the step energy $\alpha _1$, we assume that its low temperature value [@Bal05] is approximately $\alpha _1\approx 0.014$ erg/cm$^2$. This value amounts to one-tenth of surface stiffness $\alpha _0$ and in the following we can keep inequality $\alpha _1/\alpha _0\ll 1$ in mind. Moreover, this small parameter justifies all approximations that will be made further.
The uniform state of the surface holds for the electric field values as long as the contribution to the excess energy (\[f06\]) due to variations $\zeta
(\bm{r})$ from $\zeta =0$ and $n(\bm{r})$ from homogeneous value $\bar{n}$ is a positive-definite quantity. Using the following equation for Lagrange factor $$E\zeta (\bm{r})+\int\frac{n(\bm{r}')\, d^2r'}{|\bm{r}-\bm{r}'|}=\lambda$$ and putting $\delta n(\bm{r})=n(\bm{r})-\bar{n}$, we find for the variation of the excess energy $$\begin{gathered}
\delta U=\int\! d^2r\biggl(\frac{\alpha _1}{\alpha
_0}|\nabla\zeta|+\frac{(\nabla\zeta )^2}{2}+ \frac{\zeta ^2}{2} + E\,\delta n
(\bm{r} )\zeta\biggr)\nonumber
\\
+\frac{1}{2}\iint\! d^2r\, d^2r'\,\frac{\delta n(\bm{r})\delta
n(\bm{r}')}{|\bm{r}-\bm{r}'|} . \label{f07}\end{gathered}$$ To analyze it, we use the Fourier representation $$\zeta (\bm{r})=\sum\limits _{\bm{k}}\zeta
_{\bm{k}}e^{i{\bm{kr}}}\;\;\text{and}\;\; \delta n (\bm{r})=\sum\limits
_{\bm{k}}\delta n _{\bm{k}}e^{i{\bm{kr}}} ,$$ and rewrite the energy variation as $$\begin{gathered}
\delta U=\int\! d^2r\, \frac{\alpha _1}{\alpha _0}|\nabla\zeta|+
\\
\frac{1}{2}\sum\limits _{\bm{k}}\bigl[ (k^2+1)\zeta _{\bm{k}}\zeta _{\bm{k}}^{*}+
E(\delta n_{\bm{k}}\zeta _{\bm{k}}^{*}+\delta n_{\bm{k}}^{*}\zeta
_{\bm{k}})+\frac{2\pi}{k}\delta n_{\bm{k}}\delta n_{\bm{k}}^{*}\bigr] .\end{gathered}$$ Minimizing $\delta U$ over $\delta n _{\bm{k}}$ yields the optimum relation $$\label{f10}
\delta n _{\bm{k}}=-\frac{k}{2\pi}\zeta _{\bm{k}}$$ and the corresponding optimum value of the energy $$\label{f11}
\delta U=\int\frac{\alpha _1}{\alpha _0}|\nabla\zeta|\, d^2r+
\frac{1}{2}\sum\limits _{\bm{k}}\bigl(k^2+1-\frac{kE^2}{2\pi}\bigr)|\zeta
_{\bm{k}}|^2 .$$ The second term is always positive provided the inequality $E^2<2\pi(k+1/k)$ is satisfied for all wave vectors $k$. The minimum of the right-hand side of the inequality occurs at $k=k_c=1$ and corresponds to the critical field $E_c=\sqrt{4\pi}$. Thus, the crystal facet is absolutely stable at $E<E_c$.
At $E>E_c$ the stability is lost and the distortions of the homogeneous state should appear. In this regard the situation resembles the loss of stability for the rough state of the crystal surface. However, the development of the stability and the transition to unhomogeneous state differ drastically. In fact, due to positive $\alpha _1>0$ term linear in $|\nabla\zeta |$ the evolution of the crystal facet perturbations should inevitably be accompanied with overcoming some potential barrier, the barrier height being dependent on the field strength $E$. The more the field strength, the less the potential barrier height.
To proceed, let us return to the coordinate representation of Eq. (\[f11\]) $$\begin{gathered}
\delta U=\int\! d^2r\biggl(\frac{\alpha _1}{\alpha
_0}|\nabla\zeta|+\frac{(\nabla\zeta )^2}{2}+ \frac{\zeta ^2}{2}\biggr)\nonumber
\\
-\frac{1}{2}\iint\! d^2r\, d^2r'\,\frac{E^2}{(2\pi )^2}\frac{\bigl(\nabla
_{\bm{r}}\zeta (\bm{r})\nabla _{\bm{r}'}\zeta (\bm{r}')\bigr)}{|\bm{r}-\bm{r}'|}
\label{f12}\end{gathered}$$ and give a qualitative description of the matter. For this purpose, we employ a variational principle and dimensional analysis of the functional (\[f12\]). Let us represent the surface distortion $\zeta (\bm{r})$ with the aid of the trial function $f(x)$ in the axially symmetrical form as $$\label{f13}
\zeta (\bm{r})=\zeta f(r/R),$$ where $\zeta$ is a typical magnitude of distortion and $R$ is its typical size. Then we have $$\label{f14}
\delta U(\zeta , R)=\frac{\alpha _1}{\alpha _0}a|\zeta |R +b\frac{\zeta
^2}{2}+c\frac{\zeta ^2R^2}{2}-d\frac{E^2}{4\pi ^2}\frac{\zeta ^2R}{2} ,$$ and the dimensionless factors are given by $$\begin{gathered}
a=\int _0^{\infty}|f'(r)|2\pi r\, dr, \;\;\; b=\int _0^{\infty}f^{\prime\,
2}(r)2\pi r\, dr,
\\
c=\int _0^{\infty}f^2(r)2\pi r\, dr,\;\;\; d=\int d^2r\, d^2r'\frac{\bm{r\cdot
r}'}{rr'}\frac{f'(r)f'(r')}{|\bm{r}-\bm{r}'|}
\\
= \int _0^{\infty}dk\biggl(\int _0^{\infty}dr\, 2\pi rf'(r)J_1(kr)\biggr)^2 ,\end{gathered}$$ where $J_1(x)$ is the Bessel function of the first kind.
As the electric field strength exceeds the value $E_0=(8\pi ^2\sqrt{bc}/d)^{1/2}$, there appears a region of $\zeta$ and $R$ with the negative values of $\delta U$ separated always from $\delta U=0$ at $\zeta =0$ with the intermediate positive $\delta U$ values. Rewriting the excess energy $\delta U(\zeta , R)$ as $$\begin{gathered}
\label{f15}
\delta U=\frac{1}{2}\biggl(\frac{8\pi^2\alpha _1}{\alpha
_0}\biggr)^2\frac{a^2b/d^2}{E^4-E_0^4} +\frac{b}{2}\zeta ^2
\biggl(1-\sqrt{\frac{c}{b}}\,\frac{E^2}{E_0^2}R\biggr)^2
\\
-\frac{c}{2}\biggl(\frac{E^2}{E_0^2}-1\biggr) \biggl(|\zeta |R-\frac{\alpha
_1}{\alpha _0}\frac{a/c}{E^4/E_0^4 -1}\biggr)^2, \; E_0^2=\frac{8\pi
^2\sqrt{bc}}{d},\end{gathered}$$ one can readily see that the state of the crystal facet changes from the stable to metastable state at $E>E_0$ and the point $$\label{f16}
|\zeta _0|=\frac{\alpha
_1}{\alpha_0}\frac{a}{\sqrt{bc}}\,\frac{E^2E_0^2}{E^4-E_0^4}
\;\;\;\text{and}\;\;\; R_0=\sqrt{\frac{b}{c}}\,\frac{E_0^2}{E^2}$$ becomes a saddle point of the potential relief. The potential barrier height equal to $$\label{f17}
U_0= \frac{a^2}{2c}\,\frac{\alpha _1^2}{\alpha _0^2}\, \frac{E_0^4}{E^4-E_0^4}$$ must be overcome to break the flat faceting of a crystal surface.
Unfortunately, we cannot find the exact function $f(r)$ and, correspondingly, values of factors $a$, $b$, $c$ and $d$ which optimize the functional (\[f12\]). However, it is clear that the potential barrier height should be infinitely large at $E=E_c$ and thus $E_0=E_c$ for the exact solution. This entails the obvious relation $d=2\pi (bc)^{1/2}$ between coefficients for the exact solution. To estimate them, we use a trial function $f(x)=\exp (-x^2)$. The direct calculation results in $$\label{f18}
a=\pi ^{3/2},\;\; b=\pi ,\;\; c=\pi /2,\;\; d=\pi ^{5/2}/\sqrt{2},$$ and $$\label{f18a}
\frac{E_c}{E_0}=\sqrt{\frac{d}{2\pi\sqrt{bc}}}=\frac{\pi ^{1/4}}{\sqrt{2}}\approx
0.94$$ in place of unity. Hence we may expect an accuracy of our estimate within about 10%.
Let us compare the height $U_0$ of the potential barrier at the saddle point with the roughening transition temperature $T_R$ about 1.2 K. In the dimensional units we have $$\label{f19}
U_0=\frac{a}{2c}\frac{\alpha _1^2}{\varDelta\rho\, g}\frac{E_c^4}{E^4-E_c^4}\sim
1.4\times 10^{11}\frac{E_c^4}{E^4-E_c^4}\;\; (\text{in K}).$$ One may be surprised with the huge barrier height so that, unlike the rough crystal surface absolutely unstable at $E\geqslant E_c$, tens of $E_c$ should keep a crystal facet practically stable for an experimentally available time. Provided we expect a reasonable observation time of destructing the faceted state due to thermal activation mechanism, we should provide a ratio $U_0/T$ of about a few tens [@Bur; @Tan] This means that the electric field $E$ should exceed the critical one $E_c$ by a factor of about 300. The same factor certainly refers to the surface density of charges.
In the dimensional CGSE units the bending deflection $\zeta _0$ and the typical size of inhomogeneity $R_0$ are given by $$\begin{gathered}
|\zeta _0| =4\pi\frac{a}{\sqrt{bc}}\,\alpha _1\frac{E^2}{E^4-E_c^4}\sim
31\frac{\alpha _1E^2}{E^4-E_c^4} ,
\\
R_0 =\sqrt{\frac{b}{c}}\,\frac{4\pi\alpha _0}{E^2}\sim 18\frac{\alpha _0}{E^2}.
\label{f20}\end{gathered}$$ In the weak fields of few critical values the critical parameters $R_0$ and $|\zeta _0|$ prove to be of macroscopic sizes in accordance with macroscopically large height of the potential barrier.
For $E=300E_c$, we find approximately $R_0\sim 16$ nm and $|\zeta _0|\sim 2$ nm. On the whole, the electric field should be very large compared with the critical value $E_c$ in order to reduce significantly the nucleation barrier for the effective production of a few circular crystal terraces tilted with the angle about $\arctan (\alpha _1/\alpha _0)\sim 4^{\circ}$. In this sense the critical fluctuation represents a region of the crystal surface in the rough state.
From the physical point of view the angle of slope $\arctan (\alpha _1/\alpha
_0)\sim 4^{\circ}$ is determined by the competition of two contributions into the total surface energy. One originates from the regular surface term $\alpha _0\zeta
^2$ and the second does from irregular step tension term $\alpha _1|\zeta |R$. Provided $\alpha _0\zeta ^2\gg\alpha _1|\zeta |R$, the latter contribution becomes negligible and thus the interface properties resemble those in the rough surface state. On the contrary, if $\alpha _0\zeta ^2\ll\alpha _1|\zeta |R$, the dominant term linear in $|\zeta |$ is responsible for the origin of a potential barrier since the other terms quadratic in $\zeta$ are yet insignificant.
Note that the small gradient approximation we use is satisfied since $|\nabla\zeta
|\sim |\zeta _0|/R_0\sim \alpha _1/\alpha _0\ll 1$ with the exception of narrow region $E\sim E_c$. The latter remark refers also to justifying small density variations $\delta n \ll\bar{n}$ valid to the extent of smallness $|\zeta
_0|/R_0$.
Lagrangian. The quantum breaking of the crystal facet
=====================================================
The destruction of the faceted crystal surface is accompanied by overcoming some potential barrier depending on the charge surface density. There are two basic mechanisms to overcome the potential barrier. One is the thermal activation efficient at high temperatures and the second is the quantum tunneling through a potential barrier dominant at sufficiently low temperatures. In order to treat the quantum tunneling, it is necessary to involve the interface dynamics, in particular, to determine the kinetic energy of the charged interface in addition to the potential energy $U$.
As a first step, we employ the so-called metallic approximation. In this approximation it is assumed that the mobility of electrons along the superfluid-crystal He$^4$ interface is very high and the charged helium interface represents an equipotential surface so that the electric field is always normal to the interface as for a well-conducting metal. A necessary condition for such approximation assumes at least that the plasma oscillation frequency of a two-dimensional layer of electrons with effective mass $m_e$ $$\label{f20a}
\Omega _p\sim (2\pi ne^2k/m_e)^{1/2}$$ is much larger than the typical frequency $\omega$ of the gravitational-capillary or melting-crystallization waves at the same wave vector $k$. So, within our first approximation we believe that the charge density distribution $n(t,\bm{r})$ has sufficient time to accommodate to the surface distortion $\zeta (t,\bm{r})$ and is determined by the electrostatic relations in accordance with the profile $\zeta
(t,\bm{r})$.
Neglecting possible energy dissipation, we describe the charged interface dynamics using the following action $$\label{f21}
S=\int dt\, L[\zeta (t,\,\bm{r}),\dot{\zeta} (t,\,\bm{r}),n(t,\bm{r})]$$ with the Lagrangian $L$ equal to the difference between the kinetic energy functional and the potential energy functional $U$ introduced by Eq. (\[f01\]) $$\label{f22}
L=\frac{\rho _\text{eff}}{2}\!\iint\! d^2r\,
d^2r'\,\frac{\dot{\zeta}(t,\,\bm{r})\dot{\zeta}(t,\,\bm{r}')}{2\pi
|\bm{r}-\bm{r}'|} - U[\zeta(t,\,\bm{r}), n(t,\,\bm{r})] .$$ Here we ignore the compressibility of the both liquid and solid phases. Because of low temperature consideration we will also neglect the normal component density in the superfluid phase or, equivalently, difference between the superfluid density $\rho _s$ and the density of the liquid phase $\rho$. Then the effective interface density $\rho _\text{eff}$ is given by $$\rho _\text{eff}=(\rho '-\rho)^2/\rho \approx 1.9\,\text{mg/cm}^3$$ and depends on the difference $\varDelta\rho =\rho '-\rho$ between the solid density $\rho '$ and the liquid density $\rho$. For our purposes, the exact magnitude of the effective density is inessential.
Next, for convenience, let us introduce units of time equal to $(\rho
_{\text{eff}}\lambda _0^3/\alpha _0)^{1/2}\approx 3.1$ ms and measure the action in units of $(\alpha _0\rho _{\text{eff}}\lambda _0^7)^{1/2}\approx 0.62\times
10^{-5}$ erg$\cdot$s. Using the speculations and arguments bringing us to Eq. (\[f10\]) and then to Eq. (\[f12\]), we arrive at examining the following effective action $$\label{f23}
S=\int dt\, L_{\text{eff}}[\zeta (t,\,\bm{r}),\dot{\zeta} (t,\,\bm{r})]$$ with the dimensionless Lagrangian $$\begin{gathered}
\label{f23a}
L_{\text{eff}}=\frac{1}{2}\!\iint\! d^2r\,
d^2r'\,\frac{\dot{\zeta}(t,\,\bm{r})\dot{\zeta}(t,\,\bm{r}')}{2\pi
|\bm{r}-\bm{r}'|}
\\
-\int\! d^2r\biggl(\frac{\alpha _1}{\alpha _0}|\nabla\zeta|+\frac{(\nabla\zeta
)^2}{2}+ \frac{\zeta ^2}{2}\biggr)\nonumber
\\
+\frac{1}{2}\iint\! d^2r\, d^2r'\,\frac{E^2}{(2\pi )^2}\frac{\bigl(\nabla
_{\bm{r}}\zeta (t,\bm{r})\nabla _{\bm{r}'}\zeta
(t,\bm{r}')\bigr)}{|\bm{r}-\bm{r}'|}.\end{gathered}$$
Within an exponential accuracy the quantum decay rate of the metastable state is proportional to $$\label{f24}
\Gamma\propto\exp (-S_E/\hbar ),$$ where $S_E$ is the effective Euclidean action calculated at the optimum escape path. This path starts at the entrance point under the potential barrier and ends at the point at which the optimum fluctuation escapes from the barrier [@Bur; @Tan]. In other words, quantum fluctuation penetrates through the potential barrier along the path of least resistance. Before calculating the quantum rate at which the crystal facet breaks up, we must go over to the effective Euclidean action defined in imaginary time $t\rightarrow it$. We refer to books [@Qua; @Wei] for details.
As a result, we should analyze the following functional defined within the time interval $[-\hbar/2T,\,\hbar/2T]$ $$\begin{gathered}
\label{f25}
S_E=\int dt\, L_E[\zeta (t,\,\bm{r}),\dot{\zeta} (t,\,\bm{r})],
\\
L_E=\frac{1}{2}\!\iint\! d^2r\,
d^2r'\,\frac{\dot{\zeta}(t,\,\bm{r})\dot{\zeta}(t,\,\bm{r}')}{2\pi
|\bm{r}-\bm{r}'|}
\\
+\int\! d^2r\biggl(\frac{\alpha _1}{\alpha _0}|\nabla\zeta|+\frac{(\nabla\zeta
)^2}{2}+ \frac{\zeta ^2}{2}\biggr)\nonumber
\\
-\frac{1}{2}\iint\! d^2r\, d^2r'\,\frac{E^2}{(2\pi )^2}\frac{\bigl(\nabla
_{\bm{r}}\zeta (t,\bm{r})\nabla _{\bm{r}'}\zeta
(t,\bm{r}')\bigr)}{|\bm{r}-\bm{r}'|}.\end{gathered}$$ Again, the exact determination of extrema for the action $S_E$ is a rather complicated problem. We here consider only the case of zero temperature when the limits of integration over imaginary time are infinite. As before, it is convenient to take an advantage of the dimensional analysis and variational principle. We will express the surface distortion $\zeta (t,\bm{r})$ in the terms of function $f(y,x)$ with the scaled arguments as $$\label{f26}
\zeta (t,\bm{r})=\zeta f(t/\tau , r/R) .$$ Next, we calculate the action $S_E$ at zero temperature $$\begin{gathered}
S_E(\zeta ,\tau , R)= F\frac{\zeta ^2R^3}{2\tau}+ \tau\biggl(\frac{\alpha
_1}{\alpha _0}A|\zeta |R +B\frac{\zeta ^2}{2}\nonumber
\\
+C\frac{\zeta ^2R^2}{2}-D\frac{E^2}{4\pi ^2}\frac{\zeta ^2R}{2}\biggr).
\label{f27}\end{gathered}$$ The numerical factors are given by the integrals $$\begin{gathered}
\label{f28}
A=\int _{-\infty}^{\infty}dt\int _0^{\infty}|f'(t,r)|2\pi r\, dr,
\\
B=\int _{-\infty}^{\infty}dt\int _0^{\infty}f^{\prime\, 2}(t,r)2\pi r\, dr,
\\
C=\int _{-\infty}^{\infty}dt\int _0^{\infty}f^2(t,r)2\pi r\, dr,
\\
D=\int _{-\infty}^{\infty}dt\int d^2r\, d^2r'\frac{\bm{r\cdot
r}'}{rr'}\frac{f'(t,r)f'(t,r')}{|\bm{r}-\bm{r}'|}
\\
= \int _{-\infty}^{\infty}dt\int\limits _0^{\infty}dk\biggl(\int _0^{\infty}dr\,
2\pi rf'(t,r)J_1(kr)\biggr)^2 ,
\\
F=\int _{-\infty}^{\infty}dt\int d^2r\,
d^2r'\frac{\dot{f}(t,r)\dot{f}(t,r')}{2\pi|\bm{r}-\bm{r}'|}
\\
= \int _{-\infty}^{\infty}dt\int\limits _0^{\infty}\frac{dk}{2\pi}\biggl(\int
_0^{\infty}dr\, 2\pi r\dot{f}(t,r)J_0(kr)\biggr)^2 ,\end{gathered}$$ where $J_0(x)$ and $J_1(x)$ are the Bessel function of the first kind.
From the condition of vanishing derivatives in $\zeta$, $R$ and $\tau$ for $S_E$ we find the following parameters of the quantum critical fluctuation $$\begin{gathered}
\label{f29}
|\zeta _q|=\frac{A}{\sqrt{BC}}\frac{\alpha _1}{\alpha
_0}\biggl(\sqrt{1-\frac{7}{16}\frac{E_0^4}{E^4}}
+\frac{3}{4}\biggr)\frac{E^2E_0^2}{E^4-E_0^4},
\\
R_q=\frac{7}{4}\sqrt{\frac{B}{C}}\frac{E_0^2}{E^2}
\biggl(\sqrt{1-\frac{7}{16}\frac{E_0^4}{E^4}}+1\biggr)^{-1},
\\
\tau
_q=\frac{7}{2\sqrt{2}}\biggl(\frac{F^2B}{C^3}\biggr)^{1/4}\!\!\frac{E_0^3}{E\sqrt{E^4-E_0^4}}
\frac{\bigl(\sqrt{1-\frac{7}{16}\frac{E_0^4}{E^4}}+\frac{3}{4}\bigr)^{1/2}}
{\sqrt{1-\frac{7}{16}\frac{E_0^4}{E^4}} +1}.\end{gathered}$$ Here $E_0=(8\pi ^2\sqrt{BC}/D)^{1/2}$ which should coincide for the exact solution with the critical field value, i.e., $E_0=E_c=\sqrt{4\pi}$. Then we calculate the corresponding value of action $S_q$ according to $$\label{f29a}
S_q=\frac{A}{2}\frac{\alpha _1}{\alpha _0}|\zeta _q|R_q\tau _q$$ at the critical point $(\zeta _q, R_q, \tau _q)$ representing a saddle point of the functional $S_E$ (\[f27\]). Finally, we obtain $$\begin{gathered}
\label{f30}
S_q=\frac{49A^2}{16\sqrt{2}}\frac{(BF^2)^{1/4}}{C^{7/4}}\frac{\alpha _1^2}{\alpha
_0^2}\frac{\bigl(\sqrt{1-\frac{7}{16}\frac{E_0^4}{E^4}}+\frac{3}{4}\bigr)^{3/2}}
{\bigl(\sqrt{1-\frac{7}{16}\frac{E_0^4}{E^4}}+1\bigr)^2}
\\
\times\frac{E_0}{E}\biggl(\frac{E_0^4}{E^4-E_0^4}\biggr)^{3/2}.\end{gathered}$$ Like the potential barrier height, the action $S_q$ becomes infinite at the same critical field $E=E_0$.
To estimate the numerical coefficients $F$, $A$, $B$, $C$, and $D$, we choose a physically expedient trial function $f(t,r)=\exp [-(t^2+r^2)]$. The straightforward calculation gives $$\begin{gathered}
\label{f31}
A=\pi ^2,\; B=\frac{\pi ^{3/2}}{2^{1/2}},\; C=\biggl(\frac{\pi}{2}\biggr)^{3/2},\;
D=\frac{\pi^3}{2},\; F=\frac{\pi ^2}{4}\end{gathered}$$ with the same ratio $E_c/E_0$ as in (\[f18a\]).
Let us compare the action $S_q$ with the Planck constant $\hbar$. Introducing a facet capillary length $\lambda _1=(\alpha _1/\varDelta\rho\, g)^{1/2}$, we have in the dimensional units $$\begin{gathered}
\label{f32}
\frac{S_q}{\hbar}=\frac{49A^2}{16\sqrt{2}}\frac{(BF^2)^{1/4}}{C^{7/4}}
\biggl(\frac{\sqrt{\alpha _1\alpha _0}\rho_{\text{eff}}\lambda _1^7}{\hbar
^2}\biggr)^{1/2}
\\
\times\frac{\bigl(\sqrt{1-\frac{7}{16}\frac{E_c^4}{E^4}}+\frac{3}{4}\bigr)^{3/2}}
{\bigl(\sqrt{1-\frac{7}{16}\frac{E_c^4}{E^4}}+1\bigr)^2}
\frac{E_c}{E}\biggl(\frac{E_c^4}{E^4-E_c^4}\biggr)^{3/2}
\\
\approx 4\times
10^{21}\frac{\bigl(\sqrt{1-\frac{7}{16}\frac{E_c^4}{E^4}}+\frac{3}{4}\bigr)^{3/2}}
{\bigl(\sqrt{1-\frac{7}{16}\frac{E_c^4}{E^4}}+1\bigr)^2}
\frac{E_c}{E}\biggl(\frac{E_c^4}{E^4-E_c^4}\biggr)^{3/2} .\end{gathered}$$ As is seen, even for the electric fields which are dozens of times larger than the critical one $E_c$, the ratio $S_q/\hbar$ has a giant magnitude so that the crystal surface will remain in the well-defined faceted state for the practically infinite time. At $E\gg E_c$ we have an estimate $$\label{f33}
S_q/\hbar\approx 2.3\times 10^{21}(E_c/E)^7.$$ For strong $E\gg E_c$ fields, in the dimensional CGSE units the bending deflection $\zeta _0$ and the typical size of inhomogeneity $R_0$ are given by $$\begin{gathered}
\label{f34}
|\zeta _q| =4\pi\frac{7A}{4\sqrt{BC}}\frac{\alpha _1}{E^2}\sim 80\frac{\alpha
_1}{E^2} ,
\\
R_q =\frac{7}{8}\sqrt{\frac{B}{C}}\,\frac{4\pi\alpha _0}{E^2}\sim 15\frac{\alpha
_0}{E^2}.\end{gathered}$$ For $E=300E_c$, we find approximately $R_q\sim 13$ nm and $|\zeta _q|\sim 5$ nm. The estimate of the tunneling time in the strong $E\gg E_c$ fields yields $$\begin{gathered}
\label{f35}
\tau _q\approx 10^{-2}(E_c/E)^3\;\; (\text{in seconds}).\end{gathered}$$
Again, the small gradient approximation is fulfilled since $|\nabla\zeta |\sim
|\zeta _q|/R_q\sim\alpha _1/\alpha _0\ll 1$. Let us compare the plasmon frequency $\Omega _p$ with the inverse time of tunneling $\tau _q^{-1}$ in order to justify the metallic approximation. We consider the case of strong fields and take $k\sim
1/R_q$ as a typical wave vector for the spatial size of the surface distortion. Then, using (\[f20a\]), $$\label{f35a}
\Omega _p\tau _q\sim\sqrt{\frac{\rho
_{\text{eff}}}{\varDelta\rho}}\sqrt{\frac{eE_c}{m_eg}}\biggl(\frac{E_c}{E}\biggr)^{3/2}
\sim 2\times 10^{7}\biggl(\frac{E_c}{E}\biggr)^{3/2}.$$ Thus, in the fields $E=300E_c$ the fulfillment of inequality $\Omega _p\gg\tau
_q^{-1}$ evidences for the favor of the metallic approximation.
Thermal-quantum crossover temperature. The decay rate
=====================================================
Let us turn to the thermal-quantum crossover temperature $T_q$ which separates the classical thermal activation at $T>T_q$ from the quantum nucleation mechanism at lower $T<T_q$ temperatures. Here we estimate the thermal-quantum crossover temperature $T_q=T_q(E)$ as a ratio of the potential barrier height to the saddle value $S_q$ of the Euclidean action at zero temperature. In the dimensional units we have then $$\begin{gathered}
\label{f36}
T_q(E)=\frac{\hbar U_0}{S_q}=\hbar\biggl(\frac{\alpha _0}{\rho
_{\text{eff}}\lambda
_0^3}\biggr)^{1/2}\frac{8\sqrt{2}}{49}\frac{a^2C^{7/4}}{cA^2(BF^2)^{1/4}}
\\
\times\frac{\bigl(\sqrt{1-\frac{7}{16}\frac{E_c^4}{E^4}}+1\bigr)^2}
{\bigl(\sqrt{1-\frac{7}{16}\frac{E_c^4}{E^4}}+\frac{3}{4}\bigr)^{3/2}}
\frac{E\sqrt{E^4-E_c^4}}{E_c^3}\approx
\\
1.7\times 10^{-7}\frac{\bigl(\sqrt{1-\frac{7}{16}\frac{E_c^4}{E^4}}+1\bigr)^2}
{\bigl(\sqrt{1-\frac{7}{16}\frac{E_c^4}{E^4}}+\frac{3}{4}\bigr)^{3/2}}
\frac{E\sqrt{E^4-E_c^4}}{E_c^3} \; (\text{in mK}).\end{gathered}$$ Note that the thermal-quantum crossover temperature is independent of the step tension coefficient $\alpha _1$. This point is obvious since the barrier height $U_0$ and action $S_q$ are both proportional to the same factor $\alpha _1^2$.
At the electric fields comparable with the critical one $E_c$ the thermal-quantum crossover temperature, starting from its zero value at $E=E_c$, proves to be extremely small. In the strong $E\gg E_c$ fields the thermal-quantum crossover temperature grows approximately as a cube of the field $$\label{f37}
T_q(E)\approx 3\times 10^{-7}(E/E_c)^3\;\;\; (\text{in mK}).$$ For fields $E=300E_c$, we may expect a reasonable magnitude for the thermal-quantum crossover temperature of about 8 mK.
Let us consider a charged crystal facet prepared in the metastable $E>E_c$ state with adjusting thermodynamic parameters such as temperature $T$ and electric field $E$. After the lapse of some time $t_{obs}$, there will appear a nucleus of the rough state breaking the crystal faceting. Then the nucleation rate $\Gamma
=\Gamma (T,E)$ and the time of observation $t_{obs}$ are connected by the following relation $$\label{f38}
t_{obs}N_{nuc}\Gamma \simeq 1 ,$$ where $N_{nuc}$ is the total number of independent nucleation sites and $\Gamma$ is the nucleation rate at a single nucleation site. We estimate $N_{nuc}$ approximately as the total number of atoms at the crystal surface, assuming that every atom at the surface has an equal possibility to become a nucleation site within the time interval $t_{obs}$. For the crystal area of 1 cm$^2$, we put $$\label{39}
N_{nuc}\sim 10^{14} .$$
The nucleation rate $\Gamma$ can approximately be estimated as $$\label{f40}
\Gamma\sim\nu\exp(-S)$$ where $\nu$ is the attempt frequency and exponent $S$, depending on temperature, is either Arrhenius exponent $U_0/T$ or Euclidean one $S_q/\hbar$. The attempt frequency $\nu$ is associated with the surface fluctuations resulting in nonzero bending $\zeta (t,\bm{r})$ of the flat crystal facet. In general, the frequency of crystal surface fluctuations depends on the magnitude of surface bending $\zeta$ and the radius of deformation $R$ as well. This frequency can be estimated by equating the kinetic energy to the potential surface energy in Lagrangian $L$ (\[f21\]). The order-of-magnitude estimate can be represented as [@Bur11] $$\label{41}
\nu\sim\biggl(\frac{\alpha _1R+\alpha _0|\zeta |}{\rho _{\text{eff}}|\zeta
|R^3}\biggr)^{1/2}.$$
According to [@Sch], there is one optimum path, i.e., escape path which connects the entrance point with the optimum escape point and corresponds to the saddle-point value of the effective Euclidean action. In the quasiclassical approximation the main contribution to the decay rate of the metastable state is determined by such optimum escape path and its nearest vicinity. As is found above, at the optimum escape path a ratio of surface deformation $\zeta$ to its radius $R$ satisfies approximately $|\zeta |/R\sim\alpha _1/\alpha _0$. Then we arrive at $$\label{42}
\nu\sim\biggl(\frac{\alpha _1^3}{\alpha _0^2\rho _{\text{eff}}|\zeta
|^3}\biggr)^{1/2}.$$
Next, we should estimate the equilibrium fluctuations of the surface bending as a function of temperature. At high temperatures one expects the thermal activation mechanism when the average energy fluctuations should be of the order of the temperature, i.e., $\alpha _1R|\zeta|+\alpha _0\zeta ^2\sim\alpha _0\zeta ^2\sim
T$. Hence, for $T=1$ K, we expect $$\label{f43}
|\zeta |\sim 0.3\,\text{nm},\;\;\; R\sim 3\,\text{nm}\;\;\text{and}\;\;\nu\sim
5\times 10^{10}\,\text{Hz}.$$ At zero temperature the attempt frequency can be associated with the zero-point oscillations in the same potential $U=\alpha _1R|\zeta|+\alpha _0\zeta
^2\sim\alpha _0\zeta ^2$. Using $U\sim\hbar\nu (U)$ for an estimate of the ground level energy, we find $$\label{f44}
\nu\sim\biggl(\frac{\alpha _1^6}{\alpha _0\hbar ^3\rho _{\text{eff}}^2}
\biggr)^{1/7}\sim\!\! 7\times 10^{10}\,\text{Hz},\;\;\;\; U=\hbar\nu\sim
0.5\,\text{K}.$$
Note that the magnitude of the surface bending is about of the interatomic spacing and the frequency has numerically the same order of the magnitude as the Debye frequency. These magnitudes seem us reasonable. Thus, we have a relatively large preexponential factor $$\label{f45}
\nu N_{nuc}\sim 10^{24}\,\text{s}^{-1}\sim e^{55}\,\text{s}^{-1}$$ which can readily be compensated by macroscopically large potential barrier for insufficiently high density of charges. Eventually, if we wish to discover the process of the facet destruction for the time of about tens seconds, the exponents $U_0/T$ or $S_q/\hbar$ should be kept about 55.
Due to strong exponential dependence of nucleation rate $\Gamma$ on the thermodynamic parameters $T$ and $E$ the statistical dispersion of nucleation events is not large as compared with the average values of the thermodynamic parameters at which the nucleation is mainly observed. The overwhelming majority of experimental points will concentrate in the narrow region around the average values which correspond to the so-called rapid nucleation line. In essence, from the viewpoint of the time of observation the rapid nucleation line separates the metastable states into two region. One region represents the long-living states looking as stable during the experiment and the other is the short-living states which decay practically instantly.
So, for the rapid nucleation line or the breaking field $E_b$, we may expect the following behavior. Under thermal activation mechanism at high temperatures one should observe $$\label{f46}
E_b(T)\propto T^{-1/4},\;\;\; T>T_q.$$ Below the thermal-quantum crossover temperature this behavior should go over to the practically temperature-independent behavior $$\label{f47}
E_b(T)\approx\text{const},\;\;\; T<T_q.$$
In the latter connection we would like to mention a possible effect of the energy dissipation processes. As is known from the quantum dynamics of first-order phase transitions [@Bur87], the energy dissipation processes increase the effective Euclidean action and thus reduce the quantum decay rate. Accordingly, the behavior of the breaking field $E_b(T)$ in the quantum regime should grow with the temperature rise and demonstrate a maximum at the thermal-quantum crossover temperature. However, as is mentioned above, the energy dissipation in superfluid $^4$He is not large at low temperatures because of negligible density of the normal component. That is why, we expect only a slight manifestation of the energy dissipation effects in the quantum regime.
Summary
=======
To summarize, we have examined a stability of the charged crystal $^4$He surface in the atomically smooth and well-faceted state below the roughening transition temperature. Like the charged crystal $^4$He surface in the rough state, the charged crystal $^4$He facet becomes unstable at the same density of charges or corresponding critical electric field $E_c$. However, the dynamics of the transition from the initial homogeneous distribution of charges and flat crystal surface to a spatially unhomogeneous charge distribution and to a warped crystal surface proves to be qualitatively different.
In the rough surface state, as the electric field exceeds the critical value $E_c$, the homogeneous surface state becomes absolutely unstable and in this sense the development of the charge-induced instability resembles a second-order phase transition. In contrast, as the electric field exceeds the same critical value $E_c$, the homogeneous state of the atomically smooth and well-faceted crystal surface is converted into the metastable state separated with a potential barrier governed by the electric field or charge density. The barrier height is proportional to the square of the linear facet step energy.
The onset and development of the charge-induced instability at the crystal facet can be compared with the kinetics of first-order phase transitions accompanied by the nucleation and next growth of new stable phase. A nucleus of new phase here can be described as a fluctuation region of the crystal surface in the atomically rough state. The larger the charge density, the smaller the radius of the critical nucleus.
Unlike the charged rough crystal surface, in the electric fields which are tens times larger than the critical value $E_c$ the potential barrier still remains so high that the charged crystal facet, though metastable, will not break up for the experimentally obtainable uptime. To realize a breakage of the crystal facet, a few hundreds of critical value $E_c$ should be achieved. The breaking electric strength $E_b$ depends on the temperature as well as the nature of the breaking mechanism. At high temperatures the breaking dynamics is associated with the thermal activation mechanism and with the quantum tunneling through a potential barrier at sufficiently low temperatures.
ACKNOWLEDMENTS {#acknowledments .unnumbered}
==============
The author is thankful to V. L. Tsymbalenko for stimulating discussion. The work is supported in part by the RFBR Grant No. 10-02-00047a.
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|
---
abstract: 'We consider trends resulting from two formation mechanisms for short-period super-Earths: planet-planet scattering and migration. We model scenarios where these planets originate near the snow line in “cold finger” circumstellar disks. Low-mass planet-planet scattering excites planets to low periastron orbits only for lower mass stars. With long circularisation times, these planets reside on long-period eccentric orbits. Closer formation regions mean planets that reach short-period orbits by migration are most common around low-mass stars. Above $\sim$1$M_\odot$, planets massive enough to migrate to close-in orbits before the gas disk dissipates are above the critical mass for gas giant formation. Thus, there is an upper stellar mass limit for short-period super-Earths that form by migration. If disk masses are distributed as a power law, planet frequency increases with metallicity because most disks have low masses. For disk masses distributed around a relatively high mass, planet frequency decreases with increasing metallicity. As icy planets migrate, they shepherd interior objects toward the star, which grow to $\sim$1$M_\oplus$. In contrast to icy migrators, surviving shepherded planets are rocky. Upon reaching short-period orbits, planets are subject to evaporation processes. The closest planets may be reduced to rocky or icy cores. Low-mass stars have lower EUV luminosities, so the level of evaporation decreases with decreasing stellar mass.'
author:
- 'Grant M. Kennedy'
- 'Scott J. Kenyon'
title: 'Planet formation around stars of various masses: Hot super-Earths'
---
Introduction {#sec:intro}
============
With nearly 300 known extra-Solar planets, there are now several clear correlations between the properties of the planets and their host stars. The most well known trend is the increase in gas giant frequency with host star metallicity [e.g. @2005ApJ...622.1102F]. Recent radial velocity surveys suggest that giant planet frequency also increases with stellar mass [@2007ApJ...670..833J].
These trends provide tests of planet formation theories. In the core accretion model for example, gas giant planets form by coagulation of small planetesimals near the “snow line” that separates rocky and icy regions in a circumstellar disk. Once icy protoplanets reach a critical core mass, they accrete gas rapidly [@1996Icar..124...62P]. Cores benefit from extra planet building material provided by enhanced metallicities and an increase in disk masses with stellar mass. The model is thus consistent with current observations [@2004ApJ...616..567I; @2005ApJ...626.1045I; @2008ApJ...673..502K].
Gravitational instability (GI) is an alternative formation mechanism for gas giant planets, where a relatively massive disk cools enough to fragment into Jupiter-mass clumps. Although GI operates over a wide range of stellar masses, there is still debate about predicted trends with metallicity [@2007prpl.conf..607D]. Given observational biases in the current sample of extra-Solar planets, GI cannot be ruled out as a formation mechanism [@2007prpl.conf..607D].
Core accretion and GI models suggest that short-period “hot Jupiters” reside too close to their parent stars to have formed *in situ*. Thus, these planets must migrate or scatter from more distant formation regions to arrive at their final orbits [@1996Natur.380..606L; @1996Sci...274..954R]. A combination of these two mechanisms probably operates to produce the observed distribution of extra-Solar giant planets. Scattering can reproduce most of the observed eccentricity distribution, but has trouble accounting for planets in circular orbits at distances too far from their host stars for tidal circularisation [@2007astro.ph..3163F]. Migration theories can explain systems with planets in mean-motion resonances [@2002ApJ...567..596L], but they may not reproduce the observed eccentricity distribution [e.g. @2004AIPC..713..243T].
With the discovery of the first super-Earths in relatively short period orbits, migration and scattering remain possible mechanisms for planets to reach these radii [@2005Icar..177..264B; @2007ApJ...654.1110T; @2007arXiv0711.2015R]. However, the discovery of low-mass planets in systems already harbouring giant planets suggests new formation mechanisms [@2005ApJ...631L..85Z]. Because these models require gas giants, they predict trends with metallicity and stellar mass for low-mass planets similar to those for giant planets. Though some low-mass planets may have formed with help from giant planets, a flatter metallicity distribution [@2007prpl.conf..685U] and the absence of giant planets in some low-mass planet systems indicate other formation mechanisms.
Here, we consider trends that may arise in forming short-period and/or transiting icy/rocky planets in systems with no gas giants, over a range of stellar masses. The close-in planets that form are therefore the most massive in the planetary system. We first cover some background in §\[sec:background\]. In §\[sec:scattering\] we use $n$-body simulations to show that 10$M_\oplus$ planet-planet scattering is unlikely to result in transiting planets for all but the lowest mass stars. With long circularisation timescales, planets in these systems are hard to detect. We consider migration scenarios using analytic, semi-analytic and $n$-body models in §\[sec:migration\]. With migration, short-period low-mass planets most likely form around low-mass stars. Above a certain stellar mass, it is hard to form any short-period planets without giant atmospheres. Trends with metallicity depend on the disk mass distribution. Migration to short-period orbits results in significant amounts of material being shepherded inward, which affects the final structure of these systems. We discuss our results, subsequent planetary evolution, and conclude in §\[sec:summary\].
Background {#sec:background}
==========
General Picture {#sec:gen-background}
---------------
Planets form in circumstellar disks. Therefore disk structure plays a key role in setting the final configuration of planetary systems. In most planet formation models, disk structure is characterised by an outwardly decreasing radial surface density profile. This profile usually includes an increase in surface density at the “snow line,” where the temperature becomes low enough for water to freeze.
Planets form by accumulating solids in the disk. Therefore the expected increase in surface density at the snow line is often associated with the formation of gas giants like Jupiter. Forming Jupiter requires the relatively rapid growth of a $\sim$5–10$M_\oplus$ icy core, followed by a period of gas accretion [@1996Icar..124...62P]. Gas accretion must be complete before the gas disk disperses in $\sim$3Myr [e.g. @2001ApJ...553L.153H]. In the minimum mass Solar nebula model , forming the icy core rapidly requires factor of 5–10 surface density enhancements relative to the terrestrial region [@1987Icar...69..249L; @1996Icar..124...62P; @2003Icar..161..431T]. This factor is larger than the factor of 2–3 enhancements expected from Solar abundances [@2005ASPC..336...25A], or suggested by comet composition [@2005Natur.437..987K], and the factor of $\sim$4 derived in the original MMSN model [@1981PThPS..70...35H].
The need for larger surface density enhancements inspired “cold finger” disk models, which produce much larger snow line enhancements in a relatively narrow ($\lesssim$AU) radial region near the snow line [@1988Icar...75..146S; @2004ApJ...614..490C]. In this picture, a circumstellar disk has an initial equilibrium state with the water vapour (ice) concentration decreasing (increasing) beyond the snow line. As the disk diffuses and advects, water continually condenses from gas passing beyond the snow line, thus enhancing the local surface density of solids, and removing vapour phase water from the inner disk. Sublimation of planetesimals that drift inside the condensation radius by gas drag enhances this effect: the surface density beyond the snow line increases when water vapour from the sublimated planetesimals diffuses back outside the snow line [@2004ApJ...614..490C].
The first cold finger models predict a factor of $\sim$10-100 increase in the surface density of icy material in a relatively narrow region near the snow line [@1988Icar...75..146S; @2004ApJ...614..490C]. Using a more complex global disk model, @2006Icar..181..178C suggest surface density enhancements closer to 10 than 100. In their simulations, the enhancement regions are several AU wide at half the maximum planetesimal surface density.
The main differences expected for planet formation models using cold finger instead of MMSN disks are threefold. Due to the nature of the surface density enhancement: (i) fewer large planets form, (ii) large planets form in relatively low-mass disks, and (iii) planets form from material with much higher ice/rock ratios. In addition, material lost to inward planetesimal drift by gas drag [@2003Icar..161..431T] may be returned to the cold finger region, allowing continued growth. Reducing the removal of drifting planetesimals enhances growth rates, and allows formation of more massive icy planets.
Mathematical Formalism {#sec:math-background}
----------------------
In the standard coagulation model, planets grow in a circumstellar disk through repeated collisions and mergers of smaller objects [@1969QB981.S26......]. First, roughly km size planetesimals form rapidly, whether by coagulation [e.g. @2000SSRv...92..295W] or direct collapse [e.g. @1973ApJ...183.1051G]. Little knowledge of which process dominates means the size distribution of the first planetesimals is poorly constrained. Planetesimals initially grow through a rapid phase of “runaway” growth [@1996Icar..123..180K]. During the period of “oligarchic” growth that follows [@1998Icar..131..171K], protoplanetary growth rates depend on the surface density of planetesimals $\sigma_{\rm s}$, the local orbital frequency $\Omega$, the gravitational reach of the growing protoplanet, and the random velocities of the smaller planetesimals [@2001Icar..149..235I] $$\label{eq:mdot}
\dot{M_{\rm pl}} \propto \sigma_{\rm s} \, r_{\rm H}^2 \, \Omega
\, P_{\rm col}(\tilde{e},\tilde{i}) \, .$$ Here $r_{\rm H} = a \left( M_{\rm pl} / 3 M_\star \right)^{1/3}$ is the Hill radius, and $a$ is semi-major axis. The eccentricity $\tilde{e}$ and inclination $\tilde{i}$ are in units of the growing protoplanets Hill radius (i.e. $\tilde{e} = e/r_{\rm H}$). The collision probability $P_{\rm col}$ largely determines how growth proceeds: growth is fastest when planetesimals are small enough ($\lesssim$1km) to be damped by gas drag [e.g. @2004AJ....128.1348R]. In this “shear dominated” regime when $\tilde{e}$ and $\tilde{i}$ are $\lesssim$1, growth depends on Keplerian shear in the disk, rather than objects random velocities. Growth slows strongly with increasing radial distance, because $\Omega \propto a^{-3/2}$ and $\sigma_{\rm s} \propto
a^{-\delta}$, where $\delta \sim 1$–1.5.
Eventually, protoplanets accrete most of the nearby material and reach the “isolation” mass [@1987Icar...69..249L] $$\label{eq:miso1}
M_{\rm iso} = \frac{ \left( 4 \pi B \sigma_{\rm s} a^2 \right)^{3/2} }
{ \left( 3 M_\star \right)^{1/2} } \, .$$ Numerical simulations indicate that isolated oligarchs are spaced at $2B R_{\rm H} \sim 8 R_{\rm H}$ intervals [e.g. @1998Icar..131..171K]. In the terrestrial region around the Sun, the isolation mass is $\sim$0.1$M_\oplus$, and the timescale for Earth formation by the chaotic growth that follows is $\sim$10-100Myr [e.g. @2006AJ....131.1837K].
Further out in the disk, larger isolation masses allow formation of gas giant planets. The critical core mass for gas accretion depends on opacity and planetesimal accretion rates, but is $\gtrsim$10$M_\oplus$ [e.g. @2000ApJ...537.1013I; @2006ApJ...648..666R]. This mass is reached more easily further out in the disk because $M_{\rm iso}$ increases with $a$. However, growth slows rapidly with increasing radial distance; thus, there is an optimum region where cores are massive enough to accrete gas and to form giant planets before the gas disk is dissipated [@2008ApJ...673..502K]. This region is sufficiently far from the star that *in situ* formation of “hot-Jupiters” is unlikely, thus motivating theories of migration and scattering.
Migration {#sec:mig-background}
---------
Type I migration is a potential barrier to the formation of both terrestrial and giant planets [@1980ApJ...241..425G; @1997Icar..126..261W; @2002ApJ...565.1257T; @2007prpl.conf..655P]. When protoplanets reach near an Earth mass, the excitation of spiral density waves in the gaseous disk causes planets to experience a torque, and migrate inward. The timescale for a planet to spiral into the central star is [@2002ApJ...565.1257T] $$\label{eq:taumig}
\tau_{\rm mig} = \left( 2.7 + 1.1 \delta \right)^{-1}
\frac{ \left( M_\star M_\odot \right)^2 }{ M_{\rm pl} \, \sigma_{\rm
gas} \, a^2 } \frac{ h^2 }{ \Omega } \, ,$$ where $h \approx 0.05$ is the disk aspect ratio, and the stellar mass $M_\star$ is in units of Solar masses. For a planet of mass $M_{\rm
pl} = 1\,M_\oplus$ in a disk with $\sigma_{\rm gas} = 1700$g cm$^{-2}$ at 1AU around a Solar-mass star, $\tau_{\rm mig} = 1.6
\times 10^5$yr. Because this timescale is shorter than the $\sim$3Myr disk lifetime [@2001ApJ...553L.153H], and comparable with growth timescales, type I migration theory conflicts with terrestrial and giant planet formation in the Solar System [but see @2006ApJ...652L.133C].
Recent work suggests a reduced migration efficiency can resolve this problem [@2008ApJ...673..487I]. This “offset” applies to planets $\lesssim$15$M_\oplus$ and arises from corotation torques by coorbital material [@2006ApJ...652..730M]. Other ways of reducing (and even reversing) type I migration rates include turbulence arising from the magnetorotational instability [e.g. @2004MNRAS.350..849N], and eccentricity driven by planet-planet interactions [@2000MNRAS.315..823P].
If planets do not fall onto the central star, migration is a possible mechanism for producing planets on short-period orbits [@1996Natur.380..606L; @2005Icar..177..264B; @2007ApJ...654.1110T].
Scattering {#sec:scattering-background}
----------
Planet-planet scattering can also produce planets with short-period, or low periastron ($q$) orbits. Originally proposed to explain hot-Jupiters [@1996Sci...274..954R], this scenario has not been applied to low-mass planets.
Scattering favours giant planets on short-period orbits. When a gas giant scatters into a low periastron orbit, tidal interaction with the star can circularise the orbit on reasonable timescales, with $a \sim
2q$ [@1996Sci...274..954R]. For lower mass planets, long circularisation timescales make circular orbits unlikely [@2007arXiv0711.2015R]. However, if the initial scattering region is sufficiently close, as for low-mass stars, detection of low-periastron eccentric planets is possible.
We now consider two different scenarios that form short-period and/or transiting low-mass planets that begin growth near the snow line, across a range of stellar masses. When the snow line enhancement is small, many planets migrate toward close orbits. This scenario has already been studied for Solar-mass stars by @2007ApJ...654.1110T. Here, we instead consider cold finger type disks, where a few planets forming near the snow line dominate others forming elsewhere in the disk. We first consider a scattering scenario resulting from *in situ* growth, and then a migration scenario. We defer discussion of subsequent planetary evolution in final orbits to §\[sec:summary\].
Scattering {#sec:scattering}
==========
Planet-planet scattering is a likely outcome of oligarchic growth. In migration scenarios, protoplanets interact strongly with the gas disk, and they migrate to close-in orbits. However, if the gas disk disperses before planets have time to migrate, or if migration results in no net inward movement, planets form *in situ*. During oligarchic growth, protoplanets grow on orbits near the limits of dynamical stability, with damping provided by small bodies [e.g. @1988Icar...74..542S; @1998Icar..131..171K]. At later stages near isolation, their orbits can become unstable as remaining small bodies are accreted [@2004ApJ...614..497G; @2006AJ....131.1837K].
When planets start interacting dynamically, the boundary in semi-major axis between stable and unstable configurations is very sharp. Thus, two planets with orbits that become too close experience the sudden onset of a dynamical instability caused by close encounters [@1993Icar..106..247G].
In previous studies of giant planet scattering, planets begin at $\sim$AU distances from the central star, with spacings just inside the stability limit. After many interactions, one planet sometimes attains a highly eccentric orbit with a small periastron distance [e.g. @1996Sci...274..954R; @2007astro.ph..3163F]. Tidal interaction with the central star then circularises the orbit with $a
\sim 2q$.
While tidal forces can circularise gas giant orbits, the timescales for 1–10$M_\oplus$ planets on highly eccentric orbits are long [$\gtrsim$Gyr, @2007arXiv0711.2015R]. Although these planets maintain eccentric long-period orbits, transits are possible in favourable circumstances. Because planets form at shorter orbital periods around low-mass stars, these provide the best opportunity for transit observations.
Cold finger disks provide an ideal environment for oligarchic growth followed by planet-planet scattering. The width of the cold finger region allows several protoplanets to form [@2006Icar..181..178C]. Once protoplanets reach isolation, further chaotic growth may occur if their escape velocity $v_{\rm
esc}$ is less than the local Keplerian velocity $v_{\rm K}$ [$\mathcal{R} \equiv v_{\rm esc}/v_{\rm
K}$, @2004ApJ...614..497G]. In the terrestrial region of Solar-type stars, $\mathcal{R} \sim 1/4$. For gas giants, $\mathcal{R}
\gg 1$. For $M_{\rm pl} = 10\,M_\oplus$ with density $\rho = 4.5$g cm$^{-2}$, $\mathcal{R} \approx 1.3$ outside the snow line. Thus, $\sim$10$M_\oplus$ protoplanets present an approximate division between coalescence and scattering/ejection, and an order of magnitude estimate of the maximum planet mass. This mass is similar to the minimum needed for gas accretion, so scattering of super-Earths to close-in orbits appears difficult.
For less massive stars, scattering to low periastron orbits is easier. At fixed $a$, smaller $v_{\rm K}$ leads to larger $\mathcal{R}$ and a greater chance of scattering. However, the snow line also moves inward as stellar mass decreases [$a_{\rm snow}
\propto M_\star^{1-2}$, e.g. @2008ApJ...673..487I; @2008ApJ...673..502K], so scattering remains difficult. For $a_{\rm snow} \propto M_\star$, $v_K(a_{\rm snow})$ is constant for different stellar masses. However, for a fixed time period, a greater number of conjunctions for low-mass stars allows dynamical evolution to greater eccentricities.
Scattering Simulations {#sec:sim-scattering}
----------------------
To measure the likelihood of planet-planet scattering, we performed simulations over a range of stellar masses with the MERCURY integrator [@1999MNRAS.304..793C]. We initialised integrations with two 10$M_\oplus$ planets spaced near the Hill stability criterion to ensure close encounters [@1993Icar..106..247G]. This planet mass is an approximate maximum mass before cores accrete gas to become gas giants, and thus offers the best chance for scattering over coalescence. To represent a linearly stellar mass dependent snow line, the inner planet was placed at $a_{\rm in} = 3\,M_\star$AU. The outer planet begins at a random $a$ in the range 0.9–$1 \, a_{\rm in}
\left( 1 + \Delta_{\rm crit}\right)$, where $\Delta_{\rm crit} = 3
\left( M_{\rm pl}/M_\star \right)^{1/3}$ [@1993Icar..106..247G; @2007astro.ph..3163F]. Both planets begin in circular orbits with random inclinations less than 3$^\circ$; the remaining orbital elements are chosen randomly. Simulations were run with a 5day timestep for 1Gyr around stars of 0.25, 0.5, 1, and 2$M_\odot$, or halted earlier in the case of collisions (we assume perfect mergers) or ejections. A total of 520 simulations were run, 130 for each stellar mass.
Scattering Results {#sec:results-scattering}
------------------
The simulations result in three different outcomes: collisions, ejections, or survival of both planets for 1Gyr. No planets achieved periastra low enough to fall onto the central star. Most ($>85\%$) simulations resulted in collisions (Table \[tab:scattering\]). Some systems survived for the full simulation. The only ejections were for 0.25$M_\odot$.
[cccc|c]{} 0.25 & 95% & 2.5% & 2.5% & 4%\
0.5 & 95% & 0% & 5% & 0%\
1 & 94% & 0% & 6% & 0%\
2 & 84% & 0% & 16% & 0%\
With so few systems remaining after 1Gyr, we use the smallest periastron distance reached in each simulation to characterise the success of planet-planet scattering, shown in Figure \[fig:rfhist\]. As expected, the closer snow line distance for the 0.25$M_\odot$ allows smaller periastra after scattering.
For simulations of 0.25$M_\odot$ stars, 5/130 (4%) planets reach periastra less than 0.1AU. A shorter orbital period allows many more conjunctions. Thus, systems evolve further than for more massive stars. For the three ejections, the lowest periastra were reached just before a series of close encounters, which resulted in the ejection. For the three surviving systems, the lowest periastra were reached near the end of the integrations. These orbits have eccentricities $\approx$0.5, and semi-major axes $\approx$0.5AU, corresponding to an orbital period of around 260days. Circularisation times for these planets are $\sim$10Gyr [@1966Icar....5..375G; @2007arXiv0711.2015R].
Therefore, in the case of 10$M_\oplus$ planet-planet scattering, only the lowest mass stars have planets with periastra close enough for transiting orbits. However, long circularisation timescales mean these planets will likely remain on highly eccentric orbits, with periods long enough to make radial velocity and transit detections difficult.
Migration {#sec:migration}
=========
We now turn to a migration scenario. In this picture, the largest protoplanets form near the snow line by oligarchic growth. Once they reach masses of $\sim$1–10$M_\oplus$, these icy protoplanets migrate towards the central star. In a cold finger disk, migrating icy objects dominate smaller interior rocky protoplanets. Long chaotic growth timescales mean that as the icy object migrates through the terrestrial region, interior rocky objects are not accreted. They are instead scattered outward or shepherded inward. Shepherding—where interior objects are captured into mean-motion resonances—results in rocky protoplanets being pushed inward ahead of the migrating icy planet. These smaller objects merge to form large rocky planets, which are eventually accreted by the larger icy migrator or survive on an interior orbit. We assume that all objects halt their migration when they reach the inner edge of the gas disk, at $\sim$10 stellar radii.
Our goal is to calculate the growth and migration of individual protoplanets in this scenario. If several protoplanets migrate, whether they do so as a set in resonant orbits [@2007ApJ...654.1110T], or successively [e.g. @2006Icar..185..492D; @2008ApJ...673..487I], the resulting trends are similar in our picture. The main difference between outcomes is the number of icy planets on short-period orbits. The trends our models predict depend largely on disk and planet properties, not multiplicity.
When an icy protoplanet migrates, it only interacts dynamically with interior objects. Because collision cross sections are essentially geometric, the timescale for growth is much longer than the migration timescale. For example, the migration timescale for an Earth-mass planet at 1AU (several 10$^5$yr, or several 10$^6$yr if migration is less efficient) is much smaller than the chaotic growth timescale [$\tau_{\rm chaotic} \sim \rho R_{\rm pl} / \sigma_s
\Omega \sim 10^8$yr, where $R_{\rm pl}$ is the planet radius, @2004ApJ...614..497G]. Thus, the migrating protoplanet does not accrete terrestrial protoplanets; outward scattering or inward shepherding are the most likely outcomes.
The evolution of interior protoplanets depends on their random velocities. Chaotically growing objects with high eccentricities are scattered outward by the migrating protoplanet. These may interact with another migrating protoplanet or resume chaotic growth. If interior objects have finished chaotic growth and are damped by the gas disk onto more circular orbits, shepherding by capture onto resonant orbits is possible. Shepherded objects merge and form rocky planets as their orbits are pushed together by the migrating icy protoplanet. Shepherding by giant planets undergoing type II migration has been proposed as a way to form super Earth-mass planets [@2005ApJ...631L..85Z]. However, studies have yet to consider shepherding by super-Earths undergoing type I migration.
While some planets are stranded at intermediate radii as the gas disk dissipates, most planets that begin to migrate reach the inner disk edge, and might fall onto the star. Because the torque on the migrating planet changes when the disk gas surface density profile varies rapidly, as happens at the inner disk edge, this fate may be avoided [@2002ApJ...565.1257T]. Here, corotation torques affect migration, and allow for planets to cease migration before reaching the stellar surface [@2006ApJ...642..478M]. In our migration simulations, we therefore assume migration stops inside the inner disk edge [@2007ApJ...654.1110T].
In the rest of this section, we consider three models that explore different aspects of the migration scenario, and observable trends that probe stellar and disk properties. We consider the simplest scenario—when growth is so fast that planets reach isolation before migration begins—with an analytic model in §\[sec:analytic\]. As the planetesimal size increases, growth slows; the timescale becomes comparable to that for migration. The assumption made in the analytic model no longer applies, and we use a semi-analytic model to study concurrent growth and migration in §\[sec:model\]. Finally, we use $n$-body simulations in §\[sec:shepherding\] to show the shepherding effects migrating super-Earths have on terrestrial material.
An Analytic Approach {#sec:analytic}
--------------------
If we assume that protoplanets reach isolation before migration starts, then we can create a simple analytical model for our migration scenario. At isolation, protoplanets have a known migration timescale, which is shorter than the disk lifetime if they are to reach the central star. To remain in the super-Earth mass regime, the mass of a protoplanet is smaller than the critical core mass for gas accretion. Because the isolation mass changes with surface density—and thus with disk mass—only a certain range of disk masses satisfy these conditions for fixed stellar mass. To consider a range of different stars, we also consider how the snow line—where these migrating planets form—changes with stellar mass. The range of disk masses that satisfy the conditions changes with stellar mass, resulting in potentially observable trends that test migration models.
To begin, we adopt a relation for the surface density of solid material in the disk. In the standard MMSN model, $$\sigma_{\rm s} = \sigma_0 \, f_{\rm ice} \, a_{\rm AU}^{-\delta} \, ,$$ where $\sigma_0 = 8$g cm$^{-2}$, $\delta = 1$–1.5, and $a_{\rm AU}$ is $a$ in units of AU. The factor $f_{\rm ice} \sim 2$–3 is the enhancement from ice condensation beyond the snow line. This disk has a mass $\sim$0.01$M_\odot$.
To generalise this relation, we add terms to account for differences in disk mass and metallicity around stars with a range of masses. Disks around young stars have a large dispersion in mass [@2000prpl.conf..559N; @2005ApJ...631.1134A; @2007ApJ...671.1800A]. Setting the disk mass $M_{\rm disk} \propto \eta M_\star^\beta$ allows us to treat the observed trends with stellar mass—$M_{\rm disk} \propto
M_\star^\beta$, with $\beta \approx 1$—and a range ($\eta$) of disk masses at fixed stellar mass. Adopting a factor $\mathcal{M} \propto
10^{\rm [Fe/H]}$ for the metallicity of the stars and the disk yields $$\label{eq:sigma}
\sigma_{\rm s} = \sigma_0 \, \eta f_{\rm ice} \, \mathcal{M} \, M_\star^\beta
\, a_{\rm AU}^{-\delta} \, .$$ For simplicity, we combine $f_{\rm ice}$ and $\mathcal{M}$ into a single factor $\Delta = f_{\rm ice} \mathcal{M}$, which quantifies the enhancement of solid material relative to gas where these planets form. For a cold finger disk, we use $f_{\rm ice} = 10$. Thus, for typical ranges in $\mathcal{M}$ ($\sim$1/3–3) and $f_{\rm ice}$ (2–10), the plausible range of $\Delta$ is 0.6–30. We concentrate on higher $\Delta$, because these are cold finger disks.
For the surface density of the gas disk, we set $\sigma_g = 100
\sigma_s / \Delta$. Thus, the gas mass depends on $\eta$ and $\delta$, and is independent of metallicity and the enhancement in ices at the snow line. We adopt $\delta = 3/2$.
How the snow line varies with stellar mass is uncertain. The existence of gas giant planets suggests that the stages of planet formation up to isolation occur while the gas disk is still present. During these stages the snow line distance is set by viscous accretion of the gas disk. If the accretion rate onto the star is $\dot{M} \propto
M_\star^{1-2}$, then $a_{\rm snow} \propto M_\star^{6/9 - 8/9}$ [@2008ApJ...673..502K]. Later, when the star has reached the main-sequence and the gas disk has been dissipated, the main-sequence luminosity is more important and $a_{\rm snow} \propto M_\star^2$ [@2005ApJ...626.1045I]. Because we model oligarchic growth, and $\dot{M} \propto M_\star^2$ [@2005ApJ...625..906M], we adopt the snow line distance $a_{\rm snow} = 2.7 M_\star$AU. Variation of the snow line with time and stellar mass is a key component of planet formation models that consider a range of spectral types [@2008ApJ...673..502K].
Substituting our adopted surface density into the isolation mass yields $$\label{eq:miso3}
M_{\rm iso} \propto \frac{ \left( \sigma_{\rm s} \, a^2 \right)^{3/2} }
{ \left( M_\star \right)^{1/2} } =
\frac{ \left( \eta \, f_{\rm ice} \, \mathcal{M} \, M_\star \,
a^{1/2} \right)^{3/2} }
{ M_\star^{1/2} } \, .$$ The isolation mass increases with any parameter that increases the surface density. The increasing disk mass with stellar mass ($M_\star$ in numerator) is stronger than the decreasing Hill radius ($M_\star$ in denominator). Thus, at fixed $a$ the isolation mass increases with stellar mass. For our scenario, we are interested in planets that form at the snow line, so the changing snow line distance ($a = a_{\rm
snow} \propto M_\star$) makes the stellar mass dependence stronger.
Substituting $a = 2.7 M_\star$AU, equation (\[eq:miso3\]) yields the isolation mass *at the snow line* for a range of stellar and disk masses, and metallicities and snow line enhancements $$\label{eq:miso}
M_{\rm iso} = 0.12 \, \left( \Delta \, \eta \right)^{3/2}
M_\star^{7/4} M_\oplus \, .$$
Applying the same approach to type I migration yields $$\label{eq:taumig1}
\tau_{\rm mig} \propto \frac{ M_\star^2 \, h^2 \, f_{\rm mig} }
{ a^2 \, M_{\rm pl} \, \Omega \, \sigma_{\rm gas} } =
\frac{ M_\star^{1/2} \, h^2 \, f_{\rm mig} \, a }{ M_{\rm pl} \, \eta } \, ,$$ where the offset $f_{\rm mig}$ allows us to consider reduced migration rates. At fixed $a$, migration takes longer as stellar mass increases, and speeds up as planet mass increases. If planet masses vary less strongly with radial distance than $M_{\rm pl} \propto a$, then the migration timescale increases outward, and planets cannot catch up to interior ones. Even with isolated objects ($M_{\rm iso} \propto
a^{3/2}$), planets may not catch up to interior ones due to the strong slowing of growth with semi-major axis. At the snow line distance, migration slows even more strongly with increasing stellar mass due to lower gas density and slower orbital periods at larger radii.
Again substituting $a = 2.7 M_\star$AU, the timescale to migrate from the snow line to the star is $$\label{eq:tmig}
\tau_{\rm mig} = 9.1 \times 10^5 \,
\frac{ f_{\rm mig} \, M_\star^{3/2} }{ M_{\rm pl} \, \eta } \, {\rm yr}
\, ,$$ where we have set $h = 0.05$ [e.g. @2007prpl.conf..655P]. At fixed $M_\star$, massive planets in massive disks migrate to the inner disk edge fastest. The migration timescale increases with $M_\star$ because the snow line is further away.
If the migration time is shorter than the disk lifetime (i.e. $\tau_{\rm mig} \lesssim \tau_{\rm disk} \sim 1$Myr), then protoplanets reach short-period orbits. This inequality leads to $$\label{eq:miso2}
M_{\rm pl} > \frac{ 0.91 \, f_{\rm mig}
\, M_\star^{3/2} }{ \eta } \, M_\oplus \, .$$ This result yields the minimum mass for a planet to migrate to a close orbit. Substituting the isolation mass (eq. \[eq:miso\]) for $M_{\rm
pl}$ and solving for $\eta$ gives a lower relative disk mass limit of $$\label{eq:llim}
\eta > \eta_{\rm low}
= \frac{ 2.2 \, f_{\rm mig}^{2/5} }{ M_\star^{1/10} \, \Delta^{3/5}
} \, .$$ Disks more massive than this $\eta$ form protoplanets massive enough to migrate to short-period orbits before the gas disk dissipates. Planets in slightly less massive disks still migrate, but are stranded at intermediate radii as the disk disperses.
The critical $\sim$10$M_\oplus$ core mass for gas accretion provides an upper limit for the protoplanet mass. Solving $M_{\rm iso} <
10\,M_\oplus$ for $\eta$ yields $$\label{eq:ulim}
\eta < \eta_{\rm hi} = \left\{ \begin{array}{ll}
\frac{ 18.6 }{ M_\star^{7/6} \, \Delta } &
\frac{ 18.6 }{ M_\star^{7/6} \, \Delta } < 30 \\
30 & \frac{ 18.6 }{ M_\star^{7/6} \, \Delta } \geq 30
\end{array}
\right. \, ,$$ where the additional constraint of a reasonable disk mass sets $\eta
\lesssim 30$ ($M_{\rm disk} \lesssim 0.3\,M_\star$) as an upper limit [e.g. @2005ApJ...626.1045I]. Because we assume growth is fast, planetesimal accretion drops significantly at later stages. The core mass for gas accretion is then somewhat smaller [@2000ApJ...537.1013I; @2006ApJ...648..666R].
The two limits on disk mass yield a simple relation between the stellar mass, migration offset, and enhancement factor. Equating $\eta_{\rm low}$ and $\eta_{\rm hi}$, $$\label{eq:mstarmax}
M_{\star,{\rm max}} = \frac{ 7.3 }
{ \left( f_{\rm mig} \, \Delta \right)^{3/8} }$$ in units of Solar masses.[^1] This equation has a simple physical interpretation. For massive stars ($M_\star > M_{\star,{\rm max}}$), the only protoplanets massive enough to migrate to the central star before the gas disk disperses are above the critical core mass for gas accretion. These planets therefore become gas giants, rather than forming hot super-Earths. For lower stellar masses, the closer snow line distance allows planets smaller than the critical core mass to migrate to the host star. Thus, $M_{\star,{\rm max}}$ is the maximum stellar mass for hot super-Earths produced by type I migration.
Making an estimate of $M_{\star,{\rm max}}$ requires an assumed $f_{\rm mig}$ and $\Delta$. For Solar metallicity $\mathcal{M} = 1$, and a cold finger enhancement $f_{\rm ice} = 10$–20, $\Delta =
10$–20. For a migration offset $f_{\rm mig} = 10$, $M_{\star,{\rm
max}} \sim 1\,M_\odot$. Transit and radial velocity surveys routinely probe these stellar masses. Independent of the disk mass distribution, this result is therefore a simple testable prediction of hot super-Earth formation by type I migration.
Figure \[fig:f10sum\] shows the range of planet masses that reach short-period orbits for a range of stellar masses. For the analytic model (thick grey lines) the upper limit is constant at 10$M_\oplus$. The lower limit decreases as stellar mass and snow line distance decrease. The expected range of planet masses decreases with increasing stellar mass, while the average mass increases to 10$M_\oplus$, where the lines meet at $M_{\star,{\rm max}} =
1.3$$M_\odot$.
In addition to this maximum stellar mass, we can derive the probability of forming hot super-Earths around stars with $M_\star <
M_{\star,{\rm max}}$. This estimate requires an adopted distribution of $\eta$ (i.e. disk masses). If relative disk masses ($M_{\rm
disk}/M_\star$) are distributed as a power law with index $\sim$$-1.75$ [@2005ApJ...631.1134A], the (relative) probability of forming a close-in planet as a function of stellar mass for a given $\Delta$ is $$\label{eq:pprob}
P_{\rm p}(M_\star,\Delta) \propto
\int_{\eta_{\rm lo}}^{\eta_{\rm hi}} \eta^{-1.75} \, d\eta \, .$$ Alternatively, disk masses may be distributed around some “typical” relative disk mass [e.g. @2005ApJ...626.1045I] $$\label{eq:gprob}
P_{\rm g}(M_\star,\Delta) \propto \int_{\eta_{\rm lo}}^{\eta_{\rm hi}}
\exp \left( - \frac{ \left( \log(\eta) - \mu \right)^2 }{ 2 \, s^2 } \right)
d\eta$$ where we choose the standard deviation $s = 1$. This distribution is plausible because opacities may underestimate disk masses by as much as an order of magnitude, due to mass locked up in boulder size objects [@2007ApJ...671.1800A]. Therefore mm observations see disks not only with a range of masses, but in a range of evolutionary states. Unlike the case for giant planets, there is no observational anchor point, so we present these results as relative probabilities.
For a range of $\Delta$, the left panel of Figure \[fig:prob\] shows the probability distribution for the power law disk mass distribution with $f_{\rm mig} = 10$. Results are similar for $f_{\rm mig} = 1$, with the main difference that $M_{\star,{\rm max}}$ is higher (eq. \[eq:mstarmax\]). Higher $\Delta$ are most relevant here because low values describe MMSN disks, which result in many similar-mass migrating planets originating from a wide range of radii. The point where lines break and decrease toward lower stellar masses is caused by the maximum disk mass condition $\eta < 30$. In these cases the maximum short-period planet mass is not set by gas accretion, and is $<$10$M_\oplus$. This limit applies when $\Delta
\lesssim 5$ for the lower of the stellar masses we consider, so does not apply to cold finger disks with $f_{\rm ice} \gtrsim 10$ unless they have metallicity $\mathcal{M} \lesssim 0.5$.
At the lowest stellar masses, there is a clear increase in planet frequency with $\Delta$. With a power-law distribution of disk masses, the most common disks are the least massive; these require large $\Delta$ to allow them to form planets massive enough to migrate (and satisfy condition \[eq:llim\]). Near $M_{\star,{\rm max}}$, there is an optimum $\Delta$, which is a balance between the likelihood of different disk masses and the $\Delta$ needed to form close-in planets from those disks. At $M_{\star,{\rm max}}$, the only planet that reaches a short-period orbit has $M_{\rm pl} =
10\,M_\oplus$. Therefore the range of short-period planet masses decreases up to $M_{\star,{\rm max}}$. The average planet mass increases with stellar mass.
The right panel of Figure \[fig:prob\] shows the probability distribution for the Gaussian distribution with $\mu = 1$. The most common disk mass is thus $\sim$0.1$M_\star$. As $\Delta$ increases, the probability of forming a short-period planet decreases once the disk mass distribution is not truncated by the condition $\eta <
30$. In contrast to the power law distribution, the low-mass disks requiring large $\Delta$ are uncommon. Thus, as $\Delta$ increases, isolation masses are pushed over the gas accretion mass, and the likelihood of forming close-in $\lesssim$10$M_\oplus$ planets decreases. While the curves are different from the left panel, the point $M_{\star,{\rm max}}$ is the same for a given $\Delta$. With $\mu = 0$ (i.e. distributed about $M_{\rm disk} = 0.01\,M_\star$) the probability distribution is qualitatively similar to the power law disk distribution.
In summary, the simple analytical model yields testable predictions for an ensemble of super-Earths that migrate into short-period orbits from the snow line. For reasonable input parameters, we predict a maximum stellar mass $\sim$1$M_\odot$ for stars with close-in super-Earths. If circumstellar disks tend to have similar snow line enhancements, this maximum mass decreases with the metallicity of the host star. For a range of stellar masses, the frequency of hot super-Earths depends on the initial distribution of disk masses. For a power-law (Gaussian) distribution of disk masses, the model predicts more (fewer) hot super-Earths around more metal-rich stars.
To give the these trends some context, the first transiting low-mass planet orbits a star with sub-Solar mass and metallicity . The current sample of low minimum-mass planets also indicates a flatter metallicity distribution than exists for giant extra-Solar planets [@2007prpl.conf..685U]. While both disk mass distributions suggest that low stellar mass host is likely, the power law distribution argues against a low metallicity host. The Gaussian disk mass distribution, centered on a relatively high disk mass is consistent with an increasing giant planet frequency with metallicity, and a flatter or decreasing frequency for lower mass planets.
Disks with $\eta > \eta_{\rm hi}$ form gas giants. Their relative probabilities can thus be calculated by integrating equations (\[eq:pprob\]) and (\[eq:gprob\]) from $\eta_{\rm hi}$ to 30. However, because $\eta_{\rm low}$ only weakly depends on $M_\star$, giant planet frequency is roughly some constant minus the hot super-Earth frequency (i.e. generally increases with $M_\star$). This trend is essentially the result arrived at by previous theoretical studies [e.g. @2005ApJ...626.1045I; @2008ApJ...673..502K], and is at least qualitatively consistent with the observed trend [@2007ApJ...670..833J].
In constructing the above model we simplified some parameters, and assumed values for others. We now briefly consider model sensitivity to these, and whether observations may constrain them. The most uncertain simplification is how the snow line distance varies with stellar mass. Within our framework, relaxing the distance to $a_{\rm
snow} = 2.7 M_\star^\alpha$AU results in changes to Equations (\[eq:ulim\]), (\[eq:llim\]), and (\[eq:mstarmax\]) for $\alpha
= 1/2$–2 (Fig. \[fig:f10sum\] inset). A more strongly varying snow line distance ($\alpha = 2$) yields much closer $a_{\rm snow}$ and smaller $M_{\rm iso}$ (due to smaller $R_{\rm H}$) for low mass stars. A more complex snow line model could include how $a_{\rm snow}$ varies with $M_{\rm disk}$ at fixed stellar mass, or some time dependence [e.g. @2006Icar..181..178C; @2008ApJ...673..502K].
Another uncertain parameter is $\delta$, the disk surface density power-law index. While we used $\delta = 3/2$, many models also consider $\delta = 1$. With $\delta = 1$, the main results of Figure \[fig:f10sum\] are unchanged, with stronger migration accounting for lower mass planets as the snow line distance decreases. It is unlikely observations of short-period super-Earths can constrain $\alpha$ or $\delta$ based on Figure \[fig:f10sum\], because they affect lower limits to planet masses, which will be hard to detect.
The efficiency of type I migration is also unclear. Our choice of $f_{\rm mig} = 10$ is based on numerical simulations, but may also be probed by future discoveries. The maximum stellar mass $M_{\star,{\rm
max}}$ is not very sensitive to the snow line distance or disk profile, so for fixed snow line and metallicity enhancements ($\Delta$), observations probe values for $f_{\rm mig}$.
Our final major assumption is that planets form rapidly, and reach isolation before migrating. If planetesimals are small and growth is shear dominated, this assumption is generally true. With larger planetesimals however, growth is slower and planets may leave their formation regions while still growing. Planetary growth and migration are then coupled, and must be calculated simultaneously. Recently, @2006Icar..180..496C [@2006ApJ...652L.133C] showed how a semi-analytic model of oligarchic growth can take different planetesimal sizes into account, and estimate their effect on growth rates [see also @2003Icar..161..431T; @2008Icar..194..800B]. We now turn to a similar, yet simplified model to estimate the effects of planetesimal size on growth and migration.
Semi-Analytic Model {#sec:model}
-------------------
If planets grow fast enough, the isolation mass sets the range of disk masses that form migrating planets. If planetesimals are large enough, growth is not shear dominated and is slower. Migration then begins before planets reach isolation. To follow this evolution, a model treating concurrent accretion and migration is necessary. Our model tracks damping of planetesimal random velocities by gas drag and stirring by a growing protoplanet. The random velocities set how growth proceeds relative to migration, allowing comparison with the analytic model.
In the model, a single protoplanet of mass $M_{\rm pl}$ grows on a circular orbit from a planetesimal disk of small bodies of radius $r$. We adopt the accretion rate of @2001Icar..149..235I with the atmosphere enhanced accretion radius of . To account for accretion of other nearby protoplanets, the growth rate is increased by 50% [@2006ApJ...652L.133C]. Planetesimal random velocities are stirred by the growing protoplanet [@2002Icar..155..436O] and damped by gas drag [@2001Icar..149..235I]. The protoplanet accretes and stirs material within an annulus of half-width 4$R_{\rm
H}$, and undergoes type I migration at the rate derived by @2002ApJ...565.1257T, modified by the offset $f_{\rm mig}$. We use a ten times less efficient migration rate, motivated by numerical , and Monte-Carlo simulations [@2008ApJ...673..487I]. Objects have mass density $\rho = 1.5$g cm$^{-3}$ outside the snow line. Simulations are started with planetesimals in an equilibrium between protoplanet stirring and gas drag. Because we consider growth only near the snow line (see below), planetesimals do not undergo radial motions due to gas drag. Planetesimals lost to gas drag can be returned to the growth region by the cold finger mechanism [@2004ApJ...614..490C]. The system is evolved using 4th order Runge-Kutta integration with an adaptive step-size [@1992nrca.book.....P].
As before, we model protoplanets that form just outside the snow line. These are the largest objects that migrate to the central star in a cold finger disk and are largely unaffected by interior objects. However, a migrating protoplanet shepherds material inward as it migrates, and will accrete some terrestrial material. This accretion cannot be treated by the semi-analytic model, so protoplanets cease accretion once they pass inside the snow line in the semi-analytic model. We model shepherding with $n$-body simulations in §\[sec:shepherding\]. We vary $\eta$ to form 1–10$M_\oplus$ planets and use $\Delta = 10$.
Protoplanets begin with masses $1 \times 10^{-4}\,M_\oplus$, at 4$R_{\rm H}$ outside the snow line. This starting condition allows them to reach isolation if growth is faster than migration. The disk is split into 1000 equally spaced radial bins. However, because accretion inside the snow line is turned off, objects grow from material in $\sim$100 bins outside the snow line.
The snow line distance and gas disk are as in §\[sec:analytic\] (eq. \[eq:sigma\] and following text), but the surface density of the gas disk decays exponentially with an e-folding time of 1Myr. We place the inner edge of our disk at 0.2$M_\star$AU, though planets that reach a few tenths of an AU are migrating so rapidly that the exact value matters little.
To test our code, we compare growth at 5AU with Figure 1 from @2006ApJ...652L.133C. His figure compares isolation times for different $r$ with the type I migration timescale. The smallest size planetesimals allow protoplanets to reach isolation before migration starts. Growth was simulated at 5AU around a Solar-mass star, with a solid surface density of 10g cm$^2$ (so $M_{\rm iso} \approx
10\,M_\oplus$), and a gas/solids ratio of 90. Migration was not included, and the isolation time was simply compared to the analytic estimate of Equation (\[eq:taumig\]).
Figure \[fig:test\] shows growth in the absence of migration at 5AU around a Solar-mass star for a range of $r$ with similar initial conditions. The time to reach isolation is fastest for the smallest $r$ (100m), because growth is always shear dominated. For $r =
1$km, the growing protoplanet excites the small body random velocities. Growth ceases to be shear dominated at several $10^3$yr. For higher $r$, isolation takes even longer, due to the decreasing effectiveness of gas drag on larger planetesimals. Compared with Figure 1 from @2006ApJ...652L.133C, the time to reach 10$M_\oplus$ is in good agreement. Our explicit calculation of eccentricities and inclinations accounts for differences in how growth proceeds [c.f. Fig. 3 of @2006Icar..180..496C].
Models with sufficiently small planetesimals reach isolation before migration. With $f_{\rm mig} = 1$, $r \lesssim 100$m, and for $f_{\rm mig} = 10$, $r \lesssim 1$km. Thus, even with a reduced migration rate, protoplanets may still migrate before isolation.
### Semi-Analytic Model Results {#sec:results}
For the range of disk masses ($\eta$) that forms 1–10$M_\oplus$ planets around stars with masses 0.25–2$M_\odot$, Figure \[fig:f10r100m\] shows semi-major axis and mass evolution for $r =
100$m. The choice of 1$M_\oplus$ is somewhat arbitrary, but represents a rough lower limit for detection. We first describe the Solar case, and then look at differences as the stellar mass, and $r$ change.
For a Solar mass star, growth is not always fast enough for migration to occur before the gas disk is dispersed. For $\eta = 2$, the objects Hill radii increase faster than small bodies are stirred; thus growth remains shear dominated ($\tilde{e},\tilde{i} \lesssim 1$). The protoplanet successfully migrates to the inner edge of the disk. For lower $\eta$, stirring overcomes damping at several $\times$10$^3$yr and growth slows. Higher $\eta$ results in faster growth of larger objects, which migrate early enough to avoid stalling at intermediate radii. With $\eta = 1$, migration is somewhat significant, and the $\sim$3 Earth mass planet stalls at $\sim$1AU due to dissipation of the gaseous disk. For $\eta = 0.5$, the Earth-mass planet migrates little, and remains beyond the snow line. Final planet masses, and the degree of migration, are set by the initial surface density beyond the snow line.
We turn now to trends across a range of stellar masses. Because of smaller snow line distances, migration is easiest for planets in the 1–10$M_\oplus$ range around lower mass stars. Low-mass stars are the most likely to form these planets, because the range of disk masses that form them is much larger. For higher mass stars the more distant snow line makes migration unlikely for all but the most massive planets. Growth is driven out of the shear dominated regime more easily due to lower gas density at greater distances. This result confirms the maximum stellar mass $M_{\star,{\rm max}}$ described above. As in the analytic model, $M_{\star,{\rm max}}$ lies between 1–2$M_\odot$ with $\Delta = 10$, because no planet with a mass $\lesssim$10$M_\odot$ migrates significantly for 2$M_\odot$. As stellar mass increases, the relative disk mass required to form 1–10$M_\oplus$ planets decreases (eq. \[eq:miso\]).
Figure \[fig:f10r10km\] shows how growth changes if planetesimals are larger. Models again have $\Delta = 10$, but now the planetesimal radius $r = 10$km. For larger planetesimals growth is easily stirred out of the shear dominated regime by the large objects for all stellar masses. The disk masses needed to reach the same range of planet masses are higher, because planets migrate out of the accretion region before they reach isolation. For 0.25$M_\odot$ stars, the maximum $\eta = 30$ only just forms 10$M_\oplus$ planets. Again, $M_{\star,{\rm max}}$ lies between 1–2$M_\odot$, indicating that it is largely independent of planetesimal size. Though growth is slower, the results for $r = 10$km are largely the same as 100m, because the surface density can be increased to account for the slower growth.
Figure \[fig:f10sum\] also includes results from the semi-analytic model, showing the range of planet masses that reach short-period orbits for a range of stellar masses. Models were run for $\Delta =
10$, with $r = 10$m–10km and $M_\star = 0.1$–2$M_\odot$. The upper limit decreases at 0.1$M_\odot$ due to an upper limit on disk masses. Results from the analytic model are in good agreement. The difference in the lower limit arises because migration is faster at smaller radii, allowing smaller planets to reach the inner disk edge in the semi-analytic model.
In summary, using a more detailed migration model yields results similar to the simple analytic treatment in §\[sec:analytic\]. The inclusion of growth rates due to different planetesimal sizes adds another dimension due to different relative timescales for migration and accretion. The model offers more insight into how growth proceeds, and how the growth rate sets the required disk mass for forming short-period planets.
Shepherding {#sec:shepherding}
-----------
As a large body migrates inward, it captures interior objects onto mean motion resonances, and shepherds them inward. In the original scenario, a gas giant forms near the snow line, and subsequently migrates inward. As the giant migrates it shepherds interior protoplanets inward, which collide and merge to form super Earth-mass planets [@2005ApJ...631L..85Z]. Here we use $n$-body simulations to study a similar scenario, but with a low-mass planet migrating inwards from the snow line due to type I migration.
To investigate shepherding effects, we used the MERCURY integrator [@1999MNRAS.304..793C], including type I migration and damping forces . The migration rate, and eccentricity and inclination damping are reduced by a factor of $f_{\rm mig} = 10$. The inner disk edge is placed at 0.05AU, and inside this point planets cease to interact with the disk [@2007ApJ...654.1110T]. Simulations are initialised with a number of isolated protoplanets in a disk between 0.1AU and the snow line at $2.7\,M_\star$AU.
Isolation masses are calculated from Equation (\[eq:miso\]) with the half-spacing $B$ randomly varied between 3.75 and 4.25. We assume Solar metallicity and $f_{\rm ice} = 10$. One protoplanet begins beyond the snow line. This outermost protoplanet is $\approx$30 times more massive than the one immediately interior to it (eq. \[eq:miso1\]). Initial eccentricities (inclinations) are randomly distributed between 0 and 0.02 (0.5$^\circ$), and the remaining orbital elements are randomly distributed. We set the mass of the outermost planet at the middle of the range shown for $M_\star
= 0.25$, 0.5, and 1$M_\odot$ in Figure \[fig:f10sum\]; 2, 3.2, and 6$M_\oplus$ respectively. Simulations are run for 10$^8$yr with $\sim$0.3 day timesteps. Objects are allowed to collide, and are assumed to merge into a single body with no fragmentation. These simulations do not include relativistic effects, or tidal interaction with the star. See @2007ApJ...654.1110T for a more detailed study of migration to small radii, and how these effects affect final system dynamics.
### Shepherding results
Figure \[fig:shep\] shows the semi-major axis evolution resulting from these simulations. All show similar characteristics. Starting from the inner disk edge, a wave of chaotic growth moves outward [e.g. @2001Icar..152..205C; @2006AJ....131.1837K], until the number of protoplanets is reduced such that their spacing is stable. This stability is set by a balance between mutual perturbations between protoplanets, and damping by interaction with the gas disk.
When the outermost large protoplanet begins to migrate, it scatters the first objects it encounters into exterior orbits. When the interaction occurs, these objects are still undergoing eccentric chaotic growth, and are less likely to be captured onto resonances and shepherded inward. Once scattered, the outer objects slowly migrate inward. For 0.25 and 0.5$M_\odot$ the scattered planets are still relatively close to the star, and have time to set up chains of (mostly first order) resonant orbits. A few collisions occur. For 1$M_\odot$ more objects are scattered outward, which continue chaotic growth. Despite the initial disruption by the migrating object, $\sim$Earth-mass planets still form at $\sim$1AU.
When the migrating protoplanet encounters objects that have reached stable orbits, it shepherds them inward. These smaller objects accrete others as their orbits are pushed together, and several $\sim$1$M_\oplus$ rocky objects form. Shepherded objects may be accreted by the large migrator, or remain in interior resonant orbits [see @2007ApJ...654.1110T].
These simulations show that as with the gas giant case [@2005ApJ...631L..85Z], the effect of super-Earth migration on interior objects has observational consequences. Planets near the outer edge of the terrestrial region are scattered outward, while those in the inner region are shepherded to smaller radii. Shepherding results in multiple short-period planets with different compositions.
Discussion and Summary {#sec:summary}
======================
We have considered two scenarios for forming short-period $\lesssim$10$M_\oplus$ planets over a range of stellar masses: planet-planet scattering and type I migration.
Our models form planets in cold finger disks. These disks have large snow line enhancements compared to the MMSN model [@1988Icar...75..146S; @2004ApJ...614..490C]. Water vapour from the terrestrial region condenses into ices outside the snow line as the gas disk diffuses and advects. The enhancement is increased by new water vapour delivered inside the snow line by drifting icy planetesimals [@2004ApJ...614..490C]. Protoplanets forming in the cold finger regions near the snow line are much larger than others elsewhere in the disk.
We test the effectiveness of planet-planet scattering with $n$-body simulations. We consider stars with masses $M_\star =
0.25$–2$M_\odot$ and 10$M_\oplus$ planets. Planets with orbits near the limits of stability are evolved until a collision or ejection occurs, or 1Gyr. Although equal mass planet-planet scattering can produce planets with small periastra for the lowest mass stars (Fig. \[fig:rfhist\]), long circularisation times prevent them from achieving circular orbits on reasonable timescales. Thus, scattering is probably not a viable scenario for placing low-mass planets on short-period orbits for any stellar mass. For 0.25$M_\odot$, planets have periastra $\sim$0.05AU and semi-major axes $\sim$0.5AU. Though transit durations are still several hours, orbital periods of several hundred days and maximum radial velocities of a few m/s make these planets hard to detect.
Migration of icy protoplanets from the snow line is a viable mechanism for forming short-period super-Earths. Planet masses set whether they migrate to the inner disk edge before the gas disk disperses. Some planets with insufficient masses are stranded at intermediate radii as the gas disk disperses; a way to form “ocean planets” [@2003ApJ...596L.105K; @2004Icar..169..499L]. The minimum protoplanet mass for migration to a close-in orbit increases as the snow line moves out with increasing stellar mass (Fig. \[fig:f10sum\]). The maximum planet mass is $\sim$10$M_\oplus$, because above this mass they instead accrete large atmospheres and form gas giants.
Above $\sim$1$M_\odot$, the only protoplanets massive enough to migrate to close-in orbits are $\gtrsim$10$M_\oplus$ and no hot super-Earths form. This maximum stellar mass is independent of the disk mass distribution, and probes type I migration efficiency. Other uncertain parameters, such as snow line distance and disk profile do not have major observable consequences, but are not easily constrained by observations either.
For disks with similar snow line enhancements, the theory yields trends with metallicity (Fig. \[fig:prob\]). For disk masses distributed as a power law, the frequency of short-period planets increases with metallicity, because most disks have low masses. However, if disk masses are distributed around a relatively high mass, planet frequency decreases with increasing metallicity, because planets forming in the most common disks are pushed above the gas accretion mass at high metallicities. As planetesimal size increases, growth slows, and becomes longer than the migration timescale. Simulations of concurrent accretion and migration with increased planetesimal sizes require much higher disk masses to yield similar results.
As icy planets migrate from the snow line, they interact dynamically with interior rocky protoplanets (Fig, \[fig:shep\]). Protoplanets undergoing chaotic growth are scattered onto exterior orbits. Closer protoplanets on stable orbits damped by disk interaction are shepherded inward, and coalesce into a few rocky objects with masses $\sim$1$M_\oplus$. These objects may be accreted by the large migrating planet, or remain as separate planets on interior orbits. These orbits are likely near-commensurate with the icy migrators orbit [@2007ApJ...654.1110T]. If planetary systems in such configurations are found in transit surveys, compositional models may discern differences, thus confirming their origins in rocky or icy regions [@2007ApJ...665.1413V]. However, different structural models may be degenerate if the planets have atmospheres [@2007arXiv0710.4941A].
Some planets may accrete hydrogen atmospheres due to a decreased planetesimal accretion rate following isolation [@2000ApJ...537.1013I; @2006ApJ...648..666R]. To be observed as hot super-Earths requires subsequent photoevaporation . Significant photoevaporation of planets with massive atmospheres is unlikely unless the planet mass is in the $\lesssim$70$M_\oplus$ type I migration regime [@2007arXiv0711.2015R]. Thus, planets with remnant hydrogen atmospheres may form by the same migration mechanism we present here. Scattering is also a possibility for these planets to reach short-period orbits, because they have higher initial masses.
For planets originating in icy regions, their largely volatile composition has important implications for their evolution during and after formation. Icy grains may enhance growth if they stick together more easily, but also allows the possibility of large evaporation events in high energy collisions of larger objects. During the violent accretion process, and with the possible outcome of short-period orbits, melting and evaporation of ices will affect these planets [e.g. @1982Icar...52...14L; @2003ApJ...596L.105K; @2007Icar..191..453S].
After migrating to close-in orbits, initially icy/watery planets may retain large super-critical steam atmospheres, or become rocky cores stripped of volatiles entirely. @2003ApJ...596L.105K considered the existence of volatile-rich planets in the Solar habitable zone, and suggested that planets around Solar luminosity stars would be safe from evaporation at $\gtrsim$1AU but not at closer distances. He also noted that lower EUV luminosities for M dwarfs makes these stars less likely to evaporate planetary atmospheres. More recently, @2007Icar..191..453S revisited the issue, and concluded that planets $\gtrsim$6$M_\oplus$ will retain most of their water content at $\gtrsim$0.04AU from a Solar-type star. The results of both studies suggest the evaporation timescale is strongly dependent on semi-major axis. Therefore, a trend may be noticeable within the small semi-major axis range of transiting planets.
The picture that emerges is of systems with evaporated rocky planets inside $\sim$0.04AU, and steam planets somewhat outside this distance. A few stalled ocean [@2003ApJ...596L.105K; @2004Icar..169..499L] and icy planets extend through and past the habitable zone. For these planets, microlensing provides sensitivity complementary to transit and radial velocity methods at $\sim$AU distances [e.g. @2006Natur.439..437B], which will help yield trends with semi-major axis, particularly for low-mass stars.
Surveys such as the MEarth Project [@2007arXiv0709.2879N], CoRoT [@2003AdSpR..31..345B], and Kepler [@2003SPIE.4854..129B] hope to discover super-Earths by the transit method. Like those discovered by radial velocity, most planets will orbit close to their parent stars. Because they are unlikely to form *in situ*, these planets necessarily require some form of migration or scattering from their formation regions. Observed systems will thus test and inform mechanisms that form and bring planets to visible orbits.
We acknowledge support from an Australian Postgraduate Award, a Smithsonian Astrophysical Observatory pre-doctoral fellowship (GK), and the [*NASA Astrophysics Theory Program*]{} through grant NAG5-13278 and the *TPF Foundation Science Program* though grant NNG06GH25G (SK). We thank the anonymous referee for a prompt report, which improved the content of the paper. $N$-body simulations were run on computers maintained by the RSAA Computer Section at Mt Stromlo Observatory.
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[^1]: Equating (\[eq:llim\]) and (\[eq:ulim\]) has two solutions for $M_\star$ because of the upper limit of 30. The other solution is at $M_\star$ far too small to be interesting.
|
Preprint MRI-PHY/15/94, TCD-9-94\
hep-th/9411076, October 1994
**Singularities in Graviton-Dilaton System:**
Their Implications on the PPN Parameters
and the Cosmological Constant
\
Mehta Research Institute, 10 Kasturba Gandhi Marg,
Allahabad 211 002, India.
email: krama@mri.ernet.in\
> ABSTRACT. Alternatives to Einstein’s theory of general relativity can be distinguished by measuring the parametrised post Newtonian parameters. Two such parameters $\beta$ and $\gamma$, equal to one in Einstein theory, can be obtained from static spherically symmetric solutions. For the graviton-dilaton system, as in Brans-Dicke or low energy string theory, we find that if $\gamma \ne 1$ for a charge neutral point star, then there exist naked singularities. Thus, if $\gamma$ is measured to be different from one, then it cannot be explained by these theories, without implying naked singularities. We also couple a cosmological constant $\Lambda$ to the graviton-dilaton system, a la string theory. We find that static spherically symmetric solutions in low energy string theory, which describe the gravitational field of a point star in the real universe atleast upto a distance $r_* \simeq {\cal O} ({\rm pc})$, always lead to curvature singularities. These singularities are stable and much worse than the naked ones. Requiring their absence upto a distance $r_*$ implies a bound $| \Lambda | < 10^{- 102} (\frac{r_*}{{\rm pc}})^{- 2}$ in natural units. If $r_* \simeq 1 {\rm Mpc}$ then $| \Lambda | < 10^{- 114}$, and if $r_*$ extends all the way upto the edge of the universe ($10^{28} {\rm cm}$) then $| \Lambda | < 10^{- 122}$ in natural units.
**1. Introduction**
In Einstein’s theory of general relativity, the gravitational field of a point star is described by the static, spherically symmetric Schwarzschild solution. Its predictions have been verified to a very good accuracy. However, for various reasons, as described in detail in [@will], it is worthwhile to consider alternative theories of gravity. Among the popular ones are the Brans-Dicke (BD) theory, which is parametrised by a constant $\omega > 0$, and the string theory. A common feature among these generalised theories is the presence of a scalar field $\phi$, called BD scalar or dilaton. There are other generalisations of BD theory, where $\omega$ is a function of $\phi$, or the matter couplings to gravity depend on another function of $\phi$, etc. . For details see [@will]. We will consider here only BD theory and the low energy limit of the string theory.
These alternative theories can be distinguished by measuring a set of parameters called parametrised post Newtonian (PPN) parameters. Two such parameters, $\beta$ and $\gamma$ can be obtained from static spherically symmetric solutions of the graviton-dilaton system. In Einstein’s theory $\beta = \gamma = 1$. Experimentally, their measured values are given by $\frac{1}{3} (2 + 2 \gamma - \beta) = 1.003 \pm .005$ and $\gamma = 1 \pm .001$. The parameter $\beta$ is a measure of non linearity in the superposition law for gravity, and $\gamma$ is a measure of the space time curvature [@will]. In this paper, we study the static spherically symmetric solutions for BD and string theory, including only the graviton and the dilaton field. They describe the gravitational field of a point star, in these theories. We find that the only acceptable solutions all lead to the same predictions for the values of $\beta$ and $\gamma$ as in Einstein’s theory, namely $\beta = \gamma = 1$. There are more general static spherically symmetric solutions [@b]-[@tcd6], which predict $\beta = 1, \; \gamma = 1 + \epsilon$. However, these solutions always have naked curvature singularities proportional to $\epsilon^2$ and, hence, are unacceptable.
These general solutions can be better understood by coupling the electromagnetic field [@gm; @ghs]. They lead to non trivial PPN parameters for a point star of mass $M$ and charge $Q$ . In these solutions there is an inner and an outer horizon. The curvature scalar is singular at the inner horizon, but this singularity is hidden behind the outer horizon. A charge neutral star can then be obtained in two ways: in one, corresponding to the Schwarzschild solution, the PPN parameters are trivial and there is no naked singularity, while in the other, the PPN parameters are non trivial but there is a naked singularity.
Therefore neither BD nor low energy string theory can predict non trivial values for PPN parameters $\beta$ and $\gamma$, for a charge neutral star, without introducing naked singularities. Thus if naked singularities are forbidden then, for a charge neutral star, both the BD and the low energy string theory lead to the same predictions for $\beta$ and $\gamma$ as in Einstein theory. In particular, if the parameter $\gamma$ for a charge neutral point star is known to be different from one, then it cannot be explained by either BD or low energy string theory, without implying the existence of a naked singularity. In that case an alternative theory is needed that can predict a non trivial value for $\gamma$ for a charge neutral star, without any naked singularity.
In Einstein’s theory, one can also add a cosmological constant $\Lambda$. The only modifications to the static spherically symmetric solutions are that, the space time is not flat asymptotically and, for $\Lambda > 0$, it develops a new cosmological horizon [@gh]. In the second half of this paper, we couple the cosmological constant to the dilaton $\phi$, in a way analogous to the coupling of a tree level cosmological constant in low energy string theory [@tdgang].
The time dependent, expanding universe type of solutions to such system have been extensively studied in various space time dimensions [@tdgang]. In the low energy limit of the string theory, the dilaton is expected to develop a potential and acquire a mass. Hence, $\Lambda$ can be considered as a function of $\phi$, leading to a dilaton potential. Taking it to be of the form $e^{- \phi} \Lambda (\phi) = m (\phi - \phi_0)^2 + \cdots$ near the minimum, the authors of [@hh] have thoroughly analysed the implications of such a massive dilaton for the static spherically symmetric case. Depending on the choice of $m$, the system is expected to develop one, two, or three horizons. Also the static solutions to $d + d_i + 2$ dimensional gravity with a higher dimensional cosmological constant have been studied in [@wilt], where $d_i$ is the number of internal dimensions.
In this paper, we take $\Lambda$ to be a constant and analyse the static spherically symmetric solutions in $d = 4$ space time. They describe the gravitational field of point stars, and continue to do so to a very good approximation even when the stars have non relativistic velocities with respect to each other. For example, the Schwarzschild solution describes very well the gravitational effect of the sun on earth even though the earth is revolving around the sun with a speed of ${\cal O} (10 {\rm km/sec})$. Also, in Einstein’s theory, when a cosmological constant $\Lambda$ is present, the static spherically symmetric solutions still describe the gravitational field of point stars and also the redshift of distant objects [@wein]. Therefore one expects that in our present case also, when a cosmological constant $\Lambda$ is present, the static spherically symmetric solutions will describe the gravitational field of stars atleast upto a distance $r_*$, even though the real universe is not static but expanding, characterised by the Hubble constant $H_0 = 100 h_0 {\rm km/sec/Mpc}, \; 0.5 \le h_0 \le 1$. Hence, $r_*$ can reasonably be taken to be of ${\cal O} ({\rm pc})$. Therefore the study of static spherically symmetric solutions is important and physically relevent even when a cosmological constant $\Lambda$ is present.
From an analysis of such static spherically symmetric solutions, we find [@kcc] that for BD theory, they are likely to be regular outside the Schwarzschild horizon with no curvature singularities. However, for low energy string theory, the presence of a non zero cosmological constant leads to a curvature singularity, which is much worse than a naked one as explained in the text. This singularity is argued to persist when generic perturbations and higher order string effects are included. However such naked singularities have not been observed in our universe. Hence, one should require that they be absent, atleast upto a distance $r_*$, upto which the static spherically symmetric solutions analysed here are expected to describe the gravitational field of point stars. This will then impose a bound $| \Lambda | < 10^{- 102} (\frac{r_*}{{\rm pc}})^{- 2}$ in natural units. Thus if $r_* \simeq 1 {\rm Mpc}$ then $| \Lambda | < 10^{- 114}$, and if $r_*$ extends all the way upto the edge of the universe ($10^{28} {\rm cm}$) then $| \Lambda | < 10^{- 122}$ in natural units.
This paper is organised as follows. In section 2, the action and the equations of motion for the graviton and the dilaton are given, for the static spherically symmetric case. In section 3, we consider the solutions when $\Lambda = 0$, and analyse the PPN parameters and the singularities. In section 4, $\Lambda$ is taken to be non zero. We show that for low energy string theory, non zero $\Lambda$ leads to a naked curvature singularity, and give arguments for its persistence when generic perturbations and higher order string effects are included. In section 5, we conclude with a summary.
**2. Equations of motion for graviton and dilaton**
Consider the following action for graviton $(\tilde{g}_{\mu \nu})$ and dilaton $(\phi)$ fields, $$\label{starget}
S = - \frac{1}{16 \pi \kappa}
\int d^4 x \sqrt{\tilde{g}} \, e^{\phi} \,
( \tilde{R} - \tilde{a} (\tilde{\nabla} \phi)^2 + \Lambda (\phi) )$$ in the target space with coordinates $ x^{\mu}, \; \mu = 0, 1, 2, 3$, where $\kappa \; ( = 1$ in the following) is Newton’s constant. In our notation, $R_{\mu \nu \lambda \tau} = \frac{\partial^2 g_{\mu \lambda}}
{\partial x^{\nu} \partial x^{\tau}} + \cdots$. When $\tilde{a} = 1$, the action $S$ in equation (\[starget\]) corresponds to the target space effective action for low energy string theory, whose equations of motion give the $\beta$-function equations for $\tilde{g}_{\mu \nu}$ and $\phi$ in the sigma model approach to the string theory. $\Lambda (\phi)$ is the dilaton potential which, if constant, would act as a tree level cosmological constant in low energy string theory [@tdgang], given by $\Lambda = \frac{1}{2} (d + d_{int} - 10)$, which is zero for a critical string and non zero for a non critical string. The field $e^{- \frac{\phi}{2}}$ acts as a string coupling. When $\tilde{a} = - \omega $ the above action corresponds to Brans-Dicke (BD) theory, where $\omega > 0$ is the BD parameter.
In the effective action (\[starget\]), which is written in a frame (called physical frame in the following) with metric $\tilde{g}_{\mu \nu}$, the curvature term is not in the standard Einstein form. However, the standard form, where the equations of motion are often easier to analyse, can be obtained by a dilaton dependent conformal transformation $$\tilde{g}_{\mu \nu} = e^{- \phi} g_{\mu \nu}$$ to the Einstein frame with metric $g_{\mu \nu}$. The curvature scalars in these two frames are related by $$\label{rstring}
\tilde{R} = e^{\phi}
( R - 3 \nabla^2 \phi + \frac{3}{2} (\nabla \phi)^2 )$$ where $\tilde{\,}$ refers to the physical frame. The effective action now becomes $$\label{etarget}
S = - \frac{1}{16 \pi} \int d^4 x \sqrt{g} \,
( R + \frac{a}{2} (\nabla \phi)^2 + e^{- \phi} \Lambda (\phi) ) \; ,$$ where $a \equiv 3 - 2 \tilde{a} \; = 1$ for string theory and $ = 2 \omega + 3$ for BD theory. The equations of motion for $g_{\mu \nu}$ and $\phi$ that follow from this action, with $\Lambda_{\phi} \equiv \frac{\partial \Lambda}{\partial \phi}$, are $$\begin{aligned}
\label{beta}
2 R_{\mu \nu} + a \nabla_{\mu} \phi \nabla_{\nu} \phi
+ g_{\mu \nu} \Lambda e^{- \phi} & = & 0 \nonumber \\
a \nabla^2 \phi + (\Lambda - \Lambda_{\phi}) e^{- \phi} & = & 0 \; .\end{aligned}$$
There is no specified form for the function $\Lambda (\phi)$, either in Brans-Dicke theory or in string theory. However, in the low energy limit of the string theory, the dilaton is expected to acquire a mass, and consequently develop a potential of the form $e^{- \phi} \Lambda (\phi) = m (\phi - \phi_0)^2 + \cdots$ around the minimum of the potential. In two excellent papers [@hh], the implications of such a massive dilaton have been thoroughly analysed for static spherically symmetric solutions. Hence, in the following we analyse only the case where $\Lambda (\phi)$ is a constant, which corresponds to a tree level cosmological constant in low energy string theory. Furthermore, since $a \ge 1$ in string and BD theory, we also consider only $a \ge 1$.
We will look for static, spherically symmetric solutions to equations (\[beta\]). In the Schwarzschild gauge where $d s^2 = - f d t^2 + f^{- 1} d \rho^2 + r^2 d \Omega^2, \;
d \Omega^2$ being the line element on an unit sphere, and where the fields $f, \; r$, and $\phi$ depend only on $\rho$, the equations (\[beta\]) become $$\begin{aligned}
\label{rf}
\frac{(f r^2)''}{2} - 1 & = & ( f' r^2 )' \nonumber \\
= a ( \phi' f r^2 )' - \Lambda_{\phi} r^2 e^{- \phi}
& = & - \Lambda r^2 e^{- \phi} \nonumber \\
4 r'' + a r \phi'^2 & = & 0\end{aligned}$$ where $'$ denotes $\rho$-derivatives.
Sometimes, it is more convenient to work in the standard gauge where the line element is given by $d s^2 = - f d t^2 + \frac{G}{f} d r^2 + r^2 d \Omega^2$, and where the fields $f, \; G$, and $\phi$ depend only on $r$. Equations (\[beta\]) then become $$\begin{aligned}
\label{gf}
\frac{(f r^2)''}{2} - \frac{(f r^2)' G'}{4 G} - G
& = & ( f' r^2 )' - \frac{G' f' r^2}{2 G}
\nonumber \\
= a ( \phi' f r^2 )' - \frac{a \phi' G' f r^2}{2 G}
- \Lambda_{\phi} G r^2 e^{- \phi}
& = & - \Lambda G r^2 e^{- \phi} \nonumber \\
2 G' - a r G \phi'^2 & = & 0\end{aligned}$$ where $'$ denotes $r$-derivatives now in the standard gauge. The curvature scalar $\tilde{R}$ in the physical frame is given by $$\label{r}
\tilde{R} = \frac{(3 - a) f \phi'^2 e^{\phi}}{2 G}
+ \frac{(3 - 2 a) \Lambda}{a} - \frac{3 \Lambda_{\phi}}{a} \; .$$ Using (\[gf\]), it is easy to obtain the following equation for $R_1 \equiv \frac{f \phi'^2 e^{\phi}}{G}$ : $$\label{r1}
R'_1 + (\frac{4}{r} + \frac{f'}{f} - \phi') R_1
= - 2 (\Lambda - \Lambda_{\phi}) \phi' \; .$$ If $\Lambda$ is a constant then $\Lambda_{\phi} = 0$ and the second equality in (\[gf\]) can be integrated to obtain $$\label{phif}
\frac{f'}{f} - a \phi' = \frac{r_0 \sqrt{G}}{f r^2}$$ where $r_0$ is an integration constant proportional to the mass of the star.
The metric can also be written in isotropic gauge where the line element is given by $d s^2 = - f d t^2 + F (d h^2 + h^2 d \Omega^2)$ where $f$ and $F$ are functions of $h$ only. In this gauge, the observable parameters of the metric $\tilde{g}_{\mu \nu}$ in the physical frame can be extracted as follows. The mass of the star $M$ and the relevent PPN parameters $\beta$ and $\gamma$ are obtained [@will] by expanding the metric components $\tilde{f}$ and $\tilde{F}$ in the physical frame, as $h \to \infty$. These observables are defined by $$\begin{aligned}
\tilde{f} & = & 1 - \frac{2 M}{h} + \frac{2 \beta M^2}{h^2} + \cdots \\
\tilde{F} & = & 1 + \frac{2 \gamma M}{h} + \cdots \; .\end{aligned}$$ For Einstein’s theory $\beta = \gamma = 1$. The physical significance of the PPN parameters $\beta$ and $\gamma$ is that $\beta$ measures the non linearity in the superposition law of gravity, while $\gamma$ measures the space time curvature. Experimentally, $\beta$ and $\gamma$ are obtained by measuring the precession of the perihelia of the planets’ orbits and the time delay of radar echoes near the sun respectively; their measured values are given by $\frac{1}{3} (2 + 2 \gamma - \beta) = 1.003 \pm .005$ and $\gamma = 1 \pm .001$ [@will].
From now on, we will take $\Lambda (\phi) \equiv \Lambda = constant$ and $a \ge 1$.
**3. Solutions when $\Lambda$ is zero**
Consider first the solutions when $\Lambda = 0$. One then has the standard Schwarzschild solution $$\tilde{f} = 1 - \frac{\rho_0}{\rho} \; , \; \;
\tilde{r} = \rho \; , \; \; \phi = \phi_0 \; ,$$ where $\rho_0$ and $\phi_0$ are constants, which describes the gravitational field of a point star of mass $M = \frac{\rho_0}{2}$. There is a horizon at $\rho = \rho_0$ where $\tilde{g}_{tt} = \tilde{f} = 0$. The curvature scalar $\tilde{R}$ in the physical frame is regular everywhere, except at $\rho = 0$. This is the well known black hole singularity and is hidden behind the horizon. In the isotropic gauge, the solution becomes $$\tilde{f} = \left( \frac{1 - \frac{\rho_0}{4 h}}
{1 + \frac{\rho_0}{4 h}} \right)^2 \; , \; \;
\tilde{F} = \left( 1 + \frac{\rho_0}{4 h} \right)^4 \; ,$$ where $h$ and $\rho$ are related by $$\label{hrho}
\rho = h \left( 1 + \frac{\rho_0}{4 h} \right)^2 \; .$$ The PPN parameters are given by $\beta = \gamma = 1$ and are trivial.
However, there are also more general solutions [@b]-[@tcd6] where the dilaton field $\phi$ and the PPN parameters are non trivial. They are given, in the Schwarzschild gauge in the Einstein frame, by [@b; @tcd6] $$\begin{aligned}
\label{bdsoln}
f & = &
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{1 - k^2}{1 + k^2}}
\nonumber \\
r^2 & = &
\rho^2 \left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{2 k^2}{1 + k^2}}
\nonumber \\
e^{\phi - \phi_0} & = &
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{2 l}{1 + k^2}}\end{aligned}$$ where $k$ is a parameter and $l \equiv \frac{k}{\sqrt{a}}$. In the physical isotropic gauge, the metric components become $$\begin{aligned}
\tilde{f} & = &
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{1 - k^2 - 2 l}{1 + k^2}}
\nonumber \\
\tilde{F} & = & \frac{\rho^2}{h^2}
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{2 (k^2 - l)}{1 + k^2}}\end{aligned}$$ where $h$ and $\rho$ are related as in (\[hrho\]). Expanding the functions $\tilde{f}$ and $\tilde{F}$ in inverse powers of $h$, as $h \to \infty$, one gets the mass $M$ and the PPN parameters $\beta$ and $\gamma$ as $$\begin{aligned}
\label{mass}
2 M & = & \frac{1 - k^2 - 2 l}{1 + k^2} \rho_0 \nonumber \\
\beta & = & 1 \nonumber \\
\gamma & = & 1 + \frac{2 l \rho_0}{(1 + k^2) M} \; .\end{aligned}$$ The parameter $\beta$ is trivial while $\gamma$ is non trivial if $l \rho_0 \ne 0$.
The curvature scalar $\tilde{R}$ in the physical frame is given by $$\label{rtilde1}
\tilde{R} = \frac{\tilde{a} M^2 (\gamma - 1)^2 e^{\phi_0}}{\rho^4} \;
\left(1 - \frac{\rho_0}{\rho}\right)^{- \frac{1 + 3 k^2 - 2 l}{1 + k^2}}
\; .$$ In the above equations $\rho_0$ is positive, so that one obtains the standard Schwarzschild solution when $k = 0$. Also the physical mass $M$, given by (\[mass\]), must be positive which then implies that $1 - k^2 - 2 l > 0$. Hence, the metric component $\tilde{g}_{tt}$ in the physical frame vanishes at $\rho = \rho_0$. The above condition on $k$, discussed below in more detail, also implies that $1 + 3 k^2 - 2 l >0$. Hence, the curvature scalar $\tilde{R}$ in (\[rtilde1\]) becomes singular there, unless $\gamma = 1$, [*i.e.*]{} unless the PPN parameters are trivial. This singularity is naked, as will be shown presently.
We will first discuss the constraints on $k$. The positivity of the physical mass $M$ in (\[mass\]) implies that $1 - k^2 - 2 l > 0$, which restricts the parameter $k$ to be in the range $$\label{k1}
- \frac{1}{\sqrt{a}} - \sqrt{1 + \frac{1}{a}} < k <
- \frac{1}{\sqrt{a}} + \sqrt{1 + \frac{1}{a}} \; .$$ However, the above equation turns out to be only a weak constraint on $k$. A stronger one follows requiring the PPN parameter $\gamma$ to lie within the experimentally observed range $\gamma = 1 \pm .002$. In fact, from equations (\[mass\]), requiring $\gamma = 1 + \epsilon$ gives $$k = - \frac{1}{\sqrt{a}} \left( 1 + \frac{1}{\epsilon} \right)
\pm \sqrt{1 + \frac{1}{a} \left( 1 + \frac{1}{\epsilon} \right)^2} \; .$$ Taking into the account the constraint on $k$ given by equation (\[k1\]), which implies that one should take the $+$ sign for the square root above, we get $$k = - \frac{1}{\sqrt{a}} \left( 1 + \frac{1}{\epsilon} \right)
+ \sqrt{1 + \frac{1}{a} \left( 1 + \frac{1}{\epsilon} \right)^2}
\simeq - \frac{\epsilon \sqrt{a}}{2 (1 + \epsilon)} \; .$$ Hence, if $\gamma$ is required to be such that $| \gamma - 1 | \le |\epsilon|$, then one gets the following stronger constraint on $k$: $$\label{k2}
|k| < \frac{|\epsilon| \sqrt{a}}{2 (1 + \epsilon)} \; ,$$ where $|\epsilon| < .002$.
Now we will discuss the nature of the singularity at $\rho = \rho_0$.
1\. As can be seen from equation (\[rtilde1\]), the curvature scalar is singular at $\rho = \rho_0$; hence, this singularity is not a coordinate artifact and cannot be removed by any coordinate transformation.
2\. The metric on the surface $\rho = \rho_0$ has the signature $0+++$, and hence, this surface is null and the singularity is a null one.
3\. Consider an outgoing radial null geodesic, which describes an outgoing photon. Since $d \tilde{s}^2 = 0$ for such a geodesic, its equation is given by $$\frac{d t}{d \rho} =
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{k^2 - 1}{k^2 + 1}} \; ,$$ where $t$ is the external time. This gives $$\label{nullgeo}
t = \rho_* + const$$ where $\rho_*$, the analog of the ‘tortoise coordinate’, is defined by $$\rho_* = \int d \rho
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{k^2 - 1}{k^2 + 1}} \; .$$ For $k = 0$, $\rho_* = \rho + \rho_0 \ln (\rho - \rho_0)$ is the standard tortoise coordinate for Schwarzschild geometry, and it tends to $- \infty$ as $\rho \to \rho_0$. For $k \ne 0$, $\rho_*$ given above cannot be explicitly evaluated for arbitrary $k$. However, it can be shown that $\rho_*$ does not diverge as $\rho \to \rho_0$. Near $\rho_0$, let $y = \rho - \rho_0 \; \to 0$. Then, if $k \ne 0$ and obeys the bound given by equation (\[k2\]), then $$\rho_* = \rho_0 \int d y \; y^{\frac{k^2 - 1}{k^2 + 1}} + \cdots \; , \;
\; = \frac{\rho_0 (k^2 + 1)}{2 k^2} y^{\frac{2 k^2}{k^2 + 1}} + \cdots$$ where $\cdots$ denote higher order terms in $y$. The right hand side of the above equation is finite as $y \to 0$, and thus $\rho_*$ does not diverge as $\rho \to \rho_0$.
The outgoing radial null geodesic equation (\[nullgeo\]) then implies that a radially outgoing photon starting from $\rho_i \; ( \ge \rho_0 )$ at external time $t_i$ will reach an outside observer at $\rho_f \; (\rho_i < \rho_f < \infty)$ at a finite external time $t_f$ given by $$t_f - t_i = \rho_*(\rho_f) - \rho_*(\rho_i) \; .$$ Since, as shown above, $\rho_*(\rho)$ has no divergence even when $\rho = \rho_0$, it follows that a photon can travel from arbitrarily close to the singularity to an outside observer within a finite external time interval. Hence, the singularity at $\rho = \rho_0$ is naked.
4 a. Similarly a material particle can also travel from arbitrarily close to the singularity to an outside observer in a finite external time interval. This can be shown as follows. Let the line element be given by $$d \tilde{s}^2 = - g_0 d t^2 + g_1 d \rho^2 + g_2 d \Omega^2 \; ,$$ where, for our case, $$g_0 = f e^{- \phi} \; , \; \;
g_1 = \frac{e^{- \phi}}{f} \; , \; \;
g_2 = r^2 e^{- \phi}$$ with $f, \; e^{\phi}$, and $r^2$ given by equation (\[bdsoln\]). The corresponding geodesic equation for a material particle travelling radially outward, which can be derived in a standard way as in [@wein2], is given by $$\frac{d t}{d \rho} = \sqrt{\frac{g_1}{g_0 (1 + E g_0)}} \; , \; \;
\frac{d \rho}{d \tau} = const \sqrt{\frac{1 + E g_0}{g_0 g_1}}$$ where $\tau$ is the proper time (or equivalently the proper distance), $E$ is the energy of the particle which is negative in our notation, and $1 + E g_0 > 0$. For our case, these equations give $$\begin{aligned}
t & = & \int d \rho
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{k^2 - 1}{k^2 + 1}} \;
\left( 1 + E \left( 1 - \frac{\rho_0}{\rho}
\right)^{\frac{1 - k^2 - 2 l}{k^2 + 1}} \right)^{- \frac{1}{2}}
+ const \\
\tau & = & (const) \int d \rho
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{- 2 l}{k^2 + 1}} \;
\left( 1 + E \left( 1 - \frac{\rho_0}{\rho}
\right)^{\frac{1 - k^2 - 2 l}{k^2 + 1}} \right)^{- \frac{1}{2}}
+ const \; .\end{aligned}$$ Since $(1 + E g_0) > 0$, the only potential divergences in the above integrals are when $\rho \to \rho_0$. However, analysing these integrals near $\rho \to \rho_0$ as before, it can be seen that they do not diverge as $\rho \to \rho_0$. Hence, just as in the case of a photon above, it follows that a material particle can travel from arbitrarily close to the singularity to an outside observer within a finite external time interval. This again implies that the singularity at $\rho = \rho_0$ is naked.
4 b. By a similar analysis, it follows that the proper distance $\tau$ between $\rho_f$ and $\rho_i \; (\to \rho_0)$ is also finite. The integral for $\tau$ given above does not diverge since $1 + k^2 - 2 l > 0$, which follows from the constraint $1 - k^2 - 2 l > 0$ discussed before equation (\[k1\]).
5\. The curvature scalar diverges as $\rho \to \rho_0$. The ensuing tidal forces will rip away any physical apparatus as it nears $\rho_0$. However, the information about this event can be communicated to the outside observer in a finite external time since, as shown above, a photon or a material particle can travel from arbitrarily close to the singularity to an outside observer within a finite external time interval.
For these reasons, the singularity at $\rho = \rho_0$ is naked and physically unacceptable. For recent detailed discussions on naked singularities and their various general aspects, such as their definition, physical unacceptability, various scenario for their formation in Einstein’s theory, etc. , see [@psj]).
We would like to make one further remark. The situation described here is different from those corresponding to other solutions in string theory where singular null horizons appear. This is because, if and when the singularities do appear for a charge neutral point star in the later case, they are always hidden behind a horizon. For a point star with extremal charge, singular null horizons can appear, but this situation again differs from the present one in which only point stars with no charge are considered. The naked, singular ‘horizon’ occurs in our case mainly because of the requirement that the PPN parameter $\gamma$ for a charge neutral point star be non trivial, [*i.e.*]{} $\gamma \ne 1$. The motivation for this requirement has been discussed in the introduction.
One can gain more insight into the solution (\[bdsoln\]) by comparing it to that of [@gm; @ghs]. Consider, as in [@gm; @ghs], a $U(1)$ gauge field $A_{\mu}$, coupled to (\[etarget\]) through the action $$S_{em} = - \frac{1}{16 \pi} \int d^4 x \sqrt{g} \,
e^{\frac{k \phi}{\sqrt{a}}} F_{\mu \nu} F^{\mu \nu}$$ where $F_{\mu \nu} \equiv \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$. The general solution for the above system with the graviton, dilaton, and a gauge field is given in the Schwarzschild gauge in the Einstein frame, by [@gm; @ghs] $$\begin{aligned}
f & = & \left( 1 - \frac{\rho_1}{\rho} \right)
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{1 - k^2}{1 + k^2}}
\nonumber \\
r^2 & = & \rho^2
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{2 k^2}{1 + k^2}}
\nonumber \\
e^{\phi - \phi_0} & = &
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{2 l}{1 + k^2}} \nonumber \\
F_{t \rho} & = & \frac{Q}{\rho^2}\end{aligned}$$ where $l = \frac{k}{\sqrt{a}}$ and the remaining components of $F_{\mu \nu}$ are zero. In the physical isotropic gauge, the metric components become $$\begin{aligned}
\tilde{f} & = & \left( 1 - \frac{\rho_1}{\rho} \right)
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{1 - k^2 - 2 l}{1 + k^2}}
\nonumber \\
\tilde{F} & = & \frac{\rho^2}{h^2}
\left( 1 - \frac{\rho_0}{\rho} \right)^{\frac{2 (k^2 - l)}{1 + k^2}} \; ,\end{aligned}$$ and, $h$ and $\rho$ are now related by $$\rho - \rho_0 = h (1 + \frac{\rho_1 - \rho_0}{4 h})^2 \; .$$ Expanding the functions $\tilde{f}$ and $\tilde{F}$ in inverse powers of $h$ as $h \to \infty$, one gets the mass $M$, the charge $Q$, and the PPN parameters $\beta$ and $\gamma$ as $$\begin{aligned}
2 M & = & \rho_1 + \frac{1 - k^2 - 2 l}{1 + k^2} \rho_0 \\
Q^2 & = & \frac{\rho_1 \rho_0}{1 + k^2} \\
\beta & = & 1 + \frac{(1 - l) Q^2}{2 M^2} \\
\gamma & = & 1 + \frac{2 l \rho_0}{(1 + k^2) M} \; .\end{aligned}$$ The parameter $\beta$ is non trivial if the charge $Q \ne 0$ while $\gamma$ is non trivial if $l \rho_0 \ne 0$.
The curvature scalar $\tilde{R}$ in the physical frame is given by $$\label{rtilde2}
\tilde{R} = \frac{\tilde{a} M^2 (\gamma - 1)^2 e^{\phi_0}}{\rho^4} \;
\left(1 - \frac{\rho_1}{\rho}\right) \;
\left(1 - \frac{\rho_0}{\rho}\right)^{- \frac{1 + 3 k^2 - 2 l}{1 + k^2}}
\; .$$ The metric component $\tilde{g}_{tt}$ in the physical frame vanishes at $\rho = \rho_1$ and $\rho = \rho_0$. The curvature scalar $\tilde{R}$ is regular at $\rho = \rho_1$ but, since $1 + 3 k^2 - 2 l > 0$ for $a \ge 1$, it is singular at $\rho = \rho_0$ unless $\gamma = 1$. This singularity is hidden behind the horizon at $\rho_1$ if $\rho_1 > \rho_0$, and naked otherwise for the same reasons as given following equation (\[rtilde1\]).
Now, consider the charge neutral solution, [*i.e.*]{} $Q = 0$. This can be obtained by setting either $\rho_0 = 0$ or $\rho_1 = 0$. In the former case, one gets the usual Schwarzschild solution with trivial values for $\beta$ and $\gamma$. In the later case one gets the solution described in (\[bdsoln\]) where the parameter $\gamma$ is non trivial.
Thus, it can be seen from (\[rtilde1\]) and (\[rtilde2\]) that, in BD or low energy string theory, a non trivial value for the parameter $\gamma$ for a charge neutral point star implies the existence of a naked singularity. Conversely, in these theories, the absence of naked singularities necessarily implies that the PPN parameters $\beta$ and $\gamma$ for a charge neutral point star are trivial. Thus if naked singularities are forbidden then, for such a star, both the BD and the low energy string theory lead to the same predictions for $\beta$ and $\gamma$ as in Einstein theory. In particular, if the parameter $\gamma$ for such a star is found to be different from one, then it cannot be explained by either BD or low energy string theory, without implying the existence of a naked singularity. In that case an alternative theory is needed that can predict a non trivial value for $\gamma$ for such a charge neutral point star, without any naked singularity.
**4. Solutions when $\Lambda$ is non zero**
Consider now the case when $\Lambda \ne 0$. The equation involving $(\phi' f r^2)'$ in (\[gf\]) is the equation of motion for $\phi$ that follows from (\[etarget\]). However, this equation will be absent if the dilaton $\phi$ is absent. Hence, in that case, this equation is to be ignored and $\phi$ is to be set to zero in the remaining equations. The solution to (\[gf\]) is then given by $$f = 1 - \frac{r_0}{r} - \frac{\Lambda}{6} r^2 \; , \; \; G = 1 \; .$$ The curvature scalar $\tilde{R} = \Lambda$. This solution describes the static, spherically symmetric gravitational field of a point star of mass $M = \frac{r_0}{2}$ in Einstein theory, in the presence of a cosmological constant $\Lambda$ [@gh].
In the presence of both the dilaton $\phi$, and the cosmological constant $\Lambda$, the solution to equations (\[gf\]) is not known in an explicit form. Here we study this solution and its implications. The solution, required to reduce to the Schwarzschild one when $\Lambda = 0$, would describe the static, spherically symmetric gravitational field of a point star in the graviton-dilaton system (\[starget\]), in the presence of a cosmological constant $\Lambda$.
For nonzero $\Lambda$, the following general features are valid for any solution to equations (\[gf\]):
\(i) The dilaton field $\phi$ cannot be a constant. In fact, the only case where $\phi$ can be a constant for a non zero $\Lambda$ is when $\Lambda = \Lambda_{\phi}$, [*i.e.*]{}$\Lambda = \lambda e^{\phi}$. But, as can be seen from (\[etarget\]), this corresponds to pure Einstein theory with a cosmological constant $\lambda$ and a free scalar field $\phi$.
\(ii) $\ln G$, and hence $G$, strictly increases since $a \ge 1$ and consequently $(\ln G)' > 0$.
\(iii) Consider the following polynomial ansatz for the fields as $r \to \infty$. $$\begin{aligned}
f & = & A r^k + \cdots \nonumber \\
G & = & B r^l + \cdots \nonumber \\
e^{- \phi} & = & e^{- \phi_0} r^m + \cdots\end{aligned}$$ where $\cdots$ denote subleading terms in the limit $r \to \infty$ (it can be easily shown that if one of the fields has an asymptotic polynomial behaviour, then the others also have similar behaviour). Substituting these expressions into equations (\[gf\]) gives, to the leading order, $2 l = a m^2$ and $$\begin{aligned}
\label{asym}
\frac{(k + 2)}{2} (k + 1 - \frac{l}{2}) A r^k - B r^l
& = & k (k + 1 - \frac{l}{2}) A r^k \nonumber \\
= - a m (k + 1 - \frac{l}{2}) A r^k
& = & - B \Lambda e^{- \phi_0} B r^{l + m + 2} \; .\end{aligned}$$ The last two equalities above imply $k = - a m = l + m + 2$ which, together with $2 l = a m^2$, lead to $(m + 2) (m + \frac{2}{a}) = 0$. This gives the solution $(k, l, m) = (2 a, 2 a, - 2)$ or $(2, \frac{2}{a}, - \frac{2}{a})$. Using these relations and equations (\[asym\]) it follows that $$\begin{aligned}
k (k + 1 - \frac{l}{2}) A & = & - \Lambda e^{- \phi_0} B \nonumber \\
\left( (m + \frac{2}{a}) \Lambda e^{- \phi_0}
+ \frac{4}{r^{m + 2}} \right) B & = & 0 \; .\end{aligned}$$
If $a > 1$, as in BD theory, then there is always a non trivial asymptotic solution with non zero $A$ and $B$. For example, $(k, l, m) = (2, \frac{2}{a}, - \frac{2}{a})$ and $B$ arbitrary. Note that in the second relation above, the term involving $r^{m + 2}$ can be ignored to the leading order, since $m + 2 = 2 (1 - \frac{1}{a}) > 0$. Also, as can be easily checked for this solution, the curvature scalar $\tilde{R}$ in the physical frame is finite as $r \to \infty$.
However, if $a = 1$ as in low energy string theory, then the above equations are consitent only if $A = B = 0$. Hence, in this case, equations (\[gf\]) do not admit a non trivial solution where the fields are polynomials in $r$ as $r \to \infty$. A similar analysis will rule out the solutions where the fields have polynomial-logarithmic behaviour asymptotically, [*i.e.*]{}where the fields behave as $r^m (\ln^n r) (\ln^p\ln r) \ldots$ to the leading order in $r$ as $r \to \infty$.
Thus, when $a > 1$, which includes BD theory, but not the low energy string theory, a non trivial asymptotic solution for graviton and dilaton exists asymptotically, as $r \to \infty$. Therefore it is very plausible, although not proved here, that a full solution can be constructed, perhaps numerically, starting from a Schwarzschild solution near the horizon and approaching the above asymptotic form as $r \to \infty$. The curvature scalar $\tilde{R}$ in the physical frame is also likely to remain finite everywhere outside the Schwarzschild horizon.
However, for low energy string theory where $a = 1$, the situation is totally different. To start with, no non trivial solution exists for graviton and dilaton asymptotically as $r \to \infty$. To further understand the solutions to (\[gf\]), we start with the Schwarzschild solution and study how it gets modified when $\Lambda \ne 0$ (from now on we set $a = 1$). Then the expression involving $\Lambda$ in (\[gf\]) acts as a source for the fields $f, \; G$, and $\phi$, which can be solved iteratively to any order in $\Lambda$. By construction, this would reduce to the Schwarzschild solution in the limit $\Lambda \to 0$. One thus gets $$\begin{aligned}
\label{fp}
f & = & 1 - \frac{r_0}{r} - \frac{\Lambda r^2}{6}
- \frac{\Lambda^2 r^4}{120} u_2
- \frac{4 \Lambda^3 r^6}{2835} u_3 + \cdots \nonumber \\
G & = & 1 + \frac{\Lambda^2 r^4}{72} v_2
+ \frac{2 \Lambda^3 r^6}{405} v_3 + \cdots \nonumber \\
\phi & = & \phi_0 - \frac{\Lambda r^2}{6} (1 + \frac{2 r_0}{r}
+ \frac{2 r_0^2}{r^2} \ln (r - r_0)) \nonumber \\
& & - \frac{\Lambda^2 r^4}{45} w_2
- \frac{197 \Lambda^3 r^6}{45360} w_3 + \cdots\end{aligned}$$ where $\phi_0$ is a constant which can be set to zero without any physical consequence, and $u_i, \; v_i, \; w_i$ are functions of $\frac{r_0}{r}$ and $\ln r$ which tend to $1$ in the limit $\frac{r_0}{r} \ll 1$. Evaluating $u_i, \; v_i, \; w_i$ and/or further higher order terms will not illuminate the general features of the solution. Also, the series will typically have a finite radius of convergence beyond which it is meaningless. Although it is possible to construct convergent series in different intervals of $r$, it is difficult to extract general features. Hence we follow a different approach.
It turns out that one can understand the general features of the solutions using only (i) the equations (\[gf\]), (ii) the behaviour of the fields for small $r$, and (iii) their non polynomial-logarithmic behaviour in the limit $r \to \infty$.
Note that $G = 1$ for Schwarzschild solution. Let $G$ has no pole at any finite $r$. Then the requirement that any solution to (\[gf\]) reduce to the Schwarzschild one when $\Lambda = 0$, combined with the fact that $G$ is a non decreasing function, implies that $G (\infty)$, and hence, $B$ must be non zero. Then the above analysis, which excludes polynomial behaviour for the fields with non trivial coefficients, implies in particular, that the fields cannot be constant, including zero, as $r \to \infty$.
Consider first the case where $r_0 = 0$. This will describe the static, spherically symmetric gravitational field of a star of negligible mass in low energy string theory when $\Lambda \ne 0$. With $r_0 = 0$ and setting $\phi_0 = 0$, equation (\[phif\]) gives $e^{\phi} = |f|$. It also follows from (\[fp\]) that the function $f$ has a local maximum (minimum) at the origin if $\Lambda$ is positive (negative). Away from the origin, the function $f$ can\
(A) have no pole at any finite $r$ and go to either $\infty$ or a constant as $r \to \infty$, or\
(B) have a pole at a finite $r = r_p$ (its behaviour for $r > r_p$ will not be necessary for our purposes). We will also consider the case where\
(C) $f$ has a zero at $r = r_H$.
Case A: The function $f$, and hence $G$, has no pole at finite $r$. From the analysis preceding equation (\[fp\]), it is already clear that $f (\infty)$ cannot be a constant. This can also be seen as follows. A necessary condition for $f (\infty)$ to be a constant is that $f$ must have atleast one more critical point at $0 < r_c \le \infty$. Let $f' (r_1) = 0$, where $r_1 \le \infty$ is the first critical point after the origin (note that $r_1 = \infty$ corresponds to the function $f$ decreasing (increasing) to a constant monotonically if $\Lambda$ is positive (negative)). Then it follows, from the behaviour of $f$ near the origin, that $f$ must have a local minimum (maximum) at $r = r_1$, [*i.e.*]{}$f'' (r_1)$ must be positive (negative). This requirement holds good even when $r_1 = \infty$. However, from equations (\[gf\]) we get $$f'' (r_1) = - \Lambda e^{- \phi} G$$ which is negative (positive) if $\Lambda$ is positive (negative). This is in contradiction to the above condition. Therefore $f' (r) \ne 0$ for any $r > 0$, including $r = \infty$. Hence, the function $f$ obeying equations (\[gf\]) and which behaves as in (\[fp\]) near the origin, cannot be constant in the limit $r \to \infty$. From this, and the asymptotic non polynomial-logarithmic behaviour of $f$, it follows that $f (\infty) \to \infty$.
Whether these singularities are genuine or only coordinate artifacts can be decided by evaluating the curvature scalar, $\tilde{R}$, or equivalently $R_1 \equiv \frac{f \phi'^2 e^{\phi}}{G}$ which obeys the equation $$\label{r10}
R'_1 + \frac{4 R_1}{r} = - 2 \Lambda \frac{f'}{f} \; .$$ See equations (\[r\]), (\[r1\]), and (\[phif\]). It can be seen that $R_1 (\infty)$ cannot be a constant. For, if it were, then one gets $f (\infty) \to r^{- \frac{2 R_1 (\infty)}{\Lambda}}$, a polynomial behaviour for $f$ as $r \to \infty$, which is ruled out. Equation (\[r10\]) can be solved to give $$R_1 = - \frac{2 \Lambda}{r^4} \; \int dr \frac{r^4 f'}{f} \; .$$ From this it follows, as $r \to \infty$, that $\frac{f'}{f} > \frac{k}{r}$ for any constant $k$ (otherwise $R_1 (\infty) \to constant$). This implies that $f$ grows faster than any power of $r$ when $r \to \infty$. Evaluating the above integral in this limit, one then gets $R_1 (\infty) \to \infty$.
Case B: The function $f$ has a pole at a finite $r = r_p < \infty$. Then, from equation (\[r10\]) it follows, near $r = r_p$, that $$R_1 (r_p) = - 2 \Lambda \ln f (r_p) + {\cal O} (r - r_p)
\; \; \; \; \to \; \; \pm \infty \; .$$
Case C: The function $f$ has a zero at $r = r_H$. Then, from equation (\[r10\]) it follows, near $r = r_H$, that $$R_1 (r_H) = - 2 \Lambda \ln f (r_H) + {\cal O} (r - r_H)
\; \; \; \; \to \; \; \pm \infty \; .$$
Thus we see that $R_1$, and hence, the curvature scalar $\tilde{R}$ in the string frame, always diverges at one or more points $r \equiv r_s = r_p, \; r_H, \; \infty$, in low energy string theory when the cosmological constant $\Lambda \ne 0$. These singularities, which will persist even when $r_0 \ne 0$ as argued below, are naked. In fact, they are much worse, as they are created by any object, no matter how small its mass is. Thus at any point of the string target space, there will be a singularity produced by an object located at a distance $r_s$ from that point.
The above analysis also goes through when $r_0 \ne 0$ (the well known black hole singularity present now at $r = 0$, independent of $\Lambda$ and hidden behind the Schwarzschild horizon, will not concern us here). The easiest way to see it is as follows. Let the radius of convergence of the series in (\[fp\]) be $\gamma$, [*i.e.*]{} the series converges for $r < r_{con} \equiv \sqrt{\frac{\gamma}{|\Lambda|}}$ (the expansion parameter in the series is $\Lambda r^2$). Thus, for $r_0 \ll r_{con}$, its effect on the fields will be negligible by the time $r$ is near $r_{con}$, and even more so beyond $r_{con}$, as can be seen from (\[fp\]), where the functions $u_i, \; v_i, \; w_i \to 1$ in the limit $\frac{r_0}{r} \ll 1$. Hence such a non zero $r_0$ will not affect the poles and zeroes of $f, G$, and $\phi$ (which lie beyond $r_{con}$), and therefore, the curvature singularites found before will persist.
Or, one can repeat the above analysis. Now, one does not start at $r = 0$, where there is the well known black hole singularity if $r_0 \ne 0$, but at some point beyond the horizon, where the cosmological constant term, $\frac{\Lambda r^2}{6}$, in the expression for $f$ in (\[fp\]) dominates the mass term, $\frac{r_0}{r}$; that is, near when $r^3 > \frac{6 r_0}{|\Lambda|}$. This value of $r$ can be ensured to fall within the radius of convergence $r_{con}$ by choosing, for a given non zero $\Lambda$, a sufficiently small $r_0$, [*i.e.*]{}$6 r_0 < \sqrt{\frac{r^3}{|\Lambda|}}$. Then, the analysis proceeds as before. If $\Lambda$ is positive (negative), then the function $f$ will be decreasing (increasing), as $r$ is increasing beyond the value $\frac{6 r_0}{|\Lambda|}$, where the cosmological constant term in $f$ has started dominating the mass term. One can then consider the cases (A), (B), and (C) as before, and arrive at the same conclusion.
Thus, it is very likely that these singularities will also persist for any $r_0$, since the restriction on $r_0$ above is only due to the limitation of our analysis. The negligible effect of $r_0$, in the presence of a cosmological constant, is also physically reasonable since the cosmological constant can be thought of as vacuum energy density and, as $r$ increases, the vacuum energy will overwhelm any non zero mass of a star, which is proportional to $r_0$.
Similarly, one can consider a point star with charge $Q$. The fields then will be modified by the presence of terms involving $\frac{Q^2}{r^2}$, which will become negligible when $Q \ll r$. Thus, again by an analysis similar to the above, the singularities can be shown to persist even when the star is charged. Physically, the curvature singularities arise because of the run away feed back effect of the cosmological constant $\Lambda$ on the fields, as can be seen from (\[gf\]). Therefore, the effect worsens as $r$ increases. But, the effect of mass, charge, etc. of a point star decreases as $r$ increases, cannot compensate for the effects of $\Lambda$, and hence cannot remove the singularities arising due to a nonzero $\Lambda$. From this, it is also clear that any generic perturbation such as aspherical mass/charge distribution, non zero angular momentum, etc. will not remove the above singularities either, since the effects of these perturbations decrease with increasing $r$.
Thus we see that the static spherically symmetric gravitational field produced by a star in low energy string theory has a naked curvature singularity when the cosmological constant $\Lambda \ne 0$. The singularity is in fact much worse than a naked one, and is stable under generic perturbations such as the ones discussed above.
Now, as discussed in the introduction, the static spherically symmetric solution describes the gravitational field of a spherical star atleast upto a distance $r_* \simeq {\cal O} ({\rm pc})$, in our universe regardless of its non static nature. Therefore, the singularities described here must be absent atleast upto a distance $r_*$. This will then translate into a constraint on the cosmological constant $\Lambda$, in the sigma model approach to low energy string theory. If we take, somewhat arbitrarily, that the curvature becomes unacceptably strong when $| \Lambda | r^2 \simeq 1$, then requiring the absence of singularity upto a distance $r_*$ would give $$| \Lambda | r_*^2 < 1 \; ,$$ which gives the bound $$%| \Lambda | < (\frac{r_*}{{\rm pc}})^{- 2} 10^{- 102}
| \Lambda | < 10^{- 102} (\frac{r_*}{{\rm pc}})^{- 2}$$ in natural units. Thus if $r_* \simeq 1 {\rm Mpc}$ then $| \Lambda | < 10^{- 114}$, and if $r_*$ extends all the way upto the edge of the universe ($10^{28} {\rm cm}$) then $| \Lambda | < 10^{- 122}$ in natural units.
The existence of the naked singularity in low energy string theory when the cosmological constant, $\Lambda \ne 0$ also means the following. If $\Lambda$ was zero during some era in the evolution of the universe, then the mechanism (if exists) that enforces cosmic censorship - no evolution of singularities from a generic, regular, initial configuration - would also enforce the vanishing of $\Lambda$ in the long run, when the universe would be evolving sufficiently slowly for the static solutions to be applicable. Otherwise, cosmic censorship would be violated by the singularities presented above.
We now remark on the validity of the low energy effective action in string theory. This action is only perturbative and will be modified by higher order corrections in the regions of strong curvature. Hence, when these corrections are included, the singularities seen here may not be present. However, these corrections will kick in only when the curvature is strong, and the low energy effective action, and thus our analysis, is likely to remain valid until then. Therefore while the fields and the curvature may never actually become infinite, even when $\Lambda \ne 0$, in the full string action with higher order corrections, the present analysis indicates that they will become sufficiently strong as to be physically unacceptable, thus justifing the above conclusions.
**5. Conclusion**
We have analysed the static, spherically symmetric solutions to the\
graviton-dilaton system, with or without electromagnetic couplings and the cosmological constant. These solutions describe the gravitational field of a point star. The main results of the present analysis can be summarised as follows.
1\. For a charge neutral point star, neither BD nor low energy string theory predicts non trivial PPN parameters, $\beta$ and $\gamma$, without introducing naked singularities. Thus, if the naked singularities are forbidden, then these theories lead to the same predictions as in Einstein theory in the static spherically symmetric regime. In particular, if the parameter $\gamma$ for a charge neutral star is observed to be different from one, then it cannot be explained by either BD or low energy string theory, without implying the existence of a naked singularity.
2\. Upon coupling the cosmological constant $\Lambda$ as in the action (\[starget\]), in a way analogous to the coupling of a tree level cosmological constant in low energy string theory, we find the following for the static spherically symmetric solutions. For BD type theories, these solutions are likely to exist with no naked curvature singularities. However, for low energy string theory, the presence of a non zero cosmological constant leads to a curvature singularity in the universe, which is much worse than a naked singularity and is stable under generic perturbations. As discussed before, the static spherically symmetric solutions describe the gravitational field of a point star atleast upto a distance $r_* \simeq {\cal O} ({\rm pc})$, in our universe regardless of its non static nature. Therefore, the singularities described here must be absent atleast upto a distance $r_*$. This implies a bound $| \Lambda | < 10^{- 102} (\frac{r_*}{{\rm pc}})^{- 2}$ in natural units. If $r_* \simeq 1 {\rm Mpc}$ then $| \Lambda | < 10^{- 114}$, and if $r_*$ extends all the way upto the edge of the universe ($10^{28} {\rm cm}$) then $| \Lambda | < 10^{- 122}$ in natural units. We have also argued that this result, and the consequent bound on $\Lambda$, are unlikely to change even when the higher order string effects are included.
[**Note Added:**]{} After the completion of our work, we were informed by C. P. Burgess of reference [@burgess], where spherically symmetric, six parameter family of four dimensional string solutions have been studied.
Part of this work was carried out in School of Mathematics, Trinity College, Dublin and was supported by Forbairt SC/94/218. It is a pleasure to thank S. Sen for encouragement, H. S. Mani and T. R. Seshadri for many discussions. We particularly like to thank P. S. Joshi for numerous helpful communications regarding various aspects of singularity, and the referee for suggestions which, we believe, improved the quality and clarity of our paper.
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|
---
abstract: 'We perform model-independent statistical analyses of three scenarios accommodating New Physics (NP) in $\Delta F=2$ flavour-changing neutral current amplitudes. In a scenario in which NP in [[$B_d\!-\!{\,\overline{\!B}}{}_d\,$]{} mixing]{} and [[$B_s\!-\!{\,\overline{\!B}}{}_s\,$]{} mixing]{} is uncorrelated, we find the parameter point representing the Standard-Model disfavoured by 2.4 standard deviations. However, recent LHCb data on $B_s$ neutral-meson mixing forbid a good accommodation of the DØ data on the semileptonic CP asymmetry $A_{\rm SL}$. We introduce a fourth scenario with NP in both $M_{12}^{d,s}$ and $\Gamma_{12}^{d,s}$, which can accommodate all data. We discuss the viability of this possibility and emphasise the importance of separate measurements of the CP asymmetries in semileptonic $B_d$ and $B_s$ decays. All results have been obtained with the [[CKMfitter]{}]{} analysis package, featuring the frequentist statistical approach and using Rfit to handle theoretical uncertainties.'
author:
- |
A. Lenz$^{\,a}$, U. Nierste$^{\,b}$ and\
J. Charles$^{\,c}$, S. Descotes-Genon$^{\,d}$, H. Lacker$^{\,e}$, S. Monteil$^{\,f}$, V. Niess$^{\,f}$, S. T’Jampens$^{\,g}$ \[for the [[CKMfitter]{}]{} Group\]\
title: ' New Physics in $\boldsymbol{B}$–$\boldsymbol{\overline{B}}$ mixing in the light of recent LHCb data'
---
Flavour physics looks back to a quarter-century of precision studies at the B-factories with a parallel theoretical effort addressing the Standard Model (SM) predictions for the measured quantities [@Buras:2011we]. With the parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [@Cabibbo:1963yz] overconstrained by many measurements one can predict yet unmeasured quantities [@Charles:2011va]. Still, the global fit to the CKM unitarity triangle reveals some discrepancies with the SM, driven by a conflict between $B(B\to
\tau \nu)$ and $\sin(2\beta)$ measured from $B_d \to J/\Psi K$ [@Lenz:2010gu; @Lunghi:2010gvBona:2009cj]. Furthermore, in May 2010 the DØ experiment reported a deviation of the semileptonic CP asymmetry (dimuon asymmetry) in $B_{d,s}$ decays from its SM prediction [@bbln; @ln] by 3.2$\,\sigma$ [@dimuon_evidence_d0]. In June 2011 this discrepancy has increased to 3.9$\,\sigma$ [@Abazov:2011yk]. In summer 2010 the data could be interpreted in well-motivated scenarios with New Physics (NP) in [[$B\!-\!{\,\overline{\!B}}{}\,$]{} mixing]{} amplitudes [@Lenz:2010gu]. In this letter we present novel analyses which include the new data of 2011, in particular from the LHCb experiment.
[$B_q\!-\!{\,\overline{\!B}}{}_q\,$]{}$(q=d,s)$ oscillations involve the off-diagonal elements $M_{12}^q$ and $\Gamma_{12}^q$ of the $2\times 2$ mass and decay matrices, respectively. One can fix the three physical quantities $|M_{12}^q|$, $|\Gamma_{12}^q|$ and $\phi_q=\arg(-M_{12}^q/\Gamma_{12}^q)$ from the mass difference ${\ensuremath{\Delta M}}_q\simeq 2|M_{12}^q| $ among the eigenstates, their width difference ${\ensuremath{\Delta \Gamma}}_q \simeq 2\, |\Gamma_{12}^q| \cos \phi_q$ and the semileptonic CP asymmetry $$\begin{aligned}
a^q_{\rm SL} &=& {\mathrm{Im}\,}\frac{\Gamma_{12}^q}{M_{12}^q} =
\frac{|\Gamma_{12}^q|}{|M_{12}^q|} \sin \phi_q \; = \;
\frac{{\ensuremath{\Delta \Gamma}}_q}{{\ensuremath{\Delta M}}_q} \tan \phi_q . \label{defafs}\end{aligned}$$ $M_{12}^q$ is especially sensitive to NP. Therefore the two complex parameters $\Delta_s$ and $\Delta_d$, defined as $$\begin{aligned}
M_{12}^q & \!\equiv\! & M_{12}^{\text{SM},q} \cdot \Delta_q \, ,
\quad\; \Delta_q \equiv |\Delta_q| e^{i \phi^\Delta_q} ,
\quad\; q=d,s, \; \label{defdel}\end{aligned}$$ can differ substantially from the SM value $\Delta_s=\Delta_d=1$. Importantly, the NP phases $\phi^\Delta_{d,s}$ do not only affect $a^{d,s}_{\rm SL}$, but also shift the CP phases extracted from the mixing-induced CP asymmetries in $B_d \to J/\Psi K$ and $B_s \to
J/\Psi \phi$ to $2\beta+\phi^\Delta_d$ and $2\beta_s-\phi^\Delta_s$, respectively. In summer 2010 the CDF and DØ analyses of $B_s \to
J/\Psi \phi$ pointed towards a large negative value of $\phi^\Delta_s$, while simultaneously being consistent with the SM due to large errors. With a large $\phi^\Delta_s<0$ we could accommodate DØ’s large negative value for the semileptonic CP asymmetry reading $A_{\rm SL} = 0.6 a_{\rm SL}^d + 0.4 a_{\rm SL}^s$ in terms of the individual semileptonic CP asymmetries in the $B_d$ and $B_s$ systems. Moreover, the discrepancy between $B(B\to \tau \nu)$ and the mixing-induced CP asymmetry in $B_d \to J/\Psi K$ can be removed with $\phi_d^\Delta<0$. The allowed range for $\phi_d^\Delta$ implies a contribution to $A_{\rm SL}$ with the right (i.e. negative) sign. In our 2010 analysis in Ref. [@Lenz:2010gu] we have determined the preferred ranges for $\Delta_s$ and $\Delta_d$ in a simultaneous fit to the CKM parameters in three generic scenarios in which NP is confined to $\Delta F=2$ flavour-changing neutral currents. In our Scenario I we have treated $\Delta_s$, $\Delta_d$ (and three more parameters related to [[$K\!-\!{\,\overline{\!K}}{}\,$]{} mixing]{}) independently, corresponding to NP with arbitrary flavour structure. Scenario II implements minimal-flavour violation (MFV) with small bottom Yukawa coupling entailing real $\Delta_s=\Delta_d$. Scenario III covers MFV models in which $\Delta_s=\Delta_d$ is allowed to be complex. In Ref. [@Lenz:2010gu] we have found an excellent fit in Sc. I (and a good fit in Sc. 3) with all discrepancies relieved through $\Delta_{d,s}\neq 1$, while the fit has returned [[$K\!-\!{\,\overline{\!K}}{}\,$]{} mixing]{} essentially SM-like.
The recent LHCb measurement of the CP phase $\phi_s^{\psi\phi}$ from $A_{\rm CP}^{\rm mix} (B_s \to
J/\Psi \phi)$ does not permit large deviations of $\phi_s^\Delta$ from zero anymore. This trend was also confirmed by the latest CDF results [@CDF:2011af]. The current situation with the phase $2\phi_s^{\psi\phi}
\equiv -2\beta_s + \phi_s^\Delta$ and $A_{\rm SL}$ is as follows (at 68% CL): $$\begin{aligned}
2 \phi_s^{\psi\phi} = (-32^{+22}_{-21})^\circ
\quad & \mbox{D\O\ \cite{taggedphaseD0_2}} {\nonumber\\}-60^\circ \leq 2 \phi_s^{\psi\phi} \leq -2.3^\circ
\quad & \mbox{CDF \cite{CDF:2011af}} {\nonumber\\}2 \phi_s^{\psi\phi} = (-0.1\pm 5.8\pm 1.5)^\circ
\quad & \mbox{LHCb} \mbox{ $J/\psi \phi$ \cite{LHCb:2011aa}} {\nonumber\\}2 \phi_s^{\psi f_0} = (-25.2\pm 25.2\pm 1.2)^\circ
\quad & \mbox{LHCb} \mbox{ $J/\psi f_0$ \cite{LHCb:2011ab}} {\nonumber\\}A_{\rm SL} =
(-7.87 \pm 1.72 \pm 0.93 ) \cdot 10^{-3}
\quad & \mbox{D\O\ \cite{Abazov:2011yk}}
\label{exnum} \end{aligned}$$ Here $2\beta_s=2\arg (-V_{ts} V_{tb}^*/(V_{cs} V_{cb}^*))\simeq 2.2^\circ$ [@CKMfitterwebsite].
From this discussion, there is a conflict between LHCb data on $B_s\to J/\psi
\phi$ and the DØ measurement of $A_{\rm SL}$ which we cannot fully resolve in our Scenarios I, II and III. We therefore discuss a fourth scenario which also permits NP in the decay matrices $\Gamma_{12}^s$ or $\Gamma_{12}^d$.
Results for Scenarios I, II and III
===================================
In Tab. \[tab:Inputs\] we summarise the changes in the inputs compared to Tabs. 1–7 of Ref. [@Lenz:2010gu]. Following Ref. [@Charles:2011va] we have included $K_{\ell 3}$, $K_{\ell 2}$, $\pi_{\ell 2}$ (and the related $\tau$ decays) for $|V_{ud}|$ and $|V_{us}|$. Concerning the measurements of $(\phi_s,\Gamma_s)$ from $B_s\to J/\psi \phi$, we have combined the CDF and LHCb results by taking the product of their 2D profile-likelihoods [@LHCb:2011aa; @CDF:2011af]. Unfortunately, we could not obtain the corresponding likelihood from [DØ]{}. The impact of this omission is mild due to the smaller uncertainties of the CDF and LHCb results. We have neither used the LHCb result on $B_s\to J/\psi f_0$ as only $\phi_s$ (not the 2D likelihood) was provided in Ref. [@LHCb:2011ab]. But we have included the flavour-specific $B_s$ lifetime $\tau^{FS}_{B_s}$ [@HFAG11] providing an independent constraint on $\Delta\Gamma_s$. We analyse the [DØ]{} measurement of $A_{\rm SL}$ with the production fractions at 1.8-2 TeV according to Ref. [@HFAG11]: $f_s=
0.111\pm 0.014$ and $f_d=0.339\pm 0.031$, corresponding to $A_{\rm SL}=(0.532\pm 0.039) a^{d}_{\rm SL}+ (0.468\pm 0.039) a^{s}_{\rm
SL}$.
Observable Value and uncertainties Ref.
------------------------------------------------------------------------------------------- ------------------------------------ -----------------------------------
$\mathcal{B}(K \rightarrow e \nu_{e})$ $(1.584 \pm 0.020) \times 10^{-5}$ [@PDG]
$\mathcal{B}(K \rightarrow \mu \nu_{\mu})$ $0.6347\pm 0.0018$ [@Antonelli:2010yf]
$\mathcal{B}(\tau \rightarrow K \nu_{\tau})$ $0.00696\pm 0.00023$ [@Antonelli:2010yf]
$\mathcal{B}(K \!\rightarrow \mu \nu_{\mu})/\mathcal{B}(K \!\rightarrow \pi \nu_{\mu})$ $1.3344\pm 0.0041$ [@Antonelli:2010yf]
$\mathcal{B}(\tau \rightarrow K \nu_{\tau})/\mathcal{B}(\tau \rightarrow \pi \nu_{\tau})$ $(6.53\pm 0.11)\cdot 10^{-2}$ [@Banerjee:2008hg]
$\alpha$ $88.7{^{\,+4.6}_{\,-4.3}}^{\circ}$ [@CKMfitterwebsite]
$\gamma$ $(66\pm 12)^{\circ}$ [@CKMfitterwebsite]
$\Delta m_{d}$ $0.507 \pm 0.004 {\rm ps}^{-1}$ [@PDG]
$\Delta m_{s}$ $17.731\pm 0.045 {\rm ps}^{-1}$ [@Aaij:2011qx; @Abulencia:2006ze]
$A_\text{SL}$ $(-74 \pm 19)\times 10^{-4}$ [@Abazov:2011yk]
$\phi_s^{\psi\phi}$ vs. $\Delta \Gamma_{s}$ see text [@LHCb:2011aa; @CDF:2011af]
: Experimental and theoretical inputs inputs added or modified compared to Ref. [@Lenz:2010gu] and used in our fits.[]{data-label="tab:Inputs"}
$\quad$
Theoretical Parameter Value and uncertainties Ref.
----------------------------------------------- ------------------------------------ ---------------------
$f_{B_{s}}$ $229\pm2 \pm6$ MeV [@CKMfitterwebsite]
$f_{B_{s}}/f_{B_{d}}$ $1.218 \pm 0.008 \pm 0.033$ [@CKMfitterwebsite]
$\widehat{\mathcal{B}}_{B_{s}}$ $1.291 \pm 0.025 \pm 0.035$ [@CKMfitterwebsite]
${\mathcal{B}}_{B_{s}}/{\mathcal{B}}_{B_{d}}$ $1.024 \pm 0.013 \pm 0.015$ [@CKMfitterwebsite]
${\hat{{\mathcal{B}}}}_{K}$ $(0.733 \pm 0.003 \pm 0.036)$ [@CKMfitterwebsite]
$f_{K}$ $156.3 \pm 0.3 \pm 1.9 {\rm MeV}$ [@CKMfitterwebsite]
$f_{K}/f_\pi$ $1.1985\pm 0.0013\pm 0.0095$ [@CKMfitterwebsite]
$\alpha_s(M_{Z})$ $ 0.1184\pm 0 \pm 0.0007$ [@PDG]
: Experimental and theoretical inputs inputs added or modified compared to Ref. [@Lenz:2010gu] and used in our fits.[]{data-label="tab:Inputs"}
We summarise our results in Tabs. \[tab-results\] and \[tab-pulls\] and in [Fig. \[fig-Delta\_scenario1\]]{} (Sc. I) as well as [Fig. \[fig-Delta\_scenario3\]]{} (Sc. III). Even in Sc. I our fit to the data is significantly worse than in 2010 [@Lenz:2010gu]: While $\phi_d^\Delta<0 $ alleviates the discrepancy of $A_{\rm SL}$ with the SM, the LHCb result on $\phi_s^{\psi\phi}$ prevents larger contributions from the $B_s$ system to $A_{\rm SL}$. In Sc. I, we find pull values for $A_{\rm SL}$ and $\phi_s^\Delta-2\beta_s$ of 3.0$\,\sigma$ and 2.7$\,\sigma$ respectively (compared to 1.2$\,\sigma$ and 0.5$\,\sigma$ in Ref. [@Lenz:2010gu]). We do not quote pull values for $\Delta m_{d,s}$ in Sc. I, as these observables are not constrained once their experimental measurement is removed. In contrast to earlier analyses, only one solution for $\Delta_s$ survives thanks to the recent LHCb determination of $\Delta\Gamma_s>0$ [@LHCbDGsSign] entailing ${\mathrm{Re}\,}\Delta_s>0$. Tab. \[tab:pvalues\] lists the p-values for various SM hypotheses within our NP Scenarios (more information can be found in Ref. [@CKMfitterwebsite]).
![Complex parameters $\Delta_d$ (up) and $\Delta_s$ (down) in Scenario I. Here $\alpha_{\rm exp}\equiv \alpha -\phi_d^\Delta/2$. The coloured areas represent regions with ${\rm CL} < 68.3~\%$ for the individual constraints. The red area shows the region with ${\rm CL} < 68.3~\%$ for the combined fit, with the two additional contours delimiting the regions with ${\rm CL} < 95.45~\%$ and ${\rm CL} < 99.73~\%$. The $p$-value for the 2D SM hypothesis $\Delta_d=1$ ($\Delta_s=1$) is 3.0 $\sigma$ (0.0 $\sigma$). \[fig-Delta\_scenario1\]](NPmix_ReImDeltad_3contours.eps "fig:"){width="8cm"} ![Complex parameters $\Delta_d$ (up) and $\Delta_s$ (down) in Scenario I. Here $\alpha_{\rm exp}\equiv \alpha -\phi_d^\Delta/2$. The coloured areas represent regions with ${\rm CL} < 68.3~\%$ for the individual constraints. The red area shows the region with ${\rm CL} < 68.3~\%$ for the combined fit, with the two additional contours delimiting the regions with ${\rm CL} < 95.45~\%$ and ${\rm CL} < 99.73~\%$. The $p$-value for the 2D SM hypothesis $\Delta_d=1$ ($\Delta_s=1$) is 3.0 $\sigma$ (0.0 $\sigma$). \[fig-Delta\_scenario1\]](NPmix_ReImDeltas_3contours.eps "fig:"){width="8cm"}
![Constraint on the complex parameter $\Delta \equiv \Delta_d=\Delta_s$ from the fit in Scenario III with same conventions as in fig. \[fig-Delta\_scenario1\]. The $p$-value for the 2D SM hypothesis $\Delta=1$ is 2.1 $\sigma$. \[fig-Delta\_scenario3\]](NPmixScIII_ReImDelta.eps){width="8cm"}
Quantity $1 \sigma$ $3\sigma$
----------------------------------------------------------- --------------------------------- ------------------------------
\[-0.3cm\] $\mbox{Re}{(\Delta_d)}$ $0.823^{+0.143}_{-0.095}$ $0.82^{+0.54}_{-0.20}$
\[0.15cm\] $\mbox{Im}{(\Delta_d)}$ $-0.199^{+0.062}_{-0.048}$ $-0.20^{+0.18}_{-0.19}$
\[0.15cm\] $|\Delta_d|$ $0.86^{+0.14}_{-0.11}$ $0.86^{+0.55}_{-0.22}$
\[0.15cm\] $\phi^\Delta_d$ \[deg\] $-13.4^{+3.3}_{-2.0}$ $-13.4^{+12.1}_{-6.0}$
\[0.15cm\] $\mbox{Re}{(\Delta_s)}$ $0.965^{+0.133}_{-0.078}$ $0.97^{+0.30}_{-0.13}$
\[0.15cm\] $\mbox{Im}{(\Delta_s)}$ $-0.00^{+0.10}_{-0.10}$ $-0.00^{+0.32}_{-0.32}$
\[0.15cm\] $|\Delta_s|$ $0.977^{+0.121}_{-0.090}$ $0.98^{+0.29}_{-0.15}$
\[0.15cm\] $\phi^\Delta_s$ \[deg\] $-0.1^{+6.1}_{-6.1}$ $-0^{+18.}_{-18.}$
\[0.15cm\] $\phi^\Delta_d+2\beta$ \[deg\] (!) $17^{+12.}_{-13.}$ $17^{+40.}_{-55.}$
\[0.15cm\] $\phi^\Delta_s-2\beta_s$ \[deg\] (!) $-56.8^{+10.9}_{-7.0}$ $-57.^{+66.}_{-20.}$
\[0.15cm\]
\[-0.3cm\] $A_\text{SL}$ $[10^{-4}]$ (!) $-15.6^{+9.2}_{-3.9}$ $-16^{+19}_{-12}$
\[0.15cm\] $A_\text{SL}$ $[10^{-4}]$ $-17.7^{+3.9}_{-3.8}$ $-18^{+15}_{-12}$
\[0.15cm\] $a_\text{SL}^{s}-a_\text{SL}^{d}$ $[10^{-4}]$ $33.6^{+7.5}_{-8.2}$ $34^{+24}_{-32}$
\[0.15cm\] $a_\text{SL}^{d}$ $[10^{-4}]$ (!) $-33.2^{+6.6}_{-4.1}$ $-33^{+25}_{-13}$
\[0.15cm\] $a_\text{SL}^{s}$ $[10^{-4}]$ (!) $0.4^{+6.2}_{-6.3}$ $0^{+20}_{-21}$
\[0.15cm\] $\Delta\Gamma_d [\mathrm{ps}^{-1}]$ $0.00480^{+0.00070}_{-0.00129}$ $0.0048^{+0.0020}_{-0.0031}$
\[0.15cm\] $\Delta\Gamma_s [\mathrm{ps}^{-1}]$ (!) $0.155^{+0.020}_{-0.079}$ $0.155^{+0.036}_{-0.098}$
\[0.15cm\] $\Delta\Gamma_s [\mathrm{ps}^{-1}]$ $0.104^{+0.017}_{-0.016}$ $0.104^{+0.052}_{-0.041}$
\[0.15cm\]
\[-0.3cm\] $B\to \tau\nu$ $[10^{-4}]$ (!) $1.341^{+0.064}_{-0.232}$ $1.34^{+0.20}_{-0.73}$
\[0.15cm\] $B\to \tau\nu$ $[10^{-4}]$ $1.354^{+0.063}_{-0.095}$ $1.35^{+0.19}_{-0.50}$
\[0.15cm\]
: CL intervals for the results of the fits in Scenario I. The notation (!) means that the fit output represents the indirect constraint with the corresponding direct input removed.\[tab-results\]
[lcccc]{} Quantity & & Deviation & wrt&\
& SM & Sc. I & Sc. II & Sc. III\
\
$\phi^\Delta_d+2\beta$ & $2.7~\sigma$ & $2.1~\sigma$& $2.7~\sigma$ & $1.2~\sigma$\
$\phi^\Delta_s-2\beta_s$ & $0.3~\sigma$ & $2.7~\sigma$& $0.3~\sigma$ & $2.4~\sigma$\
&\
$|\epsilon_K|$ & $0.0~\sigma$ & - & $0.0~\sigma$ & -\
$\Delta m_d$ & $1.0~\sigma$ & - & $1.0~\sigma$ & $0.9~\sigma$\
$\Delta m_s$ & $0.0~\sigma$ & - & $1.0~\sigma$ & $1.3~\sigma$\
$A_\text{SL}$ & $3.7~\sigma$ & $3.0~\sigma$ & $3.7~\sigma$ & $3.0~\sigma$\
$a_\text{SL}^{d}$ & $0.9~\sigma$ & $0.3~\sigma$ & $0.8~\sigma$ & $0.4~\sigma$\
$a_\text{SL}^{s}$ & $0.2~\sigma$ & $0.2~\sigma$ & $0.2~\sigma$ & $0.0~\sigma$\
$\Delta\Gamma_s$ & $0.0~\sigma$ & $0.4~\sigma$ & $0.0~\sigma$ & $1.0~\sigma$\
&&&&\
$\mathcal{B}(B\to\tau\nu)$ & $2.8~\sigma$ & $1.1~\sigma$ & $2.8~\sigma$ & $1.7~\sigma$\
&&&&\
$\mathcal{B}(B\to\tau\nu)$, $A_\text{SL}$ & $4.3~\sigma$ & $2.8~\sigma$ & $4.2~\sigma$ & $3.4~\sigma$\
$\phi_s^\Delta-2\beta_s$, $A_\text{SL}$ & $3.3~\sigma$ & $2.7~\sigma$ & $3.3~\sigma$ & $3.2~\sigma$\
$\mathcal{B}(B\to\tau\nu)$, $\phi_s^\Delta-2\beta_s$, $A_\text{SL}$ & $4.0~\sigma$ & $2.4~\sigma$ & $3.9~\sigma$ & $3.2~\sigma$\
Hypothesis Sc. I Sc. II Sc. III
------------------------------------------------- -------------- -------------- --------------
$\mathrm{Im}{\Delta_d}=0$ $3.2 \sigma$ $2.6 \sigma$
$\mathrm{Im}{\Delta_s}=0$ $0.0 \sigma$
$\Delta_d=1$ $3.0 \sigma$ $0.6 \sigma$ $2.1 \sigma$
$\Delta_s=1$ $0.0 \sigma$
$\mathrm{Im}{\Delta_d}=\mathrm{Im}{\Delta_s}=0$ $2.8 \sigma$
$\Delta_d=\Delta_s=1$ $2.4 \sigma$
: p-values for various Standard Model hypotheses in the framework of three NP Scenarios considered. These numbers are computed from the $\chi^2$ difference with and without the hypothesis constraint, interpreted with the appropriate number of degrees of freedom.\[tab:pvalues\]
![Constraints on ${\mathrm{Im}\,}\delta_d,{\mathrm{Im}\,}\delta_s$ in Scenario IV. The 1D 68%CL intervals are ${\mathrm{Im}\,}\delta_d=0.92^{+1.13}_{-0.69},\ {\mathrm{Im}\,}\delta_s=1.2^{+1.6}_{-1.0}$. The $p$-value for the 2D SM hypothesis ${\mathrm{Im}\,}\delta_d=0.097,{\mathrm{Im}\,}\delta_s=-0.0057$ is 3.2 $\sigma$. \[fig-ScenarioIV\]](ScenarioIV_2D.eps){width="8cm"}
New Physics in $\Gamma_{12}^s$ or $\Gamma_{12}^d$
=================================================
Several authors have discussed the possibility of a sizable new CP-violating contribution to $\Gamma_{12}^s$ to explain the DØmeasurement of $A_{SL}$ [@nping12] by postulating new $B_s$ decay channels with large branching fraction. In such models also the width difference ${\ensuremath{\Delta \Gamma}}_s$ typically deviates from the SM prediction in Ref. [@dega; @ln; @Lenz:2011ti]. $\Gamma_{12}^s$ is dominated by the CKM-favoured tree-level decay $b\to c\bar{c}s$. Any competitive new decay mode will increase the total $B_s$ width, which LHCb finds as $\Gamma_s= 0.657 \pm 0.009 \pm 0.008 $ [@LHCb:2011aa], implying $\Gamma_s/\Gamma_d= 0.998 \pm 0.014 \pm 0.012 $ in excellent agreement with the SM expectation $0\leq \Gamma_s/\Gamma_d-1\leq 4\cdot 10^{-4}$ [@Lenz:2011ti]. The new interaction will open new $b\to s$ decay modes affecting precisely measured inclusive $B_d$ and $B^+$ quantities [@Lenz:2010gu]. Furthermore, new decays mediated by a particle with mass $M>M_W$ will add a term of order $M_W^4/M^4$ to $\Gamma_{12}^s/\Gamma_{12}^{{\rm SM},s}$, while $\Delta_s$ normally receives a larger contribution of order $M_W^2/M^2$. In models involving a fermion pair $(f,{\overline{f}})$ in the final state, e.g. those with an enhanced $B_s \to \tau {\overline{\tau}}$ decay [@nping12], one can solve this problem through chirality suppression. The extra contribution to $M_{12}^s$ is down by another factor of $m_f^2/M^2$, while that to $\Gamma_{12}^s$ is affected by the milder factor of $m_f^2/m_b^2$. Quantities like $\Gamma_{d,s}$ will not be chirality suppressed. Therefore it seems not possible to add large NP effects to $\Gamma_{12}^s$.
Phenomenologically it is thus much easier to postulate NP in $\Gamma_{12}^d$ rather than $\Gamma_{12}^s$, because $\Gamma_{12}^d$ is constituted by Cabibbo-suppressed decay modes like $b\to c {\overline{c}} d$. Also here chirality suppression is welcome to avoid problems with $M_{12}^d$, but inclusive decay observables like the semileptonic branching fraction or the unmeasured ${\ensuremath{\Delta \Gamma}}_d$ pose no danger. Clearly, testing this hypothesis calls for a better measurement of $a_{\rm SL}^d$. We have studied a Scenario IV including the possibility of NP in $\Gamma_{12}^{d,s}$. We stress that Sc. IV permits NP in the $|\Delta F|=1$ transitions contributing to $\Gamma_{12}^q$, but not in other $|\Delta
F|=1$ quantities entering our fits, such as ${\cal B} (B\to \tau
\nu)$. Further no new CP phase in $b\to c {\overline{c}} s$, which would change $\phi_{d,s}^\Delta$, is considered. Such a phase might further increase the hadronic uncertainty from penguin pollution, which is not an issue in the SM at the current levels of experimental precision.
Handy new parameters are $$\begin{aligned}
\!\delta_q = \frac{\Gamma_{12}^q/M_{12}^q}{{\mathrm{Re}\,}(\Gamma_{12}^{{\rm SM},q}/M_{12}^{{\rm SM},q})}, \quad q=d,s,\end{aligned}$$ ${\mathrm{Re}\,}\delta_q$, ${\mathrm{Im}\,}\delta_q$ amount to $({\ensuremath{\Delta \Gamma}}_q/{\ensuremath{\Delta M}}_q)/({\ensuremath{\Delta \Gamma}}_q^{\rm SM}/{\ensuremath{\Delta M}}_q^{\rm SM})$ and $-a_{\rm
SL}^q/({\ensuremath{\Delta \Gamma}}_q^{\rm SM}/{\ensuremath{\Delta M}}_q^{\rm SM})$, respectively. The best fit values of the SM predictions are $\delta_d^{\rm SM}=1 + 0.097\, i$ and $ \delta_s^{\rm SM}= 1 - 0.0057\, i$. ${\mathrm{Re}\,}\delta_d$ is experimentally only weakly constrained. We illustrate the correlation between ${\mathrm{Im}\,}\delta_d $ and ${\mathrm{Im}\,}\delta_s$ in [Fig. \[fig-ScenarioIV\]]{}, relegating correlations of ${\mathrm{Re}\,}\delta_s$ with ${\mathrm{Im}\,}\delta_{d,s}$ to Ref. [@CKMfitterwebsite]). The p-value of the 8D SM hypothesis $\Delta_d=\Delta_s=1$, $\delta_{d,s}=\delta_{d,s}^{\rm SM}$ is 2.6 $\sigma$.
We stress that too large values for $|\delta_s-\delta_s^{\rm SM}|$ are in conflict with other observables as explained above. We have also studied Scenario IV without NP in the $B_s$ sector ($\Delta_s=1$ and $\delta_s=\delta_{s,\mathrm{SM}}$). It could accommodate the main anomalies by improving the fit by $3.3\sigma$, but with large contributions to $\Gamma^d_{12}$: ${\mathrm{Im}\,}\delta_d=1.60^{+1.02}_{-0.76}$.
Conclusions
===========
We have performed new global fits to flavour physics data in scenarios with generic NP in the [$B_d\!-\!{\,\overline{\!B}}{}_d\,$]{} and [[$B_s\!-\!{\,\overline{\!B}}{}_s\,$]{} mixing]{} amplitudes, as defined in Ref. [@Lenz:2010gu]. Our results represent the status of the end of the year 2011. Unlike in summer 2010 the two complex NP parameters $\Delta_d$ and $\Delta_s$ (parametrising NP in $M_{12}^{d,s}$) are not sufficient to absorb all discrepancies with the SM, namely the DØ measurement of $A_{\rm SL}$ and the inconsistency between $B(B\to\tau\nu)$ and $A_{\rm CP}^{\rm mix}(B_d\to J/\Psi K)$. Still in Scenario I, which fits $\Delta_d$ and $\Delta_s$ independently, we find the SM point $\Delta_d=\Delta_s=1$ disfavoured by 2.4$\,\sigma$; this value was 3.6$\,\sigma$ in our 2010 analysis [@Lenz:2010gu] We notice that data still allow sizeable NP contributions in both $B_d$ and $B_s$ sectors up to 30-40% at the 3$\sigma$ level. The preference of Sc. I over the SM mainly stems from the fact that $B(B\to\tau\nu)$ favours $\phi_d^\Delta<0$ which alleviates the problem with $A_{\rm SL}$.
In order to fully reconcile $A_{SL}$ with $\phi_{s}^{\psi \phi}$ we have extended our study to a Scenario IV, which permits NP in both $M_{12}^{d,s}$ and $\Gamma_{12}^{d,s}$. While this scenario can accommodate all data, it is difficult to find realistic models in which the preferred NP contributions to $\Gamma_{12}^s$ (composed of Cabibbo-favoured tree-level decays) comply with other measurements. There are fewer phenomenological constraints on the Cabibbo-suppressed quantity $\Gamma_{12}^d$; a possible conflict with $M_{12}^d$ can be circumvented with chirality suppression. NP in $M_{12}^d$ and $\Gamma_{12}^d$ with the $B_s$ system essentially SM-like appears thus as an interesting possibility, requiring only a mild statistical upward fluctuation in the DØ data on $A_{\rm
SL}$. Clearly, independent measurements of $a^d_{\rm SL}$, $a^s_{\rm SL}$ and/or $a^s_{\rm SL}-a^d_{\rm SL}$ are necessary to determine whether scenarios with NP in $\Gamma_{12}^d$ and/or $\Gamma_{12}^s$ are a viable explanation of discrepancies in $\Delta F=2$ observables with respect to the Standard Model.
We thank the CDF and LHCb collaborations for providing us with the 2D profile likelihood functions needed for our analyses. A.L. is supported by DFG through a Heisenberg fellowship. U.N.acknowledges support by BMBF through grant 05H09VKF.
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